[{"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "Here's a simulation created by Khan Academy user Justin Helps that once again tries to give us an understanding of why we divide by n minus 1 to get an unbiased estimate of population variance when we're trying to calculate the sample variance. So what he does here is a simulation. It has a population that has a uniform distribution. So he says, I used a flat probabilistic distribution from 0 to 100 for my population. Then we start sampling from that population. We're going to use samples of size 50. And what we do is for each of those samples, we calculate the sample variance based on dividing by n by dividing by n minus 1 and n minus 2."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "So he says, I used a flat probabilistic distribution from 0 to 100 for my population. Then we start sampling from that population. We're going to use samples of size 50. And what we do is for each of those samples, we calculate the sample variance based on dividing by n by dividing by n minus 1 and n minus 2. And as we keep having more and more and more samples, we take the mean of the variances calculated in different ways. And we figure out what those means converge to. So that's a sample."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "And what we do is for each of those samples, we calculate the sample variance based on dividing by n by dividing by n minus 1 and n minus 2. And as we keep having more and more and more samples, we take the mean of the variances calculated in different ways. And we figure out what those means converge to. So that's a sample. Here's another sample. Here's another sample. If I sample here, then now I'm adding a bunch."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "So that's a sample. Here's another sample. Here's another sample. If I sample here, then now I'm adding a bunch. And I'm sampling continuously. And you saw something very interesting happen. When I divide by n, I get my sample variance is still, even when I'm taking the mean of many, many, many, many sample variances that I've already taken, I'm still underestimating the true variance."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "If I sample here, then now I'm adding a bunch. And I'm sampling continuously. And you saw something very interesting happen. When I divide by n, I get my sample variance is still, even when I'm taking the mean of many, many, many, many sample variances that I've already taken, I'm still underestimating the true variance. When I divide by n minus 1, it looks like I'm getting a pretty good estimate. The mean of all of my sample variances is really converged to the true variance. When I divided by n minus 2 just for kicks, it's pretty clear that I overestimated with my mean of my sample variances."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "When I divide by n, I get my sample variance is still, even when I'm taking the mean of many, many, many, many sample variances that I've already taken, I'm still underestimating the true variance. When I divide by n minus 1, it looks like I'm getting a pretty good estimate. The mean of all of my sample variances is really converged to the true variance. When I divided by n minus 2 just for kicks, it's pretty clear that I overestimated with my mean of my sample variances. I overestimated the true variance. So this gives us a pretty good sense that n minus 1 is the right thing to do. Now, this is another way and another interesting way of visualizing it."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "When I divided by n minus 2 just for kicks, it's pretty clear that I overestimated with my mean of my sample variances. I overestimated the true variance. So this gives us a pretty good sense that n minus 1 is the right thing to do. Now, this is another way and another interesting way of visualizing it. In the horizontal axis right over here, we're comparing each plot as one of our samples. And how far to the right is, how much more is that sample mean than the true mean? And when we go to the left, it's how much less is the sample mean than the true mean?"}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "Now, this is another way and another interesting way of visualizing it. In the horizontal axis right over here, we're comparing each plot as one of our samples. And how far to the right is, how much more is that sample mean than the true mean? And when we go to the left, it's how much less is the sample mean than the true mean? So for example, this sample right over here, it's all the way over to the right. The sample mean there was a lot more than the true mean. Sample mean here was a lot less than the true mean."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "And when we go to the left, it's how much less is the sample mean than the true mean? So for example, this sample right over here, it's all the way over to the right. The sample mean there was a lot more than the true mean. Sample mean here was a lot less than the true mean. Sample mean here, only a little bit more than the true mean. In the vertical axis, using this denominator, dividing by n, we calculate two different variances. One variance we use the sample mean."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "Sample mean here was a lot less than the true mean. Sample mean here, only a little bit more than the true mean. In the vertical axis, using this denominator, dividing by n, we calculate two different variances. One variance we use the sample mean. The other variance we use the population mean. And this, in the vertical axis, we compare the difference between the mean calculated with the sample mean versus the mean calculated with the population mean. So for example, this point right over here, when we calculate our mean with our sample mean, which is the normal way we do it, it significantly underestimates what the mean would have been if somehow we knew what the population mean was and we could calculate it that way."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "One variance we use the sample mean. The other variance we use the population mean. And this, in the vertical axis, we compare the difference between the mean calculated with the sample mean versus the mean calculated with the population mean. So for example, this point right over here, when we calculate our mean with our sample mean, which is the normal way we do it, it significantly underestimates what the mean would have been if somehow we knew what the population mean was and we could calculate it that way. And you get this really interesting shape. And it's something to think about. And he recommends thinking about why or what kind of a shape this actually is."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "So for example, this point right over here, when we calculate our mean with our sample mean, which is the normal way we do it, it significantly underestimates what the mean would have been if somehow we knew what the population mean was and we could calculate it that way. And you get this really interesting shape. And it's something to think about. And he recommends thinking about why or what kind of a shape this actually is. The other interesting thing is, when you look at it this way, it's pretty clear this entire graph is sitting below the horizontal axis. So we're always, when we calculate our sample variance using this formula, when we use the sample mean to do it, which we typically do, we're always getting a lower variance than when we use the population mean. Now this over here, when we divide by n minus 1, we're not always underestimating."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "And he recommends thinking about why or what kind of a shape this actually is. The other interesting thing is, when you look at it this way, it's pretty clear this entire graph is sitting below the horizontal axis. So we're always, when we calculate our sample variance using this formula, when we use the sample mean to do it, which we typically do, we're always getting a lower variance than when we use the population mean. Now this over here, when we divide by n minus 1, we're not always underestimating. Sometimes we're overestimating it. And when you take the mean of all of these variances, you converge. And here we're overestimating it a little bit more."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "Now this over here, when we divide by n minus 1, we're not always underestimating. Sometimes we're overestimating it. And when you take the mean of all of these variances, you converge. And here we're overestimating it a little bit more. And just to be clear, what we're talking about in these three graphs, let me take a screenshot of it and explain it in a little bit more depth. So just to be clear, in this red graph right over here, let me do this in a color close to, at least, so this orange, what this distance is for each of these samples, we're calculating the sample variance using, so let me, using the sample mean. And in this case, we are using n as our denominator, in this case right over here."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "And here we're overestimating it a little bit more. And just to be clear, what we're talking about in these three graphs, let me take a screenshot of it and explain it in a little bit more depth. So just to be clear, in this red graph right over here, let me do this in a color close to, at least, so this orange, what this distance is for each of these samples, we're calculating the sample variance using, so let me, using the sample mean. And in this case, we are using n as our denominator, in this case right over here. And from that, we're subtracting the sample variance, or I guess you could call this some kind of pseudo-sample variance, if we somehow knew the population mean. This isn't something that you see a lot in statistics, but it's a gauge of how much we are underestimating our sample variance, given that we don't have the true population mean at our disposal. And so this is the distance we're calculating."}, {"video_title": "Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3", "Sentence": "And in this case, we are using n as our denominator, in this case right over here. And from that, we're subtracting the sample variance, or I guess you could call this some kind of pseudo-sample variance, if we somehow knew the population mean. This isn't something that you see a lot in statistics, but it's a gauge of how much we are underestimating our sample variance, given that we don't have the true population mean at our disposal. And so this is the distance we're calculating. And you see we are always underestimating. Here, we overestimate a little bit, and we also underestimate. But when you take the mean, and when you average them all out, it converges to the actual value."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "In this video, we're going to get our bearings on the different types of studies you might statistically analyze or statistical studies. So first of all, it's worth differentiating between an experiment and an observational study. I encourage you, pause this video and think about what the difference is, at least in your head, between an experiment and an observational study. Well, you might already be familiar with experiments. You oftentimes have a hypothesis that if you do something to one group, that it might have some type of statistically significant impact on them relative to a group that you did not do it to, and you would be generally right. That is the flavor of what we're talking about when we're talking about an experiment. An experiment where actively putting people or things into a control versus treatment group."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "Well, you might already be familiar with experiments. You oftentimes have a hypothesis that if you do something to one group, that it might have some type of statistically significant impact on them relative to a group that you did not do it to, and you would be generally right. That is the flavor of what we're talking about when we're talking about an experiment. An experiment where actively putting people or things into a control versus treatment group. In the treatment group, you put the people, and you usually would want to randomly select people into the treatment group. Maybe it's a new type of medication, and maybe in the treatment group, they actually get the medication, while in the control group, which you would put people into randomly, whether they're in control or treatment, here, they might get a placebo, where they get a pill that looks just like the medication, but it really doesn't do anything. And then you wait some time, and you can see is there a statistically significant difference between the treatment group, on average, and the control group."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "An experiment where actively putting people or things into a control versus treatment group. In the treatment group, you put the people, and you usually would want to randomly select people into the treatment group. Maybe it's a new type of medication, and maybe in the treatment group, they actually get the medication, while in the control group, which you would put people into randomly, whether they're in control or treatment, here, they might get a placebo, where they get a pill that looks just like the medication, but it really doesn't do anything. And then you wait some time, and you can see is there a statistically significant difference between the treatment group, on average, and the control group. So that's what an experiment does. It's kind of this active sorting and figuring out whether some type of stimulus is able to show a difference. While an observational study, you don't actively put into groups."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "And then you wait some time, and you can see is there a statistically significant difference between the treatment group, on average, and the control group. So that's what an experiment does. It's kind of this active sorting and figuring out whether some type of stimulus is able to show a difference. While an observational study, you don't actively put into groups. Instead, you just collect data and see if you can have some insights from that data. If you can say, okay, the data, there's a population here. Can I come up with some statistics that are indicative of the population?"}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "While an observational study, you don't actively put into groups. Instead, you just collect data and see if you can have some insights from that data. If you can say, okay, the data, there's a population here. Can I come up with some statistics that are indicative of the population? I might just wanna look at averages, or I might wanna find some correlations between variables. But even when we're talking about an observational study, there are different types of it, depending on what type of data we're looking at, whether the data is backward-looking, forward-looking, or it's data that we are collecting right now based on what people think or say right now. So if we're thinking about an observational study that is looking at past data, and I could imagine doing something like this at Khan Academy, where we could look at maybe usage of Khan Academy over time."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "Can I come up with some statistics that are indicative of the population? I might just wanna look at averages, or I might wanna find some correlations between variables. But even when we're talking about an observational study, there are different types of it, depending on what type of data we're looking at, whether the data is backward-looking, forward-looking, or it's data that we are collecting right now based on what people think or say right now. So if we're thinking about an observational study that is looking at past data, and I could imagine doing something like this at Khan Academy, where we could look at maybe usage of Khan Academy over time. We have these things in our server logs, and we're able to do some analysis there. Maybe we're able to analyze and say, okay, on average, students are spending two hours per month on Khan Academy over in 2019. That would be past data, and that type of observational study would be called a retrospective study."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "So if we're thinking about an observational study that is looking at past data, and I could imagine doing something like this at Khan Academy, where we could look at maybe usage of Khan Academy over time. We have these things in our server logs, and we're able to do some analysis there. Maybe we're able to analyze and say, okay, on average, students are spending two hours per month on Khan Academy over in 2019. That would be past data, and that type of observational study would be called a retrospective study. Retro for backwards, and spective, looking. So a retrospective observational study would sample past data in order to come up with some insights. Now, you could imagine there might be the other side."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "That would be past data, and that type of observational study would be called a retrospective study. Retro for backwards, and spective, looking. So a retrospective observational study would sample past data in order to come up with some insights. Now, you could imagine there might be the other side. What if we are trying to observe things into the future? Well, here, you might take a sample of folks who you think are indicative of a population, and you might want to just track their data. So you could even consider that to be future data."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "Now, you could imagine there might be the other side. What if we are trying to observe things into the future? Well, here, you might take a sample of folks who you think are indicative of a population, and you might want to just track their data. So you could even consider that to be future data. So you pick the group, the sample, ahead of time, and then you track their data over time. I'm just gonna draw it as these little arrows that you're tracking their data. And then you see what happens."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "So you could even consider that to be future data. So you pick the group, the sample, ahead of time, and then you track their data over time. I'm just gonna draw it as these little arrows that you're tracking their data. And then you see what happens. For example, you might randomly select, hopefully a random sample of 100 women, and you wanna see in the coming year how many eggs do they eat on average per day. Well, what you would do is you selected those folks, and then you would track that data for each of them every day. And then once you have the data, you could actually do it while you're collecting it, but at the end of the study, you'll be able to see what those averages are, but you can also keep track of it while you're taking that data."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "And then you see what happens. For example, you might randomly select, hopefully a random sample of 100 women, and you wanna see in the coming year how many eggs do they eat on average per day. Well, what you would do is you selected those folks, and then you would track that data for each of them every day. And then once you have the data, you could actually do it while you're collecting it, but at the end of the study, you'll be able to see what those averages are, but you can also keep track of it while you're taking that data. And you could imagine what this was called. Instead of retrospective, we're now looking forward. So it is prospective, forward-looking observational study."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "And then once you have the data, you could actually do it while you're collecting it, but at the end of the study, you'll be able to see what those averages are, but you can also keep track of it while you're taking that data. And you could imagine what this was called. Instead of retrospective, we're now looking forward. So it is prospective, forward-looking observational study. Last but not least, some of y'all are probably thinking, what about if we're doing something now? If we go out there and we were to survey a bunch of people and say, how many eggs did you eat today? Or who are you going to vote for?"}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "So it is prospective, forward-looking observational study. Last but not least, some of y'all are probably thinking, what about if we're doing something now? If we go out there and we were to survey a bunch of people and say, how many eggs did you eat today? Or who are you going to vote for? What might we call that? Well, it's tempting to call it something with a prefix and then spective, so it all matches, but it turns out that the terminology that statisticians will typically use is a sample survey. Sample survey."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "Or who are you going to vote for? What might we call that? Well, it's tempting to call it something with a prefix and then spective, so it all matches, but it turns out that the terminology that statisticians will typically use is a sample survey. Sample survey. That right now, you're going to take a, hopefully random sample of individuals from the population that you care about, and you are just going to survey them right now and ask them, say, some questions or observe some data about them right now. So I'll leave you there. This video is to just give you a little bit of the vocabulary and a little bit of a taxonomy on the types of studies that you'll see in general, which is especially useful to know when you're exploring the world of statistics."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "Now, let's say you have a hunch that, well, maybe it is skewed towards one letter or another. How could you test this? Well, you could start with a null and alternative hypothesis, and then we can actually do a hypothesis test. So let's say that our null hypothesis is equal distribution, equal distribution of correct choices, correct choices. Or another way of thinking about it is A would be correct 25% of the time, B would be correct 25% of the time, C would be correct 25% of the time, and D would be correct 25% of the time. Now, what would be our alternative hypothesis? Well, our alternative hypothesis would be not equal distribution, not equal distribution."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "So let's say that our null hypothesis is equal distribution, equal distribution of correct choices, correct choices. Or another way of thinking about it is A would be correct 25% of the time, B would be correct 25% of the time, C would be correct 25% of the time, and D would be correct 25% of the time. Now, what would be our alternative hypothesis? Well, our alternative hypothesis would be not equal distribution, not equal distribution. Now, how are we going to actually test this? Well, we've seen this show before, at least the beginnings of the show. You have the population of all of your potential items here, and you could take a sample."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "Well, our alternative hypothesis would be not equal distribution, not equal distribution. Now, how are we going to actually test this? Well, we've seen this show before, at least the beginnings of the show. You have the population of all of your potential items here, and you could take a sample. And so let's say we take a sample of 100 items. So n is equal to 100. And let's write down the data that we get when we look at that sample."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "You have the population of all of your potential items here, and you could take a sample. And so let's say we take a sample of 100 items. So n is equal to 100. And let's write down the data that we get when we look at that sample. So this is the correct choice, correct choice. And then this would be the expected number that you would expect. And then this is the actual number."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "And let's write down the data that we get when we look at that sample. So this is the correct choice, correct choice. And then this would be the expected number that you would expect. And then this is the actual number. And if this doesn't make sense yet, we'll see it in a second. So there's four different choices. A, B, C, D. In a sample of 100, remember, in any hypothesis test, we start assuming that the null hypothesis is true."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "And then this is the actual number. And if this doesn't make sense yet, we'll see it in a second. So there's four different choices. A, B, C, D. In a sample of 100, remember, in any hypothesis test, we start assuming that the null hypothesis is true. So the expected number where A is the correct choice would be 25% of this 100. So you'd expect 25 times the A to be the correct choice, 25 times B to be the correct choice, 25 times C to be the correct choice, and 25 times D to be the correct choice. But let's say our actual results, when we look at these 100 items, we get that A is the correct choice 20 times, B is the correct choice 20 times, C is the correct choice 25 times, and D is the correct choice 35 times."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "A, B, C, D. In a sample of 100, remember, in any hypothesis test, we start assuming that the null hypothesis is true. So the expected number where A is the correct choice would be 25% of this 100. So you'd expect 25 times the A to be the correct choice, 25 times B to be the correct choice, 25 times C to be the correct choice, and 25 times D to be the correct choice. But let's say our actual results, when we look at these 100 items, we get that A is the correct choice 20 times, B is the correct choice 20 times, C is the correct choice 25 times, and D is the correct choice 35 times. So if you just look at this, you say, hey, maybe there's a higher frequency of D. But maybe you say, well, this is just a sample, and just a random chance, it might have just gotten more Ds than not. There's some probability of getting this result, even assuming that the null hypothesis is true. And that's the goal of these hypothesis tests."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "But let's say our actual results, when we look at these 100 items, we get that A is the correct choice 20 times, B is the correct choice 20 times, C is the correct choice 25 times, and D is the correct choice 35 times. So if you just look at this, you say, hey, maybe there's a higher frequency of D. But maybe you say, well, this is just a sample, and just a random chance, it might have just gotten more Ds than not. There's some probability of getting this result, even assuming that the null hypothesis is true. And that's the goal of these hypothesis tests. They say, what's the probability of getting a result at least this extreme? And if that probability is below some threshold, then we tend to reject the null hypothesis and accept an alternative. And those thresholds you have seen before, we've seen these significance levels."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "And that's the goal of these hypothesis tests. They say, what's the probability of getting a result at least this extreme? And if that probability is below some threshold, then we tend to reject the null hypothesis and accept an alternative. And those thresholds you have seen before, we've seen these significance levels. Let's say we set a significance level of 5%, 0.05. So if the probability of getting this result, or something even more different than what's expected, is less than the significance level, then we'd reject the null hypothesis. But this all leads to one really interesting question."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "And those thresholds you have seen before, we've seen these significance levels. Let's say we set a significance level of 5%, 0.05. So if the probability of getting this result, or something even more different than what's expected, is less than the significance level, then we'd reject the null hypothesis. But this all leads to one really interesting question. How do we calculate a probability of getting a result this extreme or more extreme? How do we even measure that? And this is where we're going to introduce a new statistic, and also for many of you, a new Greek letter."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "But this all leads to one really interesting question. How do we calculate a probability of getting a result this extreme or more extreme? How do we even measure that? And this is where we're going to introduce a new statistic, and also for many of you, a new Greek letter. And that is the capital Greek letter chi, which might look like an X to you, but it's a little bit curvier, and you could look up more on that. You kind of curve that part of the X. But it's a chi, not an X."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "And this is where we're going to introduce a new statistic, and also for many of you, a new Greek letter. And that is the capital Greek letter chi, which might look like an X to you, but it's a little bit curvier, and you could look up more on that. You kind of curve that part of the X. But it's a chi, not an X. And the statistic is called chi squared. And it's a way of taking the difference between the actual and the expected, and translating that into a number. And the chi squared distribution is well, I really should say distributions, are well studied."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "But it's a chi, not an X. And the statistic is called chi squared. And it's a way of taking the difference between the actual and the expected, and translating that into a number. And the chi squared distribution is well, I really should say distributions, are well studied. And we can use that to figure out what is the probability of getting a result this extreme or more extreme? And if that's lower than our significance level, we reject the null hypothesis, and it suggests the alternative. But how do we calculate the chi squared statistic here?"}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "And the chi squared distribution is well, I really should say distributions, are well studied. And we can use that to figure out what is the probability of getting a result this extreme or more extreme? And if that's lower than our significance level, we reject the null hypothesis, and it suggests the alternative. But how do we calculate the chi squared statistic here? Well, it's reasonably intuitive. What we do is, for each of these categories, in this case, it's for each of these choices, we look at the difference between the actual and the expected. So for choice A, we'd say 20 is the actual minus the expected."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "But how do we calculate the chi squared statistic here? Well, it's reasonably intuitive. What we do is, for each of these categories, in this case, it's for each of these choices, we look at the difference between the actual and the expected. So for choice A, we'd say 20 is the actual minus the expected. And then we're going to square that. And then we're going to divide by what was expected. And then we're gonna do that for choice B."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "So for choice A, we'd say 20 is the actual minus the expected. And then we're going to square that. And then we're going to divide by what was expected. And then we're gonna do that for choice B. So we're going to say the actual was 20, expected is 25, so 20 minus 25 squared over the expected over 25. Plus, then we do that for choice C, 25 minus 25, we know where that one will end up, squared over the expected over 25. And then finally, for choice D, which is going to get us 35 minus 25 squared, all of that over 25."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "And then we're gonna do that for choice B. So we're going to say the actual was 20, expected is 25, so 20 minus 25 squared over the expected over 25. Plus, then we do that for choice C, 25 minus 25, we know where that one will end up, squared over the expected over 25. And then finally, for choice D, which is going to get us 35 minus 25 squared, all of that over 25. And we are now, let's see, if we calculate this, this is going to be negative five squared, so that's going to be 25. This is going to be 25. This is going to be zero."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "And then finally, for choice D, which is going to get us 35 minus 25 squared, all of that over 25. And we are now, let's see, if we calculate this, this is going to be negative five squared, so that's going to be 25. This is going to be 25. This is going to be zero. 35 minus 25 is 10 squared, that is 100. So this is one plus one plus zero plus four. So our chi squared statistic in this example came out nice and clean, this won't always be the case, at six."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "This is going to be zero. 35 minus 25 is 10 squared, that is 100. So this is one plus one plus zero plus four. So our chi squared statistic in this example came out nice and clean, this won't always be the case, at six. So what do we make of this? Well, what we can do is then look at a chi squared distribution for the appropriate degrees of freedom, and we'll talk about that in a second, and say what is the probability of getting a chi squared statistic six or larger? And to understand what a chi squared distribution even looks like, these are multiple chi squared distributions for different values for the degrees of freedom."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "So our chi squared statistic in this example came out nice and clean, this won't always be the case, at six. So what do we make of this? Well, what we can do is then look at a chi squared distribution for the appropriate degrees of freedom, and we'll talk about that in a second, and say what is the probability of getting a chi squared statistic six or larger? And to understand what a chi squared distribution even looks like, these are multiple chi squared distributions for different values for the degrees of freedom. And to calculate the degrees of freedom, you look at the number of categories. In this case, we have four categories, and you subtract one. Now that makes a lot of sense, because if you knew how many A's, B's, and C's there are, if you knew the proportions, even the assumed proportions, you can always calculate the fourth one."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "And to understand what a chi squared distribution even looks like, these are multiple chi squared distributions for different values for the degrees of freedom. And to calculate the degrees of freedom, you look at the number of categories. In this case, we have four categories, and you subtract one. Now that makes a lot of sense, because if you knew how many A's, B's, and C's there are, if you knew the proportions, even the assumed proportions, you can always calculate the fourth one. That's why it is four minus one degrees of freedom. So in this case, our degrees of freedom are going to be equal to three. Over here, sometimes you'll see it described as k. So k is equal to three."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "Now that makes a lot of sense, because if you knew how many A's, B's, and C's there are, if you knew the proportions, even the assumed proportions, you can always calculate the fourth one. That's why it is four minus one degrees of freedom. So in this case, our degrees of freedom are going to be equal to three. Over here, sometimes you'll see it described as k. So k is equal to three. So if we look at, that's that little light blue, so we're looking at this chi squared distribution where the degree of freedom is three, and we wanna figure out what is the probability of getting a chi squared statistic that is six or greater? So we would be looking at this area right over here. And you could figure it out using a calculator, or if you're taking some type of a test, like an AP statistics exam, for example, you could use their tables they give you."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "Over here, sometimes you'll see it described as k. So k is equal to three. So if we look at, that's that little light blue, so we're looking at this chi squared distribution where the degree of freedom is three, and we wanna figure out what is the probability of getting a chi squared statistic that is six or greater? So we would be looking at this area right over here. And you could figure it out using a calculator, or if you're taking some type of a test, like an AP statistics exam, for example, you could use their tables they give you. And so a table like this could be quite useful. Remember, we're dealing with a situation where we have three degrees of freedom. We have four categories, so four minus one is three."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "And you could figure it out using a calculator, or if you're taking some type of a test, like an AP statistics exam, for example, you could use their tables they give you. And so a table like this could be quite useful. Remember, we're dealing with a situation where we have three degrees of freedom. We have four categories, so four minus one is three. And we got a chi squared value. Our chi squared statistic was six. So this right over here tells us the probability of getting a 6.25 or greater for a chi squared value is 10%."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "We have four categories, so four minus one is three. And we got a chi squared value. Our chi squared statistic was six. So this right over here tells us the probability of getting a 6.25 or greater for a chi squared value is 10%. If we go back to this chart, we just learned that this probability from 6.25 and up, when we have three degrees of freedom, that this right over here is 10%. Well, if that's 10%, then the probability, the probability of getting a chi squared value greater than or equal to six is going to be greater than 10%, greater than 10%. And we could also view this as our p-value."}, {"video_title": "Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3", "Sentence": "So this right over here tells us the probability of getting a 6.25 or greater for a chi squared value is 10%. If we go back to this chart, we just learned that this probability from 6.25 and up, when we have three degrees of freedom, that this right over here is 10%. Well, if that's 10%, then the probability, the probability of getting a chi squared value greater than or equal to six is going to be greater than 10%, greater than 10%. And we could also view this as our p-value. And so if our probability, assuming the null hypothesis, is greater than 10%, well, it's definitely going to be greater than our significance level. And because of that, we will fail to reject, fail to reject. And so this is an example of, even though in your sample you just happened to get more Ds, the probability of getting a result at least as extreme as what you saw is going to be a little bit over 10%."}, {"video_title": "Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "There's a parameter here. Let's say it's the population mean. We do not know what this is, so we take a sample. Here we're gonna take a sample of 15. So n is equal to 15, and from that sample, we can calculate a sample mean. But we also wanna construct a 98% confidence interval about that sample mean. So we're gonna go take that sample mean, and we're gonna go plus or minus some margin of error."}, {"video_title": "Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Here we're gonna take a sample of 15. So n is equal to 15, and from that sample, we can calculate a sample mean. But we also wanna construct a 98% confidence interval about that sample mean. So we're gonna go take that sample mean, and we're gonna go plus or minus some margin of error. Now, in other videos, we have talked about that we wanna use the t distribution here because we don't want to underestimate the margin of error. So it's going to be t star times the sample standard deviation divided by the square root of our sample size, which in this case was going to be 15, so the square root of n. But what they're asking us is, well, what is the appropriate critical value? What is the t star that we should use in this situation?"}, {"video_title": "Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So we're gonna go take that sample mean, and we're gonna go plus or minus some margin of error. Now, in other videos, we have talked about that we wanna use the t distribution here because we don't want to underestimate the margin of error. So it's going to be t star times the sample standard deviation divided by the square root of our sample size, which in this case was going to be 15, so the square root of n. But what they're asking us is, well, what is the appropriate critical value? What is the t star that we should use in this situation? And so we're about to look at a, I guess we call it a t table instead of a z table, but the key thing to realize is there's one extra variable to take into consideration when we're looking up the appropriate critical value on a t table, and that's this notion of degree of freedom. Sometimes it's abbreviated DF. And I'm not gonna go in-depth on degrees of freedom."}, {"video_title": "Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "What is the t star that we should use in this situation? And so we're about to look at a, I guess we call it a t table instead of a z table, but the key thing to realize is there's one extra variable to take into consideration when we're looking up the appropriate critical value on a t table, and that's this notion of degree of freedom. Sometimes it's abbreviated DF. And I'm not gonna go in-depth on degrees of freedom. It's actually a pretty deep concept, but it's this idea that you actually have a different t distribution depending on the different sample sizes, depending on the degrees of freedom. And your degree of freedom is going to be your sample size minus one. So in this situation, our degree of freedom is going to be 15 minus one."}, {"video_title": "Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And I'm not gonna go in-depth on degrees of freedom. It's actually a pretty deep concept, but it's this idea that you actually have a different t distribution depending on the different sample sizes, depending on the degrees of freedom. And your degree of freedom is going to be your sample size minus one. So in this situation, our degree of freedom is going to be 15 minus one. So in this situation, our degree of freedom is going to be equal to 14. And this isn't the first time that we have seen this. We talked a little bit about degrees of freedom when we first talked about sample standard deviations and how to have an unbiased estimate for the population standard deviation."}, {"video_title": "Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So in this situation, our degree of freedom is going to be 15 minus one. So in this situation, our degree of freedom is going to be equal to 14. And this isn't the first time that we have seen this. We talked a little bit about degrees of freedom when we first talked about sample standard deviations and how to have an unbiased estimate for the population standard deviation. And in future videos, we'll go into more advanced conversations about degrees of freedom. But for the purposes of this example, you need to know that when you're looking at the t distribution for a given degree of freedom, your degree of freedom is based on the sample size, and it's going to be your sample size minus one. When we're thinking about a confidence interval for your mean."}, {"video_title": "Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "We talked a little bit about degrees of freedom when we first talked about sample standard deviations and how to have an unbiased estimate for the population standard deviation. And in future videos, we'll go into more advanced conversations about degrees of freedom. But for the purposes of this example, you need to know that when you're looking at the t distribution for a given degree of freedom, your degree of freedom is based on the sample size, and it's going to be your sample size minus one. When we're thinking about a confidence interval for your mean. So now let's look at the t table. So we want a 98% confidence interval. And we want a degree of freedom of 14."}, {"video_title": "Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "When we're thinking about a confidence interval for your mean. So now let's look at the t table. So we want a 98% confidence interval. And we want a degree of freedom of 14. So let's get our t table out. And I actually copy and pasted this bottom part, moved it up so that you could see the whole thing here. And what's useful about this t table is they actually give our confidence levels right over here."}, {"video_title": "Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And we want a degree of freedom of 14. So let's get our t table out. And I actually copy and pasted this bottom part, moved it up so that you could see the whole thing here. And what's useful about this t table is they actually give our confidence levels right over here. So if you want a confidence level of 98%, you're going to look at this column. You're going to look at this column right over here. Another way of thinking about a confidence level of 98%, if you have a confidence level of 98%, that means you're leaving 1% unfilled in at either end of the tail."}, {"video_title": "Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And what's useful about this t table is they actually give our confidence levels right over here. So if you want a confidence level of 98%, you're going to look at this column. You're going to look at this column right over here. Another way of thinking about a confidence level of 98%, if you have a confidence level of 98%, that means you're leaving 1% unfilled in at either end of the tail. And so if you're looking at your t distribution, everything up to and including that top 1%, you would look for a tail probability of 0.01, which is, you can't see it right over there. Let me do it in a slightly brighter color, which would be that tail probability to the right. But either way, we're in this column right over here."}, {"video_title": "Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Another way of thinking about a confidence level of 98%, if you have a confidence level of 98%, that means you're leaving 1% unfilled in at either end of the tail. And so if you're looking at your t distribution, everything up to and including that top 1%, you would look for a tail probability of 0.01, which is, you can't see it right over there. Let me do it in a slightly brighter color, which would be that tail probability to the right. But either way, we're in this column right over here. We have a confidence level of 98%. And remember, our degrees of freedom, our degree of freedom here is, we have 14 degrees of freedom. And so we'll look at this row right over here."}, {"video_title": "Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "But either way, we're in this column right over here. We have a confidence level of 98%. And remember, our degrees of freedom, our degree of freedom here is, we have 14 degrees of freedom. And so we'll look at this row right over here. And so there you have it. This is our critical t value, 2.624. And so let's just go back here."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "Let's say we're trying to understand the relationship between people's height and their weight. So what we do is we go to 10 different people and we measure each of their heights and each of their weights. And so on this scatter plot here, each dot represents a person. So for example, this dot over here represents a person whose height was 60 inches or five feet tall. So that's the point 60 comma, and whose weight, which we have on the y-axis, was 125 pounds. And so when you look at this scatter plot, your eyes naturally see some type of a trend. It seems like, generally speaking, as height increases, weight increases as well."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "So for example, this dot over here represents a person whose height was 60 inches or five feet tall. So that's the point 60 comma, and whose weight, which we have on the y-axis, was 125 pounds. And so when you look at this scatter plot, your eyes naturally see some type of a trend. It seems like, generally speaking, as height increases, weight increases as well. But I said generally speaking. You definitely have circumstances where there are taller people who might weigh less. But an interesting question is, can we try to fit a line to this data?"}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "It seems like, generally speaking, as height increases, weight increases as well. But I said generally speaking. You definitely have circumstances where there are taller people who might weigh less. But an interesting question is, can we try to fit a line to this data? And this idea of trying to fit a line as closely as possible to as many of the points as possible is known as linear regression. Now, the most common technique is to try to fit a line that minimizes the squared distance to each of those points. And we're gonna talk more about that in future videos."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "But an interesting question is, can we try to fit a line to this data? And this idea of trying to fit a line as closely as possible to as many of the points as possible is known as linear regression. Now, the most common technique is to try to fit a line that minimizes the squared distance to each of those points. And we're gonna talk more about that in future videos. But for now, we wanna get an intuitive feel for that. So if you were to just eyeball it and look at a line like that, you wouldn't think that it would be a particularly good fit. It looks like most of the data sits above the line."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "And we're gonna talk more about that in future videos. But for now, we wanna get an intuitive feel for that. So if you were to just eyeball it and look at a line like that, you wouldn't think that it would be a particularly good fit. It looks like most of the data sits above the line. Similarly, something like this also doesn't look that great. Here, most of our data points are sitting below the line. But something like this actually looks very good."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "It looks like most of the data sits above the line. Similarly, something like this also doesn't look that great. Here, most of our data points are sitting below the line. But something like this actually looks very good. It looks like it's getting as close as possible to as many of the points as possible. It seems like it's describing this general trend. And so this is the actual regression line."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "But something like this actually looks very good. It looks like it's getting as close as possible to as many of the points as possible. It seems like it's describing this general trend. And so this is the actual regression line. And the equation here, we would write as, we would write y with a little hat over it. And that means that we are trying to estimate a y for a given x. It's not always going to be the actual y for a given x because as we see, sometimes the points aren't sitting on the line."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "And so this is the actual regression line. And the equation here, we would write as, we would write y with a little hat over it. And that means that we are trying to estimate a y for a given x. It's not always going to be the actual y for a given x because as we see, sometimes the points aren't sitting on the line. But we say y hat is equal to, and our y-intercept for this particular regression line, it is negative 140 plus the slope, 14 over three, times x. Now as we can see, for most of these points, given the x value of those points, the estimate that our regression line gives is different than the actual value. And that difference between the actual and the estimate from the regression line is known as the residual."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "It's not always going to be the actual y for a given x because as we see, sometimes the points aren't sitting on the line. But we say y hat is equal to, and our y-intercept for this particular regression line, it is negative 140 plus the slope, 14 over three, times x. Now as we can see, for most of these points, given the x value of those points, the estimate that our regression line gives is different than the actual value. And that difference between the actual and the estimate from the regression line is known as the residual. So let me write that down. So for example, the residual at that point, residual at that point, is going to be equal to, for a given x, the actual y value minus the estimated y value from the regression line for that same x. Or another way to think about it is, for that x value, when x is equal to 60, we're talking about the residual just at that point, it's going to be the actual y value minus our estimate of what the y value is from this regression line for that x value."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "And that difference between the actual and the estimate from the regression line is known as the residual. So let me write that down. So for example, the residual at that point, residual at that point, is going to be equal to, for a given x, the actual y value minus the estimated y value from the regression line for that same x. Or another way to think about it is, for that x value, when x is equal to 60, we're talking about the residual just at that point, it's going to be the actual y value minus our estimate of what the y value is from this regression line for that x value. So pause this video and see if you can calculate this residual, and you can visually imagine it as being this right over here. Well, to actually calculate the residual, you would take our actual value, which is 125, for that x value. Remember, we're calculating the residual for a point."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "Or another way to think about it is, for that x value, when x is equal to 60, we're talking about the residual just at that point, it's going to be the actual y value minus our estimate of what the y value is from this regression line for that x value. So pause this video and see if you can calculate this residual, and you can visually imagine it as being this right over here. Well, to actually calculate the residual, you would take our actual value, which is 125, for that x value. Remember, we're calculating the residual for a point. So it's the actual y there, minus what would be the estimated y there for that x value? Well, we could just go to this equation and say what would y hat be when x is equal to 60? Well, it's going to be equal to, let's see, we have negative 140, plus 14 over three times 60."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "Remember, we're calculating the residual for a point. So it's the actual y there, minus what would be the estimated y there for that x value? Well, we could just go to this equation and say what would y hat be when x is equal to 60? Well, it's going to be equal to, let's see, we have negative 140, plus 14 over three times 60. Let's see, 60 divided by three is 20. 20 times 14 is 280. And so all of this is going to be 140."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "Well, it's going to be equal to, let's see, we have negative 140, plus 14 over three times 60. Let's see, 60 divided by three is 20. 20 times 14 is 280. And so all of this is going to be 140. And so our residual for this point is gonna be 125 minus 140, which is negative 15. And residuals, indeed, can be negative. If your residual is negative, it means for that x value, your data point, your actual y value, is below the estimate."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "And so all of this is going to be 140. And so our residual for this point is gonna be 125 minus 140, which is negative 15. And residuals, indeed, can be negative. If your residual is negative, it means for that x value, your data point, your actual y value, is below the estimate. If we were to calculate the residual here, or if we were to calculate the residual here, our actual for that x value is above our estimate. So we would get positive residuals. And as you'll see later in your statistics career, the way that we calculate these regression lines is all about minimizing the square of these residuals."}, {"video_title": "Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "Assume that the conditions for inference were met. What is the approximate p-value for Katerina's test? So like always, pause this video and see if you can figure it out. Well, I just always like to remind ourselves what's going on here. So there's some population here. She has a null hypothesis that the mean is equal to zero, but the alternative is that it's not equal to zero. She wants to test her null hypothesis, so she takes a sample size, or she takes a sample of size six."}, {"video_title": "Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "Well, I just always like to remind ourselves what's going on here. So there's some population here. She has a null hypothesis that the mean is equal to zero, but the alternative is that it's not equal to zero. She wants to test her null hypothesis, so she takes a sample size, or she takes a sample of size six. From that, since we care about, the population parameter we care about is the population mean, she would calculate the sample mean in order to estimate that, and the sample standard deviation. And then from that, we can calculate this T-value. The T-value is going to be equal to the difference between her sample mean and the assumed, the assumed population mean from the null hypothesis, that's what this little sub zero means, it means it's the assumed mean from the null hypothesis, divided by our estimate of the standard deviation of the sampling distribution."}, {"video_title": "Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "She wants to test her null hypothesis, so she takes a sample size, or she takes a sample of size six. From that, since we care about, the population parameter we care about is the population mean, she would calculate the sample mean in order to estimate that, and the sample standard deviation. And then from that, we can calculate this T-value. The T-value is going to be equal to the difference between her sample mean and the assumed, the assumed population mean from the null hypothesis, that's what this little sub zero means, it means it's the assumed mean from the null hypothesis, divided by our estimate of the standard deviation of the sampling distribution. I say estimate because unlike when we were dealing with proportions, with proportions we can actually calculate the assumed based on the null hypothesis, sampling distribution standard deviation, but here we have to estimate it. And so it's going to be our sample standard deviation divided by the square root of N. Now in this example, they calculated all of this for us. They said, hey, this is going to be equal to 2.75."}, {"video_title": "Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "The T-value is going to be equal to the difference between her sample mean and the assumed, the assumed population mean from the null hypothesis, that's what this little sub zero means, it means it's the assumed mean from the null hypothesis, divided by our estimate of the standard deviation of the sampling distribution. I say estimate because unlike when we were dealing with proportions, with proportions we can actually calculate the assumed based on the null hypothesis, sampling distribution standard deviation, but here we have to estimate it. And so it's going to be our sample standard deviation divided by the square root of N. Now in this example, they calculated all of this for us. They said, hey, this is going to be equal to 2.75. And so we can just use that to figure out our P-value. But let's just think about what that is asking us to do. So the null hypothesis is that the mean is zero."}, {"video_title": "Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "They said, hey, this is going to be equal to 2.75. And so we can just use that to figure out our P-value. But let's just think about what that is asking us to do. So the null hypothesis is that the mean is zero. The alternative is that it is not equal to zero. So this is a situation where, if we're looking at the T-distribution right over here, it's my quick drawing of a T-distribution, if this is the mean of our T-distribution, what we care about is things that are at least 2.75 above the mean and at least 2.75 below the mean, because we care about things that are different from the mean, not just things that are greater than the mean or less than the mean. So we would look at, we would say, well, what's the probability of getting a T-value that is 2.75 or more above the mean?"}, {"video_title": "Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "So the null hypothesis is that the mean is zero. The alternative is that it is not equal to zero. So this is a situation where, if we're looking at the T-distribution right over here, it's my quick drawing of a T-distribution, if this is the mean of our T-distribution, what we care about is things that are at least 2.75 above the mean and at least 2.75 below the mean, because we care about things that are different from the mean, not just things that are greater than the mean or less than the mean. So we would look at, we would say, well, what's the probability of getting a T-value that is 2.75 or more above the mean? And similarly, what's the probability of getting a T-value that is 2.75 or less below, or 2.75 or more below the mean? So this is negative 2.75 right over there. So what we have here is a T-table, and a T-table is a little bit different than a Z-table because there's several things going on."}, {"video_title": "Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "So we would look at, we would say, well, what's the probability of getting a T-value that is 2.75 or more above the mean? And similarly, what's the probability of getting a T-value that is 2.75 or less below, or 2.75 or more below the mean? So this is negative 2.75 right over there. So what we have here is a T-table, and a T-table is a little bit different than a Z-table because there's several things going on. First of all, you have your degrees of freedom. That's just going to be your sample size minus one. So in this example, our sample size is six, so six minus one is five."}, {"video_title": "Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "So what we have here is a T-table, and a T-table is a little bit different than a Z-table because there's several things going on. First of all, you have your degrees of freedom. That's just going to be your sample size minus one. So in this example, our sample size is six, so six minus one is five. And so we are going to be, we are going to be in this row right over here. And then what you wanna do is you wanna look up your T-value. This is T distribution critical value."}, {"video_title": "Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "So in this example, our sample size is six, so six minus one is five. And so we are going to be, we are going to be in this row right over here. And then what you wanna do is you wanna look up your T-value. This is T distribution critical value. So we wanna look up 2.75 on this row. And we see 2.75, it's a little bit less than that, but that's the closest value. It's a good bit more than this right over here, but it's, so it's a little bit closer to this value than this value."}, {"video_title": "Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "This is T distribution critical value. So we wanna look up 2.75 on this row. And we see 2.75, it's a little bit less than that, but that's the closest value. It's a good bit more than this right over here, but it's, so it's a little bit closer to this value than this value. And so our tail probability, and remember, this is only giving us this probability right over here. Our tail probability is going to be between 0.025 and 0.02, and it's going to be closer than to this one. It's gonna be approximately this."}, {"video_title": "Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "It's a good bit more than this right over here, but it's, so it's a little bit closer to this value than this value. And so our tail probability, and remember, this is only giving us this probability right over here. Our tail probability is going to be between 0.025 and 0.02, and it's going to be closer than to this one. It's gonna be approximately this. It'll actually be a little bit greater, because we're gonna go a little bit in that direction, because we are less than 2.757. And so we could say this is approximately 0.02. Well, if that's 0.02 approximately, the T distribution's symmetric, this is going to be approximately 0.02."}, {"video_title": "Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "It's gonna be approximately this. It'll actually be a little bit greater, because we're gonna go a little bit in that direction, because we are less than 2.757. And so we could say this is approximately 0.02. Well, if that's 0.02 approximately, the T distribution's symmetric, this is going to be approximately 0.02. And so our P-value, which is going to be the probability of getting a T-value that is at least 2.75 above the mean and, or 2.75 below the mean, the P-value, P-value, is going to be approximately the sum of these areas, which is 0.04. And then, of course, Katarina would wanna compare that to her significance level that she set ahead of time. And if this is lower than that, then she would reject the null hypothesis, and that would suggest the alternative."}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "Sports utility vehicles, also known as SUVs, make up 12% of the vehicles she registers. Let V be the number of vehicles Amelia registers in a day until she first registers an SUV. Assume that the type of each vehicle is independent. Find the probability that Amelia registers more than four vehicles before she registers an SUV. Let's first think about what this random variable V is. It's the number of vehicles Amelia registers in a day until she registers an SUV. For example, if the first person who walks in the line or through the door has an SUV and they're trying to register it, then V would be equal to one."}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "Find the probability that Amelia registers more than four vehicles before she registers an SUV. Let's first think about what this random variable V is. It's the number of vehicles Amelia registers in a day until she registers an SUV. For example, if the first person who walks in the line or through the door has an SUV and they're trying to register it, then V would be equal to one. If the first person isn't an SUV but the second person is, then V would be equal to two, so forth and so on. So this right over here is a classic geometric random variable right over here. I'll say geometric random variable."}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "For example, if the first person who walks in the line or through the door has an SUV and they're trying to register it, then V would be equal to one. If the first person isn't an SUV but the second person is, then V would be equal to two, so forth and so on. So this right over here is a classic geometric random variable right over here. I'll say geometric random variable. We have a very clear success metric for each trial. Do we have an SUV or not? Each trial is independent."}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "I'll say geometric random variable. We have a very clear success metric for each trial. Do we have an SUV or not? Each trial is independent. They tell us that. They are independent. The probability of success in each trial is constant."}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "Each trial is independent. They tell us that. They are independent. The probability of success in each trial is constant. We have a 12% success for each new person who comes through the line. The reason why this is not a binomial random variable is that we do not have a finite number of trials. Here, we're going to keep performing trials."}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "The probability of success in each trial is constant. We have a 12% success for each new person who comes through the line. The reason why this is not a binomial random variable is that we do not have a finite number of trials. Here, we're going to keep performing trials. We're going to keep serving people in the line until we get an SUV. What we have over here, when they say find the probability that Amelia registers more than four vehicles before she registers an SUV, this is the probability that V is greater than four. I encourage you, like always, pause this video and see if you can work through it."}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "Here, we're going to keep performing trials. We're going to keep serving people in the line until we get an SUV. What we have over here, when they say find the probability that Amelia registers more than four vehicles before she registers an SUV, this is the probability that V is greater than four. I encourage you, like always, pause this video and see if you can work through it. We're going to assume that she's just not going to leave her desk or wherever the things are being registered. She's not going to leave the counter until someone shows up registering an SUV. We will just keep looking at people, I guess we could say, over multiple days, forever."}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "I encourage you, like always, pause this video and see if you can work through it. We're going to assume that she's just not going to leave her desk or wherever the things are being registered. She's not going to leave the counter until someone shows up registering an SUV. We will just keep looking at people, I guess we could say, over multiple days, forever. It will work for an infinite number of years, just for the sake of this problem, until an SUV actually shows up. Try to figure this out. I'm assuming you've had a go."}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "We will just keep looking at people, I guess we could say, over multiple days, forever. It will work for an infinite number of years, just for the sake of this problem, until an SUV actually shows up. Try to figure this out. I'm assuming you've had a go. Some of you might say, well, isn't this going to be equal to the probability that V is equal to five plus the probability that V is equal to six plus the probability that V is equal to seven, and it just goes on and on and on forever. This is actually true. You say, well, how do I calculate this?"}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "I'm assuming you've had a go. Some of you might say, well, isn't this going to be equal to the probability that V is equal to five plus the probability that V is equal to six plus the probability that V is equal to seven, and it just goes on and on and on forever. This is actually true. You say, well, how do I calculate this? It's just summing up an infinite number of things. Now, the key realization here is that one way to think about the probability that V is greater than four is this is the same thing as the probability that V is not less than or equal to four. These two things are equivalent."}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "You say, well, how do I calculate this? It's just summing up an infinite number of things. Now, the key realization here is that one way to think about the probability that V is greater than four is this is the same thing as the probability that V is not less than or equal to four. These two things are equivalent. What's the probability that V is not less than or equal to four? This might be a slightly easier thing for you to calculate. Once again, pause the video and see if you can figure it out."}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "These two things are equivalent. What's the probability that V is not less than or equal to four? This might be a slightly easier thing for you to calculate. Once again, pause the video and see if you can figure it out. What's the probability that V is not less than or equal to four? That's the same thing as the probability of first four customers or first four, I guess, people, first four, I'll say, customers, or I'll say first four cars, the customers' cars, not SUVs. This one is feeling pretty straightforward."}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "Once again, pause the video and see if you can figure it out. What's the probability that V is not less than or equal to four? That's the same thing as the probability of first four customers or first four, I guess, people, first four, I'll say, customers, or I'll say first four cars, the customers' cars, not SUVs. This one is feeling pretty straightforward. What's the probability that for each customer she goes to that they're not an SUV? That's one minus 12% or 88% or 0.88. If we want to know the probability that the first four cars are not SUVs, that's 0.88 to the fourth power."}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "This one is feeling pretty straightforward. What's the probability that for each customer she goes to that they're not an SUV? That's one minus 12% or 88% or 0.88. If we want to know the probability that the first four cars are not SUVs, that's 0.88 to the fourth power. That's all we have to calculate. Let's get our calculator out. I'm going to get 0.88, and I'm going to raise it to the fourth power."}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "If we want to know the probability that the first four cars are not SUVs, that's 0.88 to the fourth power. That's all we have to calculate. Let's get our calculator out. I'm going to get 0.88, and I'm going to raise it to the fourth power. I'm just going to round it to the nearest. Let's see. Do they tell me to round it?"}, {"video_title": "Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3", "Sentence": "I'm going to get 0.88, and I'm going to raise it to the fourth power. I'm just going to round it to the nearest. Let's see. Do they tell me to round it? Okay, I'll just round it to the nearest, I guess, well, hundredth. I'll just write it as 0.5997 is equal to or approximately equal to 0.5997. If you wanted to write this as a percentage, it would be approximately 59.97%."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "And what I want to do is think about the outliers. And to help us with that, let's actually visualize this, the distribution of actual numbers. So let us do that. So here on a number line, I have all the numbers from one to 19. And let's see. We have two ones. So I could say that's one one, and then two ones."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So here on a number line, I have all the numbers from one to 19. And let's see. We have two ones. So I could say that's one one, and then two ones. We have one six, so let's put that six there. We have got two 13s. So we're going to go up here."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So I could say that's one one, and then two ones. We have one six, so let's put that six there. We have got two 13s. So we're going to go up here. One 13 and two 13s. Let's see. We have three 14s."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So we're going to go up here. One 13 and two 13s. Let's see. We have three 14s. So 14, 14, and 14. We have a couple of 15s. 15, 15."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "We have three 14s. So 14, 14, and 14. We have a couple of 15s. 15, 15. So 15, 15. We have one 16. So that's our 16 there."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "15, 15. So 15, 15. We have one 16. So that's our 16 there. We have three 18s. One, two, three. So one, two, and then three."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So that's our 16 there. We have three 18s. One, two, three. So one, two, and then three. And then we have a 19. Then we have a 19. So when you look visually at the distribution of numbers, it looks like the meat of the distribution, so to speak, is in this area right over here."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So one, two, and then three. And then we have a 19. Then we have a 19. So when you look visually at the distribution of numbers, it looks like the meat of the distribution, so to speak, is in this area right over here. And so some people might say, OK, we have three outliers. These two ones and the six. Some people might say, well, the six is kind of close enough."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So when you look visually at the distribution of numbers, it looks like the meat of the distribution, so to speak, is in this area right over here. And so some people might say, OK, we have three outliers. These two ones and the six. Some people might say, well, the six is kind of close enough. Maybe only these two ones are outliers. And those would actually be both reasonable things to say. Now to get on the same page, statisticians will use a rule sometimes."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Some people might say, well, the six is kind of close enough. Maybe only these two ones are outliers. And those would actually be both reasonable things to say. Now to get on the same page, statisticians will use a rule sometimes. We say, well, anything that is more than 1 and 1 1 times the interquartile range from below Q1 or above Q3, well, those are going to be outliers. Well, what am I talking about? Well, actually, let's figure out the median, Q1 and Q3 here."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Now to get on the same page, statisticians will use a rule sometimes. We say, well, anything that is more than 1 and 1 1 times the interquartile range from below Q1 or above Q3, well, those are going to be outliers. Well, what am I talking about? Well, actually, let's figure out the median, Q1 and Q3 here. Then we can figure out the interquartile range. And then we can figure out by that definition what is going to be an outlier. And if that all made sense to you so far, I encourage you to pause this video and try to work through it on your own."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Well, actually, let's figure out the median, Q1 and Q3 here. Then we can figure out the interquartile range. And then we can figure out by that definition what is going to be an outlier. And if that all made sense to you so far, I encourage you to pause this video and try to work through it on your own. Or I'll do it for you right now. All right, so what's the median here? Well, the median is the middle number."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "And if that all made sense to you so far, I encourage you to pause this video and try to work through it on your own. Or I'll do it for you right now. All right, so what's the median here? Well, the median is the middle number. We have 15 numbers. So the middle number is going to be whatever number has seven on either side. So that's going to be the eighth number."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Well, the median is the middle number. We have 15 numbers. So the middle number is going to be whatever number has seven on either side. So that's going to be the eighth number. 1, 2, 3, 4, 5, 6, 7. Is that right? Yep, 6, 7."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So that's going to be the eighth number. 1, 2, 3, 4, 5, 6, 7. Is that right? Yep, 6, 7. So that's the median. And you have 1, 2, 3, 4, 5, 6, 7 numbers on the right side, too. So that is the median, sometimes called Q2."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Yep, 6, 7. So that's the median. And you have 1, 2, 3, 4, 5, 6, 7 numbers on the right side, too. So that is the median, sometimes called Q2. That is our median. Now, what is Q1? Well, Q1 is going to be the middle of this first group."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So that is the median, sometimes called Q2. That is our median. Now, what is Q1? Well, Q1 is going to be the middle of this first group. This first group has seven numbers in it. And so the middle is going to be the fourth number. It has 3 and 3."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Well, Q1 is going to be the middle of this first group. This first group has seven numbers in it. And so the middle is going to be the fourth number. It has 3 and 3. 3 to the left, 3 to the right. So that is Q1. And then Q3 is going to be the middle of this upper group."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "It has 3 and 3. 3 to the left, 3 to the right. So that is Q1. And then Q3 is going to be the middle of this upper group. Well, that also has seven numbers in it. So the middle is going to be right over there. It has 3 on either side."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "And then Q3 is going to be the middle of this upper group. Well, that also has seven numbers in it. So the middle is going to be right over there. It has 3 on either side. So that is Q3. Now, what is the interquartile range going to be? Interquartile range is going to be equal to Q3 minus Q1."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "It has 3 on either side. So that is Q3. Now, what is the interquartile range going to be? Interquartile range is going to be equal to Q3 minus Q1. The difference between 18 and 13. Between 18 and 13, well, that is going to be 18 minus 13, which is equal to 5. Now, to figure out outliers, well, outliers are going to be anything that is below."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Interquartile range is going to be equal to Q3 minus Q1. The difference between 18 and 13. Between 18 and 13, well, that is going to be 18 minus 13, which is equal to 5. Now, to figure out outliers, well, outliers are going to be anything that is below. So outliers are going to be less than our Q1 minus 1.5 times our interquartile range. And once again, this isn't some rule of the universe. This is something that statisticians have kind of said, well, if we want to have a better definition for outliers, let's just agree that it's something that's more than 1.5 times the interquartile range below Q1."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Now, to figure out outliers, well, outliers are going to be anything that is below. So outliers are going to be less than our Q1 minus 1.5 times our interquartile range. And once again, this isn't some rule of the universe. This is something that statisticians have kind of said, well, if we want to have a better definition for outliers, let's just agree that it's something that's more than 1.5 times the interquartile range below Q1. Or an outlier could be greater than Q3 plus 1.5 times the interquartile range. And once again, this is somewhat, people just decided it felt right. One could argue it should be 1.6."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "This is something that statisticians have kind of said, well, if we want to have a better definition for outliers, let's just agree that it's something that's more than 1.5 times the interquartile range below Q1. Or an outlier could be greater than Q3 plus 1.5 times the interquartile range. And once again, this is somewhat, people just decided it felt right. One could argue it should be 1.6. Or one could argue it should be 1 or 2 or whatever. But this is what people have tended to agree on. So let's think about what these numbers are."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "One could argue it should be 1.6. Or one could argue it should be 1 or 2 or whatever. But this is what people have tended to agree on. So let's think about what these numbers are. Q1, we already know. So this is going to be 13 minus 1.5 times our interquartile range. Our interquartile range here is 5."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So let's think about what these numbers are. Q1, we already know. So this is going to be 13 minus 1.5 times our interquartile range. Our interquartile range here is 5. So it's 1.5 times 5, which is 7.5. So this is 7.5. 13 minus 7.5 is what?"}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Our interquartile range here is 5. So it's 1.5 times 5, which is 7.5. So this is 7.5. 13 minus 7.5 is what? 13 minus 7 is 6. And then you subtract another 0.5 is 5.5. So we have outliers would be less than 5.5."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "13 minus 7.5 is what? 13 minus 7 is 6. And then you subtract another 0.5 is 5.5. So we have outliers would be less than 5.5. Or Q3 is 18. This is, once again, 7.5. 18 plus 7.5 is 25.5."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So we have outliers would be less than 5.5. Or Q3 is 18. This is, once again, 7.5. 18 plus 7.5 is 25.5. Or outliers greater than 25.5. So based on this, we have a numerical definition for what's an outlier. We're not just subjectively saying, oh, this feels right or that feels right."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "18 plus 7.5 is 25.5. Or outliers greater than 25.5. So based on this, we have a numerical definition for what's an outlier. We're not just subjectively saying, oh, this feels right or that feels right. And based on this, we only have two outliers, that only these two ones are less than 5.5. This is the cutoff right over here. So this dot just happened to make it."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "We're not just subjectively saying, oh, this feels right or that feels right. And based on this, we only have two outliers, that only these two ones are less than 5.5. This is the cutoff right over here. So this dot just happened to make it. And we don't have any outliers on the high side. Now, another thing to think about is drawing box and whiskers plots based on Q1, our median, our range, all the range of numbers. And you could do it either taking in consideration your outliers or not taking into consideration your outliers."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So this dot just happened to make it. And we don't have any outliers on the high side. Now, another thing to think about is drawing box and whiskers plots based on Q1, our median, our range, all the range of numbers. And you could do it either taking in consideration your outliers or not taking into consideration your outliers. So there's a couple of ways that we can do it. So let me actually clear all of this. We've figured out all of this stuff."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "And you could do it either taking in consideration your outliers or not taking into consideration your outliers. So there's a couple of ways that we can do it. So let me actually clear all of this. We've figured out all of this stuff. So let me clear all of that out. And let's actually draw a box and whiskers plot. So I'll put another, actually, let me do two here."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "We've figured out all of this stuff. So let me clear all of that out. And let's actually draw a box and whiskers plot. So I'll put another, actually, let me do two here. That's one. And then let me put another one down there. This is another."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So I'll put another, actually, let me do two here. That's one. And then let me put another one down there. This is another. Now, if we were to just draw a classic box and whiskers plot here, we would say, all right, our median's at 14. And actually, I'll do it both ways. Median's at 14."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "This is another. Now, if we were to just draw a classic box and whiskers plot here, we would say, all right, our median's at 14. And actually, I'll do it both ways. Median's at 14. Q1's at 13. Q3 is at 18. So that's the box part."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Median's at 14. Q1's at 13. Q3 is at 18. So that's the box part. And let me draw that as an actual, let me actually draw that as a box. So my best attempt. There you go."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So that's the box part. And let me draw that as an actual, let me actually draw that as a box. So my best attempt. There you go. That's the box. And this is also a box. So far, I'm doing the exact same thing."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "There you go. That's the box. And this is also a box. So far, I'm doing the exact same thing. Now, if we don't want to consider outliers, we would say, well, what's the entire range here? Well, we have things that go from 1 all the way to 19. So one way to do it is to say, hey, we start at 1."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So far, I'm doing the exact same thing. Now, if we don't want to consider outliers, we would say, well, what's the entire range here? Well, we have things that go from 1 all the way to 19. So one way to do it is to say, hey, we start at 1. And so our entire range, we go, actually, let me draw it a little bit better than that. We're going all the way from 1 to 19. Now, in this one, we're including everything."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So one way to do it is to say, hey, we start at 1. And so our entire range, we go, actually, let me draw it a little bit better than that. We're going all the way from 1 to 19. Now, in this one, we're including everything. We're including even these two outliers. But if we don't want to include those outliers, we want to make it clear that they're outliers, well, let's not include them. And what we can do instead is say, all right, including our non-outliers, we would start at 6."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Now, in this one, we're including everything. We're including even these two outliers. But if we don't want to include those outliers, we want to make it clear that they're outliers, well, let's not include them. And what we can do instead is say, all right, including our non-outliers, we would start at 6. Because 6, we're saying, is in our data set. But it is not an outlier. Let me make this look better."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "And what we can do instead is say, all right, including our non-outliers, we would start at 6. Because 6, we're saying, is in our data set. But it is not an outlier. Let me make this look better. So we are going to start at 6 and go all the way to 19. And then to say that we have these outliers, we would put this, we have outliers over there. So once again, this is a box and whiskers plot of the same data set without outliers."}, {"video_title": "Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3", "Sentence": "Here is computer output from Elise Square's regression analysis on her sample. So just to be clear what's going on, she took a sample of phones, they're not telling us exactly how many, but she took a number of phones and she found a linear relationship between processor speed and prices. So this is price right over here and this is processor speed right over here. And then she plotted her sample for every phone would be a data point. And so you see that. And then she put those data points into her computer and was able to come up with a line, a regression line for her sample. And her regression line for her sample, if we say that's going to be y is, or y hat is going to be a plus bx."}, {"video_title": "Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3", "Sentence": "And then she plotted her sample for every phone would be a data point. And so you see that. And then she put those data points into her computer and was able to come up with a line, a regression line for her sample. And her regression line for her sample, if we say that's going to be y is, or y hat is going to be a plus bx. For her sample, a is going to be 127.092, so that's that over there. And for her sample, the slope of the regression line is going to be the coefficient on speed. Another way to think about it, this x variable right over here, speed, so the coefficient on that is the slope."}, {"video_title": "Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3", "Sentence": "And her regression line for her sample, if we say that's going to be y is, or y hat is going to be a plus bx. For her sample, a is going to be 127.092, so that's that over there. And for her sample, the slope of the regression line is going to be the coefficient on speed. Another way to think about it, this x variable right over here, speed, so the coefficient on that is the slope. But we have to remind ourselves that these are estimates of maybe some true truth in the universe. If she were able to sample every phone in the market, then she would get the true population parameters. But since this is a sample, it's just an estimate."}, {"video_title": "Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3", "Sentence": "Another way to think about it, this x variable right over here, speed, so the coefficient on that is the slope. But we have to remind ourselves that these are estimates of maybe some true truth in the universe. If she were able to sample every phone in the market, then she would get the true population parameters. But since this is a sample, it's just an estimate. And just because she sees this positive linear relationship in her sample doesn't necessarily mean that this is the case for the entire population. She might have just happened to sample things that had this positive linear relationship. And so that's why she's doing this hypothesis test."}, {"video_title": "Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3", "Sentence": "But since this is a sample, it's just an estimate. And just because she sees this positive linear relationship in her sample doesn't necessarily mean that this is the case for the entire population. She might have just happened to sample things that had this positive linear relationship. And so that's why she's doing this hypothesis test. And in a hypothesis test, you actually assume that there isn't a relationship between processor speed and price. So beta right over here, this would be the true population parameter for regression on the population. So if this is the population right over here, and if somehow where it's price on the vertical axis and processor speed on the horizontal axis, and if you were able to look at the entire population, I don't know how many phones there are, but it might be billions of phones, and then do a regression line, our null hypothesis is that the slope of the regression line is going to be zero."}, {"video_title": "Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3", "Sentence": "And so that's why she's doing this hypothesis test. And in a hypothesis test, you actually assume that there isn't a relationship between processor speed and price. So beta right over here, this would be the true population parameter for regression on the population. So if this is the population right over here, and if somehow where it's price on the vertical axis and processor speed on the horizontal axis, and if you were able to look at the entire population, I don't know how many phones there are, but it might be billions of phones, and then do a regression line, our null hypothesis is that the slope of the regression line is going to be zero. So the regression line might look something like that. Where the equation of the regression line for the population, Y hat, would be alpha plus beta times X. And so our null hypothesis is that beta is equal to zero."}, {"video_title": "Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3", "Sentence": "So if this is the population right over here, and if somehow where it's price on the vertical axis and processor speed on the horizontal axis, and if you were able to look at the entire population, I don't know how many phones there are, but it might be billions of phones, and then do a regression line, our null hypothesis is that the slope of the regression line is going to be zero. So the regression line might look something like that. Where the equation of the regression line for the population, Y hat, would be alpha plus beta times X. And so our null hypothesis is that beta is equal to zero. And the alternative hypothesis, which is her suspicion, is that the true slope of the regression line is actually greater than zero. Assume that all conditions for inference have been met. At the alpha equals 0.01 level of significance, is there sufficient evidence to conclude a positive linear relationship between these variables for all mobile phones, Y?"}, {"video_title": "Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3", "Sentence": "And so our null hypothesis is that beta is equal to zero. And the alternative hypothesis, which is her suspicion, is that the true slope of the regression line is actually greater than zero. Assume that all conditions for inference have been met. At the alpha equals 0.01 level of significance, is there sufficient evidence to conclude a positive linear relationship between these variables for all mobile phones, Y? So pause this video and see if you can have a go at it. Well, in order to do this hypothesis test, we have to say, well, assuming the null hypothesis is true, assuming this is the actual slope of the population regression line, I guess you could think about it, what is the probability of us getting this result right over here? And what we can do is use this information and our estimate of the sampling distribution of the sample regression line slope, and we can come up with a t-statistic."}, {"video_title": "Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3", "Sentence": "At the alpha equals 0.01 level of significance, is there sufficient evidence to conclude a positive linear relationship between these variables for all mobile phones, Y? So pause this video and see if you can have a go at it. Well, in order to do this hypothesis test, we have to say, well, assuming the null hypothesis is true, assuming this is the actual slope of the population regression line, I guess you could think about it, what is the probability of us getting this result right over here? And what we can do is use this information and our estimate of the sampling distribution of the sample regression line slope, and we can come up with a t-statistic. And for this situation, where our alternative hypothesis is that our true population regression slope is greater than zero, our p-value can be viewed as the probability of getting a t-statistic greater than or equal to this. So getting a t-statistic greater than or equal to 2.999. Now, you could be tempted to say, hey, look, there's this column that gives us a p-value."}, {"video_title": "Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3", "Sentence": "And what we can do is use this information and our estimate of the sampling distribution of the sample regression line slope, and we can come up with a t-statistic. And for this situation, where our alternative hypothesis is that our true population regression slope is greater than zero, our p-value can be viewed as the probability of getting a t-statistic greater than or equal to this. So getting a t-statistic greater than or equal to 2.999. Now, you could be tempted to say, hey, look, there's this column that gives us a p-value. Maybe they just figured out for us that this probability is 0.004. And we have to be very, very careful here, because here, they're actually giving us, I guess you could call it a two-sided p-value. If you think of a t-distribution, and they would do it for the appropriate degrees of freedom, this is saying, what's the probability of getting a result where the absolute value is 2.999 or greater?"}, {"video_title": "Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3", "Sentence": "Now, you could be tempted to say, hey, look, there's this column that gives us a p-value. Maybe they just figured out for us that this probability is 0.004. And we have to be very, very careful here, because here, they're actually giving us, I guess you could call it a two-sided p-value. If you think of a t-distribution, and they would do it for the appropriate degrees of freedom, this is saying, what's the probability of getting a result where the absolute value is 2.999 or greater? So if this is t equals zero right here in the middle, and this is 2.999, we care about this region. We care about this right tail. This p-value right over here, this is giving us not just the right tail, but it's also saying, well, what about getting something less than negative 2.999, or including negative 2.999?"}, {"video_title": "Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3", "Sentence": "If you think of a t-distribution, and they would do it for the appropriate degrees of freedom, this is saying, what's the probability of getting a result where the absolute value is 2.999 or greater? So if this is t equals zero right here in the middle, and this is 2.999, we care about this region. We care about this right tail. This p-value right over here, this is giving us not just the right tail, but it's also saying, well, what about getting something less than negative 2.999, or including negative 2.999? So it's giving us both of these areas. So if you want the p-value for this scenario, we would just look at this. And as you can see, because this distribution is symmetric, the t-distribution is going to be symmetric, you take half of this."}, {"video_title": "Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3", "Sentence": "This p-value right over here, this is giving us not just the right tail, but it's also saying, well, what about getting something less than negative 2.999, or including negative 2.999? So it's giving us both of these areas. So if you want the p-value for this scenario, we would just look at this. And as you can see, because this distribution is symmetric, the t-distribution is going to be symmetric, you take half of this. So this is going to be equal to 0.002. And what you do in any significance test is then compare your p-value to your level of significance. And so if you look at 0.002 and compare it to 0.01, which of these is greater?"}, {"video_title": "Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3", "Sentence": "And as you can see, because this distribution is symmetric, the t-distribution is going to be symmetric, you take half of this. So this is going to be equal to 0.002. And what you do in any significance test is then compare your p-value to your level of significance. And so if you look at 0.002 and compare it to 0.01, which of these is greater? Well, at first your eyes might say, hey, two is greater than one, but this is 2,000th versus 1,000th. This is 10,000th right over here. So in this situation, our p-value is less than our level of significance."}, {"video_title": "Using a P-value to make conclusions in a test about slope AP Statistics Khan Academy.mp3", "Sentence": "And so if you look at 0.002 and compare it to 0.01, which of these is greater? Well, at first your eyes might say, hey, two is greater than one, but this is 2,000th versus 1,000th. This is 10,000th right over here. So in this situation, our p-value is less than our level of significance. And so we're saying, hey, the probability of getting a result this extreme or more extreme is so low if we assume our null hypothesis that in this situation we will reject, we will decide to reject our null hypothesis, which would suggest the alternative. So is there sufficient evidence to conclude a positive linear relationship between these variables for all mobile phones? Yes."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Let's imagine ourselves in some type of a strange casino with very strange games. And you walk up to a table, and on that table there is an empty bag. And the guy who runs the table says, look, I've got some marbles here, three green marbles, two orange marbles, and I'm going to stick them in the bag. And he literally sticks them into the empty bag to show you that it's truly three green marbles and two orange marbles. And he says, the game that I want you to play, or that if you choose to play, is you're going to look away, stick your hand in this bag. The bag is not transparent. Feel around the marbles."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "And he literally sticks them into the empty bag to show you that it's truly three green marbles and two orange marbles. And he says, the game that I want you to play, or that if you choose to play, is you're going to look away, stick your hand in this bag. The bag is not transparent. Feel around the marbles. All the marbles feel exactly the same. And if you're able to pick two green marbles, if you're able to take one marble out of the bag, it's green, you put it down on the table, then put your hand back in the bag and take another marble. And if that one is also green, then you're going to win the prize."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Feel around the marbles. All the marbles feel exactly the same. And if you're able to pick two green marbles, if you're able to take one marble out of the bag, it's green, you put it down on the table, then put your hand back in the bag and take another marble. And if that one is also green, then you're going to win the prize. You're going to win the prize of $1 if you get two greens. We say, well, this sounds like an interesting game. How much does it cost to play?"}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "And if that one is also green, then you're going to win the prize. You're going to win the prize of $1 if you get two greens. We say, well, this sounds like an interesting game. How much does it cost to play? And the guy tells you it is $0.35 to play. So obviously, fairly low stakes casino. So my question to you is, would you want to play this game?"}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "How much does it cost to play? And the guy tells you it is $0.35 to play. So obviously, fairly low stakes casino. So my question to you is, would you want to play this game? And don't put the fun factor into it. Just economically, does it make sense for you to actually play this game? Well, let's think through the probabilities a little bit."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "So my question to you is, would you want to play this game? And don't put the fun factor into it. Just economically, does it make sense for you to actually play this game? Well, let's think through the probabilities a little bit. So first of all, what's the probability that the first marble you pick is green? What's the probability that first marble is green? Actually, let me just write first green."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Well, let's think through the probabilities a little bit. So first of all, what's the probability that the first marble you pick is green? What's the probability that first marble is green? Actually, let me just write first green. Probability first green. Well, the total possible outcomes, there's five marbles here, all equally likely. So there's five possible outcomes."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Actually, let me just write first green. Probability first green. Well, the total possible outcomes, there's five marbles here, all equally likely. So there's five possible outcomes. Three of them satisfy your event that the first is green. So there's a 3 5th probability that the first is green. So you have a 3 5th chance, 3 5th probability, I should say, that after that first pick, you're kind of still in the game."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "So there's five possible outcomes. Three of them satisfy your event that the first is green. So there's a 3 5th probability that the first is green. So you have a 3 5th chance, 3 5th probability, I should say, that after that first pick, you're kind of still in the game. Now, what we really care about is your probability of winning the game. You want the first to be green and the second green. Well, let's think about this a little bit."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "So you have a 3 5th chance, 3 5th probability, I should say, that after that first pick, you're kind of still in the game. Now, what we really care about is your probability of winning the game. You want the first to be green and the second green. Well, let's think about this a little bit. What is the probability that the first is green? First, I'll just write g for green. And the second is green."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Well, let's think about this a little bit. What is the probability that the first is green? First, I'll just write g for green. And the second is green. Now, you might be tempted to say, oh, well, maybe the second being green is the same probability. It's 3 5ths. I can just multiply 3 5ths times 3 5ths, and I'll get 9 over 25."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "And the second is green. Now, you might be tempted to say, oh, well, maybe the second being green is the same probability. It's 3 5ths. I can just multiply 3 5ths times 3 5ths, and I'll get 9 over 25. Seems like a pretty straightforward thing. But the realization here is what you do with that first green marble. You don't take that first green marble out, look at it, and put it back in the bag."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "I can just multiply 3 5ths times 3 5ths, and I'll get 9 over 25. Seems like a pretty straightforward thing. But the realization here is what you do with that first green marble. You don't take that first green marble out, look at it, and put it back in the bag. So when you take that second pick, the number of green marbles that are in the bag depends on what you got on the first pick. Remember, we take the marble out. If it's a green marble, whatever marble it is, at whatever after the first pick, we leave it on the table."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "You don't take that first green marble out, look at it, and put it back in the bag. So when you take that second pick, the number of green marbles that are in the bag depends on what you got on the first pick. Remember, we take the marble out. If it's a green marble, whatever marble it is, at whatever after the first pick, we leave it on the table. We are not replacing it. So there's not any replacement here. So these are not independent events."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "If it's a green marble, whatever marble it is, at whatever after the first pick, we leave it on the table. We are not replacing it. So there's not any replacement here. So these are not independent events. Let me make this clear. Not independent. Or in particular, the second pick is dependent on the first."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "So these are not independent events. Let me make this clear. Not independent. Or in particular, the second pick is dependent on the first. If the first pick is green, then you don't have three green marbles in a bag of five. If the first pick is green, you now have two green marbles in a bag of four. So the way that we would refer to this, the probability of both of these happening, yes, it's definitely equal to the probability of the first green times, now this is kind of the new idea, the probability of the second green given, this little line right over here, just this straight up vertical line, just means given that the first was green."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Or in particular, the second pick is dependent on the first. If the first pick is green, then you don't have three green marbles in a bag of five. If the first pick is green, you now have two green marbles in a bag of four. So the way that we would refer to this, the probability of both of these happening, yes, it's definitely equal to the probability of the first green times, now this is kind of the new idea, the probability of the second green given, this little line right over here, just this straight up vertical line, just means given that the first was green. Now what is the probability that the second marble is green given that the first marble was green? Well, we draw through the scenario right over here. If the first marble is green, there are four possible outcomes, not five anymore."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "So the way that we would refer to this, the probability of both of these happening, yes, it's definitely equal to the probability of the first green times, now this is kind of the new idea, the probability of the second green given, this little line right over here, just this straight up vertical line, just means given that the first was green. Now what is the probability that the second marble is green given that the first marble was green? Well, we draw through the scenario right over here. If the first marble is green, there are four possible outcomes, not five anymore. And two of them satisfy your criteria. So the probability of the first marble being green and the second marble being green is going to be the probability that your first is green, so it's going to be 3 5ths, times the probability that the second is green given that the first was green. Now you have one less marble in the bag, and we're assuming that the first pick was green, so you only have two green marbles left."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "If the first marble is green, there are four possible outcomes, not five anymore. And two of them satisfy your criteria. So the probability of the first marble being green and the second marble being green is going to be the probability that your first is green, so it's going to be 3 5ths, times the probability that the second is green given that the first was green. Now you have one less marble in the bag, and we're assuming that the first pick was green, so you only have two green marbles left. And so what does this give us for our total probability? Well, let's see, 3 5ths times 2 4ths. Well, 2 4ths is the same thing as 1 half."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Now you have one less marble in the bag, and we're assuming that the first pick was green, so you only have two green marbles left. And so what does this give us for our total probability? Well, let's see, 3 5ths times 2 4ths. Well, 2 4ths is the same thing as 1 half. This is going to be equal to 3 5ths times 1 half, which is equal to 3 tenths. Or we could write that as 0.30. Or we could say there's a 30% chance of picking two green marbles when we are not replacing."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Well, 2 4ths is the same thing as 1 half. This is going to be equal to 3 5ths times 1 half, which is equal to 3 tenths. Or we could write that as 0.30. Or we could say there's a 30% chance of picking two green marbles when we are not replacing. So given that, let me ask you the question again. Would you want to play this game? Well, if you played this game many, many, many, many, many times, on average, you have a 30% chance of winning $1."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Or we could say there's a 30% chance of picking two green marbles when we are not replacing. So given that, let me ask you the question again. Would you want to play this game? Well, if you played this game many, many, many, many, many times, on average, you have a 30% chance of winning $1. And we haven't covered this yet, but so your expected value is really going to be 30% times $1. This gives you a little bit of a preview, which is going to be $0.30. 30% chance of winning $1."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Well, if you played this game many, many, many, many, many times, on average, you have a 30% chance of winning $1. And we haven't covered this yet, but so your expected value is really going to be 30% times $1. This gives you a little bit of a preview, which is going to be $0.30. 30% chance of winning $1. You would expect, on average, if you play this many, many, many times, that playing the game is going to give you $0.30. Now, would you want to give someone $0.35 to get, on average, $0.30? No, you would not want to play this game."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "30% chance of winning $1. You would expect, on average, if you play this many, many, many times, that playing the game is going to give you $0.30. Now, would you want to give someone $0.35 to get, on average, $0.30? No, you would not want to play this game. Now, one thing I will let you think about is, would you want to play this game if you could replace the green marble, the first pick after the first pick? If you could replace the green marble, would you want to pick? Would you want to play the game in that scenario?"}, {"video_title": "Estimating a P-value from a simulation AP Statistics Khan Academy.mp3", "Sentence": "To test her theory, Evie took a random sample of 25 students at her school, and 20% of them were vegetarians. So just from this first paragraph, some interesting things are being said. It's saying that the true population proportion, if we believe this article, of teenagers that are vegetarian, we could say that is 6%. Now, for her school, there is a null hypothesis that the proportion of students at her school that are vegetarian, so this is at her school, that the true proportion, the null would be it's just the same as the proportion of teenagers as a whole. So that would be the null hypothesis. And you can see that she's generating an alternative hypothesis, but she thinks it's higher for students at her large school. So her alternative hypothesis would be the proportion, the true population parameter for her school, school, is greater than 6%."}, {"video_title": "Estimating a P-value from a simulation AP Statistics Khan Academy.mp3", "Sentence": "Now, for her school, there is a null hypothesis that the proportion of students at her school that are vegetarian, so this is at her school, that the true proportion, the null would be it's just the same as the proportion of teenagers as a whole. So that would be the null hypothesis. And you can see that she's generating an alternative hypothesis, but she thinks it's higher for students at her large school. So her alternative hypothesis would be the proportion, the true population parameter for her school, school, is greater than 6%. And so to see whether or not you could reject the null hypothesis, you take a sample, and that's exactly what Evie did. She took a random sample of 25 students, and you calculate the sample proportion. And then you figure out what is the probability of getting a sample proportion this high or greater."}, {"video_title": "Estimating a P-value from a simulation AP Statistics Khan Academy.mp3", "Sentence": "So her alternative hypothesis would be the proportion, the true population parameter for her school, school, is greater than 6%. And so to see whether or not you could reject the null hypothesis, you take a sample, and that's exactly what Evie did. She took a random sample of 25 students, and you calculate the sample proportion. And then you figure out what is the probability of getting a sample proportion this high or greater. And if it's lower than a threshold, then you will reject your null hypothesis. And that probability, we call the p-value. The p-value is equal to the probability that your sample proportion, and she's doing this for students at her school, is going to be greater than or equal to 20% if you assumed that your null hypothesis was true."}, {"video_title": "Estimating a P-value from a simulation AP Statistics Khan Academy.mp3", "Sentence": "And then you figure out what is the probability of getting a sample proportion this high or greater. And if it's lower than a threshold, then you will reject your null hypothesis. And that probability, we call the p-value. The p-value is equal to the probability that your sample proportion, and she's doing this for students at her school, is going to be greater than or equal to 20% if you assumed that your null hypothesis was true. So if you assumed that the true proportion at your school was 6% vegetarians, but you took a sample of 25 students where you got 20%, what is the probability of getting 20% or greater for a sample of 25? Now, there's many ways to approach it, but it looks like she is using a simulation. To see how likely a sample like this was to happen by random chance alone, Evie performed a simulation."}, {"video_title": "Estimating a P-value from a simulation AP Statistics Khan Academy.mp3", "Sentence": "The p-value is equal to the probability that your sample proportion, and she's doing this for students at her school, is going to be greater than or equal to 20% if you assumed that your null hypothesis was true. So if you assumed that the true proportion at your school was 6% vegetarians, but you took a sample of 25 students where you got 20%, what is the probability of getting 20% or greater for a sample of 25? Now, there's many ways to approach it, but it looks like she is using a simulation. To see how likely a sample like this was to happen by random chance alone, Evie performed a simulation. She simulated 40 samples of n equals 25 students from a large population, where 6% of the students were vegetarian. She recorded the proportion of vegetarians in each sample. Here are the sample proportions from her 40 samples."}, {"video_title": "Estimating a P-value from a simulation AP Statistics Khan Academy.mp3", "Sentence": "To see how likely a sample like this was to happen by random chance alone, Evie performed a simulation. She simulated 40 samples of n equals 25 students from a large population, where 6% of the students were vegetarian. She recorded the proportion of vegetarians in each sample. Here are the sample proportions from her 40 samples. So what she's doing here with the simulation, this is an approximation of the sampling distribution of the sample proportions if you were to assume that your null hypothesis is true. And it says below, Evie wants to test her null hypothesis, which is that the true proportion at her school is 6% versus the alternative hypothesis that the true proportion at her school is greater than 6%, where P is the true proportion of students who are vegetarian at her school. And then they ask us, based on these simulated results, what is the approximate P value of the test?"}, {"video_title": "Estimating a P-value from a simulation AP Statistics Khan Academy.mp3", "Sentence": "Here are the sample proportions from her 40 samples. So what she's doing here with the simulation, this is an approximation of the sampling distribution of the sample proportions if you were to assume that your null hypothesis is true. And it says below, Evie wants to test her null hypothesis, which is that the true proportion at her school is 6% versus the alternative hypothesis that the true proportion at her school is greater than 6%, where P is the true proportion of students who are vegetarian at her school. And then they ask us, based on these simulated results, what is the approximate P value of the test? And they say the sample result, the sample proportion here was 20%. We saw that right over here. Well, if we assume that this is a reasonably good approximation of our sampling distribution of our sample proportions, there's 40 data points here."}, {"video_title": "Estimating a P-value from a simulation AP Statistics Khan Academy.mp3", "Sentence": "And then they ask us, based on these simulated results, what is the approximate P value of the test? And they say the sample result, the sample proportion here was 20%. We saw that right over here. Well, if we assume that this is a reasonably good approximation of our sampling distribution of our sample proportions, there's 40 data points here. And how many of these samples do we get a sample proportion that is greater than or equal to 20%? Well, you could see this is 20% right over here, 20 hundredths. And so you see we have three right over here that meet this constraint."}, {"video_title": "Estimating a P-value from a simulation AP Statistics Khan Academy.mp3", "Sentence": "Well, if we assume that this is a reasonably good approximation of our sampling distribution of our sample proportions, there's 40 data points here. And how many of these samples do we get a sample proportion that is greater than or equal to 20%? Well, you could see this is 20% right over here, 20 hundredths. And so you see we have three right over here that meet this constraint. And so that is three out of 40. So if we think this is a reasonably good approximation, we would say that our P value is going to be approximately three out of 40. That if the true population proportion for the school were 6%, if the null hypothesis were true, then approximately three out of every 40 times you would expect to get a sample with 20% or larger being vegetarians."}, {"video_title": "Estimating a P-value from a simulation AP Statistics Khan Academy.mp3", "Sentence": "And so you see we have three right over here that meet this constraint. And so that is three out of 40. So if we think this is a reasonably good approximation, we would say that our P value is going to be approximately three out of 40. That if the true population proportion for the school were 6%, if the null hypothesis were true, then approximately three out of every 40 times you would expect to get a sample with 20% or larger being vegetarians. And so three 40ths is what? Let's see. If I multiply both the numerator and the denominator by two and a half, this is approximately equal to, I say two and a half because to go from 40 to 100, and then two and a half times three would be 7.5."}, {"video_title": "Estimating a P-value from a simulation AP Statistics Khan Academy.mp3", "Sentence": "That if the true population proportion for the school were 6%, if the null hypothesis were true, then approximately three out of every 40 times you would expect to get a sample with 20% or larger being vegetarians. And so three 40ths is what? Let's see. If I multiply both the numerator and the denominator by two and a half, this is approximately equal to, I say two and a half because to go from 40 to 100, and then two and a half times three would be 7.5. So I would say this is approximately 7.5%. And this is actually a multiple choice question. And if we scroll down, we do indeed see approximately 7.5% right over there."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Note that mobile phones and unlisted numbers are not in phone books. The senator's office called those numbers until they got a response from all 100 people chosen. The poll showed that 42% of respondents were very concerned about internet privacy. What is the most concerning source of bias in this scenario? And we should also think about, well, what kind of bias is that likely to introduce? Is this likely to be an overestimate or an underestimate of the number of respondents? And maybe there is no bias here."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "What is the most concerning source of bias in this scenario? And we should also think about, well, what kind of bias is that likely to introduce? Is this likely to be an overestimate or an underestimate of the number of respondents? And maybe there is no bias here. But our choices, and no bias is not one of the choices, so you can imagine, it's going to be one of these three. So I encourage you to pause this video and think about what we just said. We're a senator, we're trying to figure out what percentage of our constituents are very concerned about internet privacy."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And maybe there is no bias here. But our choices, and no bias is not one of the choices, so you can imagine, it's going to be one of these three. So I encourage you to pause this video and think about what we just said. We're a senator, we're trying to figure out what percentage of our constituents are very concerned about internet privacy. And we go to the phone book, we sample 100 people, we keep calling them until they answer, and we get that 42% are very concerned. So what's the source of bias? All right, now let's work through this together."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "We're a senator, we're trying to figure out what percentage of our constituents are very concerned about internet privacy. And we go to the phone book, we sample 100 people, we keep calling them until they answer, and we get that 42% are very concerned. So what's the source of bias? All right, now let's work through this together. So non-response would have been the case if we selected these 100 people, and let's say only 50 people answered the phone and we didn't keep calling them. Then we'd say, well, 50 of the people who we sampled to answer our survey didn't even respond. There was a non-response there."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "All right, now let's work through this together. So non-response would have been the case if we selected these 100 people, and let's say only 50 people answered the phone and we didn't keep calling them. Then we'd say, well, 50 of the people who we sampled to answer our survey didn't even respond. There was a non-response there. What was there about those 50 people? Maybe it was something that would have skewed the survey, or actually, if we had, we'd gotten them, it would have gotten maybe get better data. But in this case, they tell us."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "There was a non-response there. What was there about those 50 people? Maybe it was something that would have skewed the survey, or actually, if we had, we'd gotten them, it would have gotten maybe get better data. But in this case, they tell us. The senator's office called those numbers until they got a response from all 100 people chosen. So the 100 people that they chose, they made sure they got a response. So non-response is not going to be an issue here."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "But in this case, they tell us. The senator's office called those numbers until they got a response from all 100 people chosen. So the 100 people that they chose, they made sure they got a response. So non-response is not going to be an issue here. All right, next choice, undercoverage. Well, undercoverage is where you're not able to sample from part of the population. A part of the population that actually might, because you didn't sample it, it might introduce bias."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So non-response is not going to be an issue here. All right, next choice, undercoverage. Well, undercoverage is where you're not able to sample from part of the population. A part of the population that actually might, because you didn't sample it, it might introduce bias. Let's think about what happened in this situation. We are a senator. We want to sample all of our constituents, but we choose, instead, we sample from the constituents who happen to be listed in the phone book."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "A part of the population that actually might, because you didn't sample it, it might introduce bias. Let's think about what happened in this situation. We are a senator. We want to sample all of our constituents, but we choose, instead, we sample from the constituents who happen to be listed in the phone book. So these are the people who happen to be, who happen to be listed in the phone book. And so we're not sampling from people who are not in the phone book, who maybe have landlines and they're unlisted, and we're not sampling from people who don't have landlines, who only have mobile phones. And you might say, well, why is that important?"}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "We want to sample all of our constituents, but we choose, instead, we sample from the constituents who happen to be listed in the phone book. So these are the people who happen to be, who happen to be listed in the phone book. And so we're not sampling from people who are not in the phone book, who maybe have landlines and they're unlisted, and we're not sampling from people who don't have landlines, who only have mobile phones. And you might say, well, why is that important? Well, think about it. People who decide not to list in the phone book, or people who don't even have a landline, some of those people might be a little bit more concerned about privacy than everyone else. They explicitly chose not to be listed."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And you might say, well, why is that important? Well, think about it. People who decide not to list in the phone book, or people who don't even have a landline, some of those people might be a little bit more concerned about privacy than everyone else. They explicitly chose not to be listed. So undercoverage is definitely a very concerning source of bias over here. We are sampling from only a subset of our entire population we care about. In particular, we're missing out on people who might care about privacy."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "They explicitly chose not to be listed. So undercoverage is definitely a very concerning source of bias over here. We are sampling from only a subset of our entire population we care about. In particular, we're missing out on people who might care about privacy. And so I would say, because of undercoverage, 42% is likely to be an underestimate of the people concerned about internet privacy. Probably a higher proportion of the people out here care about privacy, because they're unlisted or they don't even have a landline. So undercoverage, it probably introduced bias, and it implies that 42% is an under, underestimate of the percentage of the senator's constituents who care about internet privacy."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "In particular, we're missing out on people who might care about privacy. And so I would say, because of undercoverage, 42% is likely to be an underestimate of the people concerned about internet privacy. Probably a higher proportion of the people out here care about privacy, because they're unlisted or they don't even have a landline. So undercoverage, it probably introduced bias, and it implies that 42% is an under, underestimate of the percentage of the senator's constituents who care about internet privacy. Now the last question, volunteer response sampling. Well, this would be the case where you, you know, the senator, I don't know, put a billboard out or just told someone, told a bunch of people, maybe on her website, hey, vote for this, or give us your information on how much you care about internet privacy. And that would have been, the source of bias there is, well, who shows up on that website?"}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So undercoverage, it probably introduced bias, and it implies that 42% is an under, underestimate of the percentage of the senator's constituents who care about internet privacy. Now the last question, volunteer response sampling. Well, this would be the case where you, you know, the senator, I don't know, put a billboard out or just told someone, told a bunch of people, maybe on her website, hey, vote for this, or give us your information on how much you care about internet privacy. And that would have been, the source of bias there is, well, who shows up on that website? Once again, if you did, hey, come to my website and fill it out, you're filling, you're only getting information from a subset of your population who are choosing, who are volunteering. That is not the situation that she did over here. She didn't ask 100 people to volunteer."}, {"video_title": "Hypotheses for a two-sample t test AP Statistics Khan Academy.mp3", "Sentence": "Market researchers conducted a study comparing the salaries of managers at a large nationwide retail store. The researchers obtained salary and demographic data for a random sample of managers. The researchers calculated the average salary of the men in the sample and the average salary of the women in the sample. They want to test if managers who are men have a higher average salary than managers who are women. Assume that all conditions for inference have been met. Which of these is the most appropriate test and alternative hypothesis? And we can see they're talking about a paired t-test and a two-sample t-test, and then they talk about the alternative hypotheses."}, {"video_title": "Hypotheses for a two-sample t test AP Statistics Khan Academy.mp3", "Sentence": "They want to test if managers who are men have a higher average salary than managers who are women. Assume that all conditions for inference have been met. Which of these is the most appropriate test and alternative hypothesis? And we can see they're talking about a paired t-test and a two-sample t-test, and then they talk about the alternative hypotheses. So pause this video and try to figure this out on your own. So first, let's think about the difference between a paired t-test and a two-sample t-test. In a paired t-test, we're gonna construct hypotheses around a parameter for a population that will often be the mean difference."}, {"video_title": "Hypotheses for a two-sample t test AP Statistics Khan Academy.mp3", "Sentence": "And we can see they're talking about a paired t-test and a two-sample t-test, and then they talk about the alternative hypotheses. So pause this video and try to figure this out on your own. So first, let's think about the difference between a paired t-test and a two-sample t-test. In a paired t-test, we're gonna construct hypotheses around a parameter for a population that will often be the mean difference. So we have one population, so we're talking about the paired situation right over here. And so let's say we say, hey, do people run faster when they wear shorts or pants? And so for each member of the population, you could see what you would, if you really had perfect information, you would know how fast do they run with pants and how fast do they run with shorts, and then you would calculate the difference."}, {"video_title": "Hypotheses for a two-sample t test AP Statistics Khan Academy.mp3", "Sentence": "In a paired t-test, we're gonna construct hypotheses around a parameter for a population that will often be the mean difference. So we have one population, so we're talking about the paired situation right over here. And so let's say we say, hey, do people run faster when they wear shorts or pants? And so for each member of the population, you could see what you would, if you really had perfect information, you would know how fast do they run with pants and how fast do they run with shorts, and then you would calculate the difference. And then across the whole population, you could actually get that mean difference. So the mean difference of pants minus shorts. And of course, in order to estimate that or in order to do a hypothesis test around that, you would take a sample, and then you would calculate the sample mean of the difference of pants minus shorts."}, {"video_title": "Hypotheses for a two-sample t test AP Statistics Khan Academy.mp3", "Sentence": "And so for each member of the population, you could see what you would, if you really had perfect information, you would know how fast do they run with pants and how fast do they run with shorts, and then you would calculate the difference. And then across the whole population, you could actually get that mean difference. So the mean difference of pants minus shorts. And of course, in order to estimate that or in order to do a hypothesis test around that, you would take a sample, and then you would calculate the sample mean of the difference of pants minus shorts. And then you would say, hey, assuming the null hypothesis is true, you would construct some null hypothesis, likely that there is no, that this mean is zero, and you would say, hey, if the null hypothesis is true, that this is actually equal to zero, what's the probability that I got this result? If that's below your significance level, then you would reject your null hypothesis, and it would suggest the alternative that might be that, hey, maybe this mean is greater than zero. On the other hand, a two-sample t-test is where you're thinking about two different populations."}, {"video_title": "Hypotheses for a two-sample t test AP Statistics Khan Academy.mp3", "Sentence": "And of course, in order to estimate that or in order to do a hypothesis test around that, you would take a sample, and then you would calculate the sample mean of the difference of pants minus shorts. And then you would say, hey, assuming the null hypothesis is true, you would construct some null hypothesis, likely that there is no, that this mean is zero, and you would say, hey, if the null hypothesis is true, that this is actually equal to zero, what's the probability that I got this result? If that's below your significance level, then you would reject your null hypothesis, and it would suggest the alternative that might be that, hey, maybe this mean is greater than zero. On the other hand, a two-sample t-test is where you're thinking about two different populations. For example, you could be thinking about a population of men, and you could be thinking about the population of women. And you wanna compare the means between these two, say the mean salary. So you have the mean salary for men, and you have the mean salary for women."}, {"video_title": "Hypotheses for a two-sample t test AP Statistics Khan Academy.mp3", "Sentence": "On the other hand, a two-sample t-test is where you're thinking about two different populations. For example, you could be thinking about a population of men, and you could be thinking about the population of women. And you wanna compare the means between these two, say the mean salary. So you have the mean salary for men, and you have the mean salary for women. And what you're trying to do with the hypothesis test is try to come up with some conclusions about the mean difference between these two parameters. So the mean salary for men minus the mean salary for women. And our null hypothesis is usually the no news here hypothesis."}, {"video_title": "Hypotheses for a two-sample t test AP Statistics Khan Academy.mp3", "Sentence": "So you have the mean salary for men, and you have the mean salary for women. And what you're trying to do with the hypothesis test is try to come up with some conclusions about the mean difference between these two parameters. So the mean salary for men minus the mean salary for women. And our null hypothesis is usually the no news here hypothesis. And so in this situation, our null hypothesis is that there is no difference between these means, and that our alternative hypothesis in the situation that we are looking at, because they wanna test if managers who are men have a higher average salary. If they just wanted to test that whether managers who are men have a different salary, then our alternative hypothesis would look something like this, where the mean of men minus the mean of women is not equal to zero. But they aren't just testing to see if the means are different."}, {"video_title": "Hypotheses for a two-sample t test AP Statistics Khan Academy.mp3", "Sentence": "And our null hypothesis is usually the no news here hypothesis. And so in this situation, our null hypothesis is that there is no difference between these means, and that our alternative hypothesis in the situation that we are looking at, because they wanna test if managers who are men have a higher average salary. If they just wanted to test that whether managers who are men have a different salary, then our alternative hypothesis would look something like this, where the mean of men minus the mean of women is not equal to zero. But they aren't just testing to see if the means are different. They wanna see if men have a higher average salary. So instead of not equal to zero there, we would have greater than zero for our alternative hypothesis. So which choice is that?"}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "Kaito grows tomatoes in two separate fields. When the tomatoes are ready to be picked, he is curious as to whether the sizes of his tomato plants differ between the two fields. He takes a random sample of plants from each field and measures the heights of the plants. Here is a summary of the results. So what I want you to do is pause this video and conduct a two-sample t-test here. And let's assume that all of the conditions for inference are met, the random condition, the normal condition, and the independence condition. And let's assume that we are working with a significance level of 0.05."}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "Here is a summary of the results. So what I want you to do is pause this video and conduct a two-sample t-test here. And let's assume that all of the conditions for inference are met, the random condition, the normal condition, and the independence condition. And let's assume that we are working with a significance level of 0.05. So pause the video and conduct the two-sample t-test here to see whether there's evidence that the sizes of tomato plants differ between the fields. All right, now let's work through this together. So like always, let's first construct our null hypothesis."}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And let's assume that we are working with a significance level of 0.05. So pause the video and conduct the two-sample t-test here to see whether there's evidence that the sizes of tomato plants differ between the fields. All right, now let's work through this together. So like always, let's first construct our null hypothesis. And that's going to be the situation where there is no difference between the mean sizes. So that would be that the mean size in field A is equal to the mean size in field B. Now what about our alternative hypothesis?"}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "So like always, let's first construct our null hypothesis. And that's going to be the situation where there is no difference between the mean sizes. So that would be that the mean size in field A is equal to the mean size in field B. Now what about our alternative hypothesis? Well, he wants to see whether the sizes of his tomato plants differ between the two fields. He's not saying whether A is bigger than B or whether B is bigger than A. And so his alternative hypothesis would be around his suspicion that the mean of A is not equal to the mean of B, that they differ."}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "Now what about our alternative hypothesis? Well, he wants to see whether the sizes of his tomato plants differ between the two fields. He's not saying whether A is bigger than B or whether B is bigger than A. And so his alternative hypothesis would be around his suspicion that the mean of A is not equal to the mean of B, that they differ. And to do this two-sample t-test now, we assume the null hypothesis. We assume our null hypothesis. And remember, we're assuming that all of our conditions for inference are met."}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And so his alternative hypothesis would be around his suspicion that the mean of A is not equal to the mean of B, that they differ. And to do this two-sample t-test now, we assume the null hypothesis. We assume our null hypothesis. And remember, we're assuming that all of our conditions for inference are met. And then we wanna calculate a t-statistic based on this sample data that we have. And our t-statistic is going to be equal to the differences between the sample means, all of that over our estimate of the standard deviation of the sampling distribution of the difference of the sample means. This will be the sample standard deviation from sample A squared over the sample size from A plus the sample standard deviation from the B sample squared over the sample size from B."}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And remember, we're assuming that all of our conditions for inference are met. And then we wanna calculate a t-statistic based on this sample data that we have. And our t-statistic is going to be equal to the differences between the sample means, all of that over our estimate of the standard deviation of the sampling distribution of the difference of the sample means. This will be the sample standard deviation from sample A squared over the sample size from A plus the sample standard deviation from the B sample squared over the sample size from B. And let's see, we have all the numbers here to calculate it. This numerator is going to be equal to 1.3 minus 1.6, 1.3 minus 1.6, all of that over the square root of, let's see, the standard deviation, the sample standard deviation from the sample from field A is 0.5. If you square that, you're gonna get 0.25."}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "This will be the sample standard deviation from sample A squared over the sample size from A plus the sample standard deviation from the B sample squared over the sample size from B. And let's see, we have all the numbers here to calculate it. This numerator is going to be equal to 1.3 minus 1.6, 1.3 minus 1.6, all of that over the square root of, let's see, the standard deviation, the sample standard deviation from the sample from field A is 0.5. If you square that, you're gonna get 0.25. And then that's going to be over the sample size from field A over 22 plus 0.3 squared. So that is 0.3 squared is 0.09. All of that over the sample size from field B, all of that over 24."}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "If you square that, you're gonna get 0.25. And then that's going to be over the sample size from field A over 22 plus 0.3 squared. So that is 0.3 squared is 0.09. All of that over the sample size from field B, all of that over 24. The numerator is just gonna be negative 0.3, negative 0.3 divided by the square root of 0.25 divided by 22 plus 0.09 divided by 24. And that gets us negative 2.44, approximately negative 2.44. And so if you think about a t-distribution, and we'll use our calculator to figure out this probability."}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "All of that over the sample size from field B, all of that over 24. The numerator is just gonna be negative 0.3, negative 0.3 divided by the square root of 0.25 divided by 22 plus 0.09 divided by 24. And that gets us negative 2.44, approximately negative 2.44. And so if you think about a t-distribution, and we'll use our calculator to figure out this probability. So this is a t-distribution right over here. This would be the assumed mean of our t-distribution. And so we got a result that is negative, we got a t-statistic of negative 2.44."}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And so if you think about a t-distribution, and we'll use our calculator to figure out this probability. So this is a t-distribution right over here. This would be the assumed mean of our t-distribution. And so we got a result that is negative, we got a t-statistic of negative 2.44. So we're right over here. So this is negative 2.44. And so we wanna say what is the probability from this t-distribution of getting something at least this extreme?"}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And so we got a result that is negative, we got a t-statistic of negative 2.44. So we're right over here. So this is negative 2.44. And so we wanna say what is the probability from this t-distribution of getting something at least this extreme? So it would be this area, and it would also be this area. If we got 2.44 above the mean, it would also be this area. And so what I could do is I'm gonna use my calculator to figure out this probability right over here, and then I'm just gonna multiply that by two to get this one as well."}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And so we wanna say what is the probability from this t-distribution of getting something at least this extreme? So it would be this area, and it would also be this area. If we got 2.44 above the mean, it would also be this area. And so what I could do is I'm gonna use my calculator to figure out this probability right over here, and then I'm just gonna multiply that by two to get this one as well. So the probability of getting a t-value, I guess I could say where it's absolute value is greater than or equal to 2.44 is going to be approximately equal to, I'm going to go to second distribution, I'm going to go to the cumulative distribution function for our t-distribution, click that. And since I wanna think about this tail probability here, then I'm just gonna multiply it by two. The lower bound is a very, very, very negative number."}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And so what I could do is I'm gonna use my calculator to figure out this probability right over here, and then I'm just gonna multiply that by two to get this one as well. So the probability of getting a t-value, I guess I could say where it's absolute value is greater than or equal to 2.44 is going to be approximately equal to, I'm going to go to second distribution, I'm going to go to the cumulative distribution function for our t-distribution, click that. And since I wanna think about this tail probability here, then I'm just gonna multiply it by two. The lower bound is a very, very, very negative number. You could view that as functionally negative infinity. The upper bound is negative 2.44, negative 2.44. And now what's our degrees of freedom?"}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "The lower bound is a very, very, very negative number. You could view that as functionally negative infinity. The upper bound is negative 2.44, negative 2.44. And now what's our degrees of freedom? Well, if we take the conservative approach, it'll be the smaller of the two samples minus one. Well, the smaller of the two samples is 22, and so 22 minus one is 21. So put 21 in there, 221."}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And now what's our degrees of freedom? Well, if we take the conservative approach, it'll be the smaller of the two samples minus one. Well, the smaller of the two samples is 22, and so 22 minus one is 21. So put 21 in there, 221. And now I can paste, and I get that number right over there. And if I multiply that by two, because this just gives me the probability of getting something lower than that, but I also wanna think about the probability of getting something 2.44 or more above the mean of our t-distribution. So times two is going to be equal to approximately 0.024."}, {"video_title": "Two-sample t test for difference of means AP Statistics Khan Academy.mp3", "Sentence": "So put 21 in there, 221. And now I can paste, and I get that number right over there. And if I multiply that by two, because this just gives me the probability of getting something lower than that, but I also wanna think about the probability of getting something 2.44 or more above the mean of our t-distribution. So times two is going to be equal to approximately 0.024. So approximately 0.024. And what I wanna do then is compare this to my significance level. And you can see very clearly, this right over here, this is equal to our p-value."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "We're now going to explore hypothesis testing, where we're thinking about the difference between proportions of two different populations. So here it says, here are the results from a recent poll that involved sampling voters from each of two neighboring districts, District A and District B, and folks were asked whether they support a new law or not. And from each district, we took a sample of 100 voters, and then we were able to calculate the proportion from that sample that supported the law, and then here we have the combined data, including the combined proportion, and we're asked, does this suggest a significant difference between the two districts? And so this is asking for a hypothesis test, and the way we would do that is we would set up our null hypothesis, and remember, our null hypothesis is the one that we would assume that there is no difference. So we would assume that the true proportion of folks in District A that support the new law is equal to the proportion in District B that support the law, or another way to think about it is that the difference would be equal to zero. And our alternative hypothesis is that the absolute difference between the proportions is not equal to zero. And if we were doing an all-out hypothesis test, we would set a significance level, which we usually denote with an alpha."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "And so this is asking for a hypothesis test, and the way we would do that is we would set up our null hypothesis, and remember, our null hypothesis is the one that we would assume that there is no difference. So we would assume that the true proportion of folks in District A that support the new law is equal to the proportion in District B that support the law, or another way to think about it is that the difference would be equal to zero. And our alternative hypothesis is that the absolute difference between the proportions is not equal to zero. And if we were doing an all-out hypothesis test, we would set a significance level, which we usually denote with an alpha. Oftentimes, it might be a 10% significance level or a 5% significance level. Let's say we set it at a 5% significance level. And what we would do is we would say, all right, let's assume that the null hypothesis is true."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "And if we were doing an all-out hypothesis test, we would set a significance level, which we usually denote with an alpha. Oftentimes, it might be a 10% significance level or a 5% significance level. Let's say we set it at a 5% significance level. And what we would do is we would say, all right, let's assume that the null hypothesis is true. And assuming the null hypothesis is true, what is the probability of getting a difference between our sample proportions this extreme or more? And if that probability is less than our significance level, then we reject the null hypothesis, which would suggest the alternative. Now, before we go deeper into our inference, we wanna test our conditions for inference, and we've seen these many times before."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "And what we would do is we would say, all right, let's assume that the null hypothesis is true. And assuming the null hypothesis is true, what is the probability of getting a difference between our sample proportions this extreme or more? And if that probability is less than our significance level, then we reject the null hypothesis, which would suggest the alternative. Now, before we go deeper into our inference, we wanna test our conditions for inference, and we've seen these many times before. You have the random condition, where you would need to feel good that both of these samples are truly random. You would have your normal condition, which is that you would have at least 10 successes and failures in each of these samples. And we see that we do indeed have at least 10 successes and at least 10 failures in each of those samples."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "Now, before we go deeper into our inference, we wanna test our conditions for inference, and we've seen these many times before. You have the random condition, where you would need to feel good that both of these samples are truly random. You would have your normal condition, which is that you would have at least 10 successes and failures in each of these samples. And we see that we do indeed have at least 10 successes and at least 10 failures in each of those samples. And then you have your independence condition. And the independence condition, you're either sampling with replacement or you need to feel good that each of these sample sizes are no more than 10% of the entire population. And so I guess we will assume that there's at least 1,000 folks in District A and at least 1,000 folks in District B, and that would allow us to meet the independence condition."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "And we see that we do indeed have at least 10 successes and at least 10 failures in each of those samples. And then you have your independence condition. And the independence condition, you're either sampling with replacement or you need to feel good that each of these sample sizes are no more than 10% of the entire population. And so I guess we will assume that there's at least 1,000 folks in District A and at least 1,000 folks in District B, and that would allow us to meet the independence condition. And so with that out of the way, let's assume the null hypothesis. And let's just start thinking about the sampling distribution of the difference between the sample proportions, assuming that null hypothesis. So the first thing I wanna think about is what is going to be the standard deviation of the difference in the sampling distributions?"}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "And so I guess we will assume that there's at least 1,000 folks in District A and at least 1,000 folks in District B, and that would allow us to meet the independence condition. And so with that out of the way, let's assume the null hypothesis. And let's just start thinking about the sampling distribution of the difference between the sample proportions, assuming that null hypothesis. So the first thing I wanna think about is what is going to be the standard deviation of the difference in the sampling distributions? Well, we have seen in a previous video when we talked about differences of proportions, we could think about the variance. So the variance of the sampling distribution. There's a lot of notation here."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "So the first thing I wanna think about is what is going to be the standard deviation of the difference in the sampling distributions? Well, we have seen in a previous video when we talked about differences of proportions, we could think about the variance. So the variance of the sampling distribution. There's a lot of notation here. So the variance, this is going to be equal to the variance of the sampling distribution of the sample proportion from District A plus the variance of the sampling distribution of the sample proportion from District B. Now, in general, you can figure out the variance of the sampling distribution of a sample proportion with the following formula, and we've seen this before. The variance of the sampling distribution of the sample proportion is going to be equal to our true proportion times one minus our true proportion, all of that over your sample size."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "There's a lot of notation here. So the variance, this is going to be equal to the variance of the sampling distribution of the sample proportion from District A plus the variance of the sampling distribution of the sample proportion from District B. Now, in general, you can figure out the variance of the sampling distribution of a sample proportion with the following formula, and we've seen this before. The variance of the sampling distribution of the sample proportion is going to be equal to our true proportion times one minus our true proportion, all of that over your sample size. Now, in either situation, we don't know the true proportions for District A or District B. That's why we are in this, that's why we're even doing this hypothesis test to begin with. But we can try to estimate it."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "The variance of the sampling distribution of the sample proportion is going to be equal to our true proportion times one minus our true proportion, all of that over your sample size. Now, in either situation, we don't know the true proportions for District A or District B. That's why we are in this, that's why we're even doing this hypothesis test to begin with. But we can try to estimate it. Remember, we're assuming that the true proportions are equal even though we might not know what they are. And what is going to be our best estimate of that true proportion if we assume that District A and District B have no difference in terms of the number of people who support the new law? Well, the best estimate would actually be the combined sample, the combined sample proportion right over here."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "But we can try to estimate it. Remember, we're assuming that the true proportions are equal even though we might not know what they are. And what is going to be our best estimate of that true proportion if we assume that District A and District B have no difference in terms of the number of people who support the new law? Well, the best estimate would actually be the combined sample, the combined sample proportion right over here. And so to estimate these values, we use this combined sample proportion in the place of P over here. So we could say that this is going to be our combined sample proportion times one minus our combined sample proportion, all of that over our sample size. And since we're assuming that there's no difference between District A and District B, this would also apply to that right over there."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "Well, the best estimate would actually be the combined sample, the combined sample proportion right over here. And so to estimate these values, we use this combined sample proportion in the place of P over here. So we could say that this is going to be our combined sample proportion times one minus our combined sample proportion, all of that over our sample size. And since we're assuming that there's no difference between District A and District B, this would also apply to that right over there. So let me rewrite this again. The standard deviation of the sampling distribution of the difference of the sample proportions from District A and District B is going to be roughly, remember, we weren't able to calculate it exactly, but we're using this combined proportion as our best estimate. Let me do a big square root right over here, a big radical."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "And since we're assuming that there's no difference between District A and District B, this would also apply to that right over there. So let me rewrite this again. The standard deviation of the sampling distribution of the difference of the sample proportions from District A and District B is going to be roughly, remember, we weren't able to calculate it exactly, but we're using this combined proportion as our best estimate. Let me do a big square root right over here, a big radical. And so underneath that, we are going to have our estimate of this, which is 0.55 times one minus 0.55, so 0.45, over 100, plus our estimate of this, which is 0.55, it's the same thing again, times 0.45, and remember, that's because we're assuming the null hypothesis is true. All of that over this sample size, all of that over 100. And now we can get our calculator out to actually calculate it."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "Let me do a big square root right over here, a big radical. And so underneath that, we are going to have our estimate of this, which is 0.55 times one minus 0.55, so 0.45, over 100, plus our estimate of this, which is 0.55, it's the same thing again, times 0.45, and remember, that's because we're assuming the null hypothesis is true. All of that over this sample size, all of that over 100. And now we can get our calculator out to actually calculate it. And so we get the square root of 0.55 times 0.45 divided by 100. Now I could add that whole thing again, or I could just multiply by two. So times two is equal to approximately 0.07."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "And now we can get our calculator out to actually calculate it. And so we get the square root of 0.55 times 0.45 divided by 100. Now I could add that whole thing again, or I could just multiply by two. So times two is equal to approximately 0.07. So this is going to be approximately equal to 0.07. And now using this, we can calculate a z-score. And then we could think about what's the probability of getting a z-score that extreme or more?"}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "So times two is equal to approximately 0.07. So this is going to be approximately equal to 0.07. And now using this, we can calculate a z-score. And then we could think about what's the probability of getting a z-score that extreme or more? And so our z-score, or our z-value, would be equal to the difference that we got, P hat sub a minus P hat sub b, all of that over our estimate of the standard deviation of the sampling distribution of this difference between the sample proportions. So all of that over 0.07. Now this up here in yellow, it's 0.58 minus 0.52."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "And then we could think about what's the probability of getting a z-score that extreme or more? And so our z-score, or our z-value, would be equal to the difference that we got, P hat sub a minus P hat sub b, all of that over our estimate of the standard deviation of the sampling distribution of this difference between the sample proportions. So all of that over 0.07. Now this up here in yellow, it's 0.58 minus 0.52. This is going to be equal to 0.06 over 0.07. 0.07. We can get our calculator out for this again."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "Now this up here in yellow, it's 0.58 minus 0.52. This is going to be equal to 0.06 over 0.07. 0.07. We can get our calculator out for this again. And so we have 0.06 divided by 0.07 is going to be, it's approximately 0.86. So this is approximately 0.86. Now what's the probability of getting something this extreme or more extreme?"}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "We can get our calculator out for this again. And so we have 0.06 divided by 0.07 is going to be, it's approximately 0.86. So this is approximately 0.86. Now what's the probability of getting something this extreme or more extreme? Let me just make sure we can visualize it properly. So if this is our sampling distribution of the difference between our sample proportions, and we're assuming the null hypothesis, so the mean of our sampling distribution is going to be zero, is going to be zero right there. We just got a result that is less than a standard deviation above the mean."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "Now what's the probability of getting something this extreme or more extreme? Let me just make sure we can visualize it properly. So if this is our sampling distribution of the difference between our sample proportions, and we're assuming the null hypothesis, so the mean of our sampling distribution is going to be zero, is going to be zero right there. We just got a result that is less than a standard deviation above the mean. So we just got a result, so if this is one standard deviation, two standard deviations above the mean, one standard deviation below the mean, two standard deviations below the mean, we just got a result that puts us right there. And if we asked ourselves, what's the probability of getting a result at least that extreme, we would say, okay, it would be, what's the probability of getting a result all of this area right over here? And it would also be, what's this area on the other side of the mean?"}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "We just got a result that is less than a standard deviation above the mean. So we just got a result, so if this is one standard deviation, two standard deviations above the mean, one standard deviation below the mean, two standard deviations below the mean, we just got a result that puts us right there. And if we asked ourselves, what's the probability of getting a result at least that extreme, we would say, okay, it would be, what's the probability of getting a result all of this area right over here? And it would also be, what's this area on the other side of the mean? And we know that this is over 30%, because even if you just exclude one standard deviation above and below the mean, if you say anything more extreme than that, so if you put this area and this area, you're looking at roughly 31 or 32%. So the probability of getting something at least this extreme is going to be over 30%. And so it's definitely going to be higher than our significance level."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "And it would also be, what's this area on the other side of the mean? And we know that this is over 30%, because even if you just exclude one standard deviation above and below the mean, if you say anything more extreme than that, so if you put this area and this area, you're looking at roughly 31 or 32%. So the probability of getting something at least this extreme is going to be over 30%. And so it's definitely going to be higher than our significance level. It's actually completely reasonable to get a difference this extreme if we assume the null hypothesis is true. In future videos, we can go even deeper, where we can actually just look this up on a z-table to calculate these areas more precisely, and we can compare them to the significance level. But here, it's not even close."}, {"video_title": "Hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "And so it's definitely going to be higher than our significance level. It's actually completely reasonable to get a difference this extreme if we assume the null hypothesis is true. In future videos, we can go even deeper, where we can actually just look this up on a z-table to calculate these areas more precisely, and we can compare them to the significance level. But here, it's not even close. We're nowhere close to being able to reject the null hypothesis. So to answer the question, does this suggest a significant difference between the two districts? No, no it doesn't."}, {"video_title": "Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So here's the random variable X. It's a discrete random variable. It only takes on a finite number of values. Sometimes people say it takes on a countable number of values, but we see he can either make zero free throws, one, or two of the two. And the probability that he makes zero is here, one is here, two is here. And then they also give us the mean of X and the standard deviation of X. Then they tell us if the game costs Anoush $15 to play and he wins $10 per shot he makes, what are the mean and standard deviation of his net gain from playing the game N?"}, {"video_title": "Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "Sometimes people say it takes on a countable number of values, but we see he can either make zero free throws, one, or two of the two. And the probability that he makes zero is here, one is here, two is here. And then they also give us the mean of X and the standard deviation of X. Then they tell us if the game costs Anoush $15 to play and he wins $10 per shot he makes, what are the mean and standard deviation of his net gain from playing the game N? All right, so let's define a new random variable N, which is equal to his net gain, net gain. We can define N in terms of X. What is his net gain going to be?"}, {"video_title": "Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "Then they tell us if the game costs Anoush $15 to play and he wins $10 per shot he makes, what are the mean and standard deviation of his net gain from playing the game N? All right, so let's define a new random variable N, which is equal to his net gain, net gain. We can define N in terms of X. What is his net gain going to be? Well, let's see, N, it's going to be equal to 10 times however many shots he makes. So it's gonna be 10 times X. And then no matter what, he has to pay $15 to play, minus 15."}, {"video_title": "Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "What is his net gain going to be? Well, let's see, N, it's going to be equal to 10 times however many shots he makes. So it's gonna be 10 times X. And then no matter what, he has to pay $15 to play, minus 15. In fact, we could set up a little table here for the probability distribution of N. So let me make it right over here. So I'll make it look just like this one. N is equal to net gain."}, {"video_title": "Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "And then no matter what, he has to pay $15 to play, minus 15. In fact, we could set up a little table here for the probability distribution of N. So let me make it right over here. So I'll make it look just like this one. N is equal to net gain. And here we'll have the probability of N. And there's three outcomes here. The outcome that corresponds to him making zero shots, well, that would be 10 times zero minus 15, that would be a net gain of negative 15. And it would have the same probability, 0.16."}, {"video_title": "Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "N is equal to net gain. And here we'll have the probability of N. And there's three outcomes here. The outcome that corresponds to him making zero shots, well, that would be 10 times zero minus 15, that would be a net gain of negative 15. And it would have the same probability, 0.16. When he makes one shot, the net gain is gonna be 10 times one minus 15, which is negative five. But it's gonna have the same probability. He has a 48% chance of making one shot, and so it's a 48% chance of losing $5."}, {"video_title": "Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "And it would have the same probability, 0.16. When he makes one shot, the net gain is gonna be 10 times one minus 15, which is negative five. But it's gonna have the same probability. He has a 48% chance of making one shot, and so it's a 48% chance of losing $5. And then last but not least, when X is two, his net gain is going to be positive five, plus five. And so this is a 0.36 chance. So what they want us to figure out are what are the mean and standard deviation of his net gain?"}, {"video_title": "Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "He has a 48% chance of making one shot, and so it's a 48% chance of losing $5. And then last but not least, when X is two, his net gain is going to be positive five, plus five. And so this is a 0.36 chance. So what they want us to figure out are what are the mean and standard deviation of his net gain? So let's first figure out the mean of N. Well, if you scale a random variable, the corresponding mean is going to be scaled by the same amount. And if you shift a random variable, the corresponding mean is going to be shifted by the same amount. So the mean of N is going to be 10 times the mean of X, minus 15, which is equal to 10 times 1.2 minus 15."}, {"video_title": "Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So what they want us to figure out are what are the mean and standard deviation of his net gain? So let's first figure out the mean of N. Well, if you scale a random variable, the corresponding mean is going to be scaled by the same amount. And if you shift a random variable, the corresponding mean is going to be shifted by the same amount. So the mean of N is going to be 10 times the mean of X, minus 15, which is equal to 10 times 1.2 minus 15. This is 1.2. So it is 12 minus 15, which is equal to negative three. Now the standard deviation of N is going to be slightly different."}, {"video_title": "Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So the mean of N is going to be 10 times the mean of X, minus 15, which is equal to 10 times 1.2 minus 15. This is 1.2. So it is 12 minus 15, which is equal to negative three. Now the standard deviation of N is going to be slightly different. For the standard deviation, scaling matters. If you scale a random variable by a certain value, you would also scale the standard deviation by the same value. So this is going to be equal to 10 times the standard deviation of X."}, {"video_title": "Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "Now the standard deviation of N is going to be slightly different. For the standard deviation, scaling matters. If you scale a random variable by a certain value, you would also scale the standard deviation by the same value. So this is going to be equal to 10 times the standard deviation of X. Now you might say, what about the shift over here? Well, the shift should not affect the spread of the random variable. If you're scaling the random variable, well, your spread should grow by the amount that you're scaling it."}, {"video_title": "Example Transforming a discrete random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So this is going to be equal to 10 times the standard deviation of X. Now you might say, what about the shift over here? Well, the shift should not affect the spread of the random variable. If you're scaling the random variable, well, your spread should grow by the amount that you're scaling it. But by shifting it, it doesn't affect how much you disperse from the mean. So standard deviation is only affected by the scaling, but not by the shifting here. So this is going to be 10 times 0.69, which is going to, this was an approximation, so I'll say this is approximately equal to 6.9."}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "What we're going to do in this video is continue our journey trying to understand what the expected value and what the variance of a binomial variable is going to be or what the expected value or the variance of a binomial distribution is going to be, which is just the distribution of a binomial variable. And so, like in the last video, I have this binomial variable x that's defined in a very general sense. It's the number of successes from n trials, so it's a finite number of trials, where the probability of success is equal to p, so the probability is constant across the trials, for each of these independent trials. So the probability of success in one trial is not dependent on what happened in the other trials. And we also talked in that previous video where we talked about the expected value of this binomial variable, is we said, hey, it could be viewed, this binomial variable can be viewed as the sum of n of what you could really consider to be a Bernoulli variable here. So this variable, this random variable y, the probability that it's equal to one, you could view that as a success, is equal to p. The probability that it's a failure, that y is equal to zero, is one minus p. So you could view y, the outcome of y is really the, or whether y is one or zero is really whether we had a success or not in each of these trials. So if you add up n y's, then you are going to get x."}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So the probability of success in one trial is not dependent on what happened in the other trials. And we also talked in that previous video where we talked about the expected value of this binomial variable, is we said, hey, it could be viewed, this binomial variable can be viewed as the sum of n of what you could really consider to be a Bernoulli variable here. So this variable, this random variable y, the probability that it's equal to one, you could view that as a success, is equal to p. The probability that it's a failure, that y is equal to zero, is one minus p. So you could view y, the outcome of y is really the, or whether y is one or zero is really whether we had a success or not in each of these trials. So if you add up n y's, then you are going to get x. And we use that information to figure out what the expected value of x is going to be. Because the expected value of y is pretty straightforward to directly compute. Expected value of y is just probability-weighted outcomes."}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So if you add up n y's, then you are going to get x. And we use that information to figure out what the expected value of x is going to be. Because the expected value of y is pretty straightforward to directly compute. Expected value of y is just probability-weighted outcomes. So it's p times one plus one minus p, one minus p, times zero, times zero. This whole term's gonna be zero. And so the expected value of y is really just p. And so if you said the expected value of x, well that's just going to be, let me just write it over here, this is all review."}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "Expected value of y is just probability-weighted outcomes. So it's p times one plus one minus p, one minus p, times zero, times zero. This whole term's gonna be zero. And so the expected value of y is really just p. And so if you said the expected value of x, well that's just going to be, let me just write it over here, this is all review. We could say that the expected value of x is just going to be equal to, we know from our expected value properties, that's going to be equal to the sum of the expected values of these n y's. Or you could say it is n times the expected value, times the expected value of y. The expected value of y is p. So this is going to be equal to n times p. Now we're gonna do the same idea to figure out what the variance of x is going to be equal to."}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so the expected value of y is really just p. And so if you said the expected value of x, well that's just going to be, let me just write it over here, this is all review. We could say that the expected value of x is just going to be equal to, we know from our expected value properties, that's going to be equal to the sum of the expected values of these n y's. Or you could say it is n times the expected value, times the expected value of y. The expected value of y is p. So this is going to be equal to n times p. Now we're gonna do the same idea to figure out what the variance of x is going to be equal to. Because we could see, we know from our variance properties, you can't do this with standard deviation, but you could do it with variance. And then once you figure out the variance, you just take the square root for the standard deviation. The variance of x is similarly going to be the sum of the variances of these n y's."}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "The expected value of y is p. So this is going to be equal to n times p. Now we're gonna do the same idea to figure out what the variance of x is going to be equal to. Because we could see, we know from our variance properties, you can't do this with standard deviation, but you could do it with variance. And then once you figure out the variance, you just take the square root for the standard deviation. The variance of x is similarly going to be the sum of the variances of these n y's. So it's going to be similarly n times the variance, n times the variance of y. So this all boils down to what is the variance of y going to be equal to? So let me scroll over a little bit, get a little bit of more real estate, and I will figure that out right over, right over here."}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "The variance of x is similarly going to be the sum of the variances of these n y's. So it's going to be similarly n times the variance, n times the variance of y. So this all boils down to what is the variance of y going to be equal to? So let me scroll over a little bit, get a little bit of more real estate, and I will figure that out right over, right over here. All right, so we want to figure out the variance of y. So variance of y is going to be equal to what? Well here it's going to be the probability squared distances from the expected value."}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So let me scroll over a little bit, get a little bit of more real estate, and I will figure that out right over, right over here. All right, so we want to figure out the variance of y. So variance of y is going to be equal to what? Well here it's going to be the probability squared distances from the expected value. So we have a probability of p, where what is going to be our squared distance from the expected value? Well, we're gonna get a one with a probability of p. So in that case, our distance from the mean, or from the expected value, we're at one. The expected value, we already know, is equal to p. So that's, for that possible outcome, the squared distance times its probability weight."}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well here it's going to be the probability squared distances from the expected value. So we have a probability of p, where what is going to be our squared distance from the expected value? Well, we're gonna get a one with a probability of p. So in that case, our distance from the mean, or from the expected value, we're at one. The expected value, we already know, is equal to p. So that's, for that possible outcome, the squared distance times its probability weight. And then we have, actually let me scroll over a little bit, well I'll just do it right over here. Plus we have a probability of one minus p, one minus p for the other possible outcome. So in that outcome, we are at zero."}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "The expected value, we already know, is equal to p. So that's, for that possible outcome, the squared distance times its probability weight. And then we have, actually let me scroll over a little bit, well I'll just do it right over here. Plus we have a probability of one minus p, one minus p for the other possible outcome. So in that outcome, we are at zero. And the difference between zero and our expected value, well that's just going to be zero minus p. And once again, we are going to square that distance. And so this is the expression, or square that quantity. And so this is the expression for the variance of y, and we can simplify it a little bit."}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So in that outcome, we are at zero. And the difference between zero and our expected value, well that's just going to be zero minus p. And once again, we are going to square that distance. And so this is the expression, or square that quantity. And so this is the expression for the variance of y, and we can simplify it a little bit. So this is all going to be equal to, so let me just, p times one minus p squared. And then this is just going to be p squared times one minus p, plus p squared times one minus p. And let's see, we can factor out a p times one minus p. So what is that going to be left with? So if we factor out a p times one minus p here, we're just going to be left with a one minus p. And if we factor out a p times one minus p here, we're just going to have a plus p. These two cancel out."}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so this is the expression for the variance of y, and we can simplify it a little bit. So this is all going to be equal to, so let me just, p times one minus p squared. And then this is just going to be p squared times one minus p, plus p squared times one minus p. And let's see, we can factor out a p times one minus p. So what is that going to be left with? So if we factor out a p times one minus p here, we're just going to be left with a one minus p. And if we factor out a p times one minus p here, we're just going to have a plus p. These two cancel out. This is just, this whole thing is just a one. So you're left with p times one minus p, which is indeed the variance for a binomial variable. We actually proved that in other videos."}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So if we factor out a p times one minus p here, we're just going to be left with a one minus p. And if we factor out a p times one minus p here, we're just going to have a plus p. These two cancel out. This is just, this whole thing is just a one. So you're left with p times one minus p, which is indeed the variance for a binomial variable. We actually proved that in other videos. I guess it doesn't hurt to see it again. But there you have it. We know what the variance of y is."}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "We actually proved that in other videos. I guess it doesn't hurt to see it again. But there you have it. We know what the variance of y is. It is p times one minus p. And the variance of x is just n times the variance of y. So there we go. We deserve a little bit of a drum roll."}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "We know what the variance of y is. It is p times one minus p. And the variance of x is just n times the variance of y. So there we go. We deserve a little bit of a drum roll. The variance of x is equal to n times p times one minus p. So if we were to take the concrete example of the last video, where if I were to take 10 free throws, so each trial is a shot, is a free throw. So if I were to take 10 free throws and my probability of success is 0.3, I have a 30% free throw percentage, the variance that I would expect to see, so in that case, the variance, if x is the number of free throws I make after these 10 shots, my variance will be 10 times 0.3, 0.3 times one minus 0.3, so 0.7. And so that would be what?"}, {"video_title": "Variance of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "We deserve a little bit of a drum roll. The variance of x is equal to n times p times one minus p. So if we were to take the concrete example of the last video, where if I were to take 10 free throws, so each trial is a shot, is a free throw. So if I were to take 10 free throws and my probability of success is 0.3, I have a 30% free throw percentage, the variance that I would expect to see, so in that case, the variance, if x is the number of free throws I make after these 10 shots, my variance will be 10 times 0.3, 0.3 times one minus 0.3, so 0.7. And so that would be what? This right over here, so this would be equal to 10 times 0.3 times 0.7 times 0.21. So my variance in this situation is going to be equal to 2.1. It is equal to 2.1."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Miguel tracked how much sleep he got for 50 consecutive days and made a histogram of the results. Which interval contains the median sleep amount? And so they're saying is it this interval on the histogram from six to 6.5 or this one or this one or any of these? Which of these intervals contain the median? Pause this video and see if you can figure that out. All right, now let's work through this together and let's just remind ourselves how we find the median. If I had the data points 11, nine, seven, three, and two, the way that we find the median is we can order it from least to greatest, or actually you could do it from greatest to least, but let's do least to greatest."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Which of these intervals contain the median? Pause this video and see if you can figure that out. All right, now let's work through this together and let's just remind ourselves how we find the median. If I had the data points 11, nine, seven, three, and two, the way that we find the median is we can order it from least to greatest, or actually you could do it from greatest to least, but let's do least to greatest. So two, three, seven, nine, 11. And the median would be the middle number. And I have a clear middle number because I have five data points."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "If I had the data points 11, nine, seven, three, and two, the way that we find the median is we can order it from least to greatest, or actually you could do it from greatest to least, but let's do least to greatest. So two, three, seven, nine, 11. And the median would be the middle number. And I have a clear middle number because I have five data points. If I have an even number of data points, I still would wanna order them from least to greatest. So let's say that I have a one, one, three, and a seven. But here you don't have a clear middle, so the median would be the mean of the middle two numbers."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "And I have a clear middle number because I have five data points. If I have an even number of data points, I still would wanna order them from least to greatest. So let's say that I have a one, one, three, and a seven. But here you don't have a clear middle, so the median would be the mean of the middle two numbers. So in this situation, Miguel has an even number of data points. So the median would be the mean of the 25th and 26th data point. These would be the middle two data points."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "But here you don't have a clear middle, so the median would be the mean of the middle two numbers. So in this situation, Miguel has an even number of data points. So the median would be the mean of the 25th and 26th data point. These would be the middle two data points. So which interval here contains the 25th and the 26th data point? Well, we can start at the bottom. So we have, actually, let's just look at each interval and think about how many data points they have in it."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "These would be the middle two data points. So which interval here contains the 25th and the 26th data point? Well, we can start at the bottom. So we have, actually, let's just look at each interval and think about how many data points they have in it. This one has two. This one has nine. This one has 12."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So we have, actually, let's just look at each interval and think about how many data points they have in it. This one has two. This one has nine. This one has 12. And I'm just reading out how many data points there are in each of these intervals. This one has 12. This one has 11."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "This one has 12. And I'm just reading out how many data points there are in each of these intervals. This one has 12. This one has 11. I see that there. This one has two, and this one has two. So if we look at just this, we have the two lowest."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "This one has 11. I see that there. This one has two, and this one has two. So if we look at just this, we have the two lowest. If we look at the two bottom intervals combined, we have the 11 lowest. If we look at the three bottom intervals, we have the 11 plus 12, you have the 23 lowest. So this is the 23 lowest data points."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "We're told that some boxes of a certain brand of breakfast cereal include a voucher for a free video rental inside the box. The company that makes the cereal claims that a voucher can be found in 20% of boxes. However, based on their experiences eating the cereal at home, a group of students believes that the proportion of boxes with vouchers is less than 20%. This group of students purchased 65 boxes of the cereal to investigate the company's claim. The student found a total of 11 vouchers for free video rentals in the 65 boxes. Suppose it is reasonable to assume that the 65 boxes purchased by the students are a random sample of all boxes of this cereal. Based on this sample, is there support for the student's belief that the proportion of boxes with vouchers is less than 20%?"}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "This group of students purchased 65 boxes of the cereal to investigate the company's claim. The student found a total of 11 vouchers for free video rentals in the 65 boxes. Suppose it is reasonable to assume that the 65 boxes purchased by the students are a random sample of all boxes of this cereal. Based on this sample, is there support for the student's belief that the proportion of boxes with vouchers is less than 20%? Provide statistical evidence to support your answer. And so, like always, pause this video and see if you can answer it by yourself. And this actually is a question from an AP statistics exam."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "Based on this sample, is there support for the student's belief that the proportion of boxes with vouchers is less than 20%? Provide statistical evidence to support your answer. And so, like always, pause this video and see if you can answer it by yourself. And this actually is a question from an AP statistics exam. All right, now let's work through this together. And I'm going to try to model some of what you might wanna do if you were actually trying to answer this on an exam. So the first thing you might wanna say is, well, what's our null and our alternative hypothesis?"}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "And this actually is a question from an AP statistics exam. All right, now let's work through this together. And I'm going to try to model some of what you might wanna do if you were actually trying to answer this on an exam. So the first thing you might wanna say is, well, what's our null and our alternative hypothesis? Well, our null hypothesis would be, well, the reality is what the breakfast brand claims, that 20% of the boxes contain a voucher. So that would be our null hypothesis. And our alternative hypothesis would be what we suspect, that the true proportion of boxes that contain a voucher is actually less than 20%."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "So the first thing you might wanna say is, well, what's our null and our alternative hypothesis? Well, our null hypothesis would be, well, the reality is what the breakfast brand claims, that 20% of the boxes contain a voucher. So that would be our null hypothesis. And our alternative hypothesis would be what we suspect, that the true proportion of boxes that contain a voucher is actually less than 20%. Now, if you're going to do a significance test, it's good practice to set up your significance level that you're going to eventually compare your p-value to ahead of time. And so, let's say we would want to assume, assume significance level. So let me write this, significance, significance level, alpha."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "And our alternative hypothesis would be what we suspect, that the true proportion of boxes that contain a voucher is actually less than 20%. Now, if you're going to do a significance test, it's good practice to set up your significance level that you're going to eventually compare your p-value to ahead of time. And so, let's say we would want to assume, assume significance level. So let me write this, significance, significance level, alpha. Let's just go with 0.05. And then we'll wanna think about the sample. And we're gonna figure out, if we assume that the null hypothesis is true, what's the probability that we get the, the sample proportion that we do, and if that is below this significance level, then we would reject the null hypothesis."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "So let me write this, significance, significance level, alpha. Let's just go with 0.05. And then we'll wanna think about the sample. And we're gonna figure out, if we assume that the null hypothesis is true, what's the probability that we get the, the sample proportion that we do, and if that is below this significance level, then we would reject the null hypothesis. And so, what we know about the sample, we know that we took 65 boxes of cereal, n is equal to 65, they tell us that right over there. And we, from that, we can calculate what the sample proportion is. It's going to be 11 out of 65, and we can get our calculator out."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "And we're gonna figure out, if we assume that the null hypothesis is true, what's the probability that we get the, the sample proportion that we do, and if that is below this significance level, then we would reject the null hypothesis. And so, what we know about the sample, we know that we took 65 boxes of cereal, n is equal to 65, they tell us that right over there. And we, from that, we can calculate what the sample proportion is. It's going to be 11 out of 65, and we can get our calculator out. Calculators are allowed on this part of the exam. And so, what is 11 divided by 65? It gives us, and I'll just round to the nearest thousand, 169, 0.169, 0.169, I'll say approximately, because I rounded it there."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "It's going to be 11 out of 65, and we can get our calculator out. Calculators are allowed on this part of the exam. And so, what is 11 divided by 65? It gives us, and I'll just round to the nearest thousand, 169, 0.169, 0.169, I'll say approximately, because I rounded it there. Now the next thing we wanna do before we make an inference is to make sure we're meeting the conditions for inference. So I'll write this down over here. Conditions for inference, conditions for inference."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "It gives us, and I'll just round to the nearest thousand, 169, 0.169, 0.169, I'll say approximately, because I rounded it there. Now the next thing we wanna do before we make an inference is to make sure we're meeting the conditions for inference. So I'll write this down over here. Conditions for inference, conditions for inference. And this is to feel good that we are properly sampling the population, that our sampling distribution is going to be roughly normal. So the first one is random sample, that is truly a random sample. And here they tell us, it is reasonable to assume that the 65 boxes purchased by the students are a random sample."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "Conditions for inference, conditions for inference. And this is to feel good that we are properly sampling the population, that our sampling distribution is going to be roughly normal. So the first one is random sample, that is truly a random sample. And here they tell us, it is reasonable to assume that the 65 boxes purchased by the students are a random sample. So that checks that off, so I will just point that to that right over there. So that checks that off. The next one is the normal condition, that the shape is roughly normal and it isn't skewed dramatically one way or the other."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "And here they tell us, it is reasonable to assume that the 65 boxes purchased by the students are a random sample. So that checks that off, so I will just point that to that right over there. So that checks that off. The next one is the normal condition, that the shape is roughly normal and it isn't skewed dramatically one way or the other. And in order to meet that condition, the sample size times the true assumed proportion, and we're going to assume that the null hypothesis is true, and so we could say that, and we could even say that this is the proportion assumed in the null hypothesis, that's what that notation would imply. And if you're doing this on the actual test, you should explain your use of notation a little bit more than I might do for the sake of time. But this needs to be greater than or equal to 10, and n times one minus the assumed proportion needs to be greater than or equal to 10."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "The next one is the normal condition, that the shape is roughly normal and it isn't skewed dramatically one way or the other. And in order to meet that condition, the sample size times the true assumed proportion, and we're going to assume that the null hypothesis is true, and so we could say that, and we could even say that this is the proportion assumed in the null hypothesis, that's what that notation would imply. And if you're doing this on the actual test, you should explain your use of notation a little bit more than I might do for the sake of time. But this needs to be greater than or equal to 10, and n times one minus the assumed proportion needs to be greater than or equal to 10. Well, let's see, n is 65, so 65 times the assumed proportion is 0.2, that is going to be equal to 13. 13 is indeed greater than or equal to 10, so that checks off. And then we would take n, 65, times one minus the assumed proportion, so 0.8, and that is going to be equal to, let's see, that would just be 65 minus 13, which is going to be equal to 52, and that indeed is also greater than or equal to 10, so we met that condition right over there."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "But this needs to be greater than or equal to 10, and n times one minus the assumed proportion needs to be greater than or equal to 10. Well, let's see, n is 65, so 65 times the assumed proportion is 0.2, that is going to be equal to 13. 13 is indeed greater than or equal to 10, so that checks off. And then we would take n, 65, times one minus the assumed proportion, so 0.8, and that is going to be equal to, let's see, that would just be 65 minus 13, which is going to be equal to 52, and that indeed is also greater than or equal to 10, so we met that condition right over there. And then the last one is the independence, independence. We aren't sampling these boxes with replacement, so we need to feel good that they are less than 10% of the population of boxes. And they don't tell us that explicitly, but it would be good practice to just say, going to assume, assume more than, let's see, 10 times that, 650 boxes in the population, boxes in population, population, which would imply that n is less than 10%, or less than or equal to 10% of population, of population, which would allow us to check off the independence condition."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "And then we would take n, 65, times one minus the assumed proportion, so 0.8, and that is going to be equal to, let's see, that would just be 65 minus 13, which is going to be equal to 52, and that indeed is also greater than or equal to 10, so we met that condition right over there. And then the last one is the independence, independence. We aren't sampling these boxes with replacement, so we need to feel good that they are less than 10% of the population of boxes. And they don't tell us that explicitly, but it would be good practice to just say, going to assume, assume more than, let's see, 10 times that, 650 boxes in the population, boxes in population, population, which would imply that n is less than 10%, or less than or equal to 10% of population, of population, which would allow us to check off the independence condition. And so given that we've met our conditions for inference, now let's think about the sampling distribution. So the sampling distribution of the sample proportions, because that's what we're going to use to calculate AP value. So we know a few things about the sampling distribution of the sample proportions."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "And they don't tell us that explicitly, but it would be good practice to just say, going to assume, assume more than, let's see, 10 times that, 650 boxes in the population, boxes in population, population, which would imply that n is less than 10%, or less than or equal to 10% of population, of population, which would allow us to check off the independence condition. And so given that we've met our conditions for inference, now let's think about the sampling distribution. So the sampling distribution of the sample proportions, because that's what we're going to use to calculate AP value. So we know a few things about the sampling distribution of the sample proportions. We know that the mean of the sampling distribution of the sample proportions is just going to be the assumed true proportion, so that's the proportion from the null hypothesis. And we know that the standard deviation of the sampling distribution of the sample proportions, this is going to be equal to, and we've seen this in multiple videos already, this is the assumed proportion times one minus the assumed proportion from our null hypothesis divided by n, which in this case is going to be equal to 0.2 times 0.8, all of that over 65. Once again, let's get our calculator out."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "So we know a few things about the sampling distribution of the sample proportions. We know that the mean of the sampling distribution of the sample proportions is just going to be the assumed true proportion, so that's the proportion from the null hypothesis. And we know that the standard deviation of the sampling distribution of the sample proportions, this is going to be equal to, and we've seen this in multiple videos already, this is the assumed proportion times one minus the assumed proportion from our null hypothesis divided by n, which in this case is going to be equal to 0.2 times 0.8, all of that over 65. Once again, let's get our calculator out. So we're gonna have the square root of 0.2 times 0.8 divided by 65, and then close my parentheses. I get, so 0.0496. So this is approximately 0.0496."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "Once again, let's get our calculator out. So we're gonna have the square root of 0.2 times 0.8 divided by 65, and then close my parentheses. I get, so 0.0496. So this is approximately 0.0496. Now the next step is to figure out the p-value, which we can then compare to our significance level to decide whether or not to reject the null hypothesis. And in order to calculate the p-value, let's figure out our z-statistic, which is how many standard deviations above or below the mean of the sampling distribution is the sample statistic that we happen to get for this sample of 65. And we have seen this in previous videos."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "So this is approximately 0.0496. Now the next step is to figure out the p-value, which we can then compare to our significance level to decide whether or not to reject the null hypothesis. And in order to calculate the p-value, let's figure out our z-statistic, which is how many standard deviations above or below the mean of the sampling distribution is the sample statistic that we happen to get for this sample of 65. And we have seen this in previous videos. This would be equal to our sample proportion minus the assumed proportion for the population in the null hypothesis, so the difference between those, and then divided by the standard deviation of the sampling distribution of the sample proportions. This would tell us how many standard deviations are we above or below the mean of the sampling distribution. So in this particular situation, this is going to be 0.169 minus 0.2, all of that over this value right over here, which is approximately 0.0496."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "And we have seen this in previous videos. This would be equal to our sample proportion minus the assumed proportion for the population in the null hypothesis, so the difference between those, and then divided by the standard deviation of the sampling distribution of the sample proportions. This would tell us how many standard deviations are we above or below the mean of the sampling distribution. So in this particular situation, this is going to be 0.169 minus 0.2, all of that over this value right over here, which is approximately 0.0496. I can get the calculator out again. And so we have 0.169 minus 0.2. So that's how far below our sample proportion is than the mean of the sampling distribution, which is the assumed proportion from the null hypothesis, assumed population proportion."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "So in this particular situation, this is going to be 0.169 minus 0.2, all of that over this value right over here, which is approximately 0.0496. I can get the calculator out again. And so we have 0.169 minus 0.2. So that's how far below our sample proportion is than the mean of the sampling distribution, which is the assumed proportion from the null hypothesis, assumed population proportion. And then we divide that, we're gonna divide that by the standard deviation of the sampling distribution of the sample proportions. So divide that by 0.0496, and we get a z value of approximately, because remember, this is using a bunch of approximations right over here, about negative 0.625. So z is approximately negative 0.625."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "So that's how far below our sample proportion is than the mean of the sampling distribution, which is the assumed proportion from the null hypothesis, assumed population proportion. And then we divide that, we're gonna divide that by the standard deviation of the sampling distribution of the sample proportions. So divide that by 0.0496, and we get a z value of approximately, because remember, this is using a bunch of approximations right over here, about negative 0.625. So z is approximately negative 0.625. And so now we can think about the actual p-value our p-value, which is equal to the probability of getting a sample proportion that is at least as low as the one that we got. So a sample proportion that is less than or equal to the one that we got, 0.169, assuming the null hypothesis is true. So we could say assuming the null hypothesis is true, which is equal to the probability of getting a z statistic that is less than or equal to this value right over here, negative 0.625."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "So z is approximately negative 0.625. And so now we can think about the actual p-value our p-value, which is equal to the probability of getting a sample proportion that is at least as low as the one that we got. So a sample proportion that is less than or equal to the one that we got, 0.169, assuming the null hypothesis is true. So we could say assuming the null hypothesis is true, which is equal to the probability of getting a z statistic that is less than or equal to this value right over here, negative 0.625. And now we can use our calculator to actually calculate this. So what we can do is we can go to second distribution. We wanna do normal CDF."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "So we could say assuming the null hypothesis is true, which is equal to the probability of getting a z statistic that is less than or equal to this value right over here, negative 0.625. And now we can use our calculator to actually calculate this. So what we can do is we can go to second distribution. We wanna do normal CDF. So go to normal CDF. And then our lower bound is actually going to be, we could say negative infinity. Our upper bound is going to be negative, so negative 0.625, 625."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "We wanna do normal CDF. So go to normal CDF. And then our lower bound is actually going to be, we could say negative infinity. Our upper bound is going to be negative, so negative 0.625, 625. And then this is, this is, you could say, a normalized normal distribution here. So we'll just go with all of this, because we're just thinking about the z statistic right over here. Click Enter, and then click Enter."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "Our upper bound is going to be negative, so negative 0.625, 625. And then this is, this is, you could say, a normalized normal distribution here. So we'll just go with all of this, because we're just thinking about the z statistic right over here. Click Enter, and then click Enter. And then we get, this is going to be, let's say, 0.266. So this is approximately 0.266. And so let's just make sure what we just did."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "Click Enter, and then click Enter. And then we get, this is going to be, let's say, 0.266. So this is approximately 0.266. And so let's just make sure what we just did. If this right over here is the assumed sampling distribution of the sample proportions, where we are assuming that our null hypothesis is true, so the mean of our sampling distribution is going to be our assumed proportion, what we're saying is, look, we got a result over here. This is where our p hat happened to be, right over here. What's the probability of getting a result that far below the true proportion or further?"}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "And so let's just make sure what we just did. If this right over here is the assumed sampling distribution of the sample proportions, where we are assuming that our null hypothesis is true, so the mean of our sampling distribution is going to be our assumed proportion, what we're saying is, look, we got a result over here. This is where our p hat happened to be, right over here. What's the probability of getting a result that far below the true proportion or further? So this is what we calculated just now. And now when you look at this, this is almost a 27% probability. When you compare our p value, we're gonna compare our p value to our significance level."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "What's the probability of getting a result that far below the true proportion or further? So this is what we calculated just now. And now when you look at this, this is almost a 27% probability. When you compare our p value, we're gonna compare our p value to our significance level. And we see that our p value is clearly greater than our significance level. 0.266 is clearly greater than our significance level of 0.05. What we were saying is, if there was less than a 5% chance of getting the sample proportion that we got, then we would reject the null hypothesis, which would suggest the alternative."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy (2).mp3", "Sentence": "When you compare our p value, we're gonna compare our p value to our significance level. And we see that our p value is clearly greater than our significance level. 0.266 is clearly greater than our significance level of 0.05. What we were saying is, if there was less than a 5% chance of getting the sample proportion that we got, then we would reject the null hypothesis, which would suggest the alternative. But here, the probability of getting the sample proportion that we got, if we assume that the null hypothesis is true, is almost 27%. And so that's well above our significance level. So we will fail, so because of this, because of this, we fail to reject, reject our null hypothesis."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "You can never have too much practice dealing with the normal distribution, because it's really one of those super important building blocks for the rest of statistics, and really a lot of your life. So what I've done here is I've taken some sample problems. This is from ck12.org's open source flexbook, their AP Statistics flexbook. And I've taken the problems from their normal distribution chapter, so you could go to their site and actually look up these same problems. So this first problem, which of the following data sets is most likely to be normally distributed? For the other choices, explain why you believe they would not follow a normal distribution. So let's see, choice A."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "And I've taken the problems from their normal distribution chapter, so you could go to their site and actually look up these same problems. So this first problem, which of the following data sets is most likely to be normally distributed? For the other choices, explain why you believe they would not follow a normal distribution. So let's see, choice A. So this is really, my beliefs come into play. So this is unusual in the math context. It's more of a, what do I think?"}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So let's see, choice A. So this is really, my beliefs come into play. So this is unusual in the math context. It's more of a, what do I think? It's kind of an essay question. So let's see what they have here. A, the hand span, measured from the tip of the thumb to the tip of the extended fifth finger."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "It's more of a, what do I think? It's kind of an essay question. So let's see what they have here. A, the hand span, measured from the tip of the thumb to the tip of the extended fifth finger. So I think they're talking about, let me see if I can draw a hand. So if that's the index finger, and then you've got the middle finger, and then you've got your ring finger, and then you've got your pinky, and the hand will look something like that. I think they're talking about this distance, from the tip of the thumb to the tip of the extended fifth finger, which is a fancy way of saying the pinky, I think."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "A, the hand span, measured from the tip of the thumb to the tip of the extended fifth finger. So I think they're talking about, let me see if I can draw a hand. So if that's the index finger, and then you've got the middle finger, and then you've got your ring finger, and then you've got your pinky, and the hand will look something like that. I think they're talking about this distance, from the tip of the thumb to the tip of the extended fifth finger, which is a fancy way of saying the pinky, I think. They're talking about that distance right there. And they're saying, if I were to measure it of a random sample of high school seniors, what would it look like? Well, you know, how far this is, this is a combination of genetics and environmental factors."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "I think they're talking about this distance, from the tip of the thumb to the tip of the extended fifth finger, which is a fancy way of saying the pinky, I think. They're talking about that distance right there. And they're saying, if I were to measure it of a random sample of high school seniors, what would it look like? Well, you know, how far this is, this is a combination of genetics and environmental factors. Maybe how much milk you drank, or how much you hung from your pinky from a bar while you were growing up. So I would think that it is a sum of a huge number of random processes. So I would guess that it is roughly normally distributed."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "Well, you know, how far this is, this is a combination of genetics and environmental factors. Maybe how much milk you drank, or how much you hung from your pinky from a bar while you were growing up. So I would think that it is a sum of a huge number of random processes. So I would guess that it is roughly normally distributed. If I look at my own hand, and my hand I don't think has grown much since I was a high school senior. It looks like, I don't know, it looks like roughly 9 inches or so. I play guitar, maybe that helped me stretch my hand."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So I would guess that it is roughly normally distributed. If I look at my own hand, and my hand I don't think has grown much since I was a high school senior. It looks like, I don't know, it looks like roughly 9 inches or so. I play guitar, maybe that helped me stretch my hand. But it's really an essay question, so I just have to say what I feel. So I would guess that the distribution would look something like this. I don't know, I've never done this, but maybe it has a mean of 8 inches or 9 inches, and it's distributed something like this."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "I play guitar, maybe that helped me stretch my hand. But it's really an essay question, so I just have to say what I feel. So I would guess that the distribution would look something like this. I don't know, I've never done this, but maybe it has a mean of 8 inches or 9 inches, and it's distributed something like this. It's distributed something like this. So maybe it probably does look like a normal distribution. But it probably won't be a perfect, in fact, I can guarantee you it won't be a perfect normal distribution."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "I don't know, I've never done this, but maybe it has a mean of 8 inches or 9 inches, and it's distributed something like this. It's distributed something like this. So maybe it probably does look like a normal distribution. But it probably won't be a perfect, in fact, I can guarantee you it won't be a perfect normal distribution. Because one, no one can have negative length of that span. This distance can never be negative. So they're going to have a, I guess, you can have no hand, so that would maybe be counted as 0."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "But it probably won't be a perfect, in fact, I can guarantee you it won't be a perfect normal distribution. Because one, no one can have negative length of that span. This distance can never be negative. So they're going to have a, I guess, you can have no hand, so that would maybe be counted as 0. But the distribution wouldn't go into the negative domain, so it wouldn't be a perfect normal distribution on the left-hand side. It would really just end here at 0. And even on the right-hand side, there are some physically impossible hand lengths."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So they're going to have a, I guess, you can have no hand, so that would maybe be counted as 0. But the distribution wouldn't go into the negative domain, so it wouldn't be a perfect normal distribution on the left-hand side. It would really just end here at 0. And even on the right-hand side, there are some physically impossible hand lengths. No one can have a hand that's larger than the height of Earth's atmosphere or an astronomical unit. You would start touching the sun. There's some point at which it is physically impossible to get to."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "And even on the right-hand side, there are some physically impossible hand lengths. No one can have a hand that's larger than the height of Earth's atmosphere or an astronomical unit. You would start touching the sun. There's some point at which it is physically impossible to get to. And in a true normal distribution, if I were to flip a bunch of coins, there's some very, very small probability that I could get a million heads in a row. It's almost 0, but there's some probability. But in the case of hand spans, there's no way out here the probability of a human being who happens to be a high school senior having a, I don't know, one mile length hand span, that's 0."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "There's some point at which it is physically impossible to get to. And in a true normal distribution, if I were to flip a bunch of coins, there's some very, very small probability that I could get a million heads in a row. It's almost 0, but there's some probability. But in the case of hand spans, there's no way out here the probability of a human being who happens to be a high school senior having a, I don't know, one mile length hand span, that's 0. So it's not going to be a perfect normal distribution at the outliers or as we get further and further away from the mean. But I think it'll be a pretty good, in our everyday world, as good as we're going to get approximation, the normal distribution is going to be a pretty good approximation for the distribution that we see. I guess one thing that, you know, this is high school seniors, when I did this, it was kind of from my point of view as a guy."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "But in the case of hand spans, there's no way out here the probability of a human being who happens to be a high school senior having a, I don't know, one mile length hand span, that's 0. So it's not going to be a perfect normal distribution at the outliers or as we get further and further away from the mean. But I think it'll be a pretty good, in our everyday world, as good as we're going to get approximation, the normal distribution is going to be a pretty good approximation for the distribution that we see. I guess one thing that, you know, this is high school seniors, when I did this, it was kind of from my point of view as a guy. And I would argue that high school seniors, guys probably have larger hands than women. So it's possible that you actually have a bimodal distribution. So instead of having it like this, it's possible that the distribution looks like this."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "I guess one thing that, you know, this is high school seniors, when I did this, it was kind of from my point of view as a guy. And I would argue that high school seniors, guys probably have larger hands than women. So it's possible that you actually have a bimodal distribution. So instead of having it like this, it's possible that the distribution looks like this. That you have one peak for guys, maybe at 8 inches, and then maybe another peak for women at, I don't know, 7 inches, and then the distribution falls off like that. So it's also possible it could be bimodal. But in general, a normal distribution is going to be a pretty good approximation for part A of this problem."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So instead of having it like this, it's possible that the distribution looks like this. That you have one peak for guys, maybe at 8 inches, and then maybe another peak for women at, I don't know, 7 inches, and then the distribution falls off like that. So it's also possible it could be bimodal. But in general, a normal distribution is going to be a pretty good approximation for part A of this problem. Let's see what part B, what they're asking us to describe. The annual salaries of all employees of a large shipping company. So if we're talking about annual salaries, you know, we have minimum wage laws, whatnot."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "But in general, a normal distribution is going to be a pretty good approximation for part A of this problem. Let's see what part B, what they're asking us to describe. The annual salaries of all employees of a large shipping company. So if we're talking about annual salaries, you know, we have minimum wage laws, whatnot. So I would guess that any corporation, if we're talking about full-time workers at least, there's going to be some minimum salary that people have. And probably a lot of people will have that minimum salary because it'll be probably the most labor-intensive jobs. Most people are down there at the low end of the pay scale."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So if we're talking about annual salaries, you know, we have minimum wage laws, whatnot. So I would guess that any corporation, if we're talking about full-time workers at least, there's going to be some minimum salary that people have. And probably a lot of people will have that minimum salary because it'll be probably the most labor-intensive jobs. Most people are down there at the low end of the pay scale. And then you have your different middle-level managers and whatnot. And then you probably have this big gap, and then you probably have your true executives, maybe your CEO or whatnot, with, you know, if this mean right here is maybe $40,000 a year, and this is probably $80,000 where some of the mid-level managers lie. But this out here, this will probably be, you know, actually if you were to draw a real, the way I've scaled it right now, this would be $80,000, this would be about $200,000, which is actually a reasonable salary for a CEO."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "Most people are down there at the low end of the pay scale. And then you have your different middle-level managers and whatnot. And then you probably have this big gap, and then you probably have your true executives, maybe your CEO or whatnot, with, you know, if this mean right here is maybe $40,000 a year, and this is probably $80,000 where some of the mid-level managers lie. But this out here, this will probably be, you know, actually if you were to draw a real, the way I've scaled it right now, this would be $80,000, this would be about $200,000, which is actually a reasonable salary for a CEO. But the reality is that this actually might get pushed way out from there. It might look something like that. It might be way off the charts."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "But this out here, this will probably be, you know, actually if you were to draw a real, the way I've scaled it right now, this would be $80,000, this would be about $200,000, which is actually a reasonable salary for a CEO. But the reality is that this actually might get pushed way out from there. It might look something like that. It might be way off the charts. You know, let's say the CEO made $5 million in a year because he cashed in a bunch of options or something. So it would be way over here, and maybe it's the CEO and a couple of other people, the CFO or the founders. So my guess is it definitely wouldn't be a normal distribution, and it definitely would have a second, it would be bimodal."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "It might be way off the charts. You know, let's say the CEO made $5 million in a year because he cashed in a bunch of options or something. So it would be way over here, and maybe it's the CEO and a couple of other people, the CFO or the founders. So my guess is it definitely wouldn't be a normal distribution, and it definitely would have a second, it would be bimodal. You would have another peak over here for senior management up at the, unless we're, well, they're not saying, you know, if we're maybe in Europe, this would probably be closer to the left. But it won't be a perfect normal distribution, and you're not going to have any values below a certain threshold, below that kind of minimum wage level. So I would call this, when you have a tail that goes more to the right than to the left, call this a right-skewed distribution."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So my guess is it definitely wouldn't be a normal distribution, and it definitely would have a second, it would be bimodal. You would have another peak over here for senior management up at the, unless we're, well, they're not saying, you know, if we're maybe in Europe, this would probably be closer to the left. But it won't be a perfect normal distribution, and you're not going to have any values below a certain threshold, below that kind of minimum wage level. So I would call this, when you have a tail that goes more to the right than to the left, call this a right-skewed distribution. And since it has two humps right here, one there and one there, we could also say it's bimodal. I mean, it depends on what kind of company this is, but that would be my guess of a lot of large shipping companies' salaries. Let's look at choice C, or problem part C. The annual salaries of a random sample of 50 CEOs of major companies, 25 women and 25 men."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So I would call this, when you have a tail that goes more to the right than to the left, call this a right-skewed distribution. And since it has two humps right here, one there and one there, we could also say it's bimodal. I mean, it depends on what kind of company this is, but that would be my guess of a lot of large shipping companies' salaries. Let's look at choice C, or problem part C. The annual salaries of a random sample of 50 CEOs of major companies, 25 women and 25 men. The fact that they wrote this here, I think they maybe are implying that maybe men and women, you know, the gender gap has not been closed fully, and there is some discrepancy. So if it was just purely 50 CEOs of major companies, I would say it's probably close to a normal distribution. It's probably something like, well, you know, once again, there's going to be some level below which no CEO is willing to work for, although I've heard of some cases where they work for free, but they're really getting paid in other ways."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "Let's look at choice C, or problem part C. The annual salaries of a random sample of 50 CEOs of major companies, 25 women and 25 men. The fact that they wrote this here, I think they maybe are implying that maybe men and women, you know, the gender gap has not been closed fully, and there is some discrepancy. So if it was just purely 50 CEOs of major companies, I would say it's probably close to a normal distribution. It's probably something like, well, you know, once again, there's going to be some level below which no CEO is willing to work for, although I've heard of some cases where they work for free, but they're really getting paid in other ways. If you include all of those things, there's probably some base salary that all CEOs make at least that much, and then it goes up to some value, you know, the highest probability value, and then it probably has a long tail to the right. And this is if there were no gender gap. So this would just be a purely right-skewed distribution, where you have a long tail to the right."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "It's probably something like, well, you know, once again, there's going to be some level below which no CEO is willing to work for, although I've heard of some cases where they work for free, but they're really getting paid in other ways. If you include all of those things, there's probably some base salary that all CEOs make at least that much, and then it goes up to some value, you know, the highest probability value, and then it probably has a long tail to the right. And this is if there were no gender gap. So this would just be a purely right-skewed distribution, where you have a long tail to the right. Now if you assume that there's some gender gap, then you might have two humps here, which would be a bimodal distribution. So if you assume there's some gender gap, this is part c right here, then maybe there's one hump for women, and if you assume that women are less than men, then another hump for men, and there are 25 of each, so there wouldn't be necessarily more men than women, and then it would skew all the way off to the right. And in fact, I think there would probably be a chance that you have this other notion here, where you have these super CEOs or mega CEOs who make millions, while most CEOs probably just make, I'll put it in quotation marks, a few hundred thousand dollars, while there's a small subset that are way off many standard deviations to the right."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So this would just be a purely right-skewed distribution, where you have a long tail to the right. Now if you assume that there's some gender gap, then you might have two humps here, which would be a bimodal distribution. So if you assume there's some gender gap, this is part c right here, then maybe there's one hump for women, and if you assume that women are less than men, then another hump for men, and there are 25 of each, so there wouldn't be necessarily more men than women, and then it would skew all the way off to the right. And in fact, I think there would probably be a chance that you have this other notion here, where you have these super CEOs or mega CEOs who make millions, while most CEOs probably just make, I'll put it in quotation marks, a few hundred thousand dollars, while there's a small subset that are way off many standard deviations to the right. So it could even be a trimodal distribution here. So that's choice c. And so far, choice a looks like the best candidate for a pure, or the closest to being a normal distribution. Let's see what choice d is."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "And in fact, I think there would probably be a chance that you have this other notion here, where you have these super CEOs or mega CEOs who make millions, while most CEOs probably just make, I'll put it in quotation marks, a few hundred thousand dollars, while there's a small subset that are way off many standard deviations to the right. So it could even be a trimodal distribution here. So that's choice c. And so far, choice a looks like the best candidate for a pure, or the closest to being a normal distribution. Let's see what choice d is. The dates of 100 pennies taken from a cash drawer in a convenience store. 100 pennies. So that's actually an interesting experiment."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "Let's see what choice d is. The dates of 100 pennies taken from a cash drawer in a convenience store. 100 pennies. So that's actually an interesting experiment. But I would guess, and once again, this is really a question where I get to express my feelings about these things. As long as your answer is reasonable, I would say that it is right. Most pennies are newer pennies, because they go out of commission, they get traded out, they get worn out as they age, they get lost, or they get pressed at the little tourist place into those little souvenir things."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So that's actually an interesting experiment. But I would guess, and once again, this is really a question where I get to express my feelings about these things. As long as your answer is reasonable, I would say that it is right. Most pennies are newer pennies, because they go out of commission, they get traded out, they get worn out as they age, they get lost, or they get pressed at the little tourist place into those little souvenir things. I'm not even sure if that's legal, if you can do that to money legally. So my guess is that if you were to plot it, you would have a ton of pennies that are within the last few years. So the dates of 100 pennies, not their age."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "Most pennies are newer pennies, because they go out of commission, they get traded out, they get worn out as they age, they get lost, or they get pressed at the little tourist place into those little souvenir things. I'm not even sure if that's legal, if you can do that to money legally. So my guess is that if you were to plot it, you would have a ton of pennies that are within the last few years. So the dates of 100 pennies, not their age. So if we're sitting here in 2000, so if this is 2010, I would guess that right now you're not going to find any 2010 pennies, but you're probably going to find a ton of 2009 pennies, and then it probably just goes down from there. And of course, you're not going to find pennies that are older than, say, the United States, or before they even started printing pennies. So obviously this tail isn't going to go to the left forever, but my guess is you're going to have a left skewed distribution."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So the dates of 100 pennies, not their age. So if we're sitting here in 2000, so if this is 2010, I would guess that right now you're not going to find any 2010 pennies, but you're probably going to find a ton of 2009 pennies, and then it probably just goes down from there. And of course, you're not going to find pennies that are older than, say, the United States, or before they even started printing pennies. So obviously this tail isn't going to go to the left forever, but my guess is you're going to have a left skewed distribution. Where you have the bulk of the distribution on the right, but the tail goes off to the left. That's why it's called a left skewed distribution. Sometimes this is called a negatively skewed distribution, and similarly this right skewed distribution could also, or this right skewed distribution, sometimes it's called positively skewed."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So obviously this tail isn't going to go to the left forever, but my guess is you're going to have a left skewed distribution. Where you have the bulk of the distribution on the right, but the tail goes off to the left. That's why it's called a left skewed distribution. Sometimes this is called a negatively skewed distribution, and similarly this right skewed distribution could also, or this right skewed distribution, sometimes it's called positively skewed. And if you have only one hump, you don't have a multimodal distribution like this, in a left skewed distribution, your mean is going to be to the left of your median. So in this case, maybe your median might be someplace over here. But since you have this long tail to the left, your mean might be someplace over here."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "Sometimes this is called a negatively skewed distribution, and similarly this right skewed distribution could also, or this right skewed distribution, sometimes it's called positively skewed. And if you have only one hump, you don't have a multimodal distribution like this, in a left skewed distribution, your mean is going to be to the left of your median. So in this case, maybe your median might be someplace over here. But since you have this long tail to the left, your mean might be someplace over here. And likewise, in this distribution, your median, your middle value, might be someplace like this. But because it's right skewed, and for the most part it only has one big hump, this hump won't change things too much because it's small, your mean is going to be to the right of it. So that's another reason why it's called a right skewed or positively skewed distribution."}, {"video_title": "Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3", "Sentence": "So in previous videos, we talked about the claim that if I have two random variables, X and Y, that are independent, then the variance of the sum of those two random variables or the difference of those two random variables is going to be equal to the sum of the variances. So that if you have independent random variables, your variation is going to increase when you take a sum or a difference, and we built a little bit of intuition there. What I wanna talk about in this video, it's really about building even more intuition, is get a gut feeling for why this independence is important for making this claim. And to get that intuition, let's look at two random variables that are definitely random variables, but that are definitely not independent. So let's say, let's let X is equal to the number of hours that the next person you meet, so I'll say random person, random person slept yesterday. And let's say that Y is equal to the number of hours that same person, person was awake yesterday. And appreciate why these are not independent random variables."}, {"video_title": "Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3", "Sentence": "And to get that intuition, let's look at two random variables that are definitely random variables, but that are definitely not independent. So let's say, let's let X is equal to the number of hours that the next person you meet, so I'll say random person, random person slept yesterday. And let's say that Y is equal to the number of hours that same person, person was awake yesterday. And appreciate why these are not independent random variables. One of them is going to completely determine the other. If I slept eight hours yesterday, then I would have been awake for 16 hours. If I slept for 16 hours, then I would have been awake for eight hours."}, {"video_title": "Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3", "Sentence": "And appreciate why these are not independent random variables. One of them is going to completely determine the other. If I slept eight hours yesterday, then I would have been awake for 16 hours. If I slept for 16 hours, then I would have been awake for eight hours. We know that X plus Y, even though they're random variables, and there could be variation in X, and there could be variation in Y, but for any given person, remember, these are still based on that same person, X plus Y is always going to be equal to 24 hours. So these are not independent, not independent. If you're given one of the variables, it would completely determine what the other variable is."}, {"video_title": "Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3", "Sentence": "If I slept for 16 hours, then I would have been awake for eight hours. We know that X plus Y, even though they're random variables, and there could be variation in X, and there could be variation in Y, but for any given person, remember, these are still based on that same person, X plus Y is always going to be equal to 24 hours. So these are not independent, not independent. If you're given one of the variables, it would completely determine what the other variable is. The probability of getting a certain value for one variable is going to be very different given what value you got for the other variable. So they're not independent at all. So in this situation, if someone said, let's just say, for the sake of argument, that the variance of X, the variance of X, is equal to, I don't know, let's say it's equal to four, and the units for variance would be squared hours, so four hours squared."}, {"video_title": "Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3", "Sentence": "If you're given one of the variables, it would completely determine what the other variable is. The probability of getting a certain value for one variable is going to be very different given what value you got for the other variable. So they're not independent at all. So in this situation, if someone said, let's just say, for the sake of argument, that the variance of X, the variance of X, is equal to, I don't know, let's say it's equal to four, and the units for variance would be squared hours, so four hours squared. We could say that the standard deviation for X in this case would be two hours. And let's say that the variance, let's say the standard deviation of Y is also equal to two hours. And let's say that the variance of Y, variance of Y, well, it would be the square of the standard deviation, so it would be four hours, four hours squared would be our units."}, {"video_title": "Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3", "Sentence": "So in this situation, if someone said, let's just say, for the sake of argument, that the variance of X, the variance of X, is equal to, I don't know, let's say it's equal to four, and the units for variance would be squared hours, so four hours squared. We could say that the standard deviation for X in this case would be two hours. And let's say that the variance, let's say the standard deviation of Y is also equal to two hours. And let's say that the variance of Y, variance of Y, well, it would be the square of the standard deviation, so it would be four hours, four hours squared would be our units. So if we just tried to blindly say, oh, I'm just gonna apply this little expression, this claim we had, without thinking about the independence, we would try to say, well, then the variance of X plus Y, the variance of X plus Y, must be equal to the sum of their variances. So it would be four plus four. So is it equal to eight hours squared?"}, {"video_title": "Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3", "Sentence": "And let's say that the variance of Y, variance of Y, well, it would be the square of the standard deviation, so it would be four hours, four hours squared would be our units. So if we just tried to blindly say, oh, I'm just gonna apply this little expression, this claim we had, without thinking about the independence, we would try to say, well, then the variance of X plus Y, the variance of X plus Y, must be equal to the sum of their variances. So it would be four plus four. So is it equal to eight hours squared? Well, that doesn't make any sense, because we know that a random variable that is equal to X plus Y, that this is always going to be 24 hours. In fact, it's not going to have any variation. X plus Y is always gonna be 24 hours."}, {"video_title": "Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3", "Sentence": "So is it equal to eight hours squared? Well, that doesn't make any sense, because we know that a random variable that is equal to X plus Y, that this is always going to be 24 hours. In fact, it's not going to have any variation. X plus Y is always gonna be 24 hours. So for these two random variables, because they are so connected, they are not independent at all, this is actually going to be zero. There is zero variance here. X plus Y is always going to be 24, at least on Earth, where we have a 24-hour day."}, {"video_title": "Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3", "Sentence": "X plus Y is always gonna be 24 hours. So for these two random variables, because they are so connected, they are not independent at all, this is actually going to be zero. There is zero variance here. X plus Y is always going to be 24, at least on Earth, where we have a 24-hour day. I guess if someone lived on another planet or something, then it could be slightly different. And we're assuming that we have an exactly 24-hour day on Earth. So this is to give you a gut sense of why independence matters for making this claim."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So whether there was an accident in the last year for customers of all American auto insurance. Complete the following two-way table of column relative frequencies, so that's what they're talking here, this is a two-way table of column relative frequencies, if necessary, round your answers to the nearest hundred. So let's see what they're saying. They're saying, let's see, of the accidents within the last year, 28 were the people were driving an SUV, a sport utility vehicle, and 35 were in a sports car. Of the no accidents in the last year, 97 were an SUV, and 104 were a sports car. Or another way you could think of it, of the sport utility vehicles that were driven, and the total, let's see, it's 28 plus 97, which is going to be 125, of that 125, 28 had an accident within the last year, and 97 did not have an accident in the last year. Similarly, you could say of the 139 sports cars, 35 had an accident in the last year, 104 did not have an accident in the last year."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "They're saying, let's see, of the accidents within the last year, 28 were the people were driving an SUV, a sport utility vehicle, and 35 were in a sports car. Of the no accidents in the last year, 97 were an SUV, and 104 were a sports car. Or another way you could think of it, of the sport utility vehicles that were driven, and the total, let's see, it's 28 plus 97, which is going to be 125, of that 125, 28 had an accident within the last year, and 97 did not have an accident in the last year. Similarly, you could say of the 139 sports cars, 35 had an accident in the last year, 104 did not have an accident in the last year. And so what they want us to do is put those relative frequencies in here. So the way we could think about it, one right over here, this represents all of the sport utility vehicles. So one way you could think about it, that represents the whole universe of the sport utility vehicles, or at least the universe that this table shows."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Similarly, you could say of the 139 sports cars, 35 had an accident in the last year, 104 did not have an accident in the last year. And so what they want us to do is put those relative frequencies in here. So the way we could think about it, one right over here, this represents all of the sport utility vehicles. So one way you could think about it, that represents the whole universe of the sport utility vehicles, or at least the universe that this table shows. So that's really representative of the 28 plus 97. And so in each of these, we want to put the relative frequency. So this right over here is going to be 28, 28 divided by the total."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So one way you could think about it, that represents the whole universe of the sport utility vehicles, or at least the universe that this table shows. So that's really representative of the 28 plus 97. And so in each of these, we want to put the relative frequency. So this right over here is going to be 28, 28 divided by the total. Notice over here it was 28, but we want this number to be the fraction of the total. Well, the fraction of the total is going to be 28 over 97 plus 28, which of course is going to be 125. Actually, let me just write them all like that first."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So this right over here is going to be 28, 28 divided by the total. Notice over here it was 28, but we want this number to be the fraction of the total. Well, the fraction of the total is going to be 28 over 97 plus 28, which of course is going to be 125. Actually, let me just write them all like that first. This one right over here is going to be 97 over 125. And of course, when you add this one and this one, it should add up to one. Likewise, this one's going to be 35 over 139, 35 plus 104, so 139."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Actually, let me just write them all like that first. This one right over here is going to be 97 over 125. And of course, when you add this one and this one, it should add up to one. Likewise, this one's going to be 35 over 139, 35 plus 104, so 139. And this is going to be 104 over 104 plus 35, which is 139. And so let me just calculate each of them using this calculator. So let me scroll down a little bit."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Likewise, this one's going to be 35 over 139, 35 plus 104, so 139. And this is going to be 104 over 104 plus 35, which is 139. And so let me just calculate each of them using this calculator. So let me scroll down a little bit. And so if I do 28 divided by 125, I get 0.224. They said round your answers to the nearest hundredth. So this is 0.22."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So let me scroll down a little bit. And so if I do 28 divided by 125, I get 0.224. They said round your answers to the nearest hundredth. So this is 0.22. No accident within the last year, 97 divided by 125. So 97 divided by 125 is equal to, let's see, here if I round to the nearest hundredth, I'm going to round up 0.78. So this is 0.78."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So this is 0.22. No accident within the last year, 97 divided by 125. So 97 divided by 125 is equal to, let's see, here if I round to the nearest hundredth, I'm going to round up 0.78. So this is 0.78. Then 35 divided by 139, 35 divided by 139 is equal to, round to the nearest hundredth, 0.25. 0.25. And then 104 divided by 139."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So this is 0.78. Then 35 divided by 139, 35 divided by 139 is equal to, round to the nearest hundredth, 0.25. 0.25. And then 104 divided by 139. 104 divided by 139 gets me, if I round to the nearest hundredth, 0.75. 0.75. And I can check my answer, and I got it right."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And then 104 divided by 139. 104 divided by 139 gets me, if I round to the nearest hundredth, 0.75. 0.75. And I can check my answer, and I got it right. But the key thing here is to make sure we understand what's going on here. So one way to think about this is 22% of the sport utility vehicles had an accident within the last year, or you could say 0.22 of them. And you could say 78%, or 0.78, of the sport utility vehicles had no accidents."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "In the game Rock, Paper, Scissors, Kenny expects to win, tie, and lose with equal frequency. Kenny plays Rock, Paper, Scissors often, but he suspected his own games were not following that pattern. So he took a random sample of 24 games and recorded their outcomes. Here are his results. So out of the 24 games, he won four, lost 13, and tied seven times. He wants to use these results to carry out a chi-squared goodness-of-fit test to determine if the distribution of his outcomes disagrees with an even distribution. What are the values of the test statistic, the chi-squared test statistic, and p-value for Kenny's test?"}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "Here are his results. So out of the 24 games, he won four, lost 13, and tied seven times. He wants to use these results to carry out a chi-squared goodness-of-fit test to determine if the distribution of his outcomes disagrees with an even distribution. What are the values of the test statistic, the chi-squared test statistic, and p-value for Kenny's test? So pause this video and see if you can figure that out. Okay, so he's essentially just doing a hypothesis test using the chi-squared statistic because it's a hypothesis that's thinking about multiple categories. So what would his null hypothesis be?"}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "What are the values of the test statistic, the chi-squared test statistic, and p-value for Kenny's test? So pause this video and see if you can figure that out. Okay, so he's essentially just doing a hypothesis test using the chi-squared statistic because it's a hypothesis that's thinking about multiple categories. So what would his null hypothesis be? Well, his null hypothesis would be that all of the outcomes are equal probability. Outcomes equal, equal probability. And then his alternative hypothesis would be that his outcomes have not equal, not equal probability."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "So what would his null hypothesis be? Well, his null hypothesis would be that all of the outcomes are equal probability. Outcomes equal, equal probability. And then his alternative hypothesis would be that his outcomes have not equal, not equal probability. Remember, we assume that the null hypothesis is true. And then assuming if the null hypothesis is true, the probability of getting a result at least this extreme is low enough, then we would reject our null hypothesis. Another way to think about it is if our p-value is below a threshold, we would reject our null hypothesis."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "And then his alternative hypothesis would be that his outcomes have not equal, not equal probability. Remember, we assume that the null hypothesis is true. And then assuming if the null hypothesis is true, the probability of getting a result at least this extreme is low enough, then we would reject our null hypothesis. Another way to think about it is if our p-value is below a threshold, we would reject our null hypothesis. And so what he did is he took a sample of 24 games, so n is equal to 24, and then this was the data that he got. Now, before we even calculate our chi-squared statistic and figure out what's the probability of getting a chi-squared statistic that large or greater, let's make sure we meet the conditions for inference for a chi-squared goodness-of-fit test. So you've seen some of them, but some of them are a little bit different."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "Another way to think about it is if our p-value is below a threshold, we would reject our null hypothesis. And so what he did is he took a sample of 24 games, so n is equal to 24, and then this was the data that he got. Now, before we even calculate our chi-squared statistic and figure out what's the probability of getting a chi-squared statistic that large or greater, let's make sure we meet the conditions for inference for a chi-squared goodness-of-fit test. So you've seen some of them, but some of them are a little bit different. One is the random condition. I'll write them up here. The random condition."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "So you've seen some of them, but some of them are a little bit different. One is the random condition. I'll write them up here. The random condition. And that would be that this is truly a random sample of games. And it tells us right here, he took a random sample of his 24 games, so we meet that condition. The second condition, when we're talking about chi-squared hypothesis testing, is the large counts, large counts condition."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "The random condition. And that would be that this is truly a random sample of games. And it tells us right here, he took a random sample of his 24 games, so we meet that condition. The second condition, when we're talking about chi-squared hypothesis testing, is the large counts, large counts condition. And this is an important one to appreciate. This is that the expected number of each category of outcomes is at least equal to five. Now, you might say, hey, wait, wait, I only got four wins, or Kenny only got four wins out of his sample of 24."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "The second condition, when we're talking about chi-squared hypothesis testing, is the large counts, large counts condition. And this is an important one to appreciate. This is that the expected number of each category of outcomes is at least equal to five. Now, you might say, hey, wait, wait, I only got four wins, or Kenny only got four wins out of his sample of 24. But that does not violate the large counts condition. Remember, what is the expected number of wins, losses, and ties? Well, if you were assuming the null hypothesis, where the outcomes have equal probability, so the expected, the expected, I could write right over here, it would be that it's 1 3rd, 1 3rd, 1 3rd."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "Now, you might say, hey, wait, wait, I only got four wins, or Kenny only got four wins out of his sample of 24. But that does not violate the large counts condition. Remember, what is the expected number of wins, losses, and ties? Well, if you were assuming the null hypothesis, where the outcomes have equal probability, so the expected, the expected, I could write right over here, it would be that it's 1 3rd, 1 3rd, 1 3rd. And so 1 3rd of 24 is eight, eight and eight. That's what Kenny would expect. And since, because all of these are at least equal to five, we meet the large counts condition."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "Well, if you were assuming the null hypothesis, where the outcomes have equal probability, so the expected, the expected, I could write right over here, it would be that it's 1 3rd, 1 3rd, 1 3rd. And so 1 3rd of 24 is eight, eight and eight. That's what Kenny would expect. And since, because all of these are at least equal to five, we meet the large counts condition. And then the last condition is the independence condition. If we aren't sampling with replacement, then we just have to feel good that our sample size is no more than 10% of the population. And he can definitely play more than 240 games in his life, so we would assume that we meet that condition as well."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "And since, because all of these are at least equal to five, we meet the large counts condition. And then the last condition is the independence condition. If we aren't sampling with replacement, then we just have to feel good that our sample size is no more than 10% of the population. And he can definitely play more than 240 games in his life, so we would assume that we meet that condition as well. And so with that out of the way, we can actually calculate our chi-squared statistic and try to make some inference based on it. And so, let's see, our chi-squared statistic is going to be equal to, so for each category, it's going to be the difference between the expected and what he got in that sample, squared divided by the expected. So the first category is wins."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "And he can definitely play more than 240 games in his life, so we would assume that we meet that condition as well. And so with that out of the way, we can actually calculate our chi-squared statistic and try to make some inference based on it. And so, let's see, our chi-squared statistic is going to be equal to, so for each category, it's going to be the difference between the expected and what he got in that sample, squared divided by the expected. So the first category is wins. So that's going to be four minus eight, four minus eight squared over an expected number of wins of eight, plus losses, so that's 13 minus eight. 13 is how many he got, how many he lost, minus eight expected, squared over the number expected, plus he got seven ties, he would have expected eight squared, all of that over eight. And so let's see, what is this?"}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "So the first category is wins. So that's going to be four minus eight, four minus eight squared over an expected number of wins of eight, plus losses, so that's 13 minus eight. 13 is how many he got, how many he lost, minus eight expected, squared over the number expected, plus he got seven ties, he would have expected eight squared, all of that over eight. And so let's see, what is this? Four minus eight is negative four. You square that, you get 16. 13 minus eight is five."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "And so let's see, what is this? Four minus eight is negative four. You square that, you get 16. 13 minus eight is five. You square that, you get 25. Seven minus eight is negative one. Square that, you get one."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "13 minus eight is five. You square that, you get 25. Seven minus eight is negative one. Square that, you get one. And 16 divided by eight is going to be two. 25 divided by eight is going to be, let's see, that's 3 1 8, so that's 3.125. And then 1 8 is 0.125, 0.125."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "Square that, you get one. And 16 divided by eight is going to be two. 25 divided by eight is going to be, let's see, that's 3 1 8, so that's 3.125. And then 1 8 is 0.125, 0.125. You add these together, so let's see, it's gonna be two plus 3.125, 5.125, plus another.125, so that's going to be 5.25. So our chi-squared statistic is 5.25. And now to figure out our p-value, our p-value is going to be equal to the probability of getting a chi-squared statistic greater than or equal to 5.25."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "And then 1 8 is 0.125, 0.125. You add these together, so let's see, it's gonna be two plus 3.125, 5.125, plus another.125, so that's going to be 5.25. So our chi-squared statistic is 5.25. And now to figure out our p-value, our p-value is going to be equal to the probability of getting a chi-squared statistic greater than or equal to 5.25. And you could use a chi-squared table for that. And we always have to think about our degrees of freedom. We have one, two, three categories."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "And now to figure out our p-value, our p-value is going to be equal to the probability of getting a chi-squared statistic greater than or equal to 5.25. And you could use a chi-squared table for that. And we always have to think about our degrees of freedom. We have one, two, three categories. So our degrees of freedom is going to be one less than that, or three minus one, which is two. So our degrees of freedom is going to be equal to two. And that makes sense, because you know for a certain number of games, if you know the number of wins, and you know the certain number of losses, you can figure out the number of ties."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "We have one, two, three categories. So our degrees of freedom is going to be one less than that, or three minus one, which is two. So our degrees of freedom is going to be equal to two. And that makes sense, because you know for a certain number of games, if you know the number of wins, and you know the certain number of losses, you can figure out the number of ties. Or if you know any two of these categories, you can always figure out the third. So that's why you have two degrees of freedom. And so let's get out our chi-squared table."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "And that makes sense, because you know for a certain number of games, if you know the number of wins, and you know the certain number of losses, you can figure out the number of ties. Or if you know any two of these categories, you can always figure out the third. So that's why you have two degrees of freedom. And so let's get out our chi-squared table. So we have two degrees of freedom. So we are in this row. And where is 5.25?"}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "And so let's get out our chi-squared table. So we have two degrees of freedom. So we are in this row. And where is 5.25? So 5.25 is right over there. And so our probability is going to be between 0.10 and 0.05. So our p-value is going to be greater than 0.05 and less than 0.10."}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "And where is 5.25? So 5.25 is right over there. And so our probability is going to be between 0.10 and 0.05. So our p-value is going to be greater than 0.05 and less than 0.10. And so for example, if ahead of time, and he should have done this ahead of time, he set a significance level of 5%, and our p-value here is greater than 5%, which we just saw, he would fail to reject in this situation the null hypothesis. But they're not asking us that here. All they're asking us is what is our chi-squared value and what range is our p-value in?"}, {"video_title": "Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3", "Sentence": "So our p-value is going to be greater than 0.05 and less than 0.10. And so for example, if ahead of time, and he should have done this ahead of time, he set a significance level of 5%, and our p-value here is greater than 5%, which we just saw, he would fail to reject in this situation the null hypothesis. But they're not asking us that here. All they're asking us is what is our chi-squared value and what range is our p-value in? Well, let's see, 5.25 are both of these values. And we saw we got a p-value between 5% and 10%. So it is choice A right over there."}, {"video_title": "Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3", "Sentence": "Here is a computer output from a least squares regression analysis on his sample. Assume that all conditions for inference have been met. What is a 95% confidence interval for the slope of the least squares regression line? So if you feel inspired, pause the video and see if you can have a go at it. Otherwise, we'll do this together. Okay, so let's first remind ourselves what's even going on. So let's visualize the regression."}, {"video_title": "Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3", "Sentence": "So if you feel inspired, pause the video and see if you can have a go at it. Otherwise, we'll do this together. Okay, so let's first remind ourselves what's even going on. So let's visualize the regression. So our horizontal axis, or our x-axis, that would be our caffeine intake in milligrams. And then our y-axis, or our vertical axis, that would be the, I would assume it's in hours, so time studying. And Moussa here, he randomly selects 20 students."}, {"video_title": "Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3", "Sentence": "So let's visualize the regression. So our horizontal axis, or our x-axis, that would be our caffeine intake in milligrams. And then our y-axis, or our vertical axis, that would be the, I would assume it's in hours, so time studying. And Moussa here, he randomly selects 20 students. And so for each of those students, he sees how much caffeine they consumed and how much time they spent studying and plots them here. And so there'll be 20 data points. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20."}, {"video_title": "Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3", "Sentence": "And Moussa here, he randomly selects 20 students. And so for each of those students, he sees how much caffeine they consumed and how much time they spent studying and plots them here. And so there'll be 20 data points. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. He inputs these data points into a computer in order to fit a least squares regression line. And let's say the least squares regression line looks something like this. And a least squares regression line comes from trying to minimize the square distance between the line and all of these points."}, {"video_title": "Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3", "Sentence": "One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. He inputs these data points into a computer in order to fit a least squares regression line. And let's say the least squares regression line looks something like this. And a least squares regression line comes from trying to minimize the square distance between the line and all of these points. And then this is giving us information on that least squares regression line. And the most valuable things here, if we really wanna help visualize or understand the line, is what we get in this column. The constant coefficient tells us essentially what is the y-intercept here, so 2.544."}, {"video_title": "Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3", "Sentence": "And a least squares regression line comes from trying to minimize the square distance between the line and all of these points. And then this is giving us information on that least squares regression line. And the most valuable things here, if we really wanna help visualize or understand the line, is what we get in this column. The constant coefficient tells us essentially what is the y-intercept here, so 2.544. And then the coefficient on the caffeine, this is one way of thinking about, well, for every incremental increase in caffeine, how much does the time studying increase? Or you might recognize this as the slope of the least squares regression line. So this is the slope, and this would be equal to 0.164."}, {"video_title": "Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3", "Sentence": "The constant coefficient tells us essentially what is the y-intercept here, so 2.544. And then the coefficient on the caffeine, this is one way of thinking about, well, for every incremental increase in caffeine, how much does the time studying increase? Or you might recognize this as the slope of the least squares regression line. So this is the slope, and this would be equal to 0.164. Now this information right over here, it tells us how well our least squares regression line fits the data. R squared you might already be familiar with. It says how much of the variance in the y variable is explainable by the x variable."}, {"video_title": "Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3", "Sentence": "So this is the slope, and this would be equal to 0.164. Now this information right over here, it tells us how well our least squares regression line fits the data. R squared you might already be familiar with. It says how much of the variance in the y variable is explainable by the x variable. If it was one or 100%, that means all of it could be explained, and it's a very good fit. If it was zero, that means none of it can be explained. It would be a very bad fit."}, {"video_title": "Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3", "Sentence": "It says how much of the variance in the y variable is explainable by the x variable. If it was one or 100%, that means all of it could be explained, and it's a very good fit. If it was zero, that means none of it can be explained. It would be a very bad fit. Capital S, this is the standard deviation of the residuals, and it's another measure of how much these data points vary from this regression line. Now this column right over here is going to prove to be useful for answering the question at hand. This gives us the standard error of the coefficient, and the coefficient that we really care about, the statistic that we really care about, is the slope of the regression line, and this gives us the standard error for the slope of the regression line."}, {"video_title": "Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3", "Sentence": "It would be a very bad fit. Capital S, this is the standard deviation of the residuals, and it's another measure of how much these data points vary from this regression line. Now this column right over here is going to prove to be useful for answering the question at hand. This gives us the standard error of the coefficient, and the coefficient that we really care about, the statistic that we really care about, is the slope of the regression line, and this gives us the standard error for the slope of the regression line. You could view this as the estimate of the standard deviation of the sampling distribution of the slope of the regression line. Remember, we took a sample of 20 folks here, and we calculated a statistic which is the slope of the regression line. Every time you do a different sample, you will likely get a different slope, and this slope is an estimate of some true parameter in the population."}, {"video_title": "Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3", "Sentence": "This gives us the standard error of the coefficient, and the coefficient that we really care about, the statistic that we really care about, is the slope of the regression line, and this gives us the standard error for the slope of the regression line. You could view this as the estimate of the standard deviation of the sampling distribution of the slope of the regression line. Remember, we took a sample of 20 folks here, and we calculated a statistic which is the slope of the regression line. Every time you do a different sample, you will likely get a different slope, and this slope is an estimate of some true parameter in the population. This would sometimes also be called the standard error of the slope of the least squares regression line. Now these last two columns you don't have to worry about in the context of this video. This is useful if you were saying, well, assuming that there is no relationship between caffeine intake and time studying, what is the associated t statistics for the statistics that I actually calculated, and what would be the probability of getting something that extreme or more extreme, assuming that there is no association, assuming that, for example, the actual slope of the regression line is zero, and this says, well, the probability if we would assume that is actually quite low."}, {"video_title": "Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3", "Sentence": "Every time you do a different sample, you will likely get a different slope, and this slope is an estimate of some true parameter in the population. This would sometimes also be called the standard error of the slope of the least squares regression line. Now these last two columns you don't have to worry about in the context of this video. This is useful if you were saying, well, assuming that there is no relationship between caffeine intake and time studying, what is the associated t statistics for the statistics that I actually calculated, and what would be the probability of getting something that extreme or more extreme, assuming that there is no association, assuming that, for example, the actual slope of the regression line is zero, and this says, well, the probability if we would assume that is actually quite low. It's about a 1% chance that you would have gotten these results if there truly was not a relationship between caffeine intake and time studying. But with all of that out of the way, let's actually answer the question. Well, to construct a confidence interval around a statistic, you would take the value of the statistic that you calculated from your sample, so 0.164, and then it would be plus or minus a critical t value, and then this would be driven by the fact that you care about a 95% confidence interval and by the degrees of freedom, and I'll talk about that in a second, and then you would multiply that times the standard error of the statistic, and in this case, the statistic that we care about is the slope, and so this is 0.057 times 0.057, and the reason why we're using a critical t value instead of a critical z value is because our standard error of the statistic is an estimate."}, {"video_title": "Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3", "Sentence": "This is useful if you were saying, well, assuming that there is no relationship between caffeine intake and time studying, what is the associated t statistics for the statistics that I actually calculated, and what would be the probability of getting something that extreme or more extreme, assuming that there is no association, assuming that, for example, the actual slope of the regression line is zero, and this says, well, the probability if we would assume that is actually quite low. It's about a 1% chance that you would have gotten these results if there truly was not a relationship between caffeine intake and time studying. But with all of that out of the way, let's actually answer the question. Well, to construct a confidence interval around a statistic, you would take the value of the statistic that you calculated from your sample, so 0.164, and then it would be plus or minus a critical t value, and then this would be driven by the fact that you care about a 95% confidence interval and by the degrees of freedom, and I'll talk about that in a second, and then you would multiply that times the standard error of the statistic, and in this case, the statistic that we care about is the slope, and so this is 0.057 times 0.057, and the reason why we're using a critical t value instead of a critical z value is because our standard error of the statistic is an estimate. We don't actually know the standard deviation of the sampling distribution. So the last thing we have to do is figure out what is this critical t value? You can figure it out using either a calculator or using a table."}, {"video_title": "Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3", "Sentence": "Well, to construct a confidence interval around a statistic, you would take the value of the statistic that you calculated from your sample, so 0.164, and then it would be plus or minus a critical t value, and then this would be driven by the fact that you care about a 95% confidence interval and by the degrees of freedom, and I'll talk about that in a second, and then you would multiply that times the standard error of the statistic, and in this case, the statistic that we care about is the slope, and so this is 0.057 times 0.057, and the reason why we're using a critical t value instead of a critical z value is because our standard error of the statistic is an estimate. We don't actually know the standard deviation of the sampling distribution. So the last thing we have to do is figure out what is this critical t value? You can figure it out using either a calculator or using a table. I'll do it using a table, and to do that, we need to know what the degrees of freedom. Well, when you're doing this with a regression slope like we're doing right now, your degrees of freedom are going to be the number of data points you have minus two so our degrees of freedom are going to be 20 minus two which is equal to 18. I'm not gonna go into a bunch of depth right now."}, {"video_title": "Confidence interval for the slope of a regression line AP Statistics Khan Academy.mp3", "Sentence": "You can figure it out using either a calculator or using a table. I'll do it using a table, and to do that, we need to know what the degrees of freedom. Well, when you're doing this with a regression slope like we're doing right now, your degrees of freedom are going to be the number of data points you have minus two so our degrees of freedom are going to be 20 minus two which is equal to 18. I'm not gonna go into a bunch of depth right now. It actually is beyond the scope of this video for sure as to why you subtract two here, but just so that we can look it up on a table, this is our degrees of freedom. So we care about a 95% confidence level. That's equivalent to having a 2 1\u20442% tail on either side, and our degrees of freedom, it's 18, so our critical t value is 2.101, and so our 95% confidence interval is going to be 0.164 plus or minus our critical t value, 2.101, times the standard error of the statistic, times, I'll just put in parentheses, 0.057, and you could type this into a calculator if you wanted to figure out the exact values here, but the way to interpret a 95% confidence interval is that 95% of the time that you calculate a 95% confidence interval, it is going to overlap with the true value of the parameter that we are estimating."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "The amounts of fuel he uses for each part of the commute are also independent of each other. Here are summary statistics for the amount of fuel Shinji uses for each part of his commute. So when he goes to work, he uses a mean of 10 liters of fuel with a standard deviation of 1.5 liters. And on the way home, he also has a mean of 10 liters, but there is more variation, there is more spread. He has a standard deviation of two liters. Suppose that Shinji has 25 liters of fuel in his tank and he intends to drive to work and back home. What is the probability that Shinji runs out of fuel?"}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "And on the way home, he also has a mean of 10 liters, but there is more variation, there is more spread. He has a standard deviation of two liters. Suppose that Shinji has 25 liters of fuel in his tank and he intends to drive to work and back home. What is the probability that Shinji runs out of fuel? All right, this is really interesting. We have the distributions for the amount of fuel he uses to work and to home, and they say that these are normal distributions. They say that right over here, follows a normal distribution."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "What is the probability that Shinji runs out of fuel? All right, this is really interesting. We have the distributions for the amount of fuel he uses to work and to home, and they say that these are normal distributions. They say that right over here, follows a normal distribution. But here we're talking about the total amount of fuel he has to go to work and to go home. So what we wanna do is come up with a total distribution, home and back, I guess you could say. We could call this work plus home, home and back."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "They say that right over here, follows a normal distribution. But here we're talking about the total amount of fuel he has to go to work and to go home. So what we wanna do is come up with a total distribution, home and back, I guess you could say. We could call this work plus home, home and back. If you have two random variables that can be described by normal distributions, and you were to define a new random variable as their sum, the distribution of that new random variable will still be a normal distribution, and its mean will be the sum of the means of those other random variables. So the mean here, I'll say the mean of work plus home, is going to be equal to 20 liters. He will use a mean of 20 liters in the round trip."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "We could call this work plus home, home and back. If you have two random variables that can be described by normal distributions, and you were to define a new random variable as their sum, the distribution of that new random variable will still be a normal distribution, and its mean will be the sum of the means of those other random variables. So the mean here, I'll say the mean of work plus home, is going to be equal to 20 liters. He will use a mean of 20 liters in the round trip. Now for the standard deviation from home plus work, you can't just add the standard deviations, going and coming back. But because the amount of fuel going to work and the amount of fuel coming home are independent random variables, because they are independent of each other, we can add the variances. And only because they are independent can we add the variances."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "He will use a mean of 20 liters in the round trip. Now for the standard deviation from home plus work, you can't just add the standard deviations, going and coming back. But because the amount of fuel going to work and the amount of fuel coming home are independent random variables, because they are independent of each other, we can add the variances. And only because they are independent can we add the variances. So what you can say is that the variance of the combined trip is equal to the variance of going to work plus the variance of going home. So what's the variance of going to work? Well, 1.5 squared is, so this will be 1.5 squared."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "And only because they are independent can we add the variances. So what you can say is that the variance of the combined trip is equal to the variance of going to work plus the variance of going home. So what's the variance of going to work? Well, 1.5 squared is, so this will be 1.5 squared. And what's the variance coming home? Well, this is going to be two squared, two squared. Well, this is 2.25 plus four, which is equal to 6.25."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "Well, 1.5 squared is, so this will be 1.5 squared. And what's the variance coming home? Well, this is going to be two squared, two squared. Well, this is 2.25 plus four, which is equal to 6.25. So the variance on the round trip is equal to 6.25. If I were to take the square root of that, which is equal to 2.5, we can now describe the normal distribution of the round trip and use that to answer the question. So we have this normal distribution that might look something like this."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "Well, this is 2.25 plus four, which is equal to 6.25. So the variance on the round trip is equal to 6.25. If I were to take the square root of that, which is equal to 2.5, we can now describe the normal distribution of the round trip and use that to answer the question. So we have this normal distribution that might look something like this. We know its mean is 20 liters. So this is 20 liters. And we wanna know what is the probability that Shinji runs out of fuel?"}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "So we have this normal distribution that might look something like this. We know its mean is 20 liters. So this is 20 liters. And we wanna know what is the probability that Shinji runs out of fuel? Well, to run out of fuel, he would need to require more than 25 liters of fuel. So if 25 liters of fuel is right over here, so this is 25 liters of fuel, the scenario where Shinji runs out of fuel is right over here. This is where he needs more than 25 liters."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "And we wanna know what is the probability that Shinji runs out of fuel? Well, to run out of fuel, he would need to require more than 25 liters of fuel. So if 25 liters of fuel is right over here, so this is 25 liters of fuel, the scenario where Shinji runs out of fuel is right over here. This is where he needs more than 25 liters. He actually has 25 liters in his tank. So how do we figure out that area right over there? Well, we could use a z-table."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "This is where he needs more than 25 liters. He actually has 25 liters in his tank. So how do we figure out that area right over there? Well, we could use a z-table. We could say how many standard deviations above the mean is 25 liters? Well, it is five liters above the mean. So let me write this down."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "Well, we could use a z-table. We could say how many standard deviations above the mean is 25 liters? Well, it is five liters above the mean. So let me write this down. So the z here, the z is equal to 25 minus the mean, minus 20, divided by the standard deviation for, I guess you could say, this combined normal distribution. This is two standard deviations above the mean, or a z-score of plus two. So if we look at a z-table, and we look exactly two standard deviations above the mean, that will give us this area, the cumulative area below two standard deviations above the mean."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "So let me write this down. So the z here, the z is equal to 25 minus the mean, minus 20, divided by the standard deviation for, I guess you could say, this combined normal distribution. This is two standard deviations above the mean, or a z-score of plus two. So if we look at a z-table, and we look exactly two standard deviations above the mean, that will give us this area, the cumulative area below two standard deviations above the mean. And then if we subtract that from one, we will get the area that we care about. So let's get our z-table out. We care about a z-score of exactly two."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "So if we look at a z-table, and we look exactly two standard deviations above the mean, that will give us this area, the cumulative area below two standard deviations above the mean. And then if we subtract that from one, we will get the area that we care about. So let's get our z-table out. We care about a z-score of exactly two. So 2.00 is right over here,.9772. So that tells us that this area right over here is 0.9772. And so that blue area, the probability that Shinji runs out of fuel is going to be one minus 0.9772."}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "We care about a z-score of exactly two. So 2.00 is right over here,.9772. So that tells us that this area right over here is 0.9772. And so that blue area, the probability that Shinji runs out of fuel is going to be one minus 0.9772. And what is that going to be equal to? Let's see, this is going to be equal to 0.0228. Did I do that right?"}, {"video_title": "Example Analyzing distribution of sum of two normally distributed random variables Khan Academy.mp3", "Sentence": "And so that blue area, the probability that Shinji runs out of fuel is going to be one minus 0.9772. And what is that going to be equal to? Let's see, this is going to be equal to 0.0228. Did I do that right? I think I did that right. Yes, 0.0228 is the probability that Shinji runs out of fuel. If you want to think of it as a percent, it's a 2.28% chance that he runs out of fuel."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "The group has a choice of creating one large nature preserve with an area of 45 square kilometers and containing 70 at-risk species, or five small nature preserves, each with an area of three square kilometers and each containing 16 at-risk species unique to that preserve. Which choice would you recommend and why? And there's some interesting data here. Here it looks like some data they have gathered for different islands, and we have their areas, and then this is the number of species at risk in 1990, and then the species extinct by 2000. And so we can see for these various islands, we can see their areas and the proportion that got extinct, and it looks like they're plotted on this scatter plot. Now be very careful when you look at this because look at the two axes. It is the vertical axes is the proportion extinct in 2000, so it's these numbers, but the horizontal axis isn't just a straight-up area."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "Here it looks like some data they have gathered for different islands, and we have their areas, and then this is the number of species at risk in 1990, and then the species extinct by 2000. And so we can see for these various islands, we can see their areas and the proportion that got extinct, and it looks like they're plotted on this scatter plot. Now be very careful when you look at this because look at the two axes. It is the vertical axes is the proportion extinct in 2000, so it's these numbers, but the horizontal axis isn't just a straight-up area. It's the natural log of the area. And why did they do this? Well, notice, when you make the horizontal axis the natural log of the area, it looks like there's a linear relationship."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "It is the vertical axes is the proportion extinct in 2000, so it's these numbers, but the horizontal axis isn't just a straight-up area. It's the natural log of the area. And why did they do this? Well, notice, when you make the horizontal axis the natural log of the area, it looks like there's a linear relationship. But be clear, it's a linear relationship between the natural log of the area and proportion extinct in 2000. But the reason why it's valuable to do this type of transformation is now we can apply our tools of linear regression to think about what would be the proportion extinct for the 45 square kilometers versus for the five small three-kilometer islands. So pause this video and see if you can figure it out on your own."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "Well, notice, when you make the horizontal axis the natural log of the area, it looks like there's a linear relationship. But be clear, it's a linear relationship between the natural log of the area and proportion extinct in 2000. But the reason why it's valuable to do this type of transformation is now we can apply our tools of linear regression to think about what would be the proportion extinct for the 45 square kilometers versus for the five small three-kilometer islands. So pause this video and see if you can figure it out on your own. And they give us the regression data for a line that fits this data. All right, now let's work through it together and to make some space because all of it is already plotted right over here and we have our regression data. So the regression line, we know it's slope and y-intercept."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "So pause this video and see if you can figure it out on your own. And they give us the regression data for a line that fits this data. All right, now let's work through it together and to make some space because all of it is already plotted right over here and we have our regression data. So the regression line, we know it's slope and y-intercept. The y-intercept is right over here, 0.28996. So 0.2, this is, let's see, one, two, three, four, five, so 28996. It's almost two nine."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "So the regression line, we know it's slope and y-intercept. The y-intercept is right over here, 0.28996. So 0.2, this is, let's see, one, two, three, four, five, so 28996. It's almost two nine. So it's gonna be right over here would be the y-intercept. And its slope is negative 0.05 approximately. And I could eyeball it."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "It's almost two nine. So it's gonna be right over here would be the y-intercept. And its slope is negative 0.05 approximately. And I could eyeball it. It probably is going to look something like this. That's the regression line. Or another way to think about it is the regression line tells us in general the proportion, proportion and I'll just say proportion shorthand for proportion extinct is going to be equal to our y-intercept, 0.28996 minus, minus 0.05323."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "And I could eyeball it. It probably is going to look something like this. That's the regression line. Or another way to think about it is the regression line tells us in general the proportion, proportion and I'll just say proportion shorthand for proportion extinct is going to be equal to our y-intercept, 0.28996 minus, minus 0.05323. And we have to be careful here. You might be tempted to say times the area, but no, the horizontal axis here is the natural log of the area, times the natural log of the area. And so we can use this equation for both scenarios to think about what is going to be the proportion that we would expect to get extinct in either situation."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "Or another way to think about it is the regression line tells us in general the proportion, proportion and I'll just say proportion shorthand for proportion extinct is going to be equal to our y-intercept, 0.28996 minus, minus 0.05323. And we have to be careful here. You might be tempted to say times the area, but no, the horizontal axis here is the natural log of the area, times the natural log of the area. And so we can use this equation for both scenarios to think about what is going to be the proportion that we would expect to get extinct in either situation. And then how many actual species will get extinct. And then the one that maybe has fewer species that get extinct is maybe the best one. Or the one that the more that we can preserve is maybe the best one."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "And so we can use this equation for both scenarios to think about what is going to be the proportion that we would expect to get extinct in either situation. And then how many actual species will get extinct. And then the one that maybe has fewer species that get extinct is maybe the best one. Or the one that the more that we can preserve is maybe the best one. And so let's look at the two scenarios. So the first scenario is the 45 square kilometer island. And this is just one, so times one."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "Or the one that the more that we can preserve is maybe the best one. And so let's look at the two scenarios. So the first scenario is the 45 square kilometer island. And this is just one, so times one. And so what is gonna be the proportion, proportion that we would expect to go extinct based on this regression? Well it's going to be 0.28996 minus 0.05323 times the natural log of 45. And if we wanna know the actual number that go extinct, so number extinct, extinct, would be equal to the proportion, would be equal to the proportion times, how many, let's see, the 45 square kilometers and it contains 70 at-risk species."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "And this is just one, so times one. And so what is gonna be the proportion, proportion that we would expect to go extinct based on this regression? Well it's going to be 0.28996 minus 0.05323 times the natural log of 45. And if we wanna know the actual number that go extinct, so number extinct, extinct, would be equal to the proportion, would be equal to the proportion times, how many, let's see, the 45 square kilometers and it contains 70 at-risk species. So times our 70 species. And so we can get our calculator out to figure that out. So this is the proportion we would expect to go extinct in the 45 square kilometer island based on our linear regression."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "And if we wanna know the actual number that go extinct, so number extinct, extinct, would be equal to the proportion, would be equal to the proportion times, how many, let's see, the 45 square kilometers and it contains 70 at-risk species. So times our 70 species. And so we can get our calculator out to figure that out. So this is the proportion we would expect to go extinct in the 45 square kilometer island based on our linear regression. So this would be equal to, so it looks like almost 9%. And if we wanna figure out the actual number we would expect to go extinct, we would just multiply that times the number of species on that island. So times 70."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "So this is the proportion we would expect to go extinct in the 45 square kilometer island based on our linear regression. So this would be equal to, so it looks like almost 9%. And if we wanna figure out the actual number we would expect to go extinct, we would just multiply that times the number of species on that island. So times 70. And we get, so approximately about 6.11. So let me write that down. So this is going to be approximately 6.11."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "So times 70. And we get, so approximately about 6.11. So let me write that down. So this is going to be approximately 6.11. So we could say there would be approximately, if we, let's just say six extinct, this is all very approximate, extinct, and approximately 64 saved. Now let's think about the other scenario. Let's think about the scenario where we have five small nature preserves."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "So this is going to be approximately 6.11. So we could say there would be approximately, if we, let's just say six extinct, this is all very approximate, extinct, and approximately 64 saved. Now let's think about the other scenario. Let's think about the scenario where we have five small nature preserves. So it's going to be three square kilometers times five islands. And we're gonna just do the same exercise. Our proportion that goes extinct is going to be 0.28996."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "Let's think about the scenario where we have five small nature preserves. So it's going to be three square kilometers times five islands. And we're gonna just do the same exercise. Our proportion that goes extinct is going to be 0.28996. That's just the y-intercept for our regression line. Minus 0.05323, and I have a negative sign there because we have a negative slope. And this is not just times the area, it's times the natural log of the area."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "Our proportion that goes extinct is going to be 0.28996. That's just the y-intercept for our regression line. Minus 0.05323, and I have a negative sign there because we have a negative slope. And this is not just times the area, it's times the natural log of the area. It's going to be three square kilometers. Three square kilometers. And then our number extinct, our number extinct is going to be equal to our proportion that we will calculate in the line above times, let's see, five small nature preserves, each with an area of three square kilometers and each containing 16 at-risk species."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "And this is not just times the area, it's times the natural log of the area. It's going to be three square kilometers. Three square kilometers. And then our number extinct, our number extinct is going to be equal to our proportion that we will calculate in the line above times, let's see, five small nature preserves, each with an area of three square kilometers and each containing 16 at-risk species. So five times 16, if each island has 16 and there's five islands, that's going to be five times 16 is 80. So times 80. So let's figure out what this is, get the calculator out again."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "And then our number extinct, our number extinct is going to be equal to our proportion that we will calculate in the line above times, let's see, five small nature preserves, each with an area of three square kilometers and each containing 16 at-risk species. So five times 16, if each island has 16 and there's five islands, that's going to be five times 16 is 80. So times 80. So let's figure out what this is, get the calculator out again. And we are going to get, so this is going to be the proportion. It's a much higher proportion. And then we'll multiply that times our number of species."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "So let's figure out what this is, get the calculator out again. And we are going to get, so this is going to be the proportion. It's a much higher proportion. And then we'll multiply that times our number of species. So times 80 to figure out how many species will go extinct. And we have here, it's approximately 18.52. So this is approximately 18.52."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "And then we'll multiply that times our number of species. So times 80 to figure out how many species will go extinct. And we have here, it's approximately 18.52. So this is approximately 18.52. So another way to think about it is we're going to have approximately, well, if we round, let's just say 19 extinct, 19 extinct. And then if we have 19 extinct, how many are we going to save? We're going to have 61 saved."}, {"video_title": "Worked example of linear regression using transformed data AP Statistics Khan Academy.mp3", "Sentence": "So this is approximately 18.52. So another way to think about it is we're going to have approximately, well, if we round, let's just say 19 extinct, 19 extinct. And then if we have 19 extinct, how many are we going to save? We're going to have 61 saved. 61 saved. And even if you said 18 1\u20442 here and 61.5 here, on either measure, the 45 square, the big island is better. You're going to have fewer species that are extinct and more that are saved."}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "We're told that Maya and Doug are finalists in a crafting competition. For the final round, each of them spin a wheel to determine what star material must be in their craft. Maya and Doug both want to get silk as their star material. Maya will spin first, followed by Doug. What is the probability that neither contestant gets silk? Pause this video and think through this on your own before we work through this together. All right, so first let's think about what they're asking."}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "Maya will spin first, followed by Doug. What is the probability that neither contestant gets silk? Pause this video and think through this on your own before we work through this together. All right, so first let's think about what they're asking. They want to figure out the probability that neither gets silk. So I'm gonna write this in shorthand. So I'm gonna use MNS for Maya no silk."}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "All right, so first let's think about what they're asking. They want to figure out the probability that neither gets silk. So I'm gonna write this in shorthand. So I'm gonna use MNS for Maya no silk. And we're also thinking about Doug not being able to pick silk. So Maya no silk and Doug no silk. So we know that this could be viewed as the probability that Maya doesn't get silk."}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "So I'm gonna use MNS for Maya no silk. And we're also thinking about Doug not being able to pick silk. So Maya no silk and Doug no silk. So we know that this could be viewed as the probability that Maya doesn't get silk. She, after all, does get to spin this wheel first. And then we can multiply that by the probability that Doug doesn't get silk, Doug no silk, given that Maya did not get silk, Maya no silk. Now it's important to think about whether Doug's probability is independent or dependent on whether Maya got silk or not."}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "So we know that this could be viewed as the probability that Maya doesn't get silk. She, after all, does get to spin this wheel first. And then we can multiply that by the probability that Doug doesn't get silk, Doug no silk, given that Maya did not get silk, Maya no silk. Now it's important to think about whether Doug's probability is independent or dependent on whether Maya got silk or not. So let's remember, Maya will spin first, but it's not like if she picks silk that somehow silk is taken out of the running. In fact, no matter what she picks, it's not taken out of the running. Doug will then spin it again."}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "Now it's important to think about whether Doug's probability is independent or dependent on whether Maya got silk or not. So let's remember, Maya will spin first, but it's not like if she picks silk that somehow silk is taken out of the running. In fact, no matter what she picks, it's not taken out of the running. Doug will then spin it again. And so these are really two independent events. And so the probability that Doug doesn't get silk given that Maya doesn't get silk, this is going to be the same thing as the probability that just Doug doesn't get silk. It doesn't matter what happens to Maya."}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "Doug will then spin it again. And so these are really two independent events. And so the probability that Doug doesn't get silk given that Maya doesn't get silk, this is going to be the same thing as the probability that just Doug doesn't get silk. It doesn't matter what happens to Maya. And so what are each of these? Well, this is all going to be equal to the probability that Maya does not get silk. There's six pieces or six options of this wheel right over here."}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "It doesn't matter what happens to Maya. And so what are each of these? Well, this is all going to be equal to the probability that Maya does not get silk. There's six pieces or six options of this wheel right over here. Five of them entail her not getting silk on her spin. So five over six. And then similarly, when Doug goes to spin this wheel, there are six possibilities."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "The distribution of resting pulse rates of all students at Santa Maria High School was approximately normal with mean of 80 beats per minute and standard deviation of nine beats per minute. The school nurse plans to provide additional screening to students whose resting pulse rates are in the top 30% of the students who were tested. What is the minimum resting pulse rate at that school for students who will receive additional screening, round to the nearest whole number? If you feel like you know how to tackle this, I encourage you to pause this video and try to work it out. All right, now let's work this out together. They're telling us that the distribution of resting pulse rates are approximately normal. So we could use a normal distribution, and they tell us several things about this normal distribution."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "If you feel like you know how to tackle this, I encourage you to pause this video and try to work it out. All right, now let's work this out together. They're telling us that the distribution of resting pulse rates are approximately normal. So we could use a normal distribution, and they tell us several things about this normal distribution. They tell us that the mean is 80 beats per minute, so that is the mean right over there, and they tell us that the standard deviation is nine beats per minute. So on this normal distribution, we have one standard deviation above the mean, two standard deviations above the mean, so this distance right over here is nine, so this would be 89. This one right over here would be 98, and you could also go standard deviations below the mean."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "So we could use a normal distribution, and they tell us several things about this normal distribution. They tell us that the mean is 80 beats per minute, so that is the mean right over there, and they tell us that the standard deviation is nine beats per minute. So on this normal distribution, we have one standard deviation above the mean, two standard deviations above the mean, so this distance right over here is nine, so this would be 89. This one right over here would be 98, and you could also go standard deviations below the mean. This right over here would be 71. This would be 62, but what we're concerned about is the top 30% because that is who is going to be tested. So there's gonna be some value here, some threshold."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "This one right over here would be 98, and you could also go standard deviations below the mean. This right over here would be 71. This would be 62, but what we're concerned about is the top 30% because that is who is going to be tested. So there's gonna be some value here, some threshold. Let's say it is right over here, that if you are in that, if you are at that score, you have reached the minimum threshold to get additional screening. You are in the top 30%. So that means that this area right over here is going to be 30%, or 0.3."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "So there's gonna be some value here, some threshold. Let's say it is right over here, that if you are in that, if you are at that score, you have reached the minimum threshold to get additional screening. You are in the top 30%. So that means that this area right over here is going to be 30%, or 0.3. So what we can do, we can use a z-table to say for what z-score is 70% of the distribution less than that, and then we can take that z-score and use the mean and the standard deviation to come up with an actual value. In previous examples, we started with a z-score and we're looking for the percentage. This time, we're looking for the percentage."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "So that means that this area right over here is going to be 30%, or 0.3. So what we can do, we can use a z-table to say for what z-score is 70% of the distribution less than that, and then we can take that z-score and use the mean and the standard deviation to come up with an actual value. In previous examples, we started with a z-score and we're looking for the percentage. This time, we're looking for the percentage. We want it to be at least 70% and then come up with the corresponding z-score. So let's see, immediately when we look at this, and we are to the right of the mean, and so we're gonna have a positive z-score, so we're starting at 50% here. We definitely wanna get, this is 67%, 68, 69."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "This time, we're looking for the percentage. We want it to be at least 70% and then come up with the corresponding z-score. So let's see, immediately when we look at this, and we are to the right of the mean, and so we're gonna have a positive z-score, so we're starting at 50% here. We definitely wanna get, this is 67%, 68, 69. We're getting close, and on our table, this is the lowest z-score that gets us across that 70% threshold. It's at 0.7019. So it definitely crosses the threshold, and so that is a z-score of 0.53."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "We definitely wanna get, this is 67%, 68, 69. We're getting close, and on our table, this is the lowest z-score that gets us across that 70% threshold. It's at 0.7019. So it definitely crosses the threshold, and so that is a z-score of 0.53. 0.52 is too little. So we need a z-score of 0.53. Let's write that down."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "So it definitely crosses the threshold, and so that is a z-score of 0.53. 0.52 is too little. So we need a z-score of 0.53. Let's write that down. 0.53, right over there, and we just now have to figure out what value gives us a z-score of 0.53. Well, this just means 0.53 standard deviations above the mean. So to get the value, we would take our mean, and we would add 0.53 standard deviation, so 0.53 times nine, and this will get us 0.53 times nine is equal to 4.77, plus 80 is equal to 84.77."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "Let's write that down. 0.53, right over there, and we just now have to figure out what value gives us a z-score of 0.53. Well, this just means 0.53 standard deviations above the mean. So to get the value, we would take our mean, and we would add 0.53 standard deviation, so 0.53 times nine, and this will get us 0.53 times nine is equal to 4.77, plus 80 is equal to 84.77. 84.77, and they want us to round to the nearest whole number. So we will just round to 85 beats per minute. So that's the threshold."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So what I have here are 12 pieces of candy. And the ones that are colored in brown are made out of, have chocolate on the outside, and the ones that have a C on them means that they have coconut on the inside. So for example, this one over here in the top left, it's made out of chocolate on the outside, but it doesn't have coconut on the inside. While this one right over here does, is chocolate on the outside, and has coconut on the inside. While this one, whoops, I didn't want to do that. While this one, while this one right over here does not have chocolate on the outside, but it does have coconut on the inside. And this one right over here has neither chocolate nor coconut."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "While this one right over here does, is chocolate on the outside, and has coconut on the inside. While this one, whoops, I didn't want to do that. While this one, while this one right over here does not have chocolate on the outside, but it does have coconut on the inside. And this one right over here has neither chocolate nor coconut. And what I want to think about is ways to represent this information that we are looking at. And one way to do it is using a Venn diagram. So let me draw a Venn diagram."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And this one right over here has neither chocolate nor coconut. And what I want to think about is ways to represent this information that we are looking at. And one way to do it is using a Venn diagram. So let me draw a Venn diagram. So Venn diagram is one way to represent it. And the way it's typically done, my intention is that you would make a rectangle to represent the universe that you care about. In this case, it would be all the chocolates."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So let me draw a Venn diagram. So Venn diagram is one way to represent it. And the way it's typically done, my intention is that you would make a rectangle to represent the universe that you care about. In this case, it would be all the chocolates. So all the numbers inside of this should add up to the number of chocolates I have. So it should add up to 12. So that's our universe right over here."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "In this case, it would be all the chocolates. So all the numbers inside of this should add up to the number of chocolates I have. So it should add up to 12. So that's our universe right over here. And then I'll draw circles to represent the sets that I care about. So say for this one, I care about the set of the things that have chocolate. So I'll draw that with a circle."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So that's our universe right over here. And then I'll draw circles to represent the sets that I care about. So say for this one, I care about the set of the things that have chocolate. So I'll draw that with a circle. Oftentimes, you could draw them to scale, but I'm not going to draw them to scale. So that is my chocolate set. And then I'll have a coconut set."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So I'll draw that with a circle. Oftentimes, you could draw them to scale, but I'm not going to draw them to scale. So that is my chocolate set. And then I'll have a coconut set. So coconut, once again, not drawn to scale. I drew them roughly the same size, but you can see the chocolate set is bigger than the coconut set in reality. Coconut set."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And then I'll have a coconut set. So coconut, once again, not drawn to scale. I drew them roughly the same size, but you can see the chocolate set is bigger than the coconut set in reality. Coconut set. And now we can fill in the different sections. So how many of these things have chocolate but no coconut? Let's see."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Coconut set. And now we can fill in the different sections. So how many of these things have chocolate but no coconut? Let's see. We have one, two, three, four, five, six have chocolate but no coconut. So that's going to be the... Actually, let me do that in a different color because I think the colors are important. So let me do it in green."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Let's see. We have one, two, three, four, five, six have chocolate but no coconut. So that's going to be the... Actually, let me do that in a different color because I think the colors are important. So let me do it in green. So one, two, three, four, five, and six. So this section right over here is six. And once again, I'm not talking about the whole brown thing."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So let me do it in green. So one, two, three, four, five, and six. So this section right over here is six. And once again, I'm not talking about the whole brown thing. I'm talking about just this area that I've shaded in green. Now how many have chocolate and coconut? Well, that's going to be one, two, three."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And once again, I'm not talking about the whole brown thing. I'm talking about just this area that I've shaded in green. Now how many have chocolate and coconut? Well, that's going to be one, two, three. So three of them have chocolate and coconut. And notice that's this section here that's in the overlap between. Three of them go into both sets, both categories."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Well, that's going to be one, two, three. So three of them have chocolate and coconut. And notice that's this section here that's in the overlap between. Three of them go into both sets, both categories. These three have coconut and they have chocolate. How many total have chocolate? Well, six plus three, nine."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Three of them go into both sets, both categories. These three have coconut and they have chocolate. How many total have chocolate? Well, six plus three, nine. How many total have coconut? Well, we're going to have to figure that out in a second. So how many have coconut but no chocolate?"}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Well, six plus three, nine. How many total have coconut? Well, we're going to have to figure that out in a second. So how many have coconut but no chocolate? Well, there's only one with coconut and no chocolate. So that's that one right over there. And that represents this area that I'm shading in in white."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So how many have coconut but no chocolate? Well, there's only one with coconut and no chocolate. So that's that one right over there. And that represents this area that I'm shading in in white. So how many total coconut are there? Well, one plus three or four. And you see that, one, two, three, four."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And that represents this area that I'm shading in in white. So how many total coconut are there? Well, one plus three or four. And you see that, one, two, three, four. And then the last thing we'd want to fill in, because notice six plus three plus one only adds up to ten. What about the other two? Well, the other two are neither chocolate nor coconut."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And you see that, one, two, three, four. And then the last thing we'd want to fill in, because notice six plus three plus one only adds up to ten. What about the other two? Well, the other two are neither chocolate nor coconut. Actually, let me color this. So that's one, two. These are neither chocolate nor coconut."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Well, the other two are neither chocolate nor coconut. Actually, let me color this. So that's one, two. These are neither chocolate nor coconut. And I could write these two right over here. These are neither chocolate nor coconut. So that's one way to represent the information of how many chocolates, how many coconuts, and how many chocolate and coconuts, and how many neither."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "These are neither chocolate nor coconut. And I could write these two right over here. These are neither chocolate nor coconut. So that's one way to represent the information of how many chocolates, how many coconuts, and how many chocolate and coconuts, and how many neither. But there's other ways that we could do it. Another way to do it would be with a two-way table. A two-way table."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So that's one way to represent the information of how many chocolates, how many coconuts, and how many chocolate and coconuts, and how many neither. But there's other ways that we could do it. Another way to do it would be with a two-way table. A two-way table. And on one axis, say the vertical axis, we could say, let me write this. So has chocolate. Has chocolate."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "A two-way table. And on one axis, say the vertical axis, we could say, let me write this. So has chocolate. Has chocolate. I'll write chalk for short. And then I'll write no chocolate. No chocolate, chalk for short."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Has chocolate. I'll write chalk for short. And then I'll write no chocolate. No chocolate, chalk for short. And then over here, I could write coconut. I want to do that in white. I got new tools, and sometimes the color changing isn't so easy."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "No chocolate, chalk for short. And then over here, I could write coconut. I want to do that in white. I got new tools, and sometimes the color changing isn't so easy. So this is coconut. And then over here, I'll write no coconut. No coconut."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "I got new tools, and sometimes the color changing isn't so easy. So this is coconut. And then over here, I'll write no coconut. No coconut. And then let me make a little table. Let me make a table, make it clear what I'm doing here. So a line there and a line there."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "No coconut. And then let me make a little table. Let me make a table, make it clear what I'm doing here. So a line there and a line there. And then I'll add a line over here as well. And then I can just fill in the different things. So how many have this cell right over this square?"}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So a line there and a line there. And then I'll add a line over here as well. And then I can just fill in the different things. So how many have this cell right over this square? This is going to represent the number that has coconut and chocolate. Coconut and chocolate. Well, we already looked into that."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So how many have this cell right over this square? This is going to represent the number that has coconut and chocolate. Coconut and chocolate. Well, we already looked into that. That's one, two, three. That's these three right over here. So that's those three right over there."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Well, we already looked into that. That's one, two, three. That's these three right over here. So that's those three right over there. This one right over here is it has chocolate, but it doesn't have coconut. Well, that's this six right over here. It has chocolate, but it doesn't have coconut."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So that's those three right over there. This one right over here is it has chocolate, but it doesn't have coconut. Well, that's this six right over here. It has chocolate, but it doesn't have coconut. So let me write this is that six right over there. And then this box would be it has coconut, but no chocolate. Well, how many is that?"}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "It has chocolate, but it doesn't have coconut. So let me write this is that six right over there. And then this box would be it has coconut, but no chocolate. Well, how many is that? Well, coconut, no chocolate. That's that one there. And this one is going to be no coconut and no chocolate."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Well, how many is that? Well, coconut, no chocolate. That's that one there. And this one is going to be no coconut and no chocolate. And we know what that's going to be. No coconut and no chocolate is going to be two. And if we wanted to, we could even throw in totals over here."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And this one is going to be no coconut and no chocolate. And we know what that's going to be. No coconut and no chocolate is going to be two. And if we wanted to, we could even throw in totals over here. We could write, actually let me just do that just for fun. I could write total. And if I total it vertically, so three plus one, this is four."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And if we wanted to, we could even throw in totals over here. We could write, actually let me just do that just for fun. I could write total. And if I total it vertically, so three plus one, this is four. Six plus two is eight. So this four is the total number that have coconut, both the has chocolate and doesn't have chocolate. And that's the three plus one."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And if I total it vertically, so three plus one, this is four. Six plus two is eight. So this four is the total number that have coconut, both the has chocolate and doesn't have chocolate. And that's the three plus one. This eight is the total that does not have coconut. We're in no coconuts, the total of no coconut. And that, of course, is going to be the six plus this two."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And that's the three plus one. This eight is the total that does not have coconut. We're in no coconuts, the total of no coconut. And that, of course, is going to be the six plus this two. And we could total horizontally. Three plus six is nine. One plus two is three."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And that, of course, is going to be the six plus this two. And we could total horizontally. Three plus six is nine. One plus two is three. What's this nine? That's the total amount of chocolate, six plus three. What's this three?"}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "One plus two is three. What's this nine? That's the total amount of chocolate, six plus three. What's this three? This is the total amount no chocolate. That's this one plus two. Anyway, hopefully you found that interesting."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "On that exercise, there's a little interface where we can drag these around in a table to match them to the different scatter plots. And the point isn't to figure out how exactly to calculate these. We'll do that in the future, but really to get an intuition of what we're trying to measure. And the main idea is that correlation coefficients are trying to measure how well a linear model can describe the relationship between two variables. So for example, if I have, let me draw, let me do some coordinate axes here. So let's say that's one variable. Say that's my y variable."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "And the main idea is that correlation coefficients are trying to measure how well a linear model can describe the relationship between two variables. So for example, if I have, let me draw, let me do some coordinate axes here. So let's say that's one variable. Say that's my y variable. And let's say that is my x variable. And so let's say when x is low, y is low. When x is a little higher, y is a little higher."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "Say that's my y variable. And let's say that is my x variable. And so let's say when x is low, y is low. When x is a little higher, y is a little higher. When x is a little bit higher, y is higher. When x is really high, y is even higher. This one, a linear model would describe it very, very, very well."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "When x is a little higher, y is a little higher. When x is a little bit higher, y is higher. When x is really high, y is even higher. This one, a linear model would describe it very, very, very well. It's quite easy to draw a line that goes through, that essentially goes through those points. So something like this would have an r of one. r is equal to one."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "This one, a linear model would describe it very, very, very well. It's quite easy to draw a line that goes through, that essentially goes through those points. So something like this would have an r of one. r is equal to one. A linear model perfectly describes it, and it's a positive correlation. When one increases, when one variable gets larger, then the other variable is larger. When one variable is smaller, then the other variable is smaller, and vice versa."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "r is equal to one. A linear model perfectly describes it, and it's a positive correlation. When one increases, when one variable gets larger, then the other variable is larger. When one variable is smaller, then the other variable is smaller, and vice versa. Now what would an r of negative one look like? Well, that would once again be a situation where a linear model works really well, but when one variable moves up, the other one moves down, and vice versa. So let me draw my coordinates, my coordinate axes again."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "When one variable is smaller, then the other variable is smaller, and vice versa. Now what would an r of negative one look like? Well, that would once again be a situation where a linear model works really well, but when one variable moves up, the other one moves down, and vice versa. So let me draw my coordinates, my coordinate axes again. So I'm gonna try to draw a data set where the r would be negative one. So maybe when y is high, x is very low. When y becomes lower, x becomes higher."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So let me draw my coordinates, my coordinate axes again. So I'm gonna try to draw a data set where the r would be negative one. So maybe when y is high, x is very low. When y becomes lower, x becomes higher. When y becomes a good bit lower, x becomes a good bit higher. So once again, when y decreases, x increases, or as x increases, y decreases. So they're moving in opposite directions."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "When y becomes lower, x becomes higher. When y becomes a good bit lower, x becomes a good bit higher. So once again, when y decreases, x increases, or as x increases, y decreases. So they're moving in opposite directions. But you can fit a line very easily to this. So the line would look something like this. So this would have an r of negative one."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So they're moving in opposite directions. But you can fit a line very easily to this. So the line would look something like this. So this would have an r of negative one. And an r of zero, r is equal to zero, would be a data set where a line doesn't really fit very well at all. So I'll do that one really small, since I don't have much space here. So an r of zero might look something like this."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So this would have an r of negative one. And an r of zero, r is equal to zero, would be a data set where a line doesn't really fit very well at all. So I'll do that one really small, since I don't have much space here. So an r of zero might look something like this. Maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have one there, there, there, there, there. And it wouldn't necessarily be this well organized, but this gives you a sense of things. Where would you actually, how would you actually try to fit a line here?"}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So an r of zero might look something like this. Maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have one there, there, there, there, there. And it wouldn't necessarily be this well organized, but this gives you a sense of things. Where would you actually, how would you actually try to fit a line here? You could equally justify a line that looks like that, or a line that looks like that, or a line that looks like that. So there really isn't, a linear model really does not describe the relationship between the two variables that well right over here. So with that as a primer, let's see if we can tackle these scatter plots."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "Where would you actually, how would you actually try to fit a line here? You could equally justify a line that looks like that, or a line that looks like that, or a line that looks like that. So there really isn't, a linear model really does not describe the relationship between the two variables that well right over here. So with that as a primer, let's see if we can tackle these scatter plots. And the way I'm gonna do it is I'm just gonna try to eyeball what a linear model might look like. And there's different methods of trying to fit a linear model to a dataset, an imperfect dataset. I drew very perfect ones, at least for the r equals negative one and r equals one."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So with that as a primer, let's see if we can tackle these scatter plots. And the way I'm gonna do it is I'm just gonna try to eyeball what a linear model might look like. And there's different methods of trying to fit a linear model to a dataset, an imperfect dataset. I drew very perfect ones, at least for the r equals negative one and r equals one. But these are what the real world actually looks like. Nothing, very few times will things perfectly sit on a line. So for scatter plot A, if I were to try to fit a line, it would look something like, it would look something like that, if I were to try to minimize distances from these points to the line."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "I drew very perfect ones, at least for the r equals negative one and r equals one. But these are what the real world actually looks like. Nothing, very few times will things perfectly sit on a line. So for scatter plot A, if I were to try to fit a line, it would look something like, it would look something like that, if I were to try to minimize distances from these points to the line. I do see a general trend that when y is, you know, if we look at these data points over here, when y is high, x is low, and when x is high, when x is larger, y is smaller. So it looks like r is going to be less than zero, in a reasonable bit less than zero. It's going to approach this thing here."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So for scatter plot A, if I were to try to fit a line, it would look something like, it would look something like that, if I were to try to minimize distances from these points to the line. I do see a general trend that when y is, you know, if we look at these data points over here, when y is high, x is low, and when x is high, when x is larger, y is smaller. So it looks like r is going to be less than zero, in a reasonable bit less than zero. It's going to approach this thing here. And if we look at our choices, so it wouldn't be r equals 0.65, these are positive, so I wouldn't use that one or that one. And this one is almost no correlation. r equals negative 0.02, this is pretty close to zero."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "It's going to approach this thing here. And if we look at our choices, so it wouldn't be r equals 0.65, these are positive, so I wouldn't use that one or that one. And this one is almost no correlation. r equals negative 0.02, this is pretty close to zero. So I feel good with r is equal to negative 0.72. r is equal to negative 0.72. Now I want to be clear, if I didn't have these choices here, I wouldn't just be able to say, just looking at these data points, without being able to do a calculation, that r is equal to negative 0.72. I'm just basing it on the intuition that it is a negative correlation."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "r equals negative 0.02, this is pretty close to zero. So I feel good with r is equal to negative 0.72. r is equal to negative 0.72. Now I want to be clear, if I didn't have these choices here, I wouldn't just be able to say, just looking at these data points, without being able to do a calculation, that r is equal to negative 0.72. I'm just basing it on the intuition that it is a negative correlation. It seems pretty strong, you know, the pattern kind of jumps out at you that when y is large, y, x is small. When x is large, y is small. And so I like something that's approaching r equals negative one."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "I'm just basing it on the intuition that it is a negative correlation. It seems pretty strong, you know, the pattern kind of jumps out at you that when y is large, y, x is small. When x is large, y is small. And so I like something that's approaching r equals negative one. So I've used this one up already. Now scatter plot B, if I were to just try to eyeball it, once again, this is going to be imperfect, but the trend, if I were to try to fit a line, it looks something like that. So it looks like a line fits in reasonably well."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "And so I like something that's approaching r equals negative one. So I've used this one up already. Now scatter plot B, if I were to just try to eyeball it, once again, this is going to be imperfect, but the trend, if I were to try to fit a line, it looks something like that. So it looks like a line fits in reasonably well. There are some points that would still be hard to fit, and they're still pretty far from the line. And it looks like it's a positive correlation. When x is small, when y is small, x is relatively small, and vice versa."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So it looks like a line fits in reasonably well. There are some points that would still be hard to fit, and they're still pretty far from the line. And it looks like it's a positive correlation. When x is small, when y is small, x is relatively small, and vice versa. And as x grows, y grows, and when y grows, x grows. So this one's going to be positive, and it looks like it would be reasonably positive. And I have two choices here."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "When x is small, when y is small, x is relatively small, and vice versa. And as x grows, y grows, and when y grows, x grows. So this one's going to be positive, and it looks like it would be reasonably positive. And I have two choices here. So I don't know which of these it's going to be. So it's either going to be r is equal to 0.65, or r is equal to 0.84. I also get scatter plot C. Now this one's all over the place."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "And I have two choices here. So I don't know which of these it's going to be. So it's either going to be r is equal to 0.65, or r is equal to 0.84. I also get scatter plot C. Now this one's all over the place. It kind of looks like what we did over here. You know, I could, you know, what does a line look like? You could almost imagine anything."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "I also get scatter plot C. Now this one's all over the place. It kind of looks like what we did over here. You know, I could, you know, what does a line look like? You could almost imagine anything. Does it look like that? Does it look like that? Does a line look like that?"}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "You could almost imagine anything. Does it look like that? Does it look like that? Does a line look like that? These things really aren't, don't seem to, there's not a direction that you could say, well, as x increases, maybe y increases or decreases, there's no rhyme or reason here. So this looks very non-correlated. And so this one is pretty close to 0."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "Does a line look like that? These things really aren't, don't seem to, there's not a direction that you could say, well, as x increases, maybe y increases or decreases, there's no rhyme or reason here. So this looks very non-correlated. And so this one is pretty close to 0. So I feel pretty good that this is the r is equal to negative 0.02. In fact, you know, if we tried, probably the best line that could be fit would be one with a slight negative slope. So it might look something like this."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "And so this one is pretty close to 0. So I feel pretty good that this is the r is equal to negative 0.02. In fact, you know, if we tried, probably the best line that could be fit would be one with a slight negative slope. So it might look something like this. It might look something like this. And notice even when we try to fit a line, there's all sorts of points that are way off the line. So the linear model did not fit it that well."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So it might look something like this. It might look something like this. And notice even when we try to fit a line, there's all sorts of points that are way off the line. So the linear model did not fit it that well. So r is equal to negative 0.02. So we use that one. And so now we have scatterplot D. So that's going to use one of the other positive correlations."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So the linear model did not fit it that well. So r is equal to negative 0.02. So we use that one. And so now we have scatterplot D. So that's going to use one of the other positive correlations. And it does look like, you know, there is a positive correlation when y is low, x is low, and when x is high, y is high, and vice versa. And so we could try to fit something that looks something like that. But it's still not as good as that one."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "And so now we have scatterplot D. So that's going to use one of the other positive correlations. And it does look like, you know, there is a positive correlation when y is low, x is low, and when x is high, y is high, and vice versa. And so we could try to fit something that looks something like that. But it's still not as good as that one. You could see the points that we're trying to fit, there's several points that are still pretty far away from our model. So the model is not fitting it that well. So I would say scatterplot B is a better fit."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "But it's still not as good as that one. You could see the points that we're trying to fit, there's several points that are still pretty far away from our model. So the model is not fitting it that well. So I would say scatterplot B is a better fit. A linear model works better for scatterplot B than it works for scatterplot D. So I would give the higher r to scatterplot B, and the lower r, r equals 0.65 to scatterplot D. r is equal to 0.65. And once again, that's because with a linear model, it looks like there's a trend, but there's several data points that really, more data points are way off the line in scatterplot D than in the case of scatterplot B. There's a few that are still way off the line, but these are even more off of the line in D."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "Let's say I go to 16 students, and I ask them to measure how many glasses of water they drink per day for the last 30 days, and then to average it. And so this data point right over here tells us one student drank an average of 0.5 glasses of water per day. That person is probably very dehydrated. This person drank 8.1 glasses of water per day on average for the last 30 days. They are better hydrated. If we want to visualize that, we can set up a frequency histogram, where we can create some categories. So this first category would be for data points that are greater than or equal to zero and less than one."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "This person drank 8.1 glasses of water per day on average for the last 30 days. They are better hydrated. If we want to visualize that, we can set up a frequency histogram, where we can create some categories. So this first category would be for data points that are greater than or equal to zero and less than one. And we can see that two data points fall into that category, and that's why the bar right over here for that category is up to two. This category right over here is greater than or equal to three and less than four. Notice there are four data points in that category, and on this frequency histogram, the height of the bar is indeed four."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "So this first category would be for data points that are greater than or equal to zero and less than one. And we can see that two data points fall into that category, and that's why the bar right over here for that category is up to two. This category right over here is greater than or equal to three and less than four. Notice there are four data points in that category, and on this frequency histogram, the height of the bar is indeed four. So this is a nice way of looking at a distribution, but you might be more concerned with what percentage of my data falls into each of these categories. And that becomes especially interesting if we have many, many, many, many data points. And if we had 1,600,432,507 data points, well, just knowing the absolute number that fit into each category isn't so useful."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "Notice there are four data points in that category, and on this frequency histogram, the height of the bar is indeed four. So this is a nice way of looking at a distribution, but you might be more concerned with what percentage of my data falls into each of these categories. And that becomes especially interesting if we have many, many, many, many data points. And if we had 1,600,432,507 data points, well, just knowing the absolute number that fit into each category isn't so useful. The percent that fits into each category is a lot more useful. And so for that, we could set up a relative frequency histogram. So notice, this is representing the same data."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "And if we had 1,600,432,507 data points, well, just knowing the absolute number that fit into each category isn't so useful. The percent that fits into each category is a lot more useful. And so for that, we could set up a relative frequency histogram. So notice, this is representing the same data. But in that first category, instead of the bar height being two, the bar height is now 12.5%. Why is that? Because two of the 16 data points fall into this category."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "So notice, this is representing the same data. But in that first category, instead of the bar height being two, the bar height is now 12.5%. Why is that? Because two of the 16 data points fall into this category. 2 16ths is 1 8th, which is 12.5%. And this one right over here, notice, instead of the height being four, for four data points, it's now 25%. But these are saying the same thing."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "Because two of the 16 data points fall into this category. 2 16ths is 1 8th, which is 12.5%. And this one right over here, notice, instead of the height being four, for four data points, it's now 25%. But these are saying the same thing. Four out of the 16 data points fall into this category. 4 16ths is 1 4th, which is 25%. So both of these types of histograms are really useful, and you will see them used all of the time."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "But these are saying the same thing. Four out of the 16 data points fall into this category. 4 16ths is 1 4th, which is 25%. So both of these types of histograms are really useful, and you will see them used all of the time. But there are also cases where you have many, many, many more data points, and you want more granular categories. So what you could do is, well, let's just make our categories a little more granular. So for example, instead of them being one glass of water wide, maybe you make them half a glass of water wide."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "So both of these types of histograms are really useful, and you will see them used all of the time. But there are also cases where you have many, many, many more data points, and you want more granular categories. So what you could do is, well, let's just make our categories a little more granular. So for example, instead of them being one glass of water wide, maybe you make them half a glass of water wide. So this first category could be greater than or equal to zero, and less than 0.5. And that will give you a clearer picture. And I'm now assuming a world where we have more than 16 data points."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "So for example, instead of them being one glass of water wide, maybe you make them half a glass of water wide. So this first category could be greater than or equal to zero, and less than 0.5. And that will give you a clearer picture. And I'm now assuming a world where we have more than 16 data points. Maybe we have 16 million data points. This would be percentages on the left-hand side. But maybe that isn't good enough for you."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "And I'm now assuming a world where we have more than 16 data points. Maybe we have 16 million data points. This would be percentages on the left-hand side. But maybe that isn't good enough for you. Maybe you wanna get even more granular. So you make everything, each category, a quarter of a glass. But maybe that doesn't satisfy you."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "But maybe that isn't good enough for you. Maybe you wanna get even more granular. So you make everything, each category, a quarter of a glass. But maybe that doesn't satisfy you. You wanna get more and more and more granular. Well, you could imagine where this is going. You could get to a point where you're approaching an infinite number of categories, and each category is infinitely thin, is super, super thin, to a point that if you just connect the tops of the bars, that you will actually get a curve."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "But maybe that doesn't satisfy you. You wanna get more and more and more granular. Well, you could imagine where this is going. You could get to a point where you're approaching an infinite number of categories, and each category is infinitely thin, is super, super thin, to a point that if you just connect the tops of the bars, that you will actually get a curve. And this type of curve is something that we actually use in the statistics. And as promised at the beginning of the video, this is the density curve we talk about. And what's valuable about a density curve, it is a visualization of a distribution where the data points can take on any value in a continuum."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "You could get to a point where you're approaching an infinite number of categories, and each category is infinitely thin, is super, super thin, to a point that if you just connect the tops of the bars, that you will actually get a curve. And this type of curve is something that we actually use in the statistics. And as promised at the beginning of the video, this is the density curve we talk about. And what's valuable about a density curve, it is a visualization of a distribution where the data points can take on any value in a continuum. They're not just thrown into these coarse buckets. So how would you interpret something like this? If you look over the entire interval from zero, let's say, to nine, assuming no one drank more than an average of nine glasses per day, even in our 16 million data points, well then the area under the curve over that interval is going to be 100%, or 1.0."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "And what's valuable about a density curve, it is a visualization of a distribution where the data points can take on any value in a continuum. They're not just thrown into these coarse buckets. So how would you interpret something like this? If you look over the entire interval from zero, let's say, to nine, assuming no one drank more than an average of nine glasses per day, even in our 16 million data points, well then the area under the curve over that interval is going to be 100%, or 1.0. This is going to be true for any density curve, that the entire area of the curve is 100%. It represents all of the data points. A density curve will also never take on a negative value."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "If you look over the entire interval from zero, let's say, to nine, assuming no one drank more than an average of nine glasses per day, even in our 16 million data points, well then the area under the curve over that interval is going to be 100%, or 1.0. This is going to be true for any density curve, that the entire area of the curve is 100%. It represents all of the data points. A density curve will also never take on a negative value. You won't see the curve dip down and do something strange like that. Now with that out of the way, let's think about how we would make use of it. If I wanted to know what percentage of my data falls between two and four glasses, well I would look at that interval."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "A density curve will also never take on a negative value. You won't see the curve dip down and do something strange like that. Now with that out of the way, let's think about how we would make use of it. If I wanted to know what percentage of my data falls between two and four glasses, well I would look at that interval. I'd go from two to four. I would look at this interval right over here, and I would try to figure out the area under the curve here. And this area is going to be greater than or equal to zero and less than or equal to 100%."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "If I wanted to know what percentage of my data falls between two and four glasses, well I would look at that interval. I'd go from two to four. I would look at this interval right over here, and I would try to figure out the area under the curve here. And this area is going to be greater than or equal to zero and less than or equal to 100%. When I eyeball it right over here, it looks like it's about 40% of the entire area under the curve. So just eyeballing it, I would say roughly 40% of my data falls into this interval. If I were to ask you what percentage of the data is greater than three, well then you would be looking at this area, and it looks like it is about 50%, but once again, I am estimating it."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "And this area is going to be greater than or equal to zero and less than or equal to 100%. When I eyeball it right over here, it looks like it's about 40% of the entire area under the curve. So just eyeballing it, I would say roughly 40% of my data falls into this interval. If I were to ask you what percentage of the data is greater than three, well then you would be looking at this area, and it looks like it is about 50%, but once again, I am estimating it. But you can start to see how even with estimation, a density curve could be useful. In the real world, statisticians will often have tables that might represent the information for the density curve. They might have computer programs or some type of automated tool, and there are also well-known density curves, the famous bell curve that we will study later on where there's a lot of precise data and a lot of tools to exactly figure out the areas."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "If I were to ask you what percentage of the data is greater than three, well then you would be looking at this area, and it looks like it is about 50%, but once again, I am estimating it. But you can start to see how even with estimation, a density curve could be useful. In the real world, statisticians will often have tables that might represent the information for the density curve. They might have computer programs or some type of automated tool, and there are also well-known density curves, the famous bell curve that we will study later on where there's a lot of precise data and a lot of tools to exactly figure out the areas. The last thing I'd like to cover is a key misconception for density curves. If I were to ask you approximately what percentage of my data is exactly three glasses of water per day, and when I say exactly, I mean exactly the number 3.000 with zeros just going on and on forever, the exact number three. So you might be tempted to just say, okay, this is three."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "They might have computer programs or some type of automated tool, and there are also well-known density curves, the famous bell curve that we will study later on where there's a lot of precise data and a lot of tools to exactly figure out the areas. The last thing I'd like to cover is a key misconception for density curves. If I were to ask you approximately what percentage of my data is exactly three glasses of water per day, and when I say exactly, I mean exactly the number 3.000 with zeros just going on and on forever, the exact number three. So you might be tempted to just say, okay, this is three. Let me see the corresponding point on the curve. It looks like it is about 0.2 or a little higher than that, so maybe you would say a little bit more than 20% or approximately 20%. And what I would say to you is this is wrong."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "So you might be tempted to just say, okay, this is three. Let me see the corresponding point on the curve. It looks like it is about 0.2 or a little higher than that, so maybe you would say a little bit more than 20% or approximately 20%. And what I would say to you is this is wrong. Remember, the percentage of the data in an interval is not the height of the curve. It is the area under the curve in that interval. And if we're just talking about one precise value, like exactly the number three, there is no area under the curve."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "And what I would say to you is this is wrong. Remember, the percentage of the data in an interval is not the height of the curve. It is the area under the curve in that interval. And if we're just talking about one precise value, like exactly the number three, there is no area under the curve. This vertical line that I just drew over the number of three has no width, and this actually makes sense in the real world. Even if you were to look at 16 million people, it is very unlikely that even anyone would drink exactly three glasses of water per day. I'm talking about not one atom more or one atom less than three glasses."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "And if we're just talking about one precise value, like exactly the number three, there is no area under the curve. This vertical line that I just drew over the number of three has no width, and this actually makes sense in the real world. Even if you were to look at 16 million people, it is very unlikely that even anyone would drink exactly three glasses of water per day. I'm talking about not one atom more or one atom less than three glasses. There might be many people between 2.9 and 3.1, but no one is exactly three glasses a day. When someone says I'm drinking three glasses of water per day, that'd be a rough estimate. They're probably 3.0001 or 2.99999 or 3.15 or whatever else."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "I'm talking about not one atom more or one atom less than three glasses. There might be many people between 2.9 and 3.1, but no one is exactly three glasses a day. When someone says I'm drinking three glasses of water per day, that'd be a rough estimate. They're probably 3.0001 or 2.99999 or 3.15 or whatever else. And so instead, you could say what percentage falls in the interval maybe that is greater than or equal to 2.9 and less than or equal to 3.1. And so once you have an interval, then you actually can look at the area. So we're gonna go from 2.9 to 3.1."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "They're probably 3.0001 or 2.99999 or 3.15 or whatever else. And so instead, you could say what percentage falls in the interval maybe that is greater than or equal to 2.9 and less than or equal to 3.1. And so once you have an interval, then you actually can look at the area. So we're gonna go from 2.9 to 3.1. So now we have an interval that actually has width, and so it'd be roughly the size of this yellow area that I'm shading in right over here. And we can approximate it with a rectangle, even though the top of this curve isn't flat, so we could say, look, it's approximately like a rectangle that is 0.2 high. And what's the width?"}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy (2).mp3", "Sentence": "So we're gonna go from 2.9 to 3.1. So now we have an interval that actually has width, and so it'd be roughly the size of this yellow area that I'm shading in right over here. And we can approximate it with a rectangle, even though the top of this curve isn't flat, so we could say, look, it's approximately like a rectangle that is 0.2 high. And what's the width? The width here, if we're going from 2.9 to 3.1, the width is going to be 0.2 wide. And so we could approximate this area by approximating this rectangle, the area of the rectangle. 0.2 times 0.2, that would give us an area of 0.04."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Amelia only likes to use the drive-through for restaurants where the average wait time is in the bottom 10% for that town. What is the maximum average wait time for restaurants where Amelia likes to use the drive-through? Round to the nearest whole second. Like always, if you feel like you can tackle this, pause this video and try to do so. I'm assuming you paused it. Now let's work through this together. So let's think about what's going on."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Like always, if you feel like you can tackle this, pause this video and try to do so. I'm assuming you paused it. Now let's work through this together. So let's think about what's going on. They're telling us that the distribution of average wait times is approximately normal. So let's get a visualization of a normal distribution. And they tell us several things about this normal distribution."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So let's think about what's going on. They're telling us that the distribution of average wait times is approximately normal. So let's get a visualization of a normal distribution. And they tell us several things about this normal distribution. They tell us that the mean is 185 seconds. So that's 185 there. The standard deviation is 11 seconds."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And they tell us several things about this normal distribution. They tell us that the mean is 185 seconds. So that's 185 there. The standard deviation is 11 seconds. So for example, this is going to be 11 more than the mean. So this would be 196 seconds. This would be another 11."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "The standard deviation is 11 seconds. So for example, this is going to be 11 more than the mean. So this would be 196 seconds. This would be another 11. Each of these dotted lines are one standard deviation more. So this would be 207. This would be 11 seconds less than the mean."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "This would be another 11. Each of these dotted lines are one standard deviation more. So this would be 207. This would be 11 seconds less than the mean. So this would be 174, and so on and so forth. And we wanna find the maximum average wait time for restaurants where Amelia likes to use the drive-through. Well, what are those restaurants?"}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "This would be 11 seconds less than the mean. So this would be 174, and so on and so forth. And we wanna find the maximum average wait time for restaurants where Amelia likes to use the drive-through. Well, what are those restaurants? That's where the average wait time is in the bottom 10% for that town. So how do we think about it? Well, there's going to be some value."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Well, what are those restaurants? That's where the average wait time is in the bottom 10% for that town. So how do we think about it? Well, there's going to be some value. Let me mark it off right over here in this red color. So we're gonna have some threshold value right over here where this is anything that level or lower is going to be in the bottom 10%. Well, another way to think about it is this is the largest wait time for which you are still in the bottom 10%."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Well, there's going to be some value. Let me mark it off right over here in this red color. So we're gonna have some threshold value right over here where this is anything that level or lower is going to be in the bottom 10%. Well, another way to think about it is this is the largest wait time for which you are still in the bottom 10%. And so this area right over here is going to be 10% of the total, or it's going to be 0.10. So the way we can tackle this is we can get up a z-table and figure out what z-score gives us a proportion of only 0.10 being less than that z-score. And then using that z-score, we can figure out this value, the actual wait time."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Well, another way to think about it is this is the largest wait time for which you are still in the bottom 10%. And so this area right over here is going to be 10% of the total, or it's going to be 0.10. So the way we can tackle this is we can get up a z-table and figure out what z-score gives us a proportion of only 0.10 being less than that z-score. And then using that z-score, we can figure out this value, the actual wait time. So let's get our z-table out. And since we know that this is below the mean, the mean would be the 50th percentile, we know we're gonna have a negative z-score. So I'm gonna take out the part of the table that has the negative z-scores on it."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And then using that z-score, we can figure out this value, the actual wait time. So let's get our z-table out. And since we know that this is below the mean, the mean would be the 50th percentile, we know we're gonna have a negative z-score. So I'm gonna take out the part of the table that has the negative z-scores on it. And remember, we're looking for 10%, but we don't wanna go beyond 10%. We wanna be sure that that value is within the 10th percentile, that any higher will be out of the 10th percentile. So let's see, when we have these really negative z's, so far it only doesn't even get to the first percentile yet."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So I'm gonna take out the part of the table that has the negative z-scores on it. And remember, we're looking for 10%, but we don't wanna go beyond 10%. We wanna be sure that that value is within the 10th percentile, that any higher will be out of the 10th percentile. So let's see, when we have these really negative z's, so far it only doesn't even get to the first percentile yet. So let's scroll down a little bit. And let's remember as we do so that this is zero in the hundredths place, one, two, three, four, five, six, seven, eight, nine. So let's remember those columns."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So let's see, when we have these really negative z's, so far it only doesn't even get to the first percentile yet. So let's scroll down a little bit. And let's remember as we do so that this is zero in the hundredths place, one, two, three, four, five, six, seven, eight, nine. So let's remember those columns. So let's see, if we are at a z-score of negative 1.28, remember, this is, the hundredths is zero, one, two, three, four, five, six, seven, eight. So this right over here is a z-score of negative 1.28. And that's a little bit crossing the 10th percentile."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So let's remember those columns. So let's see, if we are at a z-score of negative 1.28, remember, this is, the hundredths is zero, one, two, three, four, five, six, seven, eight. So this right over here is a z-score of negative 1.28. And that's a little bit crossing the 10th percentile. But if we get a little bit more negative than that, we are in the 10th percentile. So this is negative 1.29. And this does seem to be the highest z-score for which we are within the 10th percentile."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And that's a little bit crossing the 10th percentile. But if we get a little bit more negative than that, we are in the 10th percentile. So this is negative 1.29. And this does seem to be the highest z-score for which we are within the 10th percentile. So negative 1.29 is our z-score. So this is z equals negative 1.29. And if we wanna figure out the actual value for that, we would start with the mean, which is 185."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And this does seem to be the highest z-score for which we are within the 10th percentile. So negative 1.29 is our z-score. So this is z equals negative 1.29. And if we wanna figure out the actual value for that, we would start with the mean, which is 185. And then we would say, well, we wanna go 1.29 standard deviations below the mean. The negative says we're going below the mean. So we could say minus 1.29 times the standard deviation."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And if we wanna figure out the actual value for that, we would start with the mean, which is 185. And then we would say, well, we wanna go 1.29 standard deviations below the mean. The negative says we're going below the mean. So we could say minus 1.29 times the standard deviation. And they tell us up here the standard deviation is 11 seconds. So it's going to be 1.29 times 11. And this is going to be equal to 1.29 times 11 is equal to 14.19."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So we could say minus 1.29 times the standard deviation. And they tell us up here the standard deviation is 11 seconds. So it's going to be 1.29 times 11. And this is going to be equal to 1.29 times 11 is equal to 14.19. And then I'll make that negative and then add that to 185. Plus 185 is equal to 170.81. 170.81."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And this is going to be equal to 1.29 times 11 is equal to 14.19. And then I'll make that negative and then add that to 185. Plus 185 is equal to 170.81. 170.81. Now they say round to the nearest whole second. There's a couple of ways to think about it. If you really wanna ensure that you're not gonna cross the 10th percentile, you might wanna round to the nearest second that is below this threshold."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "170.81. Now they say round to the nearest whole second. There's a couple of ways to think about it. If you really wanna ensure that you're not gonna cross the 10th percentile, you might wanna round to the nearest second that is below this threshold. So you might say that this is approximately 170 seconds. If you were to just round normally, this would go to 171. But just by doing that, you might have crossed the threshold."}, {"video_title": "Example constructing and interpreting a confidence interval for p AP Statistics Khan Academy.mp3", "Sentence": "We're told Della has over 500 songs on her mobile phone, and she wants to estimate what proportion of the songs are by a female artist. She takes a simple random sample, that's what SRS stands for, of 50 songs on her phone, and finds that 20 of the songs sampled are by a female artist. Based on this sample, which of the following is a 99% confidence interval for the proportion of songs on her phone that are by a female artist? So like always, pause this video and see if you can figure it out on your own. Della has a library of 500 songs right over here, and she's trying to figure out the proportion that are sung by a female artist. She doesn't have the time to go through all 500 songs to figure out the true population proportion, P, so instead she takes a sample of 50 songs, N is equal to 50, and from that she calculates a sample proportion, which we could denote with P hat, and she finds that 20 out of the 50 are sung by a female. 20 out of the 50, which is the same thing, is 0.4, and then she wants to construct a 99% confidence interval."}, {"video_title": "Example constructing and interpreting a confidence interval for p AP Statistics Khan Academy.mp3", "Sentence": "So like always, pause this video and see if you can figure it out on your own. Della has a library of 500 songs right over here, and she's trying to figure out the proportion that are sung by a female artist. She doesn't have the time to go through all 500 songs to figure out the true population proportion, P, so instead she takes a sample of 50 songs, N is equal to 50, and from that she calculates a sample proportion, which we could denote with P hat, and she finds that 20 out of the 50 are sung by a female. 20 out of the 50, which is the same thing, is 0.4, and then she wants to construct a 99% confidence interval. So before we even go about constructing the confidence interval, you wanna check to make sure that we're making some valid assumptions, we're using a valid technique. So before we actually calculate the confidence interval, let's just make sure that our sampling distribution is not distorted in some way, and so that we can, with confidence, make a confidence interval. So the first condition is to make sure that your sample is truly random, and they tell us that it's a simple random sample, so we'll take their word for it."}, {"video_title": "Example constructing and interpreting a confidence interval for p AP Statistics Khan Academy.mp3", "Sentence": "20 out of the 50, which is the same thing, is 0.4, and then she wants to construct a 99% confidence interval. So before we even go about constructing the confidence interval, you wanna check to make sure that we're making some valid assumptions, we're using a valid technique. So before we actually calculate the confidence interval, let's just make sure that our sampling distribution is not distorted in some way, and so that we can, with confidence, make a confidence interval. So the first condition is to make sure that your sample is truly random, and they tell us that it's a simple random sample, so we'll take their word for it. The next condition is to assume that your sampling distribution of the sample proportions is approximately normal, and there, you wanna be confident, or you wanna see that in your sample, you have at least 10 successes and at least 10 failures. Well, here, we have 20 successes, which means, well, 50 minus 20, we have 30 failures. So both of those are more than 10, and so meets that condition."}, {"video_title": "Example constructing and interpreting a confidence interval for p AP Statistics Khan Academy.mp3", "Sentence": "So the first condition is to make sure that your sample is truly random, and they tell us that it's a simple random sample, so we'll take their word for it. The next condition is to assume that your sampling distribution of the sample proportions is approximately normal, and there, you wanna be confident, or you wanna see that in your sample, you have at least 10 successes and at least 10 failures. Well, here, we have 20 successes, which means, well, 50 minus 20, we have 30 failures. So both of those are more than 10, and so meets that condition. And then the last condition is, sometimes it'll called the independence test or the independence rule or the 10% rule. If you were doing this sample with replacement, so if she were to look at one song, test whether it's a female or not, and then put it back in her pile and then look at another song, then each of those observations would truly be independent. But we don't know that."}, {"video_title": "Example constructing and interpreting a confidence interval for p AP Statistics Khan Academy.mp3", "Sentence": "So both of those are more than 10, and so meets that condition. And then the last condition is, sometimes it'll called the independence test or the independence rule or the 10% rule. If you were doing this sample with replacement, so if she were to look at one song, test whether it's a female or not, and then put it back in her pile and then look at another song, then each of those observations would truly be independent. But we don't know that. In fact, we'll assume that she didn't do it with the replacement, and so if you don't do it with the replacement, you can assume rough independence for each observation of a song if this is no more than 10% of the population. And so it looks like it is exactly 10% of the population, so Della just squeezes through on our independence test right over there. So with that out of the way, let's just think about what the confidence interval is going to be."}, {"video_title": "Example constructing and interpreting a confidence interval for p AP Statistics Khan Academy.mp3", "Sentence": "But we don't know that. In fact, we'll assume that she didn't do it with the replacement, and so if you don't do it with the replacement, you can assume rough independence for each observation of a song if this is no more than 10% of the population. And so it looks like it is exactly 10% of the population, so Della just squeezes through on our independence test right over there. So with that out of the way, let's just think about what the confidence interval is going to be. Well, it's going to be her sample proportion, plus or minus, there's going to be some critical value, and this critical value is going to be dictated by our confidence level we wanna have, and then that critical value times the standard deviation of the sampling distribution of the sample proportions, which we don't know. And so instead of having that, we use the standard error of the sample proportion, and in this case, it would be p hat times one minus p hat, all of that over n, our sample size, all of that over 50. So what's this going to be?"}, {"video_title": "Example constructing and interpreting a confidence interval for p AP Statistics Khan Academy.mp3", "Sentence": "So with that out of the way, let's just think about what the confidence interval is going to be. Well, it's going to be her sample proportion, plus or minus, there's going to be some critical value, and this critical value is going to be dictated by our confidence level we wanna have, and then that critical value times the standard deviation of the sampling distribution of the sample proportions, which we don't know. And so instead of having that, we use the standard error of the sample proportion, and in this case, it would be p hat times one minus p hat, all of that over n, our sample size, all of that over 50. So what's this going to be? We're gonna get p hat, our sample proportion here, is 0.4 plus or minus, I'll save the z star here, our critical value, for a little bit. We're gonna use a z table for that. And so we're gonna have 0.4 right over there, one minus 0.4 is times 0.6, all of that over 50."}, {"video_title": "Example constructing and interpreting a confidence interval for p AP Statistics Khan Academy.mp3", "Sentence": "So what's this going to be? We're gonna get p hat, our sample proportion here, is 0.4 plus or minus, I'll save the z star here, our critical value, for a little bit. We're gonna use a z table for that. And so we're gonna have 0.4 right over there, one minus 0.4 is times 0.6, all of that over 50. So we can already look at some choices that look interesting here. This choice and this choice both look interesting, and the main thing we have to reason through is which one has a correct critical value. Do we wanna go 1.96 standard errors above and below our sample proportion, or do we wanna go 2.576 standard errors above and below our sample proportion?"}, {"video_title": "Example constructing and interpreting a confidence interval for p AP Statistics Khan Academy.mp3", "Sentence": "And so we're gonna have 0.4 right over there, one minus 0.4 is times 0.6, all of that over 50. So we can already look at some choices that look interesting here. This choice and this choice both look interesting, and the main thing we have to reason through is which one has a correct critical value. Do we wanna go 1.96 standard errors above and below our sample proportion, or do we wanna go 2.576 standard errors above and below our sample proportion? And the key is the 99% confidence level. Now, if we have a 99% confidence level, one way to think about it is, so let me just do my best shot at drawing a normal distribution here. And so if you want a 99% confidence level, that means you wanna contain the 99%, the middle 99% under the curve right over here, that area."}, {"video_title": "Example constructing and interpreting a confidence interval for p AP Statistics Khan Academy.mp3", "Sentence": "Do we wanna go 1.96 standard errors above and below our sample proportion, or do we wanna go 2.576 standard errors above and below our sample proportion? And the key is the 99% confidence level. Now, if we have a 99% confidence level, one way to think about it is, so let me just do my best shot at drawing a normal distribution here. And so if you want a 99% confidence level, that means you wanna contain the 99%, the middle 99% under the curve right over here, that area. And so if this is 99%, then this right over here is going to be 0.5%, and this right over here is 0.5%. We want the z value that's going to leave 0.5% above it. And so that's actually going to be 99.5% is what we wanna look up on the table."}, {"video_title": "Example constructing and interpreting a confidence interval for p AP Statistics Khan Academy.mp3", "Sentence": "And so if you want a 99% confidence level, that means you wanna contain the 99%, the middle 99% under the curve right over here, that area. And so if this is 99%, then this right over here is going to be 0.5%, and this right over here is 0.5%. We want the z value that's going to leave 0.5% above it. And so that's actually going to be 99.5% is what we wanna look up on the table. And that's because many z tables, including the one that you might see on something like an AP Stats exam, they will have the area up to and including, up to and including a certain value. And so they're not going to leave this free right over here. So let's just look up 99.5% on our z table."}, {"video_title": "Example constructing and interpreting a confidence interval for p AP Statistics Khan Academy.mp3", "Sentence": "And so that's actually going to be 99.5% is what we wanna look up on the table. And that's because many z tables, including the one that you might see on something like an AP Stats exam, they will have the area up to and including, up to and including a certain value. And so they're not going to leave this free right over here. So let's just look up 99.5% on our z table. All right, so let me move this down so you can see it. All right, that's our z table. Let's see, we're at 99 point, okay, it's gonna be right in this area right over here."}, {"video_title": "Example constructing and interpreting a confidence interval for p AP Statistics Khan Academy.mp3", "Sentence": "So let's just look up 99.5% on our z table. All right, so let me move this down so you can see it. All right, that's our z table. Let's see, we're at 99 point, okay, it's gonna be right in this area right over here. And so that is 2.5, looks like 2.57 or 2.58 around that. And so this right over here is about 2.57. It's between 2.57 and 2.58, which gives us enough information to answer this question."}, {"video_title": "Example constructing and interpreting a confidence interval for p AP Statistics Khan Academy.mp3", "Sentence": "Let's see, we're at 99 point, okay, it's gonna be right in this area right over here. And so that is 2.5, looks like 2.57 or 2.58 around that. And so this right over here is about 2.57. It's between 2.57 and 2.58, which gives us enough information to answer this question. It's definitely not going to be this one right over here. We have 2.576, which is indeed between 2.57 and 2.58. So let's remind ourselves."}, {"video_title": "Example constructing and interpreting a confidence interval for p AP Statistics Khan Academy.mp3", "Sentence": "It's between 2.57 and 2.58, which gives us enough information to answer this question. It's definitely not going to be this one right over here. We have 2.576, which is indeed between 2.57 and 2.58. So let's remind ourselves. We've been able to construct our confidence interval right over here. But what does that actually mean? That means that if we were to repeatedly take samples of size 50 and repeatedly use this technique to construct confidence intervals, that roughly 99% of those intervals constructed this way are going to contain our true population parameter."}, {"video_title": "Mean of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "So the expected value of X, which I could also denote as the mean of our random variable X, let's say I expect to see three dogs a day, and similarly for the cats, the expected value of Y is equal to, I could also denote that as the mean of Y, is going to be equal to, and this is just for the sake of argument, let's say I expect to see four cats a day, and in previous videos, we defined how do you take the mean of a random variable, or the expected value of a random variable. What we're going to think about now is, what would be the expected value of X plus Y? Or another way of saying that, the mean of the sum of these two random variables. Well, it turns out, and I'm not proving it just yet, that the mean of the sum of random variables is equal to the sum of the means. So this is going to be equal to the mean of random variable X plus the mean of random variable Y. And so in this particular case, if I were to say, well what's the expected number of dogs and cats that I would see in a given day? Well, I would add these two means."}, {"video_title": "Mean of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, it turns out, and I'm not proving it just yet, that the mean of the sum of random variables is equal to the sum of the means. So this is going to be equal to the mean of random variable X plus the mean of random variable Y. And so in this particular case, if I were to say, well what's the expected number of dogs and cats that I would see in a given day? Well, I would add these two means. It would be three plus four, it would be equal to seven. So in this particular case, it is equal to three plus four, which is equal to seven. And similarly, if I were to ask you the difference, if I were to say, well what's the, how many more cats in a given day would I expect to see than dogs, so the expected value of Y minus X?"}, {"video_title": "Mean of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, I would add these two means. It would be three plus four, it would be equal to seven. So in this particular case, it is equal to three plus four, which is equal to seven. And similarly, if I were to ask you the difference, if I were to say, well what's the, how many more cats in a given day would I expect to see than dogs, so the expected value of Y minus X? What would that be? Well, intuitively, you might say, well, hey, if we can add random, if the expected value of the sum is the sum of the expected values, then the expected value or the mean of the difference will be the differences of the means, and that is absolutely true. So this is the same thing as the mean of Y minus X, which is equal to the mean of Y, is going to be equal to the mean of Y minus the mean of X minus the mean of X, and in this particular case, it would be equal to four minus three, minus three is equal to one."}, {"video_title": "Mean of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "And similarly, if I were to ask you the difference, if I were to say, well what's the, how many more cats in a given day would I expect to see than dogs, so the expected value of Y minus X? What would that be? Well, intuitively, you might say, well, hey, if we can add random, if the expected value of the sum is the sum of the expected values, then the expected value or the mean of the difference will be the differences of the means, and that is absolutely true. So this is the same thing as the mean of Y minus X, which is equal to the mean of Y, is going to be equal to the mean of Y minus the mean of X minus the mean of X, and in this particular case, it would be equal to four minus three, minus three is equal to one. So another way of thinking about this intuitively is I would expect to see on a given day one more cat than dogs. Now, the example that I've just used, this is discrete random variables, on a given day, I wouldn't see 2.2 dogs or pi dogs. The expected value itself does not have to be a whole number because you could, of course, average it over many days, but this same idea that the mean of a sum is the same thing as the sum of means and that the mean of a difference of random variables is the same as the difference of the means."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "And so what we're going to do is go look at the people who took the test. We're going to plot for each person the amount that they studied and their score. So for example, this data point is someone who studied an hour and they got a one on the test. And then we're going to fit a regression line. And this blue regression line is the actual regression line for these four data points. And here is the equation for that regression line. Now there's a couple of things to keep in mind."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "And then we're going to fit a regression line. And this blue regression line is the actual regression line for these four data points. And here is the equation for that regression line. Now there's a couple of things to keep in mind. Normally when you're doing this type of analysis, you would do it with far more than four data points. The reason why I kept this to four is because we're actually going to calculate how good a fit this regression line is by hand. And typically you would not do it by hand."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "Now there's a couple of things to keep in mind. Normally when you're doing this type of analysis, you would do it with far more than four data points. The reason why I kept this to four is because we're actually going to calculate how good a fit this regression line is by hand. And typically you would not do it by hand. We have computers for that. Now the way that we're going to measure how good a fit this regression line is to the data has several names. One name is the standard deviation of the residuals."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "And typically you would not do it by hand. We have computers for that. Now the way that we're going to measure how good a fit this regression line is to the data has several names. One name is the standard deviation of the residuals. Another name is the root mean square deviation, sometimes abbreviated RMSD. Sometimes it's called root mean square error. So what we're going to do is, is for every point, we're going to calculate the residual."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "One name is the standard deviation of the residuals. Another name is the root mean square deviation, sometimes abbreviated RMSD. Sometimes it's called root mean square error. So what we're going to do is, is for every point, we're going to calculate the residual. And then we're going to square it, and then we're going to add up the sum of those squared residuals. So we're going to take the sum of the residuals, residuals squared, and then we're going to divide that by the number of data points we have minus two. And we can talk in future videos or a more advanced statistics class of why you divide by two."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "So what we're going to do is, is for every point, we're going to calculate the residual. And then we're going to square it, and then we're going to add up the sum of those squared residuals. So we're going to take the sum of the residuals, residuals squared, and then we're going to divide that by the number of data points we have minus two. And we can talk in future videos or a more advanced statistics class of why you divide by two. But it's related to the idea that what we're calculating here is a statistic, and we're trying to estimate a true parameter as best as possible, and n minus two actually does the trick for us. But to calculate the root mean square deviation, we would then take a square root of this. And some of you might recognize strong parallels between this and how we calculated sample standard deviation early in our statistics career, and I encourage you to think about it."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "And we can talk in future videos or a more advanced statistics class of why you divide by two. But it's related to the idea that what we're calculating here is a statistic, and we're trying to estimate a true parameter as best as possible, and n minus two actually does the trick for us. But to calculate the root mean square deviation, we would then take a square root of this. And some of you might recognize strong parallels between this and how we calculated sample standard deviation early in our statistics career, and I encourage you to think about it. But let's actually calculate it by hand, as I mentioned earlier in this video, to see how things actually play out. So to do that, I'm going to give ourselves a little table here. So let's say that is our x value in that column."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "And some of you might recognize strong parallels between this and how we calculated sample standard deviation early in our statistics career, and I encourage you to think about it. But let's actually calculate it by hand, as I mentioned earlier in this video, to see how things actually play out. So to do that, I'm going to give ourselves a little table here. So let's say that is our x value in that column. Let's make this our y value. Let's make this y hat, which is going to be equal to 2.5x minus two. And then let's make this the residual squared, which is going to be our y value minus our y hat value, our actual minus our estimate for that given x squared."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "So let's say that is our x value in that column. Let's make this our y value. Let's make this y hat, which is going to be equal to 2.5x minus two. And then let's make this the residual squared, which is going to be our y value minus our y hat value, our actual minus our estimate for that given x squared. And then we're going to sum them all up, divide by n minus two, and take the square root. So first let's do this data point. So that's the point one comma one, one comma one."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "And then let's make this the residual squared, which is going to be our y value minus our y hat value, our actual minus our estimate for that given x squared. And then we're going to sum them all up, divide by n minus two, and take the square root. So first let's do this data point. So that's the point one comma one, one comma one. Now what is the estimate from our regression line? Well for that x value, when x is equal to one, it's gonna be 2.5 times one minus two. So it's gonna be 2.5 times one minus two, which is equal to 0.5."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "So that's the point one comma one, one comma one. Now what is the estimate from our regression line? Well for that x value, when x is equal to one, it's gonna be 2.5 times one minus two. So it's gonna be 2.5 times one minus two, which is equal to 0.5. And so our residual squared is going to be one minus 0.5, one minus 0.5 squared, which is equal to, that's gonna be 0.5 squared, which is going to be 0.25. All right, let's do the next data point. We have this one right over here."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "So it's gonna be 2.5 times one minus two, which is equal to 0.5. And so our residual squared is going to be one minus 0.5, one minus 0.5 squared, which is equal to, that's gonna be 0.5 squared, which is going to be 0.25. All right, let's do the next data point. We have this one right over here. It is two comma two. Now our estimate from the regression line when x equals two is going to be equal to 2.5 times our x value, times two, minus two, which is going to be equal to three. And so our residual squared is going to be two minus three, two minus three squared, which is negative one squared, which is going to be equal to one."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "We have this one right over here. It is two comma two. Now our estimate from the regression line when x equals two is going to be equal to 2.5 times our x value, times two, minus two, which is going to be equal to three. And so our residual squared is going to be two minus three, two minus three squared, which is negative one squared, which is going to be equal to one. Then we can go to this point. So that's the point two comma three, two comma three. Now our estimate from our regression line is going to be 2.5 times our x value, times two, minus two, which is going to be equal to three."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "And so our residual squared is going to be two minus three, two minus three squared, which is negative one squared, which is going to be equal to one. Then we can go to this point. So that's the point two comma three, two comma three. Now our estimate from our regression line is going to be 2.5 times our x value, times two, minus two, which is going to be equal to three. And so our residual here is going to be zero. And you can see that that point sits on the regression line. So it's going to be three minus three, three minus three squared, which is equal to zero."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "Now our estimate from our regression line is going to be 2.5 times our x value, times two, minus two, which is going to be equal to three. And so our residual here is going to be zero. And you can see that that point sits on the regression line. So it's going to be three minus three, three minus three squared, which is equal to zero. And then last but not least, we have this point right over here. When x is three, our y value, this person studied three hours and they got a six on the test. So y is equal to six."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "So it's going to be three minus three, three minus three squared, which is equal to zero. And then last but not least, we have this point right over here. When x is three, our y value, this person studied three hours and they got a six on the test. So y is equal to six. And so our estimate from the regression line, you could say what you would have expected to get based on that regression line, is 2.5 times our x value, times three, minus two is equal to 5.5. And so our residual squared is six minus 5.5 squared, minus 5.5 squared. So it's.5 squared, which is 0.25."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "So y is equal to six. And so our estimate from the regression line, you could say what you would have expected to get based on that regression line, is 2.5 times our x value, times three, minus two is equal to 5.5. And so our residual squared is six minus 5.5 squared, minus 5.5 squared. So it's.5 squared, which is 0.25. So now the next step, let me take the sum of all of these squared residuals. So this is, let me just write it this way. Actually, let me just do it like this."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "So it's.5 squared, which is 0.25. So now the next step, let me take the sum of all of these squared residuals. So this is, let me just write it this way. Actually, let me just do it like this. So the sum of the residuals, residuals squared, is equal to, if I just sum all of this up, it's going to be 1.5, 1.5. 1.5, 1.5. And then if I divide that by n minus two, so if I divide by n minus two, that's going to be equal to, I have four data points, so I'm gonna divide by four minus two, so I'm gonna divide by two."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "Actually, let me just do it like this. So the sum of the residuals, residuals squared, is equal to, if I just sum all of this up, it's going to be 1.5, 1.5. 1.5, 1.5. And then if I divide that by n minus two, so if I divide by n minus two, that's going to be equal to, I have four data points, so I'm gonna divide by four minus two, so I'm gonna divide by two. And then I'm gonna wanna take the square root of that. Then we'll take the square root of that. And so this is going to get us 1.5 over two is the same thing as 3.4."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "And then if I divide that by n minus two, so if I divide by n minus two, that's going to be equal to, I have four data points, so I'm gonna divide by four minus two, so I'm gonna divide by two. And then I'm gonna wanna take the square root of that. Then we'll take the square root of that. And so this is going to get us 1.5 over two is the same thing as 3.4. So it's the square root of 3.4, or the square root of three over two. And you could use a calculator to figure what that is, to figure out what that is as a decimal. But this gives us a sense of how good a fit this regression line is."}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "And so this is going to get us 1.5 over two is the same thing as 3.4. So it's the square root of 3.4, or the square root of three over two. And you could use a calculator to figure what that is, to figure out what that is as a decimal. But this gives us a sense of how good a fit this regression line is. The closer this is to zero, the better the fit of the regression line. The further away from zero, the worse fit. And what would be the units for the root mean square deviation?"}, {"video_title": "Standard deviation of residuals or root mean square deviation (RMSD) AP Statistics Khan Academy.mp3", "Sentence": "But this gives us a sense of how good a fit this regression line is. The closer this is to zero, the better the fit of the regression line. The further away from zero, the worse fit. And what would be the units for the root mean square deviation? Well, it would be in terms of whatever your units are for your y-axis. In this case, it would be the score on the test. And that's one of the other values of this calculation, of taking the square root of the sum of the squares of the residuals, dividing by n minus two."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "Consider the following story. Bob is in a room and he has two coins. One fair coin and one double-sided coin. He picks one at random, flips it, and shouts the result. Heads! Now what is the probability that he flipped the fair coin? To answer this question, we need only rewind and grow a tree."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "He picks one at random, flips it, and shouts the result. Heads! Now what is the probability that he flipped the fair coin? To answer this question, we need only rewind and grow a tree. The first event, he picks one of two coins. So our tree grows two branches, leading to two equally likely outcomes, fair or unfair. The next event, he flips the coin."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "To answer this question, we need only rewind and grow a tree. The first event, he picks one of two coins. So our tree grows two branches, leading to two equally likely outcomes, fair or unfair. The next event, he flips the coin. We grow again. If he had the fair coin, we know this flip can result in two equally likely outcomes, heads and tails. While the unfair coin results in two outcomes, both heads."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "The next event, he flips the coin. We grow again. If he had the fair coin, we know this flip can result in two equally likely outcomes, heads and tails. While the unfair coin results in two outcomes, both heads. Our tree is finished and we see it has four leaves, representing four equally likely outcomes. The final step, new evidence. He says, heads!"}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "While the unfair coin results in two outcomes, both heads. Our tree is finished and we see it has four leaves, representing four equally likely outcomes. The final step, new evidence. He says, heads! Whenever we gain evidence, we must trim our tree. We cut any branch leading to tails because we know tails did not occur. And that is it."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "He says, heads! Whenever we gain evidence, we must trim our tree. We cut any branch leading to tails because we know tails did not occur. And that is it. So the probability that he chose the fair coin is the one fair outcome leading to heads, divided by the three possible outcomes leading to heads, or one third. What happens if he flips again and reports, heads! Remember, after each event, our tree grows."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "And that is it. So the probability that he chose the fair coin is the one fair outcome leading to heads, divided by the three possible outcomes leading to heads, or one third. What happens if he flips again and reports, heads! Remember, after each event, our tree grows. The fair coin leaves result in two equally likely outcomes, heads and tails. The unfair coin leaves result in two equally likely outcomes, heads and heads. After we hear the second, heads!"}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "Remember, after each event, our tree grows. The fair coin leaves result in two equally likely outcomes, heads and tails. The unfair coin leaves result in two equally likely outcomes, heads and heads. After we hear the second, heads! We cut any branches leading to tails. Therefore, the probability the coin is fair after two heads in a row is the one fair outcome leading to heads, divided by all possible outcome leading to heads, or one fifth. Notice our confidence in the fair coin is dropping as more heads occur, though realize it will never reach zero."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "After we hear the second, heads! We cut any branches leading to tails. Therefore, the probability the coin is fair after two heads in a row is the one fair outcome leading to heads, divided by all possible outcome leading to heads, or one fifth. Notice our confidence in the fair coin is dropping as more heads occur, though realize it will never reach zero. No matter how many flips occur, we can never be 100% certain the coin is unfair. In fact, all conditional probability questions can be solved by growing trees. Let's do one more to be sure."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "Notice our confidence in the fair coin is dropping as more heads occur, though realize it will never reach zero. No matter how many flips occur, we can never be 100% certain the coin is unfair. In fact, all conditional probability questions can be solved by growing trees. Let's do one more to be sure. Bob has three coins. Two are fair. One is biased, which is weighted to land heads two thirds of the time and tails one third."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "Let's do one more to be sure. Bob has three coins. Two are fair. One is biased, which is weighted to land heads two thirds of the time and tails one third. He chooses a coin at random and flips it. Heads! Now, what is the probability he chose the biased coin?"}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "One is biased, which is weighted to land heads two thirds of the time and tails one third. He chooses a coin at random and flips it. Heads! Now, what is the probability he chose the biased coin? Let's rewind and build a tree. The first event, choosing the coin, can lead to three equally likely outcomes, fair coin, fair coin, and unfair coin. The next event, the coin is flipped."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "Now, what is the probability he chose the biased coin? Let's rewind and build a tree. The first event, choosing the coin, can lead to three equally likely outcomes, fair coin, fair coin, and unfair coin. The next event, the coin is flipped. Each fair coin leads to two equally likely leaves, heads and tails. The biased coin leads to three equally likely leaves, two representing heads and one representing tails. Now, the trick is to always make sure our tree is balanced, meaning an equal amount of leaves growing out of each branch."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "The next event, the coin is flipped. Each fair coin leads to two equally likely leaves, heads and tails. The biased coin leads to three equally likely leaves, two representing heads and one representing tails. Now, the trick is to always make sure our tree is balanced, meaning an equal amount of leaves growing out of each branch. To do this, we simply scale up the number of branches to the least common multiple. For two and three, this is six. And finally, we label our leaves."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "Now, the trick is to always make sure our tree is balanced, meaning an equal amount of leaves growing out of each branch. To do this, we simply scale up the number of branches to the least common multiple. For two and three, this is six. And finally, we label our leaves. The fair coin now splits into six equally likely leaves, three heads and three tails. For the biased coin, we now have two tail leaves and four head leaves, and that is it. When Bob shows the result, heads!"}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "And finally, we label our leaves. The fair coin now splits into six equally likely leaves, three heads and three tails. For the biased coin, we now have two tail leaves and four head leaves, and that is it. When Bob shows the result, heads! this new evidence allows us to trim all branches, leading to tails, since tails did not occur. So, the probability that he chose the biased coin, given heads occur? Well, four leaves can come from the biased coin, divided by all possible leaves."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "When Bob shows the result, heads! this new evidence allows us to trim all branches, leading to tails, since tails did not occur. So, the probability that he chose the biased coin, given heads occur? Well, four leaves can come from the biased coin, divided by all possible leaves. Four divided by ten, or 40%. When in doubt, it's always possible to answer conditional probability questions by Bayes' theorem. It tells us the probability of event A, given some new evidence B."}, {"video_title": "Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3", "Sentence": "Faye read an article that said 26% of Americans can speak more than one language. She was curious if this figure was higher in her city, so she tested. Her null hypothesis is that the proportion in her city is the same as all Americans, 26%. Her alternative hypothesis is it's actually greater than 26%, where P represents the proportion of people in her city that can speak more than one language. She found that 40 of 120 people sampled could speak more than one language. So what's going on is, here's the population of her city. She took a sample."}, {"video_title": "Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3", "Sentence": "Her alternative hypothesis is it's actually greater than 26%, where P represents the proportion of people in her city that can speak more than one language. She found that 40 of 120 people sampled could speak more than one language. So what's going on is, here's the population of her city. She took a sample. Her sample size is 120. And then she calculates her sample proportion, which is 40 out of 120. And this is going to be equal to 1 3rd, which is approximately equal to 0.33."}, {"video_title": "Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3", "Sentence": "She took a sample. Her sample size is 120. And then she calculates her sample proportion, which is 40 out of 120. And this is going to be equal to 1 3rd, which is approximately equal to 0.33. And then she calculates the test statistic for these results was z is approximately equal to 1.83. We do this in other videos, but just as a reminder of how she gets this, she's really trying to say, well, how many standard deviations above the assumed proportion, remember, when we're doing the significance test, we're assuming that the null hypothesis is true, and then we figure out, well, what's the probability of getting something at least this extreme or more? And then if it's below a threshold, then we would reject the null hypothesis, which would suggest the alternative."}, {"video_title": "Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3", "Sentence": "And this is going to be equal to 1 3rd, which is approximately equal to 0.33. And then she calculates the test statistic for these results was z is approximately equal to 1.83. We do this in other videos, but just as a reminder of how she gets this, she's really trying to say, well, how many standard deviations above the assumed proportion, remember, when we're doing the significance test, we're assuming that the null hypothesis is true, and then we figure out, well, what's the probability of getting something at least this extreme or more? And then if it's below a threshold, then we would reject the null hypothesis, which would suggest the alternative. But that's what this z statistic is, is, well, how many standard deviations above the assumed proportion is that? So the z statistic, and we did this in previous videos, you would find the difference between this, what we got for our sample, our sample proportion, and the assumed true proportion, so 0.33 minus 0.26, all of that over the standard deviation of the sampling distribution of the sample proportions. And we've seen that in previous videos."}, {"video_title": "Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3", "Sentence": "And then if it's below a threshold, then we would reject the null hypothesis, which would suggest the alternative. But that's what this z statistic is, is, well, how many standard deviations above the assumed proportion is that? So the z statistic, and we did this in previous videos, you would find the difference between this, what we got for our sample, our sample proportion, and the assumed true proportion, so 0.33 minus 0.26, all of that over the standard deviation of the sampling distribution of the sample proportions. And we've seen that in previous videos. That is just going to be the assumed proportion, so it would be just this. It'd be the assumed population proportion times one minus the assumed population proportion over n. In this particular situation, that would be 0.26 times one minus 0.26, all of that over our n, that's our sample size, 120. And if you calculate this, this should give us approximately 1.83."}, {"video_title": "Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3", "Sentence": "And we've seen that in previous videos. That is just going to be the assumed proportion, so it would be just this. It'd be the assumed population proportion times one minus the assumed population proportion over n. In this particular situation, that would be 0.26 times one minus 0.26, all of that over our n, that's our sample size, 120. And if you calculate this, this should give us approximately 1.83. So they did all of that for us. And they say, assuming that the necessary conditions are met, they're talking about the necessary conditions to assume that the sampling distribution of the sample proportions is roughly normal, and that's the random condition, the normal condition, the independence condition that we have talked about in the past. What is the approximate p-value?"}, {"video_title": "Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3", "Sentence": "And if you calculate this, this should give us approximately 1.83. So they did all of that for us. And they say, assuming that the necessary conditions are met, they're talking about the necessary conditions to assume that the sampling distribution of the sample proportions is roughly normal, and that's the random condition, the normal condition, the independence condition that we have talked about in the past. What is the approximate p-value? Well, this p-value, this is the p-value would be equal to the probability of in a normal distribution, we're assuming that the sampling distribution is normal because we met the necessary conditions. So in a normal distribution, what is the probability of getting a z greater than or equal to 1.83? So to help us visualize this, imagine, let's visualize what the sampling distribution would look like, we're assuming it is roughly normal."}, {"video_title": "Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3", "Sentence": "What is the approximate p-value? Well, this p-value, this is the p-value would be equal to the probability of in a normal distribution, we're assuming that the sampling distribution is normal because we met the necessary conditions. So in a normal distribution, what is the probability of getting a z greater than or equal to 1.83? So to help us visualize this, imagine, let's visualize what the sampling distribution would look like, we're assuming it is roughly normal. The mean of the sampling distribution right over here would be the assumed population proportion. So that would be p-naught, when we put that little zero there, that means the assumed population proportion from the null hypothesis, and that's 0.26. And this result that we got from our sample is 1.83 standard deviations above the mean of the sampling distribution, so 1.83."}, {"video_title": "Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3", "Sentence": "So to help us visualize this, imagine, let's visualize what the sampling distribution would look like, we're assuming it is roughly normal. The mean of the sampling distribution right over here would be the assumed population proportion. So that would be p-naught, when we put that little zero there, that means the assumed population proportion from the null hypothesis, and that's 0.26. And this result that we got from our sample is 1.83 standard deviations above the mean of the sampling distribution, so 1.83. So that would be 1.83 standard deviations. And so what we wanna do, this probability is this area under our normal curve right over here. So now let's get our z table."}, {"video_title": "Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3", "Sentence": "And this result that we got from our sample is 1.83 standard deviations above the mean of the sampling distribution, so 1.83. So that would be 1.83 standard deviations. And so what we wanna do, this probability is this area under our normal curve right over here. So now let's get our z table. So notice, this z table gives us the area to the left of a certain z value. We wanted it to the right of a certain z value, but a normal distribution is symmetric, so instead of saying anything greater than or equal to 1.83 standard deviations above the mean, we could say anything less than or equal to 1.83 standard deviations below the mean. So this is negative 1.83."}, {"video_title": "Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3", "Sentence": "So now let's get our z table. So notice, this z table gives us the area to the left of a certain z value. We wanted it to the right of a certain z value, but a normal distribution is symmetric, so instead of saying anything greater than or equal to 1.83 standard deviations above the mean, we could say anything less than or equal to 1.83 standard deviations below the mean. So this is negative 1.83. And so we could look at that on this z table right over here. Negative one, let me, negative 1.8, negative 1.83 is this right over here. So 0.0336."}, {"video_title": "Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3", "Sentence": "So this is negative 1.83. And so we could look at that on this z table right over here. Negative one, let me, negative 1.8, negative 1.83 is this right over here. So 0.0336. So there we have it. So this is approximately 0.0336, 0.0336, or a little over 3% or a little less than 4%. And so what Faye would then do is compare that to the significance level that she should have set before conducting this significance test."}, {"video_title": "Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3", "Sentence": "So 0.0336. So there we have it. So this is approximately 0.0336, 0.0336, or a little over 3% or a little less than 4%. And so what Faye would then do is compare that to the significance level that she should have set before conducting this significance test. And so if her significance level was, say, 5%, well then in that situation, since this is lower than that significance level, she would be able to reject the null hypothesis. She would say, hey, the probability of getting this result, assuming that the null hypothesis is true, is below my threshold, it's quite low, and so I will reject it, and it would suggest the alternative. However, if her significance level was lower than this, for whatever reason, if she had, say, a 1% significance level, then she would fail to reject the null hypothesis."}, {"video_title": "How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3", "Sentence": "But then when you go back to your log, you notice that some blue ink spilled over one of the ages and you forgot how old that child is. And at first you're really worried, your whole system of keeping records seems to, you know, you've lost information. But then you remember that every time you wrote down a new age that month, you recalculated the mean. And so you have the mean here of being four, the mean age is four for the six children. So given that, given that you know the mean and that you know five out of six of the ages, can you figure out what the sixth age is? And I encourage you to pause the video and try to figure it out on your own. So assuming you've had a shot at it, so let's just call this missing age, let's call that question mark."}, {"video_title": "How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And so you have the mean here of being four, the mean age is four for the six children. So given that, given that you know the mean and that you know five out of six of the ages, can you figure out what the sixth age is? And I encourage you to pause the video and try to figure it out on your own. So assuming you've had a shot at it, so let's just call this missing age, let's call that question mark. So let's just think about how do we calculate, how would we calculate a mean if we knew what question mark is? Well, we would take the total, we would take the total of ages, of ages, we would then divide that by the number of children, we would then divide that by the number of ages that we had, and then that would be equal to, that would be equal to the mean. Or another way to think about it, if you multiply both sides times the number of ages, the number of ages on that side and the number of, of ages on that side, then this is going to cancel with that, and we're going to be left with the total, the total is going to be equal to, is going to be equal to the mean times the number of ages."}, {"video_title": "How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So assuming you've had a shot at it, so let's just call this missing age, let's call that question mark. So let's just think about how do we calculate, how would we calculate a mean if we knew what question mark is? Well, we would take the total, we would take the total of ages, of ages, we would then divide that by the number of children, we would then divide that by the number of ages that we had, and then that would be equal to, that would be equal to the mean. Or another way to think about it, if you multiply both sides times the number of ages, the number of ages on that side and the number of, of ages on that side, then this is going to cancel with that, and we're going to be left with the total, the total is going to be equal to, is going to be equal to the mean times the number of ages. Mean times, and I'll just write times the number, times number of data points, or number of ages. So maybe we can use this information, because we're just going to have this missing question mark here, and we know the mean and we know the number of ages, so we just have to solve for the question mark, so let's do that. So let's go back to the beginning here, just so that this makes sense with some numbers."}, {"video_title": "How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Or another way to think about it, if you multiply both sides times the number of ages, the number of ages on that side and the number of, of ages on that side, then this is going to cancel with that, and we're going to be left with the total, the total is going to be equal to, is going to be equal to the mean times the number of ages. Mean times, and I'll just write times the number, times number of data points, or number of ages. So maybe we can use this information, because we're just going to have this missing question mark here, and we know the mean and we know the number of ages, so we just have to solve for the question mark, so let's do that. So let's go back to the beginning here, just so that this makes sense with some numbers. The total of ages, that's going to be five plus two plus question mark, plus question mark, plus two, this two, plus two, plus four, plus eight. We're going to divide by the number of ages. We're going to divide it by the number of ages."}, {"video_title": "How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So let's go back to the beginning here, just so that this makes sense with some numbers. The total of ages, that's going to be five plus two plus question mark, plus question mark, plus two, this two, plus two, plus four, plus eight. We're going to divide by the number of ages. We're going to divide it by the number of ages. Well, we have six ages here. One, two, three, four, five, six. Six ages, and that's going to be equal to the mean."}, {"video_title": "How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We're going to divide it by the number of ages. Well, we have six ages here. One, two, three, four, five, six. Six ages, and that's going to be equal to the mean. This is going to be equal to the mean. The mean here is four. This is just how you calculate the mean."}, {"video_title": "How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Six ages, and that's going to be equal to the mean. This is going to be equal to the mean. The mean here is four. This is just how you calculate the mean. Let's see if we can simplify this. Five plus two is seven. Let me do this, that's the wrong color."}, {"video_title": "How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3", "Sentence": "This is just how you calculate the mean. Let's see if we can simplify this. Five plus two is seven. Let me do this, that's the wrong color. Five plus two is seven. Two plus four is six, plus eight is 14. 14, and then seven plus 14 is 21."}, {"video_title": "How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let me do this, that's the wrong color. Five plus two is seven. Two plus four is six, plus eight is 14. 14, and then seven plus 14 is 21. We're left with 21 plus question mark over six is equal to four. Now we can do what we did when we just wrote it all out. We can multiply both sides times the number of ages, the number of data points we have."}, {"video_title": "How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3", "Sentence": "14, and then seven plus 14 is 21. We're left with 21 plus question mark over six is equal to four. Now we can do what we did when we just wrote it all out. We can multiply both sides times the number of ages, the number of data points we have. We can multiply both sides times six. We can multiply both sides, both sides times six. Six on that side, six on this side."}, {"video_title": "How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We can multiply both sides times the number of ages, the number of data points we have. We can multiply both sides times six. We can multiply both sides, both sides times six. Six on that side, six on this side. Six in the numerator, six in the denominator, those cancel. All we're left is, on the left-hand side, we're left with 21 plus question mark. All of these other green numbers, those are simplified, five plus two plus two plus four plus eight is 21, and we still have the question mark."}, {"video_title": "How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Six on that side, six on this side. Six in the numerator, six in the denominator, those cancel. All we're left is, on the left-hand side, we're left with 21 plus question mark. All of these other green numbers, those are simplified, five plus two plus two plus four plus eight is 21, and we still have the question mark. We get 21 plus question mark. I'm going to do that green color. 21 plus this question mark."}, {"video_title": "How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3", "Sentence": "All of these other green numbers, those are simplified, five plus two plus two plus four plus eight is 21, and we still have the question mark. We get 21 plus question mark. I'm going to do that green color. 21 plus this question mark. The thing that we're trying to solve for, the missing number, is going to be equal to, is going to be equal to four times six. What's four times six? That's 24."}, {"video_title": "How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3", "Sentence": "21 plus this question mark. The thing that we're trying to solve for, the missing number, is going to be equal to, is going to be equal to four times six. What's four times six? That's 24. What's the question mark? 21 plus what is equal to 24? We can, of course, you might just, well, it's going to be three, or if you want to, you could say, well, question mark is going to be, question mark is going to be equal to, is going to be equal to 24 minus 21, which is, of course, three."}, {"video_title": "How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3", "Sentence": "That's 24. What's the question mark? 21 plus what is equal to 24? We can, of course, you might just, well, it's going to be three, or if you want to, you could say, well, question mark is going to be, question mark is going to be equal to, is going to be equal to 24 minus 21, which is, of course, three. Which, of course, let me just write this down. The question mark is equal to three. The missing age, you were able to figure it out based on the information you had, because you had the mean, you were able to figure out that behind the splotch, that behind the splotch, you had a three."}, {"video_title": "Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "It's a discrete random variable. It can only take on a finite number of values, and I defined it as the number of workouts I might do in a week. And we calculated the expected value of our random variable X, which you could also denote as the mean of X, and we used the Greek letter mu, which we use for population mean. And all we did is it's the probability weighted sum of the various outcomes. And we got for this random variable with this probability distribution, we got an expected value or a mean of 2.1. What we're going to do now is extend this idea to measuring spread. And so we're going to think about what is the variance of this random variable, and then we could take the square root of that to find out what is the standard deviation."}, {"video_title": "Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "And all we did is it's the probability weighted sum of the various outcomes. And we got for this random variable with this probability distribution, we got an expected value or a mean of 2.1. What we're going to do now is extend this idea to measuring spread. And so we're going to think about what is the variance of this random variable, and then we could take the square root of that to find out what is the standard deviation. The way we are going to do this has parallels with the way that we've calculated variance in the past. So the variance of our random variable X, what we're going to do is take the difference between each outcome and the mean, square that difference, and then we're going to multiply it by the probability of that outcome. So for example, for this first data point, you're going to have zero minus 2.1 squared times the probability of getting zero times 0.1."}, {"video_title": "Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "And so we're going to think about what is the variance of this random variable, and then we could take the square root of that to find out what is the standard deviation. The way we are going to do this has parallels with the way that we've calculated variance in the past. So the variance of our random variable X, what we're going to do is take the difference between each outcome and the mean, square that difference, and then we're going to multiply it by the probability of that outcome. So for example, for this first data point, you're going to have zero minus 2.1 squared times the probability of getting zero times 0.1. Then you're going to get plus one minus 2.1 squared times the probability that you get one, times 0.15. Then you're going to get plus two minus 2.1 squared times the probability that you get a two, times 0.4. Then you have plus three minus 2.1 squared times 0.25, and then last but not least, you have plus four minus 2.1 squared times 0.1."}, {"video_title": "Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "So for example, for this first data point, you're going to have zero minus 2.1 squared times the probability of getting zero times 0.1. Then you're going to get plus one minus 2.1 squared times the probability that you get one, times 0.15. Then you're going to get plus two minus 2.1 squared times the probability that you get a two, times 0.4. Then you have plus three minus 2.1 squared times 0.25, and then last but not least, you have plus four minus 2.1 squared times 0.1. So once again, the difference between each outcome and the mean, we square it and we multiply times the probability of that outcome. So this is going to be negative 2.1 squared, which is just 2.1 squared, so I'll just write this as 2.1 squared times 0.1. That's the first term."}, {"video_title": "Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "Then you have plus three minus 2.1 squared times 0.25, and then last but not least, you have plus four minus 2.1 squared times 0.1. So once again, the difference between each outcome and the mean, we square it and we multiply times the probability of that outcome. So this is going to be negative 2.1 squared, which is just 2.1 squared, so I'll just write this as 2.1 squared times 0.1. That's the first term. And then we're going to have plus, one minus 2.1 is negative 1.1, and then we're going to square that. So that's just going to be the same thing as 1.1 squared, which is 1.21, but I'll just write it out. 1.1 squared times 0.15, and then this is going to be two minus 2.1 is negative 0.1 when you square it is going to be equal to, so plus 0.01."}, {"video_title": "Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "That's the first term. And then we're going to have plus, one minus 2.1 is negative 1.1, and then we're going to square that. So that's just going to be the same thing as 1.1 squared, which is 1.21, but I'll just write it out. 1.1 squared times 0.15, and then this is going to be two minus 2.1 is negative 0.1 when you square it is going to be equal to, so plus 0.01. If you have negative 0.1 times negative 0.1, that's 0.01 times 0.4 times 0.4, and then plus, this is going to be 0.9 squared, so that is 0.81 times 0.25, and then we're almost there. This is going to be plus 1.9 squared, 1.9 squared times 0.1, and we get 1.19. So this is all going to be equal to 1.19, and if we want to get the standard deviation for this random variable, and we would denote that with the Greek letter sigma, the standard deviation for the random variable X is going to be equal to the square root of the variance."}, {"video_title": "Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "1.1 squared times 0.15, and then this is going to be two minus 2.1 is negative 0.1 when you square it is going to be equal to, so plus 0.01. If you have negative 0.1 times negative 0.1, that's 0.01 times 0.4 times 0.4, and then plus, this is going to be 0.9 squared, so that is 0.81 times 0.25, and then we're almost there. This is going to be plus 1.9 squared, 1.9 squared times 0.1, and we get 1.19. So this is all going to be equal to 1.19, and if we want to get the standard deviation for this random variable, and we would denote that with the Greek letter sigma, the standard deviation for the random variable X is going to be equal to the square root of the variance. The square root of 1.19, which is equal to, let's just get the calculator back here. So we are just going to take the square root of, I'll just type it again, 1.19, and that gives us, so it's approximately 1.09. Approximately 1.09."}, {"video_title": "Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "So this is all going to be equal to 1.19, and if we want to get the standard deviation for this random variable, and we would denote that with the Greek letter sigma, the standard deviation for the random variable X is going to be equal to the square root of the variance. The square root of 1.19, which is equal to, let's just get the calculator back here. So we are just going to take the square root of, I'll just type it again, 1.19, and that gives us, so it's approximately 1.09. Approximately 1.09. So let's see if this makes sense. Let me put this all on a number line right over here. So you have the outcome zero, one, two, three, and four."}, {"video_title": "Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "Approximately 1.09. So let's see if this makes sense. Let me put this all on a number line right over here. So you have the outcome zero, one, two, three, and four. So you have a 10% chance of getting a zero. So I will draw that like this. Let's just say this is a height of 10%."}, {"video_title": "Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "So you have the outcome zero, one, two, three, and four. So you have a 10% chance of getting a zero. So I will draw that like this. Let's just say this is a height of 10%. You have a 15% chance of getting a one, so that'll be 1.5 times higher. So it'll look something like this. You have a 40% chance of getting a two, so that's going to be like this."}, {"video_title": "Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "Let's just say this is a height of 10%. You have a 15% chance of getting a one, so that'll be 1.5 times higher. So it'll look something like this. You have a 40% chance of getting a two, so that's going to be like this. So you have a 40% chance of getting a two. You have a 25% chance of getting a three, so it'll look like this. And then you have a 10% chance of getting a four."}, {"video_title": "Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "You have a 40% chance of getting a two, so that's going to be like this. So you have a 40% chance of getting a two. You have a 25% chance of getting a three, so it'll look like this. And then you have a 10% chance of getting a four. So it'll look like that. So this is a visualization of this discrete probability distribution where I didn't draw a vertical axis here, but this would be.1, this would be.15, this is.25, and that is.4. And then we see that the mean is at 2.1."}, {"video_title": "Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "And then you have a 10% chance of getting a four. So it'll look like that. So this is a visualization of this discrete probability distribution where I didn't draw a vertical axis here, but this would be.1, this would be.15, this is.25, and that is.4. And then we see that the mean is at 2.1. The mean is at 2.1, which makes sense. Even though this random variable only takes on integer values, you can have a mean that takes on a non-integer value. And then the standard deviation is 1.09."}, {"video_title": "Variance and standard deviation of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "And then we see that the mean is at 2.1. The mean is at 2.1, which makes sense. Even though this random variable only takes on integer values, you can have a mean that takes on a non-integer value. And then the standard deviation is 1.09. So 1.09 above the mean is going to get us close to 3.2. And 1.09 below the mean is gonna get us close to one. And so this all, at least intuitively, feels reasonable."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "In this video, we're going to talk about random sampling, which we've already talked about in other videos. And we're going to compare what we already know about simple random sampling to a new type of random sampling that we're going to introduce in this video, and that is systematic random sampling. So let's look at an example. Let's say that there is a concert that is happening, and we expect approximately 10,000 people to attend the concert. And we want to randomly sample people at the concert. Maybe we want to do a study on how do people get to the concert? How do people get to the concert?"}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "Let's say that there is a concert that is happening, and we expect approximately 10,000 people to attend the concert. And we want to randomly sample people at the concert. Maybe we want to do a study on how do people get to the concert? How do people get to the concert? Do they drive and park? Do they ride with a friend? Do they take an Uber or a cab of some kind?"}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "How do people get to the concert? Do they drive and park? Do they ride with a friend? Do they take an Uber or a cab of some kind? And so we want to find a random sample, ideally without bias, to survey people. So there's a couple of ways you could do it. You could try to do a simple random sample."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "Do they take an Uber or a cab of some kind? And so we want to find a random sample, ideally without bias, to survey people. So there's a couple of ways you could do it. You could try to do a simple random sample. And that might be a case of if you could somehow get the names of all 10,000 people and put them into a big bowl like this, and then let's say you want to sample 100 people. Let's say you want to sample approximately 100 people. You could just mix up all the names that may be on these little pieces of paper, 10,000 of them, and then pull them out and pull out a random sample of 100 of them."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "You could try to do a simple random sample. And that might be a case of if you could somehow get the names of all 10,000 people and put them into a big bowl like this, and then let's say you want to sample 100 people. Let's say you want to sample approximately 100 people. You could just mix up all the names that may be on these little pieces of paper, 10,000 of them, and then pull them out and pull out a random sample of 100 of them. That would be a simple random sample. But you could already imagine there might be some logistic difficulties of doing this. How are you going to get the 10,000 names?"}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "You could just mix up all the names that may be on these little pieces of paper, 10,000 of them, and then pull them out and pull out a random sample of 100 of them. That would be a simple random sample. But you could already imagine there might be some logistic difficulties of doing this. How are you going to get the 10,000 names? You're gonna write them on a piece of paper. That's gonna be a, you'd have to really mix some goods so it's truly random who you're picking out. So are there other ways of doing a random sample?"}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "How are you going to get the 10,000 names? You're gonna write them on a piece of paper. That's gonna be a, you'd have to really mix some goods so it's truly random who you're picking out. So are there other ways of doing a random sample? And as you can imagine, yes, there are. And that's where systematic random sampling is useful. One way to think about systematic random sampling is you're going to randomly sample a subset of the people who are maybe walking into the concert."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "So are there other ways of doing a random sample? And as you can imagine, yes, there are. And that's where systematic random sampling is useful. One way to think about systematic random sampling is you're going to randomly sample a subset of the people who are maybe walking into the concert. So let's say people get to the concert and they start forming a line to get into the concert. What you wanna do in systematic random sampling is randomly pick your first person. There's a bunch of ways that you could do that."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "One way to think about systematic random sampling is you're going to randomly sample a subset of the people who are maybe walking into the concert. So let's say people get to the concert and they start forming a line to get into the concert. What you wanna do in systematic random sampling is randomly pick your first person. There's a bunch of ways that you could do that. Let's say you have a random number generator that'll generate a number from one to 100, and that's going to be the first person you survey if that random number generator generates a 37, then you're going to start with the 37th person in line. So you pick that first person randomly, you survey them. And remember, our goal is to sample about 100 people out of 10,000."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "There's a bunch of ways that you could do that. Let's say you have a random number generator that'll generate a number from one to 100, and that's going to be the first person you survey if that random number generator generates a 37, then you're going to start with the 37th person in line. So you pick that first person randomly, you survey them. And remember, our goal is to sample about 100 people out of 10,000. So we wanna roughly sample one out of every 100 people. And so what you do there is once you have that first person that you're sampling, you then sample every 100th person after that. That's called sometimes the sample interval."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "And remember, our goal is to sample about 100 people out of 10,000. So we wanna roughly sample one out of every 100 people. And so what you do there is once you have that first person that you're sampling, you then sample every 100th person after that. That's called sometimes the sample interval. And the reason why 100 people is because if you sample every 100th person after that, you're going to roughly get 100 people in your sample out of a total of 10,000. So this is going to be after 100, you're going to sample someone else. And then after another 100, you're going to sample someone else."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "That's called sometimes the sample interval. And the reason why 100 people is because if you sample every 100th person after that, you're going to roughly get 100 people in your sample out of a total of 10,000. So this is going to be after 100, you're going to sample someone else. And then after another 100, you're going to sample someone else. Now, the reason why this is useful is you could say, okay, that first person was random, and then every person after that. It doesn't seem like there'd be any bias for why they would be the 100th person after that first person. You don't wanna just do the first 100 people because those might be the early birds, the people who maybe disproportionately went parking or planned early or had some bias in some way."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "And then after another 100, you're going to sample someone else. Now, the reason why this is useful is you could say, okay, that first person was random, and then every person after that. It doesn't seem like there'd be any bias for why they would be the 100th person after that first person. You don't wanna just do the first 100 people because those might be the early birds, the people who maybe disproportionately went parking or planned early or had some bias in some way. So you do wanna make sure that you're getting both the beginning, the middle, and the end of the line, which this thing helps. Now, we have to be careful. Even systematic random sampling is not foolproof."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "You don't wanna just do the first 100 people because those might be the early birds, the people who maybe disproportionately went parking or planned early or had some bias in some way. So you do wanna make sure that you're getting both the beginning, the middle, and the end of the line, which this thing helps. Now, we have to be careful. Even systematic random sampling is not foolproof. There's a situation where inadvertently, even this system has bias. Let's say that this is the arena. This is a top view of the arena right over here."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "Even systematic random sampling is not foolproof. There's a situation where inadvertently, even this system has bias. Let's say that this is the arena. This is a top view of the arena right over here. And this is the line of people coming in. And this is where you are standing, and you are counting every 100th person. But maybe, and let's say there's a tree right over here, and maybe there's a road."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "This is a top view of the arena right over here. And this is the line of people coming in. And this is where you are standing, and you are counting every 100th person. But maybe, and let's say there's a tree right over here, and maybe there's a road. I'm making this quite elaborate. So maybe there is a road right over here. And a lot of people, maybe all of the people who are walking or taking a cab are coming from this direction."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "But maybe, and let's say there's a tree right over here, and maybe there's a road. I'm making this quite elaborate. So maybe there is a road right over here. And a lot of people, maybe all of the people who are walking or taking a cab are coming from this direction. And maybe all of the people from the parking lot are coming from this direction. And maybe you have a police officer right over here who is doing crowd control, who lets 50 of these people in, followed by 50 of these people in. Well, in that situation, every 100th, you might end up just sampling one side or the other."}, {"video_title": "Comparing P-values to different significance levels AP Statistics Khan Academy.mp3", "Sentence": "And we're gonna talk about two things. The different conclusions you might make based on the different significance levels that you might set, and also why it's important to set your significance levels ahead of time, before you conduct an experiment and calculate the p-values, for frankly, ethical purposes. So to help us get this, let's look at a scenario right over here, which tells us, Rahim heard that spinning, rather than flipping a penny, raises the probability above 50% that the penny lands showing heads. That's actually quite fascinating, if that's true. He tested this by spinning 10 different pennies, 10 times each, so that would be a total of 100 spins. His hypotheses were, his null hypothesis is that, by spinning, your proportion doesn't change, rather, versus flipping, it's still 50%. And his alternative hypothesis is that, by spinning, your proportion of heads is greater than 50%, where p is the true proportion of spins that a penny would land showing heads."}, {"video_title": "Comparing P-values to different significance levels AP Statistics Khan Academy.mp3", "Sentence": "That's actually quite fascinating, if that's true. He tested this by spinning 10 different pennies, 10 times each, so that would be a total of 100 spins. His hypotheses were, his null hypothesis is that, by spinning, your proportion doesn't change, rather, versus flipping, it's still 50%. And his alternative hypothesis is that, by spinning, your proportion of heads is greater than 50%, where p is the true proportion of spins that a penny would land showing heads. In his 100 spins, the penny landed showing heads in 59 spins. Rahim calculated that the statistic, so this is the sample proportion here, it's 59 out of 100 were heads, so that's 0.59, or 59 hundredths, and he calculated, had an associated p-value of approximately 0.036. So, based on this scenario, if ahead of time, Rahim had set his significance level at 0.05, what conclusions would he now make?"}, {"video_title": "Comparing P-values to different significance levels AP Statistics Khan Academy.mp3", "Sentence": "And his alternative hypothesis is that, by spinning, your proportion of heads is greater than 50%, where p is the true proportion of spins that a penny would land showing heads. In his 100 spins, the penny landed showing heads in 59 spins. Rahim calculated that the statistic, so this is the sample proportion here, it's 59 out of 100 were heads, so that's 0.59, or 59 hundredths, and he calculated, had an associated p-value of approximately 0.036. So, based on this scenario, if ahead of time, Rahim had set his significance level at 0.05, what conclusions would he now make? And, while you're pausing it, think about how that may or may not have been different if he set his significance levels ahead of time at 0.01. Pause the video and try to figure that out. So, let's, first of all, remind ourselves what a p-value even is."}, {"video_title": "Comparing P-values to different significance levels AP Statistics Khan Academy.mp3", "Sentence": "So, based on this scenario, if ahead of time, Rahim had set his significance level at 0.05, what conclusions would he now make? And, while you're pausing it, think about how that may or may not have been different if he set his significance levels ahead of time at 0.01. Pause the video and try to figure that out. So, let's, first of all, remind ourselves what a p-value even is. You could view it as the probability of getting a sample proportion at least this large if you assume that the null hypothesis is true. And if that is low enough, if it is some, if it's below some threshold, which is our significance level, then we will reject the null hypothesis. And so, in this scenario, we do see that 0.036, our p-value, is indeed less than alpha."}, {"video_title": "Comparing P-values to different significance levels AP Statistics Khan Academy.mp3", "Sentence": "So, let's, first of all, remind ourselves what a p-value even is. You could view it as the probability of getting a sample proportion at least this large if you assume that the null hypothesis is true. And if that is low enough, if it is some, if it's below some threshold, which is our significance level, then we will reject the null hypothesis. And so, in this scenario, we do see that 0.036, our p-value, is indeed less than alpha. It is indeed less than 0.05. And because of that, we would reject the null hypothesis. And in everyday language, rejecting the null hypothesis is rejecting the notion that the true proportion of spins that a penny would land showing heads is 50%."}, {"video_title": "Comparing P-values to different significance levels AP Statistics Khan Academy.mp3", "Sentence": "And so, in this scenario, we do see that 0.036, our p-value, is indeed less than alpha. It is indeed less than 0.05. And because of that, we would reject the null hypothesis. And in everyday language, rejecting the null hypothesis is rejecting the notion that the true proportion of spins that a penny would land showing heads is 50%. And if you reject your null hypothesis, you could also say that suggests our alternative hypothesis that the true proportion of spins that a penny would land showing heads is greater than 50%. Now, what about the situation where our significance level was lower? Well, in this situation, our p-value, our probability of getting that sample statistic if we assumed our null hypothesis were true, in this situation, it's greater than or equal to, and it's greater than in this particular situation, than our threshold, than our significance level."}, {"video_title": "Comparing P-values to different significance levels AP Statistics Khan Academy.mp3", "Sentence": "And in everyday language, rejecting the null hypothesis is rejecting the notion that the true proportion of spins that a penny would land showing heads is 50%. And if you reject your null hypothesis, you could also say that suggests our alternative hypothesis that the true proportion of spins that a penny would land showing heads is greater than 50%. Now, what about the situation where our significance level was lower? Well, in this situation, our p-value, our probability of getting that sample statistic if we assumed our null hypothesis were true, in this situation, it's greater than or equal to, and it's greater than in this particular situation, than our threshold, than our significance level. And so here, we would say that we fail, fail to reject our null hypothesis. So we're failing to reject this right over here, and it will not help us suggest our alternative hypothesis. And so, because of the difference between what you would conclude given this change in significance levels, that's why it's really important to set these levels ahead of time, because you could imagine it's human nature."}, {"video_title": "Comparing P-values to different significance levels AP Statistics Khan Academy.mp3", "Sentence": "Well, in this situation, our p-value, our probability of getting that sample statistic if we assumed our null hypothesis were true, in this situation, it's greater than or equal to, and it's greater than in this particular situation, than our threshold, than our significance level. And so here, we would say that we fail, fail to reject our null hypothesis. So we're failing to reject this right over here, and it will not help us suggest our alternative hypothesis. And so, because of the difference between what you would conclude given this change in significance levels, that's why it's really important to set these levels ahead of time, because you could imagine it's human nature. If you're a researcher of some kind, you want to have an interesting result. You want to discover something. You wanna be able to tell your friends, hey, my alternative hypothesis, it actually is suggested."}, {"video_title": "Comparing P-values to different significance levels AP Statistics Khan Academy.mp3", "Sentence": "And so, because of the difference between what you would conclude given this change in significance levels, that's why it's really important to set these levels ahead of time, because you could imagine it's human nature. If you're a researcher of some kind, you want to have an interesting result. You want to discover something. You wanna be able to tell your friends, hey, my alternative hypothesis, it actually is suggested. We can reject the assumption, the status quo. I found something that actually makes a difference. And so it's very tempting for a researcher to calculate your p-values and then say, oh, well, maybe no one will notice if I then set my significance values so that it's just high enough so that I can reject my null hypothesis."}, {"video_title": "Comparing P-values to different significance levels AP Statistics Khan Academy.mp3", "Sentence": "You wanna be able to tell your friends, hey, my alternative hypothesis, it actually is suggested. We can reject the assumption, the status quo. I found something that actually makes a difference. And so it's very tempting for a researcher to calculate your p-values and then say, oh, well, maybe no one will notice if I then set my significance values so that it's just high enough so that I can reject my null hypothesis. If you did that, that would be very unethical. In future videos, we'll start thinking about the question of, okay, if I'm doing it ahead of time, if I'm setting my significance level ahead of time, how do I decide to set the threshold? When should it be 1 100th?"}, {"video_title": "Comparing P-values to different significance levels AP Statistics Khan Academy.mp3", "Sentence": "And so it's very tempting for a researcher to calculate your p-values and then say, oh, well, maybe no one will notice if I then set my significance values so that it's just high enough so that I can reject my null hypothesis. If you did that, that would be very unethical. In future videos, we'll start thinking about the question of, okay, if I'm doing it ahead of time, if I'm setting my significance level ahead of time, how do I decide to set the threshold? When should it be 1 100th? When should it be 5 100ths? When should it be 10 100ths? Or when should it be something else?"}, {"video_title": "Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And so let's say we can set that, and let's make that 60% of the gumballs are green. But let's say someone else comes along and they don't actually know the proportion of gumballs that are green, but they can take samples. And so let's say they take samples of 50 at a time. And so they draw a sample. The sample proportion right over here actually just happened to be 0.6. But then they could draw another sample. This time the sample proportion is 0.52, or 52% of those 50 gumballs happened to be green."}, {"video_title": "Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And so they draw a sample. The sample proportion right over here actually just happened to be 0.6. But then they could draw another sample. This time the sample proportion is 0.52, or 52% of those 50 gumballs happened to be green. Now you could say, all right, well, these are all different estimates, but for any given estimate, how confident are we that a certain range around that estimate actually contains the true population proportion? And so if we look at this tab right over here, that's what confidence intervals are good for. And in a previous video, we talked about how you calculate the confidence interval."}, {"video_title": "Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "This time the sample proportion is 0.52, or 52% of those 50 gumballs happened to be green. Now you could say, all right, well, these are all different estimates, but for any given estimate, how confident are we that a certain range around that estimate actually contains the true population proportion? And so if we look at this tab right over here, that's what confidence intervals are good for. And in a previous video, we talked about how you calculate the confidence interval. What we wanna do is say, well, there's a 95% chance, and we get that from this confidence level, and 95% is the confidence level people typically use. And so there's a 95% chance that whatever our sample proportion is, that it's within two standard deviations of the true proportion, or that the true proportion is going to be contained in an interval that are two standard deviations on either side of our sample proportion. Well, if you don't know the true proportion, the way that you estimate the standard deviation is with the standard error, which we've done in previous videos."}, {"video_title": "Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And in a previous video, we talked about how you calculate the confidence interval. What we wanna do is say, well, there's a 95% chance, and we get that from this confidence level, and 95% is the confidence level people typically use. And so there's a 95% chance that whatever our sample proportion is, that it's within two standard deviations of the true proportion, or that the true proportion is going to be contained in an interval that are two standard deviations on either side of our sample proportion. Well, if you don't know the true proportion, the way that you estimate the standard deviation is with the standard error, which we've done in previous videos. And so this is two standard errors to the right and two standard errors to the left of our sample proportion. And our confidence interval is this entire interval going from this left point to this right point. And as we draw more samples, you can see it's not obvious, but our intervals change depending on what our actual sample proportion is, because we use our sample proportion to calculate our confidence interval, because we're assuming whoever's doing the sampling does not actually know the true population proportion."}, {"video_title": "Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Well, if you don't know the true proportion, the way that you estimate the standard deviation is with the standard error, which we've done in previous videos. And so this is two standard errors to the right and two standard errors to the left of our sample proportion. And our confidence interval is this entire interval going from this left point to this right point. And as we draw more samples, you can see it's not obvious, but our intervals change depending on what our actual sample proportion is, because we use our sample proportion to calculate our confidence interval, because we're assuming whoever's doing the sampling does not actually know the true population proportion. Now, what's interesting here about this simulation is that we can see what percentage of the time does our confidence interval, does it actually contain the true parameter? So let me just draw 25 samples at a time. And so you can see here that right now, 93%, for 93% of our samples, did our confidence interval actually contain our population parameter?"}, {"video_title": "Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And as we draw more samples, you can see it's not obvious, but our intervals change depending on what our actual sample proportion is, because we use our sample proportion to calculate our confidence interval, because we're assuming whoever's doing the sampling does not actually know the true population proportion. Now, what's interesting here about this simulation is that we can see what percentage of the time does our confidence interval, does it actually contain the true parameter? So let me just draw 25 samples at a time. And so you can see here that right now, 93%, for 93% of our samples, did our confidence interval actually contain our population parameter? And we can keep sampling over here. And we can see the more samples that we take, it really is approaching that close to 95% of the time, our confidence interval does indeed contain the true parameter. And so once again, we did all that math in the previous video, but here you can see that confidence intervals, calculated the way that we've calculated them, actually do a pretty good job of what they claim to do."}, {"video_title": "Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And so you can see here that right now, 93%, for 93% of our samples, did our confidence interval actually contain our population parameter? And we can keep sampling over here. And we can see the more samples that we take, it really is approaching that close to 95% of the time, our confidence interval does indeed contain the true parameter. And so once again, we did all that math in the previous video, but here you can see that confidence intervals, calculated the way that we've calculated them, actually do a pretty good job of what they claim to do. That if we calculate a confidence interval based on a confidence level of 95%, that it is indeed the case that roughly 95% of the time, the true parameter, the population proportion, will be contained in that interval. And I could just draw more and more and more samples. And we can actually see that happening."}, {"video_title": "Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And so once again, we did all that math in the previous video, but here you can see that confidence intervals, calculated the way that we've calculated them, actually do a pretty good job of what they claim to do. That if we calculate a confidence interval based on a confidence level of 95%, that it is indeed the case that roughly 95% of the time, the true parameter, the population proportion, will be contained in that interval. And I could just draw more and more and more samples. And we can actually see that happening. Every now and then for sure, you get a sample where even when you calculate your confidence interval, the true parameter, the true population proportion is not contained. But that is the exception that happens very infrequently. 95% of the time, your true population parameter is contained in that interval."}, {"video_title": "Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And we can actually see that happening. Every now and then for sure, you get a sample where even when you calculate your confidence interval, the true parameter, the true population proportion is not contained. But that is the exception that happens very infrequently. 95% of the time, your true population parameter is contained in that interval. Now another interesting thing to see is if we increase our sample size, our confidence interval is going to get narrower. So if we increase our sample size, we'll just make it 200. Now let's draw some samples."}, {"video_title": "Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Aubrey wanted to see if there's a connection between the time a given exam takes place and the average score of this exam. She collected data about exams from the previous year. Plot the data in a scatter plot, and let's see, they give us a couple of rows here. This is the class, then they give us the period of the day that the class happened, and then they give us the average score on an exam. And one could, you know, we have to be a little careful with this study, maybe there's some correlation depending on what subject is taught during what period, but let's just use her data, at least just based on her data see if, see, well, definitely do what they're asking us, plot a scatter plot, and then see if there is any connection. So let's see, on the vertical, on the horizontal axis, we have period, and on this investigation, this exploration she's doing, she's trying to see, well, does the period of the day somehow drive average scores? So that's why period is on the horizontal axis, and the thing that's being, the thing that's driving is on the horizontal, the thing that's being driven is on the vertical."}, {"video_title": "Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3", "Sentence": "This is the class, then they give us the period of the day that the class happened, and then they give us the average score on an exam. And one could, you know, we have to be a little careful with this study, maybe there's some correlation depending on what subject is taught during what period, but let's just use her data, at least just based on her data see if, see, well, definitely do what they're asking us, plot a scatter plot, and then see if there is any connection. So let's see, on the vertical, on the horizontal axis, we have period, and on this investigation, this exploration she's doing, she's trying to see, well, does the period of the day somehow drive average scores? So that's why period is on the horizontal axis, and the thing that's being, the thing that's driving is on the horizontal, the thing that's being driven is on the vertical. So let's plot each of these points. Period one, average score 93. Period one, average score 93, right over there."}, {"video_title": "Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3", "Sentence": "So that's why period is on the horizontal axis, and the thing that's being, the thing that's driving is on the horizontal, the thing that's being driven is on the vertical. So let's plot each of these points. Period one, average score 93. Period one, average score 93, right over there. Period six, 87. Period six, 87. 80, oh, that's not the right place, and then we can move it if we want."}, {"video_title": "Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Period one, average score 93, right over there. Period six, 87. Period six, 87. 80, oh, that's not the right place, and then we can move it if we want. 87, right over there. Period two, 70. Two, period two, 70."}, {"video_title": "Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3", "Sentence": "80, oh, that's not the right place, and then we can move it if we want. 87, right over there. Period two, 70. Two, period two, 70. Period four, 62. Four, four and 62, right over there. Period four and 86."}, {"video_title": "Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Two, period two, 70. Period four, 62. Four, four and 62, right over there. Period four and 86. Period four and again an 86, that's right over there. Period one, 73. Period one, 73."}, {"video_title": "Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Period four and 86. Period four and again an 86, that's right over there. Period one, 73. Period one, 73. Period three, average score of 73 as well. Period three, 73. Period one, 80, average score of 80."}, {"video_title": "Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Period one, 73. Period three, average score of 73 as well. Period three, 73. Period one, 80, average score of 80. So period one, average score of 80. And then period three, average score of 96. Period three, average score of 96."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "The scatter plot below displays a set of bivariate data along with its least squares regression line. Consider removing the outlier at 95 comma one. So 95 comma one, we're talking about that outlier right over there, and calculating a new least squares regression line. What effects would removing the outlier have? Choose all answers that apply. Like always, pause this video and see if you could figure it out. Well let's see, even with this outlier here, we have an upward sloping regression line."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "What effects would removing the outlier have? Choose all answers that apply. Like always, pause this video and see if you could figure it out. Well let's see, even with this outlier here, we have an upward sloping regression line. And so it looks like our r already is going to be greater than zero, and of course it's going to be less than one. So our r is going to be greater than zero and less than one. We know it's not going to be equal one because then we would go perfectly through all of the dots."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "Well let's see, even with this outlier here, we have an upward sloping regression line. And so it looks like our r already is going to be greater than zero, and of course it's going to be less than one. So our r is going to be greater than zero and less than one. We know it's not going to be equal one because then we would go perfectly through all of the dots. And it's clear that this point right over here is indeed an outlier. The residual between this point and the line is quite high. We have a pretty big distance right over here."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "We know it's not going to be equal one because then we would go perfectly through all of the dots. And it's clear that this point right over here is indeed an outlier. The residual between this point and the line is quite high. We have a pretty big distance right over here. It would be a negative residual. And so this point is definitely bringing down the r, and it's definitely bringing down the slope of the regression line. If we were to remove this point, we're more likely to have a line that looks something like this, in which case it looks like we would get a much, much, much, much better fit."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "We have a pretty big distance right over here. It would be a negative residual. And so this point is definitely bringing down the r, and it's definitely bringing down the slope of the regression line. If we were to remove this point, we're more likely to have a line that looks something like this, in which case it looks like we would get a much, much, much, much better fit. The only reason why the line isn't doing that is it's trying to get close to this point right over here. So if we remove this outlier, our r would increase. So r would increase."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "If we were to remove this point, we're more likely to have a line that looks something like this, in which case it looks like we would get a much, much, much, much better fit. The only reason why the line isn't doing that is it's trying to get close to this point right over here. So if we remove this outlier, our r would increase. So r would increase. And also the slope of our line would increase. And slope would increase. We'd have a better fit to this positively correlated data."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "So r would increase. And also the slope of our line would increase. And slope would increase. We'd have a better fit to this positively correlated data. And we would no longer have this point dragging the slope down anymore. So let's see which choices apply. The coefficient of determination, r squared, would increase."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "We'd have a better fit to this positively correlated data. And we would no longer have this point dragging the slope down anymore. So let's see which choices apply. The coefficient of determination, r squared, would increase. Well, if r would increase, then squaring that value would increase as well, so I will circle that. The coefficient, the correlation coefficient r would get close to zero. No, in fact, it would get closer to one because we would have a better fit here."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "The coefficient of determination, r squared, would increase. Well, if r would increase, then squaring that value would increase as well, so I will circle that. The coefficient, the correlation coefficient r would get close to zero. No, in fact, it would get closer to one because we would have a better fit here. And so I will rule that out. The slope of the least squares regression line would increase. Yes, indeed, this point, this outlier is pulling it down."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "No, in fact, it would get closer to one because we would have a better fit here. And so I will rule that out. The slope of the least squares regression line would increase. Yes, indeed, this point, this outlier is pulling it down. If you take it out, it'll allow the slope to increase. So I will circle that as well. Let's do another example."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "Yes, indeed, this point, this outlier is pulling it down. If you take it out, it'll allow the slope to increase. So I will circle that as well. Let's do another example. The scatter plot below displays a set of bivariate data along with its least squares regression line. Same idea. Consider removing the outlier 10, negative 18, so we're talking about that point there, and calculating a new least squares regression line."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "Let's do another example. The scatter plot below displays a set of bivariate data along with its least squares regression line. Same idea. Consider removing the outlier 10, negative 18, so we're talking about that point there, and calculating a new least squares regression line. So what would happen this time? So as is, without removing this outlier, we have a negative slope for the regression line. So we're dealing with a negative r. So we already know that negative one is less than r, which is less than zero."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "Consider removing the outlier 10, negative 18, so we're talking about that point there, and calculating a new least squares regression line. So what would happen this time? So as is, without removing this outlier, we have a negative slope for the regression line. So we're dealing with a negative r. So we already know that negative one is less than r, which is less than zero. Without even removing the outlier, we know it's not going to be negative one. If r was exactly negative one, then it would be a downward-sloping line that went exactly through all of the points. But if we remove this point, what's going to happen?"}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "So we're dealing with a negative r. So we already know that negative one is less than r, which is less than zero. Without even removing the outlier, we know it's not going to be negative one. If r was exactly negative one, then it would be a downward-sloping line that went exactly through all of the points. But if we remove this point, what's going to happen? Well, this least squares regression is being pulled down here by this outlier. So if you were to remove this point, the least squares regression line could move up on the left-hand side, and so you'll probably have a line that looks more like that. And I'm just hand-drawing it, but even what I hand-drew looks like a better fit for the leftover points."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "But if we remove this point, what's going to happen? Well, this least squares regression is being pulled down here by this outlier. So if you were to remove this point, the least squares regression line could move up on the left-hand side, and so you'll probably have a line that looks more like that. And I'm just hand-drawing it, but even what I hand-drew looks like a better fit for the leftover points. And so clearly, the new line that I drew after removing the outlier, this has a more negative slope. So removing the outlier would decrease r. r would get closer to negative one. It would be closer to being a perfect negative correlation."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "And I'm just hand-drawing it, but even what I hand-drew looks like a better fit for the leftover points. And so clearly, the new line that I drew after removing the outlier, this has a more negative slope. So removing the outlier would decrease r. r would get closer to negative one. It would be closer to being a perfect negative correlation. And also, it would decrease the slope. Decrease the slope, which choices match that? The coefficient of determination r squared would decrease."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "It would be closer to being a perfect negative correlation. And also, it would decrease the slope. Decrease the slope, which choices match that? The coefficient of determination r squared would decrease. So let's be very careful. r was already negative. If we decrease it, it's going to become more negative."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "The coefficient of determination r squared would decrease. So let's be very careful. r was already negative. If we decrease it, it's going to become more negative. If you square something that is more negative, it's not going to become smaller. Let's say before you remove the data point, r was, I'm just gonna make up a value, let's say it was negative 0.4, and then after removing the outlier, r becomes more negative, and it's going to be equal to negative 0.5. Well, if you square this, this would be positive 0.16, while this would be positive 0.25."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "If we decrease it, it's going to become more negative. If you square something that is more negative, it's not going to become smaller. Let's say before you remove the data point, r was, I'm just gonna make up a value, let's say it was negative 0.4, and then after removing the outlier, r becomes more negative, and it's going to be equal to negative 0.5. Well, if you square this, this would be positive 0.16, while this would be positive 0.25. So if r is already negative, and if you make it more negative, it would not decrease r squared. It actually would increase r squared. So I will rule this one out."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "Well, if you square this, this would be positive 0.16, while this would be positive 0.25. So if r is already negative, and if you make it more negative, it would not decrease r squared. It actually would increase r squared. So I will rule this one out. The slope of the least squares regression line would increase. No, it's going to decrease. It's going to be a stronger negative correlation."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "So I will rule this one out. The slope of the least squares regression line would increase. No, it's going to decrease. It's going to be a stronger negative correlation. Rule that one out. The y-intercept of the least squares regression line would increase. Yes, by getting rid of this outlier, you could think of it as the left side of this line is going to increase, or another way to think about it, the slope of this line is going to decrease."}, {"video_title": "Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3", "Sentence": "It's going to be a stronger negative correlation. Rule that one out. The y-intercept of the least squares regression line would increase. Yes, by getting rid of this outlier, you could think of it as the left side of this line is going to increase, or another way to think about it, the slope of this line is going to decrease. It's going to become more negative. We know that the least squares regression line will always go through the mean of both variables. So we're just gonna pivot around the mean of both variables, which would mean that the y-intercept will go higher."}, {"video_title": "Conclusion for a two-sample t test using a P-value AP Statistics Khan Academy.mp3", "Sentence": "The sociologist obtained a random sample of women from each country. Here are the results of their test. So you can see a 100-person sample from France, a 100-person sample from Switzerland. They actually don't have to be the same sample size. We have our sample means, our sample standard deviations. You have the standard error of the mean, which for each sample would be our estimate of the standard deviation of the sampling distribution of the sample mean. And here it says t-test for the means of these different populations being different."}, {"video_title": "Conclusion for a two-sample t test using a P-value AP Statistics Khan Academy.mp3", "Sentence": "They actually don't have to be the same sample size. We have our sample means, our sample standard deviations. You have the standard error of the mean, which for each sample would be our estimate of the standard deviation of the sampling distribution of the sample mean. And here it says t-test for the means of these different populations being different. And just to make sure we can make sense of this, let's just remind ourselves what's going on. The null hypothesis is that there's no difference in the mean number of babies that women in France have versus the mean number of babies that women in Switzerland have. That would be our null hypothesis, the no-news-here hypothesis, and our alternative would be that they are different."}, {"video_title": "Conclusion for a two-sample t test using a P-value AP Statistics Khan Academy.mp3", "Sentence": "And here it says t-test for the means of these different populations being different. And just to make sure we can make sense of this, let's just remind ourselves what's going on. The null hypothesis is that there's no difference in the mean number of babies that women in France have versus the mean number of babies that women in Switzerland have. That would be our null hypothesis, the no-news-here hypothesis, and our alternative would be that they are different. And that's what we have right over here. It's a t-test to see if we have evidence that would suggest our alternative hypothesis. And so what we do is we assume the null hypothesis."}, {"video_title": "Conclusion for a two-sample t test using a P-value AP Statistics Khan Academy.mp3", "Sentence": "That would be our null hypothesis, the no-news-here hypothesis, and our alternative would be that they are different. And that's what we have right over here. It's a t-test to see if we have evidence that would suggest our alternative hypothesis. And so what we do is we assume the null hypothesis. From that, you're able to calculate a t-statistic. And then from that t-statistic and the degrees of freedom, you are able to calculate a p-value. And if that p-value is below your significance level, then you'd say, hey, this was a pretty unlikely scenario."}, {"video_title": "Conclusion for a two-sample t test using a P-value AP Statistics Khan Academy.mp3", "Sentence": "And so what we do is we assume the null hypothesis. From that, you're able to calculate a t-statistic. And then from that t-statistic and the degrees of freedom, you are able to calculate a p-value. And if that p-value is below your significance level, then you'd say, hey, this was a pretty unlikely scenario. Let me reject the null hypothesis, which would suggest the alternative. But if your p-value is greater than your significance level, then you would fail to reject your null hypothesis, and so you would not have sufficient evidence to conclude the alternative. So what's going on over here?"}, {"video_title": "Conclusion for a two-sample t test using a P-value AP Statistics Khan Academy.mp3", "Sentence": "And if that p-value is below your significance level, then you'd say, hey, this was a pretty unlikely scenario. Let me reject the null hypothesis, which would suggest the alternative. But if your p-value is greater than your significance level, then you would fail to reject your null hypothesis, and so you would not have sufficient evidence to conclude the alternative. So what's going on over here? You really just have to compare this value to this value. It says at the alpha is equal to 0.05 level of significance, is there sufficient evidence to conclude that there is a difference in the average number of babies women in each country have? Well, we can see that our p-value, 0.13, is greater than our alpha value, 0.05."}, {"video_title": "Conclusion for a two-sample t test using a P-value AP Statistics Khan Academy.mp3", "Sentence": "So what's going on over here? You really just have to compare this value to this value. It says at the alpha is equal to 0.05 level of significance, is there sufficient evidence to conclude that there is a difference in the average number of babies women in each country have? Well, we can see that our p-value, 0.13, is greater than our alpha value, 0.05. And so because of that, we fail to reject our null hypothesis. And to answer their question, no, there is not sufficient evidence to conclude that there is a difference. There's not sufficient evidence to reject the null hypothesis and suggest the alternative."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "And we've already done some hypothesis testing with the chi-squared statistic. And we've even done some hypothesis testing based on two-way tables. And now we're going to extend that by thinking about a chi-squared test for association between two variables. So let's say that we suspect that someone's foot length is related to their hand length, that these things are not independent. Well, what we can do is set up a hypothesis test. And remember, the null hypothesis in a hypothesis test is to always assume no news. So what we could say is here is that there's no association, no association between foot and hand length."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "So let's say that we suspect that someone's foot length is related to their hand length, that these things are not independent. Well, what we can do is set up a hypothesis test. And remember, the null hypothesis in a hypothesis test is to always assume no news. So what we could say is here is that there's no association, no association between foot and hand length. Another way to think about it is that they are independent. And oftentimes what we're doing is called a chi-squared test for independence. And then our alternative hypothesis would be our suspicion there is an association."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "So what we could say is here is that there's no association, no association between foot and hand length. Another way to think about it is that they are independent. And oftentimes what we're doing is called a chi-squared test for independence. And then our alternative hypothesis would be our suspicion there is an association. There is an association. So foot and hand length are not independent. So what we can then do is go to a population and we can randomly sample it."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "And then our alternative hypothesis would be our suspicion there is an association. There is an association. So foot and hand length are not independent. So what we can then do is go to a population and we can randomly sample it. And so let's say we randomly sample 100 folks. And for all of those 100 folks, we figure out whether their right hand is longer, their left hand is longer, or both hands are the same. And we also do that for the feet."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "So what we can then do is go to a population and we can randomly sample it. And so let's say we randomly sample 100 folks. And for all of those 100 folks, we figure out whether their right hand is longer, their left hand is longer, or both hands are the same. And we also do that for the feet. And we tabulate all of the data. And this is the data that we actually get. Now it's worth thinking about this for a second on how what we just did is different from a chi-squared test for homogeneity."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "And we also do that for the feet. And we tabulate all of the data. And this is the data that we actually get. Now it's worth thinking about this for a second on how what we just did is different from a chi-squared test for homogeneity. In a chi-squared test for homogeneity, we sample from two different populations, or we look at two different groups, and we see whether the distribution of a certain variable amongst those two different groups is the same. Here we are just sampling from one group, but we're thinking about two different variables for that one group. We're thinking about feet length and we're thinking about hand length."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "Now it's worth thinking about this for a second on how what we just did is different from a chi-squared test for homogeneity. In a chi-squared test for homogeneity, we sample from two different populations, or we look at two different groups, and we see whether the distribution of a certain variable amongst those two different groups is the same. Here we are just sampling from one group, but we're thinking about two different variables for that one group. We're thinking about feet length and we're thinking about hand length. And so you can see here that 11 folks had both their right hand longer and their right foot longer. Three folks had their right hand longer but their left foot was longer. And then eight folks had their right hand longer but both feet were the same."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "We're thinking about feet length and we're thinking about hand length. And so you can see here that 11 folks had both their right hand longer and their right foot longer. Three folks had their right hand longer but their left foot was longer. And then eight folks had their right hand longer but both feet were the same. Likewise, we had nine people where their left foot and hand was longer, but you had two people where the left hand was longer but the right foot was longer. And we could go through all of these. But to do our chi-squared test, we would have said what would be the expected value of each of these data points if we assumed that the null hypothesis was true, that there was no association between foot and hand length."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "And then eight folks had their right hand longer but both feet were the same. Likewise, we had nine people where their left foot and hand was longer, but you had two people where the left hand was longer but the right foot was longer. And we could go through all of these. But to do our chi-squared test, we would have said what would be the expected value of each of these data points if we assumed that the null hypothesis was true, that there was no association between foot and hand length. So to help us do that, I'm gonna make a total of our columns here and also a total of our rows here. Let me draw a line here so we know what is going on. And so what are the total number of people who had a longer right hand?"}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "But to do our chi-squared test, we would have said what would be the expected value of each of these data points if we assumed that the null hypothesis was true, that there was no association between foot and hand length. So to help us do that, I'm gonna make a total of our columns here and also a total of our rows here. Let me draw a line here so we know what is going on. And so what are the total number of people who had a longer right hand? Well, it's gonna be 11 plus three plus eight, which is 22. The total number of people who had a longer left hand is two plus nine plus 14, which is 25. And then the total number of people whose hands had the same length, 12 plus 13 plus 28, 25 plus 28, that is 53."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "And so what are the total number of people who had a longer right hand? Well, it's gonna be 11 plus three plus eight, which is 22. The total number of people who had a longer left hand is two plus nine plus 14, which is 25. And then the total number of people whose hands had the same length, 12 plus 13 plus 28, 25 plus 28, that is 53. And then if I were to total this column, 22 plus 25 is 47 plus 53, we get 100 right over here. And then if we total the number of people who had a longer right foot, 11 plus two plus 12, that's 13 plus 12, that is 25. Longer left foot, three plus nine plus 13, that's also 25."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "And then the total number of people whose hands had the same length, 12 plus 13 plus 28, 25 plus 28, that is 53. And then if I were to total this column, 22 plus 25 is 47 plus 53, we get 100 right over here. And then if we total the number of people who had a longer right foot, 11 plus two plus 12, that's 13 plus 12, that is 25. Longer left foot, three plus nine plus 13, that's also 25. And then we could either add these up and we would get 50 or we could say, hey, 25 plus 25 plus what is 100? Well, that is going to be equal to 50. Now, to figure out these expected values, remember, we're going to figure out the expected values assuming that the null hypothesis is true, assuming that these distributions are independent, that feet length and hand length are independent variables."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "Longer left foot, three plus nine plus 13, that's also 25. And then we could either add these up and we would get 50 or we could say, hey, 25 plus 25 plus what is 100? Well, that is going to be equal to 50. Now, to figure out these expected values, remember, we're going to figure out the expected values assuming that the null hypothesis is true, assuming that these distributions are independent, that feet length and hand length are independent variables. Well, if they are independent, which we are assuming, then our best estimate is that 22% have a longer right hand and our best estimate is that 25% have a longer right foot. And so out of 100, you would expect 0.22 times 0.25 times 100 to have a longer right hand and foot. I'm just multiplying the probabilities, which you would do if these were independent variables."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "Now, to figure out these expected values, remember, we're going to figure out the expected values assuming that the null hypothesis is true, assuming that these distributions are independent, that feet length and hand length are independent variables. Well, if they are independent, which we are assuming, then our best estimate is that 22% have a longer right hand and our best estimate is that 25% have a longer right foot. And so out of 100, you would expect 0.22 times 0.25 times 100 to have a longer right hand and foot. I'm just multiplying the probabilities, which you would do if these were independent variables. And so 0.22 times 0.25, let's see, 1 1\u20444 of 22 is 5 1\u20442, so this is going to be equal to 5.5. Now, what number would you expect to have a longer right hand but a longer left foot? So that would be 0.22 times 0.25 times 100."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "I'm just multiplying the probabilities, which you would do if these were independent variables. And so 0.22 times 0.25, let's see, 1 1\u20444 of 22 is 5 1\u20442, so this is going to be equal to 5.5. Now, what number would you expect to have a longer right hand but a longer left foot? So that would be 0.22 times 0.25 times 100. Well, we already calculated what that would be. That would be 5.5. And then to figure out the expected number that it would have a longer right hand but both feet would be the same length, we could multiply 22 out of 100 times 50 out of 100 times 100, which is going to be 1\u20442 of 22, which is equal to 11."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "So that would be 0.22 times 0.25 times 100. Well, we already calculated what that would be. That would be 5.5. And then to figure out the expected number that it would have a longer right hand but both feet would be the same length, we could multiply 22 out of 100 times 50 out of 100 times 100, which is going to be 1\u20442 of 22, which is equal to 11. And we can keep going. This value right over here would be 0.25 times 0.25 times 100, 25 times 25 is 625, so that would be 6.25. This value right over here would be 0.25 times 0.25 times 100, which is again 6.25."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "And then to figure out the expected number that it would have a longer right hand but both feet would be the same length, we could multiply 22 out of 100 times 50 out of 100 times 100, which is going to be 1\u20442 of 22, which is equal to 11. And we can keep going. This value right over here would be 0.25 times 0.25 times 100, 25 times 25 is 625, so that would be 6.25. This value right over here would be 0.25 times 0.25 times 100, which is again 6.25. And then this value right over here, couple of ways we can get it. We can multiply 0.25 times 50 times 100, which would get us to 12.5. Or we could have said this plus this plus this has to equal 25, so this would be 12.5."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "This value right over here would be 0.25 times 0.25 times 100, which is again 6.25. And then this value right over here, couple of ways we can get it. We can multiply 0.25 times 50 times 100, which would get us to 12.5. Or we could have said this plus this plus this has to equal 25, so this would be 12.5. And now this expected value we can figure out because 5.5 plus 6.25 plus this is going to equal 25. So let's see, 5.5 plus 6.25 is 11.75. 11.75 plus 13.25 is equal to 25."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "Or we could have said this plus this plus this has to equal 25, so this would be 12.5. And now this expected value we can figure out because 5.5 plus 6.25 plus this is going to equal 25. So let's see, 5.5 plus 6.25 is 11.75. 11.75 plus 13.25 is equal to 25. Same thing over here. This would be 13.25, because this is 11.75 plus 13.25 is equal to 25. If we add these two together, we get 26.5."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "11.75 plus 13.25 is equal to 25. Same thing over here. This would be 13.25, because this is 11.75 plus 13.25 is equal to 25. If we add these two together, we get 26.5. 26.5 plus what is equal to 53? Would be equal to another 26.5. Now once you figure out all of your expected values, that's a good time to test your conditions."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "If we add these two together, we get 26.5. 26.5 plus what is equal to 53? Would be equal to another 26.5. Now once you figure out all of your expected values, that's a good time to test your conditions. The first condition is that you took a random sample, so let's assume we had done that. The second condition is that your expected value for any of the data points has to be at least equal to five. And we can see that all of our expected values are at least equal to five."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "Now once you figure out all of your expected values, that's a good time to test your conditions. The first condition is that you took a random sample, so let's assume we had done that. The second condition is that your expected value for any of the data points has to be at least equal to five. And we can see that all of our expected values are at least equal to five. The actual data points we got do not have to be equal to five. So it's okay that we got a two here because the expected value here is five or larger. And then the last condition is the independence condition that either we are sampling with replacement or that we have to feel comfortable that our sample size is no more than 10% of the population."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "And we can see that all of our expected values are at least equal to five. The actual data points we got do not have to be equal to five. So it's okay that we got a two here because the expected value here is five or larger. And then the last condition is the independence condition that either we are sampling with replacement or that we have to feel comfortable that our sample size is no more than 10% of the population. So let's assume that that happened as well. So assuming we met all of those conditions, we are ready to calculate our chi-squared statistic. And so what we're going to do is for every data point, we're gonna find the difference between the data point, 11 minus the expected, minus 5.5, squared over the expected."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "And then the last condition is the independence condition that either we are sampling with replacement or that we have to feel comfortable that our sample size is no more than 10% of the population. So let's assume that that happened as well. So assuming we met all of those conditions, we are ready to calculate our chi-squared statistic. And so what we're going to do is for every data point, we're gonna find the difference between the data point, 11 minus the expected, minus 5.5, squared over the expected. So I did that one. Now I'll do this one. So plus three minus 5.5 squared over 5.5 plus, and I'll do this one, eight minus 11 squared over 11."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "And so what we're going to do is for every data point, we're gonna find the difference between the data point, 11 minus the expected, minus 5.5, squared over the expected. So I did that one. Now I'll do this one. So plus three minus 5.5 squared over 5.5 plus, and I'll do this one, eight minus 11 squared over 11. Then I'll do this one. Two minus 6.25 squared over 6.25. And I'll keep doing it."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "So plus three minus 5.5 squared over 5.5 plus, and I'll do this one, eight minus 11 squared over 11. Then I'll do this one. Two minus 6.25 squared over 6.25. And I'll keep doing it. I'm gonna do it for all nine of these data points. And I actually calculated this ahead of time to save some time. And so if you do this for all nine of the data points, you're going to get a chi-squared statistic of 11.942."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "And I'll keep doing it. I'm gonna do it for all nine of these data points. And I actually calculated this ahead of time to save some time. And so if you do this for all nine of the data points, you're going to get a chi-squared statistic of 11.942. Now before we calculate the p-value, we're gonna have to think about what are our degrees of freedom. Now we have a three by three table here. So one way to think about it, it's the number of rows minus one times the number of columns minus one."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "And so if you do this for all nine of the data points, you're going to get a chi-squared statistic of 11.942. Now before we calculate the p-value, we're gonna have to think about what are our degrees of freedom. Now we have a three by three table here. So one way to think about it, it's the number of rows minus one times the number of columns minus one. This is two times two, which is equal to four. Another way to think about it is if you know four of these data points and you know the totals, then you could figure out the other five data points. And so now we are ready to calculate a p-value."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "So one way to think about it, it's the number of rows minus one times the number of columns minus one. This is two times two, which is equal to four. Another way to think about it is if you know four of these data points and you know the totals, then you could figure out the other five data points. And so now we are ready to calculate a p-value. And you could do that using a calculator and you could do that using a chi-squared table. But let's say we did it using a calculator. And we get a p-value of 0.018."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "And so now we are ready to calculate a p-value. And you could do that using a calculator and you could do that using a chi-squared table. But let's say we did it using a calculator. And we get a p-value of 0.018. And just to remind ourselves what this is, this is the probability of getting a chi-squared statistic at least this large or larger. And so next, we do what we always do with hypothesis testing. We compare this to our significance level."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "And we get a p-value of 0.018. And just to remind ourselves what this is, this is the probability of getting a chi-squared statistic at least this large or larger. And so next, we do what we always do with hypothesis testing. We compare this to our significance level. And we actually should have set our significance level from the beginning. So let's just assume that when we set up our hypotheses here, we also said that we want a significance level of 0.05. You really should do this before you calculate all of this."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "We compare this to our significance level. And we actually should have set our significance level from the beginning. So let's just assume that when we set up our hypotheses here, we also said that we want a significance level of 0.05. You really should do this before you calculate all of this. But then you compare your p-value to your significance level. And we see that this p-value is a good bit less than our significance level. And so one way to think about it is we got all these expected values, assuming that the null hypothesis was true."}, {"video_title": "Chi-square test for association (independence) AP Statistics Khan Academy.mp3", "Sentence": "You really should do this before you calculate all of this. But then you compare your p-value to your significance level. And we see that this p-value is a good bit less than our significance level. And so one way to think about it is we got all these expected values, assuming that the null hypothesis was true. But the probability of getting a result this extreme or more extreme is less than 2%, which is lower than our significance level. And so this will lead us to reject our null hypothesis. And it suggests to us that there is an association between hand length and foot length."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Suppose that 5% of all their applicants are actually using illegal drugs and we randomly select an applicant. Given the applicant tests positive, what is the probability that they are actually on drugs? So let's work through this together. So first, let's just make sure we understand what they're telling us. So there is this drug test for the job applicants and then the test has a false positive rate of 2%. What does that mean? That means that in 2% of the cases when it should have read negative, that the person didn't do the drugs, it actually read positive."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "So first, let's just make sure we understand what they're telling us. So there is this drug test for the job applicants and then the test has a false positive rate of 2%. What does that mean? That means that in 2% of the cases when it should have read negative, that the person didn't do the drugs, it actually read positive. It is a false positive. It should have read negative, but it read positive. Another way to think about it, if someone did not do drugs and you take this test, there's a 2% chance saying that you did do the illegal drugs."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "That means that in 2% of the cases when it should have read negative, that the person didn't do the drugs, it actually read positive. It is a false positive. It should have read negative, but it read positive. Another way to think about it, if someone did not do drugs and you take this test, there's a 2% chance saying that you did do the illegal drugs. They also say that there is a false negative rate of 1%. What does that mean? That means that 1% of the time, if someone did actually take the illegal drugs, it'll say that they didn't."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Another way to think about it, if someone did not do drugs and you take this test, there's a 2% chance saying that you did do the illegal drugs. They also say that there is a false negative rate of 1%. What does that mean? That means that 1% of the time, if someone did actually take the illegal drugs, it'll say that they didn't. It is falsely giving a negative result when it should have given a positive one. And then they say that 5% of all their applicants are actually using illegal drugs. So there's several ways that we can think about it."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "That means that 1% of the time, if someone did actually take the illegal drugs, it'll say that they didn't. It is falsely giving a negative result when it should have given a positive one. And then they say that 5% of all their applicants are actually using illegal drugs. So there's several ways that we can think about it. One of the easiest ways to conceptualize, let's just make up a large number of applicants and I'll use a number where it's fairly straightforward to do the mathematics. So let's say that we start off with 10,000 applicants. And so I will both talk in absolute numbers, and I just made this number up."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "So there's several ways that we can think about it. One of the easiest ways to conceptualize, let's just make up a large number of applicants and I'll use a number where it's fairly straightforward to do the mathematics. So let's say that we start off with 10,000 applicants. And so I will both talk in absolute numbers, and I just made this number up. It could have been 1,000, it could have been 100,000, but I like this number because it's easy to do the math. It's better than saying 9,785. And so this is also going to be 100% of the applicants."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "And so I will both talk in absolute numbers, and I just made this number up. It could have been 1,000, it could have been 100,000, but I like this number because it's easy to do the math. It's better than saying 9,785. And so this is also going to be 100% of the applicants. Now they give us some crucial information here. They tell us that 5% of all their applicants are actually using illegal drugs. So we can immediately break this 10,000 group into the ones that are doing the drugs and the ones that are not."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "And so this is also going to be 100% of the applicants. Now they give us some crucial information here. They tell us that 5% of all their applicants are actually using illegal drugs. So we can immediately break this 10,000 group into the ones that are doing the drugs and the ones that are not. So 5% are actually on the drugs. 95% are not on the drugs. So what's 5% of 10,000?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "So we can immediately break this 10,000 group into the ones that are doing the drugs and the ones that are not. So 5% are actually on the drugs. 95% are not on the drugs. So what's 5% of 10,000? So that would be 500. So 500 on drugs. On drugs."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "So what's 5% of 10,000? So that would be 500. So 500 on drugs. On drugs. And so once again, this is 5% of our original population. And then how many are not on drugs? Well, 9,500 not."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "On drugs. And so once again, this is 5% of our original population. And then how many are not on drugs? Well, 9,500 not. Not on drugs. And once again, this is 95% of our group of applicants. So now let's administer the test."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Well, 9,500 not. Not on drugs. And once again, this is 95% of our group of applicants. So now let's administer the test. So what is going to happen when we administer the test to the people who are on drugs? Well, the test ideally would give a positive result. It would say positive for all of them, but we know that it's not a perfect test."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "So now let's administer the test. So what is going to happen when we administer the test to the people who are on drugs? Well, the test ideally would give a positive result. It would say positive for all of them, but we know that it's not a perfect test. It's going to give negative for some of them. It will falsely give a negative result for some of them. And we know that because it has a false negative rate of 1%."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "It would say positive for all of them, but we know that it's not a perfect test. It's going to give negative for some of them. It will falsely give a negative result for some of them. And we know that because it has a false negative rate of 1%. And so of these 500, 99% is going to get the correct result in that they're going to test positive. So what is 99% of 500? Well, let's see, that would be 495."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "And we know that because it has a false negative rate of 1%. And so of these 500, 99% is going to get the correct result in that they're going to test positive. So what is 99% of 500? Well, let's see, that would be 495. 495 are going to test positive. I will just use a positive right over there. And then we're going to have 1%, 1%, which is five, are going to test negative."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Well, let's see, that would be 495. 495 are going to test positive. I will just use a positive right over there. And then we're going to have 1%, 1%, which is five, are going to test negative. They are going to falsely test negative. This is the false negative rate. And so if we say what percent of our original applicant pool is on drugs and tests positive?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "And then we're going to have 1%, 1%, which is five, are going to test negative. They are going to falsely test negative. This is the false negative rate. And so if we say what percent of our original applicant pool is on drugs and tests positive? Well, 495 over 10,000, this is 4.95%. What percent is of the original applicant pool that is on drugs but tests negative for drugs? The test says that, hey, they're not taking drugs."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "And so if we say what percent of our original applicant pool is on drugs and tests positive? Well, 495 over 10,000, this is 4.95%. What percent is of the original applicant pool that is on drugs but tests negative for drugs? The test says that, hey, they're not taking drugs. Well, this is going to be five out of 10,000, which is 0.05%. Another way that you can get these percentages, if you take 5% and multiply by 1%, you're going to get 0.05%, 5 hundredths of a percent. If you take 5% and multiply by 99%, you're going to get 4.95%."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "The test says that, hey, they're not taking drugs. Well, this is going to be five out of 10,000, which is 0.05%. Another way that you can get these percentages, if you take 5% and multiply by 1%, you're going to get 0.05%, 5 hundredths of a percent. If you take 5% and multiply by 99%, you're going to get 4.95%. Now let's keep going. Now let's go to the folks who aren't taking the drugs. And this is where the false positive rate is going to come into effect."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "If you take 5% and multiply by 99%, you're going to get 4.95%. Now let's keep going. Now let's go to the folks who aren't taking the drugs. And this is where the false positive rate is going to come into effect. So we have a false positive rate of 2%. So 2% are going to test positive. What's 2% of 9,500?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "And this is where the false positive rate is going to come into effect. So we have a false positive rate of 2%. So 2% are going to test positive. What's 2% of 9,500? It's 190 would test positive, even though they're not on drugs. This is the false positive rate. So they are testing positive."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "What's 2% of 9,500? It's 190 would test positive, even though they're not on drugs. This is the false positive rate. So they are testing positive. And then the other 98% will correctly come out negative. And so the other 98%, so 9,500 minus 190, that's going to be 9,310 will correctly test negative. Now what percent of the original applicant pool is this?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "So they are testing positive. And then the other 98% will correctly come out negative. And so the other 98%, so 9,500 minus 190, that's going to be 9,310 will correctly test negative. Now what percent of the original applicant pool is this? Well, 190 is 1.9%. And we could calculate it by 190 over 10,000, or you could just say 2% of 95% is 1.9%. Once again, multiply the path along the tree."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Now what percent of the original applicant pool is this? Well, 190 is 1.9%. And we could calculate it by 190 over 10,000, or you could just say 2% of 95% is 1.9%. Once again, multiply the path along the tree. What percent is 9,310? Well, that is going to be 93.10%. You could say this is 9,310 over 10,000, or you can multiply by the path on our probability tree here."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Once again, multiply the path along the tree. What percent is 9,310? Well, that is going to be 93.10%. You could say this is 9,310 over 10,000, or you can multiply by the path on our probability tree here. 95% times 98% gets us to 93.10%. But now I think we are ready to answer the question. Given that the applicant tests positive, what is the probability that they are actually on drugs?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "You could say this is 9,310 over 10,000, or you can multiply by the path on our probability tree here. 95% times 98% gets us to 93.10%. But now I think we are ready to answer the question. Given that the applicant tests positive, what is the probability that they are actually on drugs? So let's look at the first part. Given the applicant tests positive. So which applicants actually tested positive?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Given that the applicant tests positive, what is the probability that they are actually on drugs? So let's look at the first part. Given the applicant tests positive. So which applicants actually tested positive? You have these 495 here tested positive, correctly tested positive, and then you have these 190 right over here incorrectly tested positive, but they did test positive. So how many tested positive? Well, we have 495 plus 190 tested positive."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "So which applicants actually tested positive? You have these 495 here tested positive, correctly tested positive, and then you have these 190 right over here incorrectly tested positive, but they did test positive. So how many tested positive? Well, we have 495 plus 190 tested positive. That's the total number that tested positive. And then which of them were actually on the drugs? Well, of the ones that tested positive, 495 were actually on the drugs."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Well, we have 495 plus 190 tested positive. That's the total number that tested positive. And then which of them were actually on the drugs? Well, of the ones that tested positive, 495 were actually on the drugs. We have 495 divided by 495 plus 190 is equal to 0.7226, so we could say approximately 72%. Approximately 72%. Now this is really interesting."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Well, of the ones that tested positive, 495 were actually on the drugs. We have 495 divided by 495 plus 190 is equal to 0.7226, so we could say approximately 72%. Approximately 72%. Now this is really interesting. Given the applicant tests positive, what is the probability that they are actually on drugs? When you look at these false positive and false negative rates, they seem quite low. But now when we actually did the calculation, the probability that someone's actually on drugs, it's high, but it's not that high."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Now this is really interesting. Given the applicant tests positive, what is the probability that they are actually on drugs? When you look at these false positive and false negative rates, they seem quite low. But now when we actually did the calculation, the probability that someone's actually on drugs, it's high, but it's not that high. It's not like if someone were to test positive that you'd say, oh, they are definitely taking the drugs. And you could also get to this result just by using the percentages. For example, you could think in terms of what percentage of the original applicants end up testing positive?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "But now when we actually did the calculation, the probability that someone's actually on drugs, it's high, but it's not that high. It's not like if someone were to test positive that you'd say, oh, they are definitely taking the drugs. And you could also get to this result just by using the percentages. For example, you could think in terms of what percentage of the original applicants end up testing positive? Well, that's 4.95% plus 1.9%. 4.95, we'll just do it in terms of percent, plus 1.9%, and of them, what percentage were actually on the drugs? Well, that was the 4.95%."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "For example, you could think in terms of what percentage of the original applicants end up testing positive? Well, that's 4.95% plus 1.9%. 4.95, we'll just do it in terms of percent, plus 1.9%, and of them, what percentage were actually on the drugs? Well, that was the 4.95%. And notice, this would give you the exact same result. Now there's an interesting takeaway here. Because this is saying, of the people that test positive, 72% are actually on the drugs."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Well, that was the 4.95%. And notice, this would give you the exact same result. Now there's an interesting takeaway here. Because this is saying, of the people that test positive, 72% are actually on the drugs. You could think about it the other way around. Of the people who test positive, 4.95 plus 1.90, what percentage aren't on drugs? Well, that was 1.90."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Because this is saying, of the people that test positive, 72% are actually on the drugs. You could think about it the other way around. Of the people who test positive, 4.95 plus 1.90, what percentage aren't on drugs? Well, that was 1.90. And this comes out to be approximately 28%. 100% minus 72%. And so, if we were in a court of law, and let's say the prosecuting attorney, let's say I got tested positive for drugs, and the prosecuting attorney says, look, this test is very good."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Well, that was 1.90. And this comes out to be approximately 28%. 100% minus 72%. And so, if we were in a court of law, and let's say the prosecuting attorney, let's say I got tested positive for drugs, and the prosecuting attorney says, look, this test is very good. It only has a false positive rate of 2%. Sal, and Sal tested positive, he is probably taking drugs. A jury who doesn't really understand this well, or go through the trouble that we just did, might say, oh yeah, Sal probably took the drugs."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "And so, if we were in a court of law, and let's say the prosecuting attorney, let's say I got tested positive for drugs, and the prosecuting attorney says, look, this test is very good. It only has a false positive rate of 2%. Sal, and Sal tested positive, he is probably taking drugs. A jury who doesn't really understand this well, or go through the trouble that we just did, might say, oh yeah, Sal probably took the drugs. But when we look at this, even if I test positive using this test, there's a 28% chance that I'm not taking drugs, that I was just in this false positive group. And the reason why this number is a good bit larger than this number is because when we looked at the original division between those who take drugs and don't take drugs, most don't take the illegal drugs. And so 2% of this larger group of the ones that don't take the drugs, well, this is actually a fairly large number relative to the percentage that do take the drugs and test positive."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "A jury who doesn't really understand this well, or go through the trouble that we just did, might say, oh yeah, Sal probably took the drugs. But when we look at this, even if I test positive using this test, there's a 28% chance that I'm not taking drugs, that I was just in this false positive group. And the reason why this number is a good bit larger than this number is because when we looked at the original division between those who take drugs and don't take drugs, most don't take the illegal drugs. And so 2% of this larger group of the ones that don't take the drugs, well, this is actually a fairly large number relative to the percentage that do take the drugs and test positive. So I will leave you there. This is fascinating, not just for this particular case, but you will see analysis like this all the time when we're looking at whether a certain medication is effective or a certain procedure is effective. It's important to be able to do this analysis."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "If the sum of the rolls is 10 or greater, you win. If it is less than 10, you lose. What is the probability of winning three rolls to 10? So there are several ways that you can approach this. The way we're going to tackle it in this video is we're going to try to come up with an experimental probability. We're going to do many experiments trying to win three rolls to 10 and figure out the proportion that we actually win. And the more experiments we try, the better, the more likely that we're gonna get a good approximation of the actual probability."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So there are several ways that you can approach this. The way we're going to tackle it in this video is we're going to try to come up with an experimental probability. We're going to do many experiments trying to win three rolls to 10 and figure out the proportion that we actually win. And the more experiments we try, the better, the more likely that we're gonna get a good approximation of the actual probability. So let's do that. And to help us, I'm going to have a computer generate a string of random digits from zero to nine. And the way that we're going to use this is, remember, we're rolling a fair six-sided die."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "And the more experiments we try, the better, the more likely that we're gonna get a good approximation of the actual probability. So let's do that. And to help us, I'm going to have a computer generate a string of random digits from zero to nine. And the way that we're going to use this is, remember, we're rolling a fair six-sided die. So the outcome could be one, two, three, four, five, or six for each roll. In this random number list that the computer has generated, I do get digits from one to six, but I also get the digits seven, eight, nine, and zero. And so what I'm going to do for each experiment, I'm gonna start at the top left and I'm gonna consider each digit a roll."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "And the way that we're going to use this is, remember, we're rolling a fair six-sided die. So the outcome could be one, two, three, four, five, or six for each roll. In this random number list that the computer has generated, I do get digits from one to six, but I also get the digits seven, eight, nine, and zero. And so what I'm going to do for each experiment, I'm gonna start at the top left and I'm gonna consider each digit a roll. If it gives me an invalid result for a six-sided die, so if it's a zero, an eight, a seven, or a nine, I will just ignore that. I will just say, well, that wasn't a valid roll. It's like you roll the die and it fell off the table or something like that."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "And so what I'm going to do for each experiment, I'm gonna start at the top left and I'm gonna consider each digit a roll. If it gives me an invalid result for a six-sided die, so if it's a zero, an eight, a seven, or a nine, I will just ignore that. I will just say, well, that wasn't a valid roll. It's like you roll the die and it fell off the table or something like that. So let's do that. Let's do multiple experiments of taking three rolls, sum them up, and we'll see how many we can do to figure out an experimental probability of winning Pascal's game. So let me set up a little table here."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "It's like you roll the die and it fell off the table or something like that. So let's do that. Let's do multiple experiments of taking three rolls, sum them up, and we'll see how many we can do to figure out an experimental probability of winning Pascal's game. So let me set up a little table here. So I want space to show the sum. So this is going to be the experiments. Experiment."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So let me set up a little table here. So I want space to show the sum. So this is going to be the experiments. Experiment. So let me write the sum. And over here, we're gonna say, did we win? All right, so let's start with experiment one."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Experiment. So let me write the sum. And over here, we're gonna say, did we win? All right, so let's start with experiment one. So our first roll, we got a one. Our second roll, we got a five. We're doing quite well."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "All right, so let's start with experiment one. So our first roll, we got a one. Our second roll, we got a five. We're doing quite well. And then our third roll, we got a six. Did we win? Well, one plus five plus six is 12."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "We're doing quite well. And then our third roll, we got a six. Did we win? Well, one plus five plus six is 12. Yes, we won. Let's do another experiment. This is going to be experiment two."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Well, one plus five plus six is 12. Yes, we won. Let's do another experiment. This is going to be experiment two. We can just keep going here. These are random digits. So we have a six in our first roll."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "This is going to be experiment two. We can just keep going here. These are random digits. So we have a six in our first roll. We got a two in our second roll. We got a four in our third roll. Did we win?"}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So we have a six in our first roll. We got a two in our second roll. We got a four in our third roll. Did we win? Yes, once again, this summed up to 12. So we won. All right, let's do another experiment."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Did we win? Yes, once again, this summed up to 12. So we won. All right, let's do another experiment. So experiment number three. So this first thing is invalid. So this is our first roll."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "All right, let's do another experiment. So experiment number three. So this first thing is invalid. So this is our first roll. We got a six. And then this is invalid. Our second roll, we get a three."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So this is our first roll. We got a six. And then this is invalid. Our second roll, we get a three. This is invalid, that is invalid, that is invalid. And then in our third roll, we got a two. So we squeaked by, this adds up to 11."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Our second roll, we get a three. This is invalid, that is invalid, that is invalid. And then in our third roll, we got a two. So we squeaked by, this adds up to 11. Yes, that looks like a win. All right, let's do our fourth experiment here. So our first roll, we got a one."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So we squeaked by, this adds up to 11. Yes, that looks like a win. All right, let's do our fourth experiment here. So our first roll, we got a one. This is invalid. Second roll, we got a two. This is invalid."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So our first roll, we got a one. This is invalid. Second roll, we got a two. This is invalid. Third roll, we get a five. Did we win? One plus two plus five is eight."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "This is invalid. Third roll, we get a five. Did we win? One plus two plus five is eight. No, we did not win. So that was our first non-win. So let's keep going."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "One plus two plus five is eight. No, we did not win. So that was our first non-win. So let's keep going. This is interesting. All right, this is invalid. So we're going to have, so this is trial five."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So let's keep going. This is interesting. All right, this is invalid. So we're going to have, so this is trial five. We are going to have four plus three plus one. Four plus three plus one adds up to eight. Did we win?"}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So we're going to have, so this is trial five. We are going to have four plus three plus one. Four plus three plus one adds up to eight. Did we win? No. Let's just keep going here. So I'm gonna keep going with my table where I have experiment."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Did we win? No. Let's just keep going here. So I'm gonna keep going with my table where I have experiment. I'll do five more trials. Experiment, sum, and do we win? Let me make the table."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So I'm gonna keep going with my table where I have experiment. I'll do five more trials. Experiment, sum, and do we win? Let me make the table. This is just a continuation of the table we had before. I don't wanna go below the page because I wanna be able to look at our random numbers here. So we are on to experiment six."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Let me make the table. This is just a continuation of the table we had before. I don't wanna go below the page because I wanna be able to look at our random numbers here. So we are on to experiment six. Experiment six, we are getting a three in the first roll, a three in the second roll. This isn't looking good. And then a two in our third roll."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So we are on to experiment six. Experiment six, we are getting a three in the first roll, a three in the second roll. This isn't looking good. And then a two in our third roll. Did we win? No, this is less than 10. Now we go to experiment seven."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "And then a two in our third roll. Did we win? No, this is less than 10. Now we go to experiment seven. Experiment seven, we get a two in our first roll. This is invalid. We get a three in our second roll, plus three."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Now we go to experiment seven. Experiment seven, we get a two in our first roll. This is invalid. We get a three in our second roll, plus three. And we get a one in our third roll, so plus one. Once again, we did not win. Now we go to experiment, we will go to experiment eight."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "We get a three in our second roll, plus three. And we get a one in our third roll, so plus one. Once again, we did not win. Now we go to experiment, we will go to experiment eight. We get a one in our first roll. We get a three in our second roll. This is invalid."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Now we go to experiment, we will go to experiment eight. We get a one in our first roll. We get a three in our second roll. This is invalid. The die fell off the table. We can think of it that way. And then in our third roll, we get a five, plus five."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "This is invalid. The die fell off the table. We can think of it that way. And then in our third roll, we get a five, plus five. Did we win? No, this adds up to nine. So we had a string of wins to begin with, but now we're getting a string of non-wins."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "And then in our third roll, we get a five, plus five. Did we win? No, this adds up to nine. So we had a string of wins to begin with, but now we're getting a string of non-wins. All right, now let's go to experiment nine. So we get a six in our first roll. We get a four in our second roll."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So we had a string of wins to begin with, but now we're getting a string of non-wins. All right, now let's go to experiment nine. So we get a six in our first roll. We get a four in our second roll. And then these are all invalid. And then we get a five in our third roll. Did we win here?"}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "We get a four in our second roll. And then these are all invalid. And then we get a five in our third roll. Did we win here? Yes, we won over here. This is definitely going to be greater than 10. This is 15 here."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Did we win here? Yes, we won over here. This is definitely going to be greater than 10. This is 15 here. All right, last experiment, or at least for this video, last experiment. You could keep going. In fact, I encourage you to after this to see if you can get a more accurate, a better approximation of the theoretical probability of winning by doing more experiments to calculate an experimental probability."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "This is 15 here. All right, last experiment, or at least for this video, last experiment. You could keep going. In fact, I encourage you to after this to see if you can get a more accurate, a better approximation of the theoretical probability of winning by doing more experiments to calculate an experimental probability. So here, experiment 10. First roll, we get a five. Second roll, we get a two."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "In fact, I encourage you to after this to see if you can get a more accurate, a better approximation of the theoretical probability of winning by doing more experiments to calculate an experimental probability. So here, experiment 10. First roll, we get a five. Second roll, we get a two. This is invalid, invalid, invalid. Then we get a six. Here, we definitely won."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Second roll, we get a two. This is invalid, invalid, invalid. Then we get a six. Here, we definitely won. So with 10 trials, based on 10 trials, or 10 experiments, what is our experimental probability of winning this game? Well, out of the 10 experiments, how many did we win? Well, it looks like we won one, two, three, four, five."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Here, we definitely won. So with 10 trials, based on 10 trials, or 10 experiments, what is our experimental probability of winning this game? Well, out of the 10 experiments, how many did we win? Well, it looks like we won one, two, three, four, five. So based on just these 10 experiments, we got a pretty clean 50%. So do you think the theoretical probability is actually 50%? Maybe you'd wanna continue running these experiments over and over."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "I encourage you to pause this video, read it, and see if you can figure out, is this a sample study? Is it an observational study? Is it an experiment? And then also think about what type of conclusions can you make based on the information in this study. All right, now let's work on this together. British researchers were interested in the relationship between farmers' approach to their cows and cows' milk yield. They prepared a survey questionnaire regarding the farmers' perception of the cows' mental capacity, the treatment they give to the cows, and the cows' yield."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "And then also think about what type of conclusions can you make based on the information in this study. All right, now let's work on this together. British researchers were interested in the relationship between farmers' approach to their cows and cows' milk yield. They prepared a survey questionnaire regarding the farmers' perception of the cows' mental capacity, the treatment they give to the cows, and the cows' yield. The survey was filled by all the farms in Great Britain. After analyzing their results, they found that on farms where cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. All right, so they're making a connection between two variables."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "They prepared a survey questionnaire regarding the farmers' perception of the cows' mental capacity, the treatment they give to the cows, and the cows' yield. The survey was filled by all the farms in Great Britain. After analyzing their results, they found that on farms where cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. All right, so they're making a connection between two variables. One was whether cows called by name, whether cows named, all right, whether cows named, and this would be a categorical variable because for any given farmer, it's gonna be a yes or a no that the cows are named. And so they're trying to form a connection between whether the cows are named and milk yield. And this would be a quantitative variable because you're measuring it in terms of number of liters."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "All right, so they're making a connection between two variables. One was whether cows called by name, whether cows named, all right, whether cows named, and this would be a categorical variable because for any given farmer, it's gonna be a yes or a no that the cows are named. And so they're trying to form a connection between whether the cows are named and milk yield. And this would be a quantitative variable because you're measuring it in terms of number of liters. Milk yield, whether we are drawing a connection. And they're able to draw some form of a connection. They're saying, hey, when the cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "And this would be a quantitative variable because you're measuring it in terms of number of liters. Milk yield, whether we are drawing a connection. And they're able to draw some form of a connection. They're saying, hey, when the cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. So first, let's just think about what type of statistical study this is. And we could think, okay, is this a sample study? Is this a sample study?"}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "They're saying, hey, when the cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. So first, let's just think about what type of statistical study this is. And we could think, okay, is this a sample study? Is this a sample study? Is this an observational study? Observational, or is this an experiment? Now, a sample study, an experiment, a sample study, you would be trying to estimate a parameter for a broader population."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "Is this a sample study? Is this an observational study? Observational, or is this an experiment? Now, a sample study, an experiment, a sample study, you would be trying to estimate a parameter for a broader population. Here, it's not so much that they're estimating the parameter they're trying to see the connection between two variables. And that brings us to observational study because that's what an observational study is all about. Can we draw a connection?"}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "Now, a sample study, an experiment, a sample study, you would be trying to estimate a parameter for a broader population. Here, it's not so much that they're estimating the parameter they're trying to see the connection between two variables. And that brings us to observational study because that's what an observational study is all about. Can we draw a connection? Can we draw a positive or a negative correlation between variables based on observations? So we've surveyed a population here, the farmers in Great Britain, and we are able to draw some type of connection between these variables. And so this is clearly an observational study."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "Can we draw a connection? Can we draw a positive or a negative correlation between variables based on observations? So we've surveyed a population here, the farmers in Great Britain, and we are able to draw some type of connection between these variables. And so this is clearly an observational study. Now, this is not an experiment. If there was an experiment, we would take the farmers and we would randomly assign them into one or two groups. And in one group, we would say, don't name, no name, no naming."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "And so this is clearly an observational study. Now, this is not an experiment. If there was an experiment, we would take the farmers and we would randomly assign them into one or two groups. And in one group, we would say, don't name, no name, no naming. And in the other group, we would say, name your cows. And then we would wait some period of time and we would see the average milk production going into the experiment in the no naming group and the naming group. And then we will see, we wait some period of time, six months, a year, and then we will see the average milk production after either not naming or naming the cows for six months."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "And in one group, we would say, don't name, no name, no naming. And in the other group, we would say, name your cows. And then we would wait some period of time and we would see the average milk production going into the experiment in the no naming group and the naming group. And then we will see, we wait some period of time, six months, a year, and then we will see the average milk production after either not naming or naming the cows for six months. So that's not what occurred here. Here, we just did the survey to everybody and we just asked them this question and we were able to find this connection between whether the cows were named and the actual milk yield. So clearly, not an experiment."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "And then we will see, we wait some period of time, six months, a year, and then we will see the average milk production after either not naming or naming the cows for six months. So that's not what occurred here. Here, we just did the survey to everybody and we just asked them this question and we were able to find this connection between whether the cows were named and the actual milk yield. So clearly, not an experiment. This was an observational study. Now, the next thing is what can we conclude here? We know when, they're telling us that when the cows were named, it looks like there was a 258 liter higher yield on average."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "So clearly, not an experiment. This was an observational study. Now, the next thing is what can we conclude here? We know when, they're telling us that when the cows were named, it looks like there was a 258 liter higher yield on average. So the conclusion that we can strictly make here is like, well, for farmers in Great Britain, there is a correlation, a positive correlation between whether cows are named and the milk yield. So that we can say for sure. So let me write that down."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "We know when, they're telling us that when the cows were named, it looks like there was a 258 liter higher yield on average. So the conclusion that we can strictly make here is like, well, for farmers in Great Britain, there is a correlation, a positive correlation between whether cows are named and the milk yield. So that we can say for sure. So let me write that down. So for Great Britain, for Great Britain farmers, Great Britain farmers, we have a positive correlation. Positive correlation between naming cows between naming cows and milk yield. And milk yield."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "So let me write that down. So for Great Britain, for Great Britain farmers, Great Britain farmers, we have a positive correlation. Positive correlation between naming cows between naming cows and milk yield. And milk yield. That's pretty much what we can say here. Now, some people might be tempted to try to draw causality. You'll see this all the time where you see these observational studies and people try to hint that maybe there's a causal relationship here."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "And milk yield. That's pretty much what we can say here. Now, some people might be tempted to try to draw causality. You'll see this all the time where you see these observational studies and people try to hint that maybe there's a causal relationship here. Maybe the naming is actually what makes the milk yield go up or maybe it's the other way. The cows produce a lot of milk, the farmers like them more and they wanna name them. It's like, hey, that's my high milk producing cow."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "You'll see this all the time where you see these observational studies and people try to hint that maybe there's a causal relationship here. Maybe the naming is actually what makes the milk yield go up or maybe it's the other way. The cows produce a lot of milk, the farmers like them more and they wanna name them. It's like, hey, that's my high milk producing cow. So there's a lot of temptation to say naming, that maybe there's a causality that naming causes more milk, more milk, or that maybe more milk causes naming. You or the farmers really like that cow so they start naming them or whatever it might be. But you can't make this causal relationship based on this observational study."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "It's like, hey, that's my high milk producing cow. So there's a lot of temptation to say naming, that maybe there's a causality that naming causes more milk, more milk, or that maybe more milk causes naming. You or the farmers really like that cow so they start naming them or whatever it might be. But you can't make this causal relationship based on this observational study. You might have been able to do it with a well-constructed experiment but not with an observational study. And that's because there could be some confounding variable that is driving both of them. So for example, that confounding variable might just be a nice farmer."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "But you can't make this causal relationship based on this observational study. You might have been able to do it with a well-constructed experiment but not with an observational study. And that's because there could be some confounding variable that is driving both of them. So for example, that confounding variable might just be a nice farmer. A nice farmer. And we can define nice in a lot of ways. They're gentle."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "So for example, that confounding variable might just be a nice farmer. A nice farmer. And we can define nice in a lot of ways. They're gentle. And a nice farmer is more likely to name and a nice farmer is more likely to get, it gets a higher yield. And the reason why this is a confounding variable, if you were to control for that, if you just take, well, let's just control for nice farmers and then see if naming makes a difference, it might not make a difference. If the farmer is petting the cows and treating them humanely and doing other things, it might not matter whether the farmer names them or not."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "They're gentle. And a nice farmer is more likely to name and a nice farmer is more likely to get, it gets a higher yield. And the reason why this is a confounding variable, if you were to control for that, if you just take, well, let's just control for nice farmers and then see if naming makes a difference, it might not make a difference. If the farmer is petting the cows and treating them humanely and doing other things, it might not matter whether the farmer names them or not. Likewise, if you take some less nice farmers who hit their cows and they have really inhumane conditions, it might not make a difference whether they name the cows or not. And so it's very important that you, from the observational studies, you might, if they're well-constructed, you might be able to make a, you might be able to say there's a correlation. But you won't be able to make a, drive a causal or make a causal conclusion."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "However, based on their experiences eating the cereal at home, a group of students believes that the proportion of boxes with vouchers is less than 20%. This group of students purchased 65 boxes of the cereal to investigate the company's claim. The student found a total of 11 vouchers for free video rentals in the 65 boxes. Suppose it is reasonable to assume that the 65 boxes purchased by the students are a random sample of all boxes of this cereal. Based on this sample, is there support for the student's belief that the proportion of boxes with vouchers is less than 20%? Provide statistical evidence to support your answer. And so, like always, pause this video and see if you can answer it by yourself."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "Suppose it is reasonable to assume that the 65 boxes purchased by the students are a random sample of all boxes of this cereal. Based on this sample, is there support for the student's belief that the proportion of boxes with vouchers is less than 20%? Provide statistical evidence to support your answer. And so, like always, pause this video and see if you can answer it by yourself. And this actually is a question from an AP statistics exam. All right, now let's work through this together. And I'm going to try to model some of what you might wanna do if you were actually trying to answer this on an exam."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "And so, like always, pause this video and see if you can answer it by yourself. And this actually is a question from an AP statistics exam. All right, now let's work through this together. And I'm going to try to model some of what you might wanna do if you were actually trying to answer this on an exam. So the first thing you might wanna say is, well, what's our null and our alternative hypothesis? Well, our null hypothesis would be, well, the reality is what the breakfast brand claims, that 20% of the boxes contain a voucher. So that would be our null hypothesis."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "And I'm going to try to model some of what you might wanna do if you were actually trying to answer this on an exam. So the first thing you might wanna say is, well, what's our null and our alternative hypothesis? Well, our null hypothesis would be, well, the reality is what the breakfast brand claims, that 20% of the boxes contain a voucher. So that would be our null hypothesis. And our alternative hypothesis would be what we suspect, that the true proportion of boxes that contain a voucher is actually less than 20%. Now, if you're going to do a significance test, it's good practice to set up your significance level that you're going to eventually compare your p-value to ahead of time. And so, let's say we would want to assume, assume significance level."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "So that would be our null hypothesis. And our alternative hypothesis would be what we suspect, that the true proportion of boxes that contain a voucher is actually less than 20%. Now, if you're going to do a significance test, it's good practice to set up your significance level that you're going to eventually compare your p-value to ahead of time. And so, let's say we would want to assume, assume significance level. So let me write this, significance, significance level, alpha. Let's just go with 0.05. And then we'll wanna think about the sample."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "And so, let's say we would want to assume, assume significance level. So let me write this, significance, significance level, alpha. Let's just go with 0.05. And then we'll wanna think about the sample. And we're gonna figure out, if we assume that the null hypothesis is true, what's the probability that we get the, the sample proportion that we do, and if that is below this significance level, then we would reject the null hypothesis. And so, what we know about the sample, we know that we took 65 boxes of cereal, n is equal to 65, they tell us that right over there. And we, from that, we can calculate what the sample proportion is."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "And then we'll wanna think about the sample. And we're gonna figure out, if we assume that the null hypothesis is true, what's the probability that we get the, the sample proportion that we do, and if that is below this significance level, then we would reject the null hypothesis. And so, what we know about the sample, we know that we took 65 boxes of cereal, n is equal to 65, they tell us that right over there. And we, from that, we can calculate what the sample proportion is. It's going to be 11 out of 65, and we can get our calculator out. Calculators are allowed on this part of the exam. And so, what is 11 divided by 65?"}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "And we, from that, we can calculate what the sample proportion is. It's going to be 11 out of 65, and we can get our calculator out. Calculators are allowed on this part of the exam. And so, what is 11 divided by 65? It gives us, and I'll just round to the nearest thousand, 169, 0.169, 0.169, I'll say approximately, because I rounded it there. Now the next thing we wanna do before we make an inference is to make sure we're meeting the conditions for inference. So I'll write this down over here."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "And so, what is 11 divided by 65? It gives us, and I'll just round to the nearest thousand, 169, 0.169, 0.169, I'll say approximately, because I rounded it there. Now the next thing we wanna do before we make an inference is to make sure we're meeting the conditions for inference. So I'll write this down over here. Conditions for inference, conditions for inference. And this is to feel good that we are properly sampling the population, that our sampling distribution is going to be roughly normal. So the first one is random sample, that is truly a random sample."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "So I'll write this down over here. Conditions for inference, conditions for inference. And this is to feel good that we are properly sampling the population, that our sampling distribution is going to be roughly normal. So the first one is random sample, that is truly a random sample. And here they tell us, it is reasonable to assume that the 65 boxes purchased by the students are a random sample. So that checks that off, so I will just point that to that right over there. So that checks that off."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "So the first one is random sample, that is truly a random sample. And here they tell us, it is reasonable to assume that the 65 boxes purchased by the students are a random sample. So that checks that off, so I will just point that to that right over there. So that checks that off. The next one is the normal condition, that the shape is roughly normal and it isn't skewed dramatically one way or the other. And in order to meet that condition, the sample size times the true assumed proportion, and we're going to assume that the null hypothesis is true, and so we could say that, and we could even say that this is the proportion assumed in the null hypothesis, that's what that notation would imply. And if you're doing this on the actual test, you should explain your use of notation a little bit more than I might do for the sake of time."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "So that checks that off. The next one is the normal condition, that the shape is roughly normal and it isn't skewed dramatically one way or the other. And in order to meet that condition, the sample size times the true assumed proportion, and we're going to assume that the null hypothesis is true, and so we could say that, and we could even say that this is the proportion assumed in the null hypothesis, that's what that notation would imply. And if you're doing this on the actual test, you should explain your use of notation a little bit more than I might do for the sake of time. But this needs to be greater than or equal to 10, and n times one minus the assumed proportion needs to be greater than or equal to 10. Well, let's see, n is 65, so 65 times the assumed proportion is 0.2, that is going to be equal to 13. 13 is indeed greater than or equal to 10, so that checks off."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "And if you're doing this on the actual test, you should explain your use of notation a little bit more than I might do for the sake of time. But this needs to be greater than or equal to 10, and n times one minus the assumed proportion needs to be greater than or equal to 10. Well, let's see, n is 65, so 65 times the assumed proportion is 0.2, that is going to be equal to 13. 13 is indeed greater than or equal to 10, so that checks off. And then we would take n, 65, times one minus the assumed proportion, so 0.8, and that is going to be equal to, let's see, that would just be 65 minus 13, which is going to be equal to 52, and that indeed is also greater than or equal to 10, so we met that condition right over there. And then the last one is the independence, independence. We aren't sampling these boxes with replacement, so we need to feel good that they are less than 10% of the population of boxes."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "13 is indeed greater than or equal to 10, so that checks off. And then we would take n, 65, times one minus the assumed proportion, so 0.8, and that is going to be equal to, let's see, that would just be 65 minus 13, which is going to be equal to 52, and that indeed is also greater than or equal to 10, so we met that condition right over there. And then the last one is the independence, independence. We aren't sampling these boxes with replacement, so we need to feel good that they are less than 10% of the population of boxes. And they don't tell us that explicitly, but it would be good practice to just say, going to assume, assume more than, let's see, 10 times that, 650 boxes in the population, boxes in population, population, which would imply that n is less than 10%, or less than or equal to 10% of population, of population, which would allow us to check off the independence condition. And so given that we've met our conditions for inference, now let's think about the sampling distribution. So the sampling distribution of the sample proportions, because that's what we're going to use to calculate AP value."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "We aren't sampling these boxes with replacement, so we need to feel good that they are less than 10% of the population of boxes. And they don't tell us that explicitly, but it would be good practice to just say, going to assume, assume more than, let's see, 10 times that, 650 boxes in the population, boxes in population, population, which would imply that n is less than 10%, or less than or equal to 10% of population, of population, which would allow us to check off the independence condition. And so given that we've met our conditions for inference, now let's think about the sampling distribution. So the sampling distribution of the sample proportions, because that's what we're going to use to calculate AP value. So we know a few things about the sampling distribution of the sample proportions. We know that the mean of the sampling distribution of the sample proportions is just going to be the assumed true proportion, so that's the proportion from the null hypothesis. And we know that the standard deviation of the sampling distribution of the sample proportions, this is going to be equal to, and we've seen this in multiple videos already, this is the assumed proportion times one minus the assumed proportion from our null hypothesis divided by n, which in this case is going to be equal to 0.2 times 0.8, all of that over 65."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "So the sampling distribution of the sample proportions, because that's what we're going to use to calculate AP value. So we know a few things about the sampling distribution of the sample proportions. We know that the mean of the sampling distribution of the sample proportions is just going to be the assumed true proportion, so that's the proportion from the null hypothesis. And we know that the standard deviation of the sampling distribution of the sample proportions, this is going to be equal to, and we've seen this in multiple videos already, this is the assumed proportion times one minus the assumed proportion from our null hypothesis divided by n, which in this case is going to be equal to 0.2 times 0.8, all of that over 65. Once again, let's get our calculator out. So we're gonna have the square root of 0.2 times 0.8 divided by 65, and then close my parentheses. I get, so 0.0496."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "And we know that the standard deviation of the sampling distribution of the sample proportions, this is going to be equal to, and we've seen this in multiple videos already, this is the assumed proportion times one minus the assumed proportion from our null hypothesis divided by n, which in this case is going to be equal to 0.2 times 0.8, all of that over 65. Once again, let's get our calculator out. So we're gonna have the square root of 0.2 times 0.8 divided by 65, and then close my parentheses. I get, so 0.0496. So this is approximately 0.0496. Now the next step is to figure out the p-value, which we can then compare to our significance level to decide whether or not to reject the null hypothesis. And in order to calculate the p-value, let's figure out our z-statistic, which is how many standard deviations above or below the mean of the sampling distribution is the sample statistic that we happen to get for this sample of 65."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "I get, so 0.0496. So this is approximately 0.0496. Now the next step is to figure out the p-value, which we can then compare to our significance level to decide whether or not to reject the null hypothesis. And in order to calculate the p-value, let's figure out our z-statistic, which is how many standard deviations above or below the mean of the sampling distribution is the sample statistic that we happen to get for this sample of 65. And we have seen this in previous videos. This would be equal to our sample proportion minus the assumed proportion for the population in the null hypothesis, so the difference between those, and then divided by the standard deviation of the sampling distribution of the sample proportions. This would tell us how many standard deviations are we above or below the mean of the sampling distribution."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "And in order to calculate the p-value, let's figure out our z-statistic, which is how many standard deviations above or below the mean of the sampling distribution is the sample statistic that we happen to get for this sample of 65. And we have seen this in previous videos. This would be equal to our sample proportion minus the assumed proportion for the population in the null hypothesis, so the difference between those, and then divided by the standard deviation of the sampling distribution of the sample proportions. This would tell us how many standard deviations are we above or below the mean of the sampling distribution. So in this particular situation, this is going to be 0.169 minus 0.2, all of that over this value right over here, which is approximately 0.0496. I can get the calculator out again. And so we have 0.169 minus 0.2."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "This would tell us how many standard deviations are we above or below the mean of the sampling distribution. So in this particular situation, this is going to be 0.169 minus 0.2, all of that over this value right over here, which is approximately 0.0496. I can get the calculator out again. And so we have 0.169 minus 0.2. So that's how far below our sample proportion is than the mean of the sampling distribution, which is the assumed proportion from the null hypothesis, assumed population proportion. And then we divide that, we're gonna divide that by the standard deviation of the sampling distribution of the sample proportions. So divide that by 0.0496, and we get a z value of approximately, because remember, this is using a bunch of approximations right over here, about negative 0.625."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "And so we have 0.169 minus 0.2. So that's how far below our sample proportion is than the mean of the sampling distribution, which is the assumed proportion from the null hypothesis, assumed population proportion. And then we divide that, we're gonna divide that by the standard deviation of the sampling distribution of the sample proportions. So divide that by 0.0496, and we get a z value of approximately, because remember, this is using a bunch of approximations right over here, about negative 0.625. So z is approximately negative 0.625. And so now we can think about the actual p-value our p-value, which is equal to the probability of getting a sample proportion that is at least as low as the one that we got. So a sample proportion that is less than or equal to the one that we got, 0.169, assuming the null hypothesis is true."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "So divide that by 0.0496, and we get a z value of approximately, because remember, this is using a bunch of approximations right over here, about negative 0.625. So z is approximately negative 0.625. And so now we can think about the actual p-value our p-value, which is equal to the probability of getting a sample proportion that is at least as low as the one that we got. So a sample proportion that is less than or equal to the one that we got, 0.169, assuming the null hypothesis is true. So we could say assuming the null hypothesis is true, which is equal to the probability of getting a z statistic that is less than or equal to this value right over here, negative 0.625. And now we can use our calculator to actually calculate this. So what we can do is we can go to second distribution."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "So a sample proportion that is less than or equal to the one that we got, 0.169, assuming the null hypothesis is true. So we could say assuming the null hypothesis is true, which is equal to the probability of getting a z statistic that is less than or equal to this value right over here, negative 0.625. And now we can use our calculator to actually calculate this. So what we can do is we can go to second distribution. We wanna do normal CDF. So go to normal CDF. And then our lower bound is actually going to be, we could say negative infinity."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "So what we can do is we can go to second distribution. We wanna do normal CDF. So go to normal CDF. And then our lower bound is actually going to be, we could say negative infinity. Our upper bound is going to be negative, so negative 0.625, 625. And then this is, this is, you could say, a normalized normal distribution here. So we'll just go with all of this, because we're just thinking about the z statistic right over here."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "And then our lower bound is actually going to be, we could say negative infinity. Our upper bound is going to be negative, so negative 0.625, 625. And then this is, this is, you could say, a normalized normal distribution here. So we'll just go with all of this, because we're just thinking about the z statistic right over here. Click Enter, and then click Enter. And then we get, this is going to be, let's say, 0.266. So this is approximately 0.266."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "So we'll just go with all of this, because we're just thinking about the z statistic right over here. Click Enter, and then click Enter. And then we get, this is going to be, let's say, 0.266. So this is approximately 0.266. And so let's just make sure what we just did. If this right over here is the assumed sampling distribution of the sample proportions, where we are assuming that our null hypothesis is true, so the mean of our sampling distribution is going to be our assumed proportion, what we're saying is, look, we got a result over here. This is where our p hat happened to be, right over here."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "So this is approximately 0.266. And so let's just make sure what we just did. If this right over here is the assumed sampling distribution of the sample proportions, where we are assuming that our null hypothesis is true, so the mean of our sampling distribution is going to be our assumed proportion, what we're saying is, look, we got a result over here. This is where our p hat happened to be, right over here. What's the probability of getting a result that far below the true proportion or further? So this is what we calculated just now. And now when you look at this, this is almost a 27% probability."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "This is where our p hat happened to be, right over here. What's the probability of getting a result that far below the true proportion or further? So this is what we calculated just now. And now when you look at this, this is almost a 27% probability. When you compare our p value, we're gonna compare our p value to our significance level. And we see that our p value is clearly greater than our significance level. 0.266 is clearly greater than our significance level of 0.05."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "And now when you look at this, this is almost a 27% probability. When you compare our p value, we're gonna compare our p value to our significance level. And we see that our p value is clearly greater than our significance level. 0.266 is clearly greater than our significance level of 0.05. What we were saying is, if there was less than a 5% chance of getting the sample proportion that we got, then we would reject the null hypothesis, which would suggest the alternative. But here, the probability of getting the sample proportion that we got, if we assume that the null hypothesis is true, is almost 27%. And so that's well above our significance level."}, {"video_title": "Significance test for a proportion free response example AP Statistics Khan Academy.mp3", "Sentence": "0.266 is clearly greater than our significance level of 0.05. What we were saying is, if there was less than a 5% chance of getting the sample proportion that we got, then we would reject the null hypothesis, which would suggest the alternative. But here, the probability of getting the sample proportion that we got, if we assume that the null hypothesis is true, is almost 27%. And so that's well above our significance level. So we will fail, so because of this, because of this, we fail to reject, reject our null hypothesis. And from that, we can say, not enough evidence to suggest our alternative hypothesis. And if you have time, you might wanna say, there's not enough evidence to suggest that less than 20% of the boxes have the free video rental voucher that they talk about in the original problem description."}, {"video_title": "Sampling distribution of the difference in sample means Probability example Khan Academy.mp3", "Sentence": "And we figured out the mean of that sampling distribution, we figured out the standard deviation of that sampling distribution, and we were able to establish that it is normal. So that the sampling distribution of the difference of sample means will look something like this. So it is normal. So it is going to have that classic bell shape like that. We know its mean is at five grams. So this is five grams. And we know that a standard deviation is roughly 0.79 grams."}, {"video_title": "Sampling distribution of the difference in sample means Probability example Khan Academy.mp3", "Sentence": "So it is going to have that classic bell shape like that. We know its mean is at five grams. So this is five grams. And we know that a standard deviation is roughly 0.79 grams. So this is one standard deviation above, maybe two standard deviations, three. This is one standard deviation below, two standard deviations below, three standard deviations below. This right over here would be 5.79 grams."}, {"video_title": "Sampling distribution of the difference in sample means Probability example Khan Academy.mp3", "Sentence": "And we know that a standard deviation is roughly 0.79 grams. So this is one standard deviation above, maybe two standard deviations, three. This is one standard deviation below, two standard deviations below, three standard deviations below. This right over here would be 5.79 grams. This over here would be 4.21 grams. Now, if we wanna find the probability that the mean weights from the samples are more than six grams apart, well, remember, this is the difference between the sample mean from A and the sample mean from B. And so there's one situation where A is more than six grams larger than B, or our sample mean from A is more than six grams larger than the sample mean from B."}, {"video_title": "Sampling distribution of the difference in sample means Probability example Khan Academy.mp3", "Sentence": "This right over here would be 5.79 grams. This over here would be 4.21 grams. Now, if we wanna find the probability that the mean weights from the samples are more than six grams apart, well, remember, this is the difference between the sample mean from A and the sample mean from B. And so there's one situation where A is more than six grams larger than B, or our sample mean from A is more than six grams larger than the sample mean from B. And that would be this area right over here. If six is right around here, it would be that area. But there's also a possibility that the sample mean from B will be larger, will be more than six grams larger than the sample mean from A."}, {"video_title": "Sampling distribution of the difference in sample means Probability example Khan Academy.mp3", "Sentence": "And so there's one situation where A is more than six grams larger than B, or our sample mean from A is more than six grams larger than the sample mean from B. And that would be this area right over here. If six is right around here, it would be that area. But there's also a possibility that the sample mean from B will be larger, will be more than six grams larger than the sample mean from A. So if you really were to extend this far to the left, and you're really not gonna be able to see it much like this. So this is five, maybe this gets us to about zero right over here, and then you're going to get negative six someplace out here. There is some area under the curve where this difference is more negative than negative six, or another way to think about it, it's less than negative six."}, {"video_title": "Sampling distribution of the difference in sample means Probability example Khan Academy.mp3", "Sentence": "But there's also a possibility that the sample mean from B will be larger, will be more than six grams larger than the sample mean from A. So if you really were to extend this far to the left, and you're really not gonna be able to see it much like this. So this is five, maybe this gets us to about zero right over here, and then you're going to get negative six someplace out here. There is some area under the curve where this difference is more negative than negative six, or another way to think about it, it's less than negative six. So to figure out this probability, we need to calculate both of these areas. Now you could do that with a Z table, and we've done that in many examples before, or we could use some type of online tool or old online calculator. So this is at a site, stapplet.com slash normal.html."}, {"video_title": "Sampling distribution of the difference in sample means Probability example Khan Academy.mp3", "Sentence": "There is some area under the curve where this difference is more negative than negative six, or another way to think about it, it's less than negative six. So to figure out this probability, we need to calculate both of these areas. Now you could do that with a Z table, and we've done that in many examples before, or we could use some type of online tool or old online calculator. So this is at a site, stapplet.com slash normal.html. And what we can do is, we know we have a normal distribution that has a mean of five grams, the standard deviation is 0.79, and then we can plot the distribution, there we have there, and then we can calculate the area. Well, we could calculate the area outside of a region. And then that region, the left boundary, would be negative six, and then the right boundary would be six."}, {"video_title": "Sampling distribution of the difference in sample means Probability example Khan Academy.mp3", "Sentence": "So this is at a site, stapplet.com slash normal.html. And what we can do is, we know we have a normal distribution that has a mean of five grams, the standard deviation is 0.79, and then we can plot the distribution, there we have there, and then we can calculate the area. Well, we could calculate the area outside of a region. And then that region, the left boundary, would be negative six, and then the right boundary would be six. And then if we calculate that area, it is, you see the right boundary right over here, or the part that's greater than the right boundary, but there's a little bit of, be negligible, that's far to the left that we're not seeing as well. But there you have it, that area, which is mostly the one that you're seeing here visualized, you're not seeing the one on the left, it is 10.28%, the combination of the two. And so that's our probability."}, {"video_title": "Reading bar charts comparing two sets of data Pre-Algebra Khan Academy.mp3", "Sentence": "This is the scores on midterm and final exams. So this axis, the vertical axis, is the scores. And then it's by student. And the blue bar is the midterm. And the yellow bar is the final. And the question they ask us is, by how many points did Nadia's score improve from the midterm to the final exam? So let's look at Nadia."}, {"video_title": "Reading bar charts comparing two sets of data Pre-Algebra Khan Academy.mp3", "Sentence": "And the blue bar is the midterm. And the yellow bar is the final. And the question they ask us is, by how many points did Nadia's score improve from the midterm to the final exam? So let's look at Nadia. So this is who we're talking about, Nadia. And we care about how many points did she improve from the midterm to the final. Midterm is blue, final is yellow."}, {"video_title": "Reading bar charts comparing two sets of data Pre-Algebra Khan Academy.mp3", "Sentence": "So let's look at Nadia. So this is who we're talking about, Nadia. And we care about how many points did she improve from the midterm to the final. Midterm is blue, final is yellow. So in the midterm, it looks like she scored, and if I were to eyeball it, it looks like 75 points. And on the final, it looks like she scored 80. Looks like she scored 85 points."}, {"video_title": "Reading bar charts comparing two sets of data Pre-Algebra Khan Academy.mp3", "Sentence": "Midterm is blue, final is yellow. So in the midterm, it looks like she scored, and if I were to eyeball it, it looks like 75 points. And on the final, it looks like she scored 80. Looks like she scored 85 points. So it looks like her score improved by 10 points. 10 points. Let's try one more."}, {"video_title": "Reading bar charts comparing two sets of data Pre-Algebra Khan Academy.mp3", "Sentence": "Looks like she scored 85 points. So it looks like her score improved by 10 points. 10 points. Let's try one more. How many students improve their scores from the midterm to the final exam? So to improve from the midterm to the final, that means that the yellow bar for a given student, which is the final, is going to be higher than the midterm bar. That's the only way you can improve from the midterm to the final."}, {"video_title": "Reading bar charts comparing two sets of data Pre-Algebra Khan Academy.mp3", "Sentence": "Let's try one more. How many students improve their scores from the midterm to the final exam? So to improve from the midterm to the final, that means that the yellow bar for a given student, which is the final, is going to be higher than the midterm bar. That's the only way you can improve from the midterm to the final. So Brandon improved from the midterm to the final. Vanessa improved from the midterm to the final. Daniel improved from the midterm to the final."}, {"video_title": "Reading bar charts comparing two sets of data Pre-Algebra Khan Academy.mp3", "Sentence": "That's the only way you can improve from the midterm to the final. So Brandon improved from the midterm to the final. Vanessa improved from the midterm to the final. Daniel improved from the midterm to the final. Kevin improved from the midterm to the final. William got a lower score on the final than the midterm. So he did not improve."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "Let's say that we've come up with a new pill that we think has a good chance of helping people with diabetes control their blood sugar. When someone has diabetes, their blood sugar is unusually high, that damages their body in a bunch of different ways. So we want to conduct an experiment to test if this pill really can help people lower their blood sugar. So the first thing we need to think about is how do we even measure or test whether people's blood sugar is getting lowered? Well, for our experiment, what's typically done is we measure folks' hemoglobin A1c. You don't have to worry too much about this in the context of statistics, but a hemoglobin A1c test is a way that's typically used to measure your average blood sugar over the last three months. And we have whole videos on Khan Academy explaining how that works."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "So the first thing we need to think about is how do we even measure or test whether people's blood sugar is getting lowered? Well, for our experiment, what's typically done is we measure folks' hemoglobin A1c. You don't have to worry too much about this in the context of statistics, but a hemoglobin A1c test is a way that's typically used to measure your average blood sugar over the last three months. And we have whole videos on Khan Academy explaining how that works. So our hope would be that our pill lowers people's blood sugar, which shows up as a lowered A1c. Now we have terms for this. The thing that is causing something else to change, we call this the explanatory variable."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "And we have whole videos on Khan Academy explaining how that works. So our hope would be that our pill lowers people's blood sugar, which shows up as a lowered A1c. Now we have terms for this. The thing that is causing something else to change, we call this the explanatory variable. Explanatory variable. And the thing that might get changed by that explanatory variable, depending on whether you take the pill or not, we call that our response. Response variable."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "The thing that is causing something else to change, we call this the explanatory variable. Explanatory variable. And the thing that might get changed by that explanatory variable, depending on whether you take the pill or not, we call that our response. Response variable. So now let's actually conduct the experiment. So what we would do is we would go to the population, population of diabetics, and we would wanna take a random sample from that population of diabetics, a reasonably large one, and later in statistics we talk about what a good-sized sample might be. But let's say that we randomly sample, randomly sample 100 folks."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "Response variable. So now let's actually conduct the experiment. So what we would do is we would go to the population, population of diabetics, and we would wanna take a random sample from that population of diabetics, a reasonably large one, and later in statistics we talk about what a good-sized sample might be. But let's say that we randomly sample, randomly sample 100 folks. So we randomly sample 100 folks from that population of diabetics. And then you would want to assign these folks randomly to two different groups. One would be your control group, and this would be the group of people who won't take the new medicine."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "But let's say that we randomly sample, randomly sample 100 folks. So we randomly sample 100 folks from that population of diabetics. And then you would want to assign these folks randomly to two different groups. One would be your control group, and this would be the group of people who won't take the new medicine. And then you would have your treatment group. These are the groups of folks who will be given the new medicine, the treatment group. Now in some cases you can just randomly assign these 100 folks between these two groups."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "One would be your control group, and this would be the group of people who won't take the new medicine. And then you would have your treatment group. These are the groups of folks who will be given the new medicine, the treatment group. Now in some cases you can just randomly assign these 100 folks between these two groups. And one way to do it is you could give all of them a random number between one and 100, and then the top 50 go into treatment, and the bottom 50 go into the control, or you could use a computer to randomly assign folks. Now sometimes you might wanna be a little bit more sophisticated than that. For example, there might be evidence that someone's sex might somehow influence how they respond to a drug."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "Now in some cases you can just randomly assign these 100 folks between these two groups. And one way to do it is you could give all of them a random number between one and 100, and then the top 50 go into treatment, and the bottom 50 go into the control, or you could use a computer to randomly assign folks. Now sometimes you might wanna be a little bit more sophisticated than that. For example, there might be evidence that someone's sex might somehow influence how they respond to a drug. So what you could do is something called block design, where let's say this group just happens to have 60 females and 40 males. Well in block design, you can randomly assign, but you can do it in a way that you can ensure that both of these groups have the same proportions of male and female. So for example, if you have 60 females here, you can ensure that 30 of them end up in the control, and 30 end up in the treatment."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "For example, there might be evidence that someone's sex might somehow influence how they respond to a drug. So what you could do is something called block design, where let's say this group just happens to have 60 females and 40 males. Well in block design, you can randomly assign, but you can do it in a way that you can ensure that both of these groups have the same proportions of male and female. So for example, if you have 60 females here, you can ensure that 30 of them end up in the control, and 30 end up in the treatment. But you would assign those 60 females randomly between these two groups, and similarly, you can do block design. Of these 40 males, 20 end up in the control, and 20 end up in the treatment. So once you have folks in both of these groups, what you would probably want to do is measure their A1c at the beginning."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "So for example, if you have 60 females here, you can ensure that 30 of them end up in the control, and 30 end up in the treatment. But you would assign those 60 females randomly between these two groups, and similarly, you can do block design. Of these 40 males, 20 end up in the control, and 20 end up in the treatment. So once you have folks in both of these groups, what you would probably want to do is measure their A1c at the beginning. You could view that as a baseline. And then over the course of the experiment, you would give the pill to the treatment group, and in the control group, you might be saying, oh, we just wouldn't do anything. But the best practice is actually to give a pill that looks just like the real thing to the control group."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "So once you have folks in both of these groups, what you would probably want to do is measure their A1c at the beginning. You could view that as a baseline. And then over the course of the experiment, you would give the pill to the treatment group, and in the control group, you might be saying, oh, we just wouldn't do anything. But the best practice is actually to give a pill that looks just like the real thing to the control group. This is known as a placebo. And the reason why we do that is there's definitely evidence that when people think they're taking a pill that might help them, that even psychologically, it can have an effect on them, and sometimes it helps them. This is known as the placebo effect."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "But the best practice is actually to give a pill that looks just like the real thing to the control group. This is known as a placebo. And the reason why we do that is there's definitely evidence that when people think they're taking a pill that might help them, that even psychologically, it can have an effect on them, and sometimes it helps them. This is known as the placebo effect. And not only would you give both groups a pill that looks the same, even though this one in the treatment group actually has the medicine in it, you also would not want to tell folks which group they are in. When you don't tell them which group they are in, that's known as a blind experiment. And you probably also don't wanna tell the people who are administering the experiment which group they are administering, and that's called a double blind."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "This is known as the placebo effect. And not only would you give both groups a pill that looks the same, even though this one in the treatment group actually has the medicine in it, you also would not want to tell folks which group they are in. When you don't tell them which group they are in, that's known as a blind experiment. And you probably also don't wanna tell the people who are administering the experiment which group they are administering, and that's called a double blind. So even the doctors or the nurses that are administering the experiment, when they're giving a pill to the control group, they don't know that that pill is the placebo. And you might say, well, why is it important for an experiment to be blind or especially double blind? Well, that avoids, one, any type of psychological effect from the point of the patient, or from the, say, the caregivers in this situation."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "And you probably also don't wanna tell the people who are administering the experiment which group they are administering, and that's called a double blind. So even the doctors or the nurses that are administering the experiment, when they're giving a pill to the control group, they don't know that that pill is the placebo. And you might say, well, why is it important for an experiment to be blind or especially double blind? Well, that avoids, one, any type of psychological effect from the point of the patient, or from the, say, the caregivers in this situation. So they don't kind of give it away. They don't tell these folks, hey, you're actually just pretending to take a pill. And so that ensures that we minimize the amount of influence or bias that might happen."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "Well, that avoids, one, any type of psychological effect from the point of the patient, or from the, say, the caregivers in this situation. So they don't kind of give it away. They don't tell these folks, hey, you're actually just pretending to take a pill. And so that ensures that we minimize the amount of influence or bias that might happen. You might even have a triple blind experiment where even the folks who are analyzing the eventual data from this experiment don't know whether they're analyzing the data from the control or the treatment. They just compare the two different groups. But anyway, you do, you, people take the medicine and the placebo over the course of the experiment."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "And so that ensures that we minimize the amount of influence or bias that might happen. You might even have a triple blind experiment where even the folks who are analyzing the eventual data from this experiment don't know whether they're analyzing the data from the control or the treatment. They just compare the two different groups. But anyway, you do, you, people take the medicine and the placebo over the course of the experiment. Maybe this lasts for three months. And then you would wanna measure their A1c later. And then you would see their change in the A1c."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "But anyway, you do, you, people take the medicine and the placebo over the course of the experiment. Maybe this lasts for three months. And then you would wanna measure their A1c later. And then you would see their change in the A1c. Now, if you saw that there wasn't really a difference in the change in A1c between the control and the treatment group, then you'd say, well, that probably means that my pill didn't work. Now, if you do get a greater reduction in the treatment group and you do the statistical analysis, which we will learn in statistics, and you show that, hey, there's a very low probability this happened purely due to chance, well, then you've got something. You could probably conclude that there is a causal connection between taking the pill and lowering your A1c level."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "And then you would see their change in the A1c. Now, if you saw that there wasn't really a difference in the change in A1c between the control and the treatment group, then you'd say, well, that probably means that my pill didn't work. Now, if you do get a greater reduction in the treatment group and you do the statistical analysis, which we will learn in statistics, and you show that, hey, there's a very low probability this happened purely due to chance, well, then you've got something. You could probably conclude that there is a causal connection between taking the pill and lowering your A1c level. But once again, you cannot be 100% sure. And so this is why it's very important for people to be able to replicate your experiment. Because what you'd wanna do, either yourself or other researchers, might wanna conduct the experiment with different sample sizes in different countries and different populations, maybe with different ages at different times of year, to ensure that they continue to see this result."}, {"video_title": "Confidence interval for hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "Data from a recent semester show that 80% of students passed the in-person setting and 75% of students passed the online setting. They were willing to treat these as representative samples of all students who may take each setting of the course. The teachers used those results to make a 95% confidence interval to estimate the difference between the proportion of students who pass in each setting of the course. So this is a 95% confidence interval for the difference between the proportion who pass the in-person course and the online course. The resulting interval was approximately, went from negative 0.04 to 0.14. And just to make sure we understand what this is saying, this is saying 95% of the time that you go through this, because we're talking about a 95% confidence interval, 95% of the time you take these samples and then you construct a confidence interval for the difference in proportions that it will actually contain the true proportion. They want to use this interval to test their null hypothesis that the true proportions are the same versus their alternative hypothesis that their true proportions are different."}, {"video_title": "Confidence interval for hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "So this is a 95% confidence interval for the difference between the proportion who pass the in-person course and the online course. The resulting interval was approximately, went from negative 0.04 to 0.14. And just to make sure we understand what this is saying, this is saying 95% of the time that you go through this, because we're talking about a 95% confidence interval, 95% of the time you take these samples and then you construct a confidence interval for the difference in proportions that it will actually contain the true proportion. They want to use this interval to test their null hypothesis that the true proportions are the same versus their alternative hypothesis that their true proportions are different. Assume that all conditions for inference have been met. Based on the interval, what do we know about the corresponding p-value and conclusion to their test? So pause this video and try to figure it out on your own."}, {"video_title": "Confidence interval for hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "They want to use this interval to test their null hypothesis that the true proportions are the same versus their alternative hypothesis that their true proportions are different. Assume that all conditions for inference have been met. Based on the interval, what do we know about the corresponding p-value and conclusion to their test? So pause this video and try to figure it out on your own. All right, so what's interesting here is we're going to use a confidence interval to think about a hypothesis test. And remember, in a hypothesis test, we assume that our null hypothesis is true. We'll assume this, and there's another way we could write it."}, {"video_title": "Confidence interval for hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "So pause this video and try to figure it out on your own. All right, so what's interesting here is we're going to use a confidence interval to think about a hypothesis test. And remember, in a hypothesis test, we assume that our null hypothesis is true. We'll assume this, and there's another way we could write it. We could write it like this, that the difference between the in-person and the online true proportions is equal to zero. These are equivalent statements. In a hypothesis test, we will assume that this is true."}, {"video_title": "Confidence interval for hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "We'll assume this, and there's another way we could write it. We could write it like this, that the difference between the in-person and the online true proportions is equal to zero. These are equivalent statements. In a hypothesis test, we will assume that this is true. And then in a traditional hypothesis test, we set some significance level. And so let's say we set that significance level at 5%, and that is a very typical significance level. And if the results we get, if the probability of getting the results that we do get for the difference in the sample proportions is less than 5%, we say, hey, that's pretty unlikely."}, {"video_title": "Confidence interval for hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "In a hypothesis test, we will assume that this is true. And then in a traditional hypothesis test, we set some significance level. And so let's say we set that significance level at 5%, and that is a very typical significance level. And if the results we get, if the probability of getting the results that we do get for the difference in the sample proportions is less than 5%, we say, hey, that's pretty unlikely. We're gonna reject the null hypothesis, which will suggest the alternative. But here we have something interesting. We have a confidence interval."}, {"video_title": "Confidence interval for hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "And if the results we get, if the probability of getting the results that we do get for the difference in the sample proportions is less than 5%, we say, hey, that's pretty unlikely. We're gonna reject the null hypothesis, which will suggest the alternative. But here we have something interesting. We have a confidence interval. And it turns out that if the sum of your confidence level and your significance level is equal to 100%, and you're doing a two-sided hypothesis test, so you're thinking about, well, our alternative hypothesis isn't just that the in-person is greater than the online or that it's less than the online. It's that they are different. So we have a two-sided hypothesis test."}, {"video_title": "Confidence interval for hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "We have a confidence interval. And it turns out that if the sum of your confidence level and your significance level is equal to 100%, and you're doing a two-sided hypothesis test, so you're thinking about, well, our alternative hypothesis isn't just that the in-person is greater than the online or that it's less than the online. It's that they are different. So we have a two-sided hypothesis test. In these situations, you can actually make some inferences about your p-value from your confidence interval. Think about it this way. We are assuming our null hypothesis is true when we do this hypothesis test."}, {"video_title": "Confidence interval for hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "So we have a two-sided hypothesis test. In these situations, you can actually make some inferences about your p-value from your confidence interval. Think about it this way. We are assuming our null hypothesis is true when we do this hypothesis test. And so when we construct a 95% confidence interval, we would expect that 95% of confidence intervals, of confidence intervals, would overlap, overlap with zero. Where did I get zero from? Remember, this is a confidence interval for the difference in proportions."}, {"video_title": "Confidence interval for hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "We are assuming our null hypothesis is true when we do this hypothesis test. And so when we construct a 95% confidence interval, we would expect that 95% of confidence intervals, of confidence intervals, would overlap, overlap with zero. Where did I get zero from? Remember, this is a confidence interval for the difference in proportions. And our null hypothesis is that the true difference in proportions is zero. So 95% of the time that we do this, if we assume that the null hypothesis is true, we will overlap with zero. Or another way you could think about it, 5% of confidence intervals, confidence intervals, would not overlap, overlap with zero."}, {"video_title": "Confidence interval for hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "Remember, this is a confidence interval for the difference in proportions. And our null hypothesis is that the true difference in proportions is zero. So 95% of the time that we do this, if we assume that the null hypothesis is true, we will overlap with zero. Or another way you could think about it, 5% of confidence intervals, confidence intervals, would not overlap, overlap with zero. And so if you are in a situation where you go through this process, you try to construct a 95% confidence interval, and you don't overlap with your assumed difference of the true proportions from your null hypothesis, well, in this situation, your p-value is going to be less than your 5% significance level. So in this situation, you would reject, you would reject your null hypothesis. And in this first situation, your p-value is going to be greater than or equal to your alpha level, and you would fail to reject, fail to reject."}, {"video_title": "Confidence interval for hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "Or another way you could think about it, 5% of confidence intervals, confidence intervals, would not overlap, overlap with zero. And so if you are in a situation where you go through this process, you try to construct a 95% confidence interval, and you don't overlap with your assumed difference of the true proportions from your null hypothesis, well, in this situation, your p-value is going to be less than your 5% significance level. So in this situation, you would reject, you would reject your null hypothesis. And in this first situation, your p-value is going to be greater than or equal to your alpha level, and you would fail to reject, fail to reject. So what's the situation here? Well, our interval actually does include the assumed difference in true proportions from the null hypothesis. So that means, assuming the null hypothesis, we are in this first scenario."}, {"video_title": "Confidence interval for hypothesis test for difference in proportions AP Statistics Khan Academy.mp3", "Sentence": "And in this first situation, your p-value is going to be greater than or equal to your alpha level, and you would fail to reject, fail to reject. So what's the situation here? Well, our interval actually does include the assumed difference in true proportions from the null hypothesis. So that means, assuming the null hypothesis, we are in this first scenario. This is one of the 95% of confidence intervals where we actually did overlap with the true parameter that we are trying to estimate. And so in that situation, our p-value is going to be greater than or equal to our alpha, which in this case is 5%. And so we fail to reject the null hypothesis."}, {"video_title": "Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "Now right over here, this table describes the probability distribution for X. And as you can see, X can take on only a finite number of values, zero, one, two, three, or four, and so because there's a finite number of values here, we would call this a discrete random variable. And you can see that this is a valid probability distribution because the combined probability is one. .1 plus 0.15 plus 0.4 plus 0.25 plus 0.1 is one, and none of these are negative probabilities, which wouldn't have made sense. But what we care about in this video is the notion of an expected value of a discrete random variable, which you would just note this way. And one way to think about it is, once we calculate the expected value of this variable, of this random variable, that in a given week, that would give you a sense of the expected number of workouts. This is also sometimes referred to as the mean of a random variable."}, {"video_title": "Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": ".1 plus 0.15 plus 0.4 plus 0.25 plus 0.1 is one, and none of these are negative probabilities, which wouldn't have made sense. But what we care about in this video is the notion of an expected value of a discrete random variable, which you would just note this way. And one way to think about it is, once we calculate the expected value of this variable, of this random variable, that in a given week, that would give you a sense of the expected number of workouts. This is also sometimes referred to as the mean of a random variable. This right over here is the Greek letter mu, which is often used to denote the mean, so this is the mean of the random variable X. But how do we actually compute it? To compute this, we essentially just take the weighted sum of the various outcomes, and we weight them by the probabilities."}, {"video_title": "Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "This is also sometimes referred to as the mean of a random variable. This right over here is the Greek letter mu, which is often used to denote the mean, so this is the mean of the random variable X. But how do we actually compute it? To compute this, we essentially just take the weighted sum of the various outcomes, and we weight them by the probabilities. So for example, this is going to be, the first outcome here is zero, and we'll weight it by its probability of 0.1. So it's zero times 0.1, plus the next outcome is one, and it would be weighted by its probability of 0.15, so plus one times 0.15, plus the next outcome is two, it has a probability of 0.4, plus two times 0.4, plus the outcome three has a probability of 0.25, plus three times 0.25, and then last but not least, we have the outcome four workouts in a week that has a probability of 0.1, plus four times 0.1. Well, we can simplify this a little bit."}, {"video_title": "Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "To compute this, we essentially just take the weighted sum of the various outcomes, and we weight them by the probabilities. So for example, this is going to be, the first outcome here is zero, and we'll weight it by its probability of 0.1. So it's zero times 0.1, plus the next outcome is one, and it would be weighted by its probability of 0.15, so plus one times 0.15, plus the next outcome is two, it has a probability of 0.4, plus two times 0.4, plus the outcome three has a probability of 0.25, plus three times 0.25, and then last but not least, we have the outcome four workouts in a week that has a probability of 0.1, plus four times 0.1. Well, we can simplify this a little bit. Zero times anything is just a zero, so one times 0.15 is 0.15, two times 0.4 is 0.8, three times 0.25 is 0.75, and then four times 0.1 is 0.4. And so we just have to add up these numbers. So we get 0.15 plus 0.8 plus 0.75 plus 0.4, and let's say 0.4, 0.75, 0.8, let's add them all together, and so let's see, five plus five is 10, and then this is two plus eight is 10, plus seven is 17, plus four is 21."}, {"video_title": "Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "Well, we can simplify this a little bit. Zero times anything is just a zero, so one times 0.15 is 0.15, two times 0.4 is 0.8, three times 0.25 is 0.75, and then four times 0.1 is 0.4. And so we just have to add up these numbers. So we get 0.15 plus 0.8 plus 0.75 plus 0.4, and let's say 0.4, 0.75, 0.8, let's add them all together, and so let's see, five plus five is 10, and then this is two plus eight is 10, plus seven is 17, plus four is 21. So we get all of this is going to be equal to 2.1. So one way to think about it is, the expected value of x, the expected number of workouts for me in a week, given this probability distribution, is 2.1. Now you might be saying, wait, hold on a second."}, {"video_title": "Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "So we get 0.15 plus 0.8 plus 0.75 plus 0.4, and let's say 0.4, 0.75, 0.8, let's add them all together, and so let's see, five plus five is 10, and then this is two plus eight is 10, plus seven is 17, plus four is 21. So we get all of this is going to be equal to 2.1. So one way to think about it is, the expected value of x, the expected number of workouts for me in a week, given this probability distribution, is 2.1. Now you might be saying, wait, hold on a second. All of the outcomes here are whole numbers. How can you have 2.1 workouts in a week? What is 0.1 of a workout?"}, {"video_title": "Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "Now you might be saying, wait, hold on a second. All of the outcomes here are whole numbers. How can you have 2.1 workouts in a week? What is 0.1 of a workout? Well, this isn't saying that in a given week, you would expect me to work out exactly 2.1 times, but this is valuable because you could say, well, in 10 weeks, you would expect me to do roughly 21 workouts. Sometimes I might do zero workouts, sometimes one, sometimes two, sometimes three, sometimes four, but in 100 weeks, you might expect me to do 210 workouts. So even for a random variable that can only take on integer values, you can still have a non-integer expected value, and it is still useful."}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "And right over here, we have its probability distribution. And I've drawn it as a bell curve, as a normal distribution right over here, but it could have many other distributions, but for the visualization's sake, it's a normal one in this example. And I've also drawn the mean of this distribution right over here, and I've also drawn one standard deviation above the mean and one standard deviation below the mean. What we're going to do in this video is think about how does this distribution, and in particular, how does the mean and the standard deviation get affected if we were to add to this random variable or if we were to scale this random variable. So let's first think about what would happen if we have another random variable, which is equal to, let's call this random variable y, which is equal to whatever the random variable x is, and we're going to add a constant. So let's say we add, so we're gonna add some constant here. I'll do a lowercase k. This is not a random variable."}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "What we're going to do in this video is think about how does this distribution, and in particular, how does the mean and the standard deviation get affected if we were to add to this random variable or if we were to scale this random variable. So let's first think about what would happen if we have another random variable, which is equal to, let's call this random variable y, which is equal to whatever the random variable x is, and we're going to add a constant. So let's say we add, so we're gonna add some constant here. I'll do a lowercase k. This is not a random variable. This is a constant. It could be the number 10. So if these are random heights of people walking out of the mall, well, you're just gonna add 10 inches to their height for some reason."}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "I'll do a lowercase k. This is not a random variable. This is a constant. It could be the number 10. So if these are random heights of people walking out of the mall, well, you're just gonna add 10 inches to their height for some reason. Maybe you wanna figure out, well, the distribution of people's heights with helmets on or plumed hats or whatever it might be, how would that affect, how would the mean of y and the standard deviation of y relate to x? So we could visualize that. So what the distribution of y would look like, so instead of this, instead of the center of the distribution, instead of the mean here being right at this point, it's going to be shifted up by k. In fact, we can shift, the entire distribution would be shifted to the right by k in this example."}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "So if these are random heights of people walking out of the mall, well, you're just gonna add 10 inches to their height for some reason. Maybe you wanna figure out, well, the distribution of people's heights with helmets on or plumed hats or whatever it might be, how would that affect, how would the mean of y and the standard deviation of y relate to x? So we could visualize that. So what the distribution of y would look like, so instead of this, instead of the center of the distribution, instead of the mean here being right at this point, it's going to be shifted up by k. In fact, we can shift, the entire distribution would be shifted to the right by k in this example. And maybe k is quite large. Maybe it looks something like that. This is my distribution for my random variable y here."}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "So what the distribution of y would look like, so instead of this, instead of the center of the distribution, instead of the mean here being right at this point, it's going to be shifted up by k. In fact, we can shift, the entire distribution would be shifted to the right by k in this example. And maybe k is quite large. Maybe it looks something like that. This is my distribution for my random variable y here. And you can see that the distribution has just shifted to the right by k. So we have moved to the right by k. We would have moved to the left if k was negative or if we were subtracting k. And so this clearly changes the mean. The mean is going to now be k larger. So we can write that down."}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "This is my distribution for my random variable y here. And you can see that the distribution has just shifted to the right by k. So we have moved to the right by k. We would have moved to the left if k was negative or if we were subtracting k. And so this clearly changes the mean. The mean is going to now be k larger. So we can write that down. We can say that the mean of our random variable y is equal to the mean of x, the mean of x of our random variable x plus k, plus k. You see that right over here. But has the standard deviation changed? Well, remember, standard deviation is a way of measuring typical spread from the mean, and that won't change."}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "So we can write that down. We can say that the mean of our random variable y is equal to the mean of x, the mean of x of our random variable x plus k, plus k. You see that right over here. But has the standard deviation changed? Well, remember, standard deviation is a way of measuring typical spread from the mean, and that won't change. So for our random variable x, this length right over here is one standard deviation. Well, that's also going to be the same as one standard deviation here. This is one standard deviation here."}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, remember, standard deviation is a way of measuring typical spread from the mean, and that won't change. So for our random variable x, this length right over here is one standard deviation. Well, that's also going to be the same as one standard deviation here. This is one standard deviation here. This is going to be the same as our standard deviation for our random variable y. And so we can say the standard deviation of y, of our random variable y, is equal to the standard deviation of our random variable x. So if you just add to a random variable, it would change the mean, but not the standard deviation."}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "This is one standard deviation here. This is going to be the same as our standard deviation for our random variable y. And so we can say the standard deviation of y, of our random variable y, is equal to the standard deviation of our random variable x. So if you just add to a random variable, it would change the mean, but not the standard deviation. You see it visually here. Now, what if you were to scale a random variable? So what if I have another random variable?"}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "So if you just add to a random variable, it would change the mean, but not the standard deviation. You see it visually here. Now, what if you were to scale a random variable? So what if I have another random variable? I don't know, let's call it z. And let's say z is equal to some constant, some constant times x. And so remember, this isn't, the k is not a random variable, it's just gonna be a number."}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "So what if I have another random variable? I don't know, let's call it z. And let's say z is equal to some constant, some constant times x. And so remember, this isn't, the k is not a random variable, it's just gonna be a number. It could be, say, the number two. Well, let's think about what would happen. So let me redraw the distribution for our random variable x."}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "And so remember, this isn't, the k is not a random variable, it's just gonna be a number. It could be, say, the number two. Well, let's think about what would happen. So let me redraw the distribution for our random variable x. So let's see, if k were two, what would happen is, is this distribution would be scaled out, it would be stretched out by two, and since the area always has to be one, it would actually be flattened down by a scale of two as well, so it still has the same area. So I can do that with my little drawing tool here. Let me try to, first I'm going to stretch it out by, whoops, first, actually, I'll make it shorter by a factor of two, but more importantly, it is going to be stretched out by a factor of two."}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "So let me redraw the distribution for our random variable x. So let's see, if k were two, what would happen is, is this distribution would be scaled out, it would be stretched out by two, and since the area always has to be one, it would actually be flattened down by a scale of two as well, so it still has the same area. So I can do that with my little drawing tool here. Let me try to, first I'm going to stretch it out by, whoops, first, actually, I'll make it shorter by a factor of two, but more importantly, it is going to be stretched out by a factor of two. So let me align the axes here so that we can appreciate this. So it's going to look something like this. It's going to look something like this when you scale the random variable."}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "Let me try to, first I'm going to stretch it out by, whoops, first, actually, I'll make it shorter by a factor of two, but more importantly, it is going to be stretched out by a factor of two. So let me align the axes here so that we can appreciate this. So it's going to look something like this. It's going to look something like this when you scale the random variable. This is what the distribution of our random variable z is going to look like. I'll do it in the z's color so that it's clear. And so you can see two things."}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "It's going to look something like this when you scale the random variable. This is what the distribution of our random variable z is going to look like. I'll do it in the z's color so that it's clear. And so you can see two things. One, the mean for sure shifted. The mean here for sure got pushed out. It definitely got scaled up."}, {"video_title": "Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3", "Sentence": "And so you can see two things. One, the mean for sure shifted. The mean here for sure got pushed out. It definitely got scaled up. But also, we see that the standard deviations got scaled, that the standard deviation right over here of z, that this has been scaled, it actually turns out that it's been scaled by a factor of k. So this is going to be equal to k times the standard deviation of our random variable x. And it turns out that our mean right over here, so let me write that too, that our mean of our random variable z is going to be equal to, that's also going to be scaled up, times, or it's gonna be k times the mean of our random variable x. So the big takeaways here, if you have one random variable that's constructed by adding a constant to another random variable, it's going to shift the mean by that constant, but it's not going to affect the standard deviation."}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "They wondered if this held true in their own town. So they took a sample of 200 residents to test the null hypothesis, is that the unemployment rate is the same as the national one versus the alternative hypothesis, which is that the unemployment rate is not the same as the national, where P is the proportion of residents in the town that are unemployed. The sample included 22 residents who were unemployed. Assuming that the conditions for inference have been met, and so that's the random, normal, and independence conditions that we've talked about in previous videos, identify the correct test statistic for this significance test. So let me just, I like to rewrite everything just to make sure I've understood what's going on. We have a null hypothesis that the true proportion of unemployed people in our town, that's what this P represents, is the same as the national unemployment. And remember, a null hypothesis tends to be the no news here, nothing to report, so to speak."}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Assuming that the conditions for inference have been met, and so that's the random, normal, and independence conditions that we've talked about in previous videos, identify the correct test statistic for this significance test. So let me just, I like to rewrite everything just to make sure I've understood what's going on. We have a null hypothesis that the true proportion of unemployed people in our town, that's what this P represents, is the same as the national unemployment. And remember, a null hypothesis tends to be the no news here, nothing to report, so to speak. And we have our alternative hypothesis that no, the true unemployment in this town is different, is different than 8%. And so what we would do is we would set some type of a significance level. We would assume that the mayor of the town sets it."}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "And remember, a null hypothesis tends to be the no news here, nothing to report, so to speak. And we have our alternative hypothesis that no, the true unemployment in this town is different, is different than 8%. And so what we would do is we would set some type of a significance level. We would assume that the mayor of the town sets it. Let's say he sets or she sets a significance level of 0.5. And then what we wanna do is conduct the experiment. So this is the entire population of the town."}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "We would assume that the mayor of the town sets it. Let's say he sets or she sets a significance level of 0.5. And then what we wanna do is conduct the experiment. So this is the entire population of the town. They take a sample of 200 people. So this is our sample. N is equal to 200."}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "So this is the entire population of the town. They take a sample of 200 people. So this is our sample. N is equal to 200. Since it met the independence condition, we'll assume that this is less than 10% of the population. And we calculate a sample statistic here. And it would be, since we care about the true population proportion, the sample statistic we would care about is the sample proportion."}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "N is equal to 200. Since it met the independence condition, we'll assume that this is less than 10% of the population. And we calculate a sample statistic here. And it would be, since we care about the true population proportion, the sample statistic we would care about is the sample proportion. And we figure out that it is 22 out of the 200 people in the sample are unemployed. So this is 0.11. Now the next step is, assuming the null hypothesis is true, what is the probability of getting a result this far away or further from the assumed population proportion?"}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "And it would be, since we care about the true population proportion, the sample statistic we would care about is the sample proportion. And we figure out that it is 22 out of the 200 people in the sample are unemployed. So this is 0.11. Now the next step is, assuming the null hypothesis is true, what is the probability of getting a result this far away or further from the assumed population proportion? And if that probability is lower than alpha, then we would reject the null hypothesis, which would suggest the alternative. But how do you figure out this probability? Well, one way to think about it is we could say how many standard deviations away from the true proportion, the assumed proportion, is it?"}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Now the next step is, assuming the null hypothesis is true, what is the probability of getting a result this far away or further from the assumed population proportion? And if that probability is lower than alpha, then we would reject the null hypothesis, which would suggest the alternative. But how do you figure out this probability? Well, one way to think about it is we could say how many standard deviations away from the true proportion, the assumed proportion, is it? And then we could say, what's the probability of getting that many standard deviations or further from the true proportion? We could use a z-table to do that. And so what we wanna do is figure out the number of standard deviations."}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Well, one way to think about it is we could say how many standard deviations away from the true proportion, the assumed proportion, is it? And then we could say, what's the probability of getting that many standard deviations or further from the true proportion? We could use a z-table to do that. And so what we wanna do is figure out the number of standard deviations. And so that would be a z-statistic. And so how do we figure it out? Well, we can figure out the difference between the sample proportion here and the assumed population proportion."}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "And so what we wanna do is figure out the number of standard deviations. And so that would be a z-statistic. And so how do we figure it out? Well, we can figure out the difference between the sample proportion here and the assumed population proportion. So that would be 0.11 minus 0.08 divided by the standard deviation of the sampling distribution of the sample proportions. And we can figure that out. Remember, all that is is, and sometimes we say, well, we don't know what the population proportion is."}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Well, we can figure out the difference between the sample proportion here and the assumed population proportion. So that would be 0.11 minus 0.08 divided by the standard deviation of the sampling distribution of the sample proportions. And we can figure that out. Remember, all that is is, and sometimes we say, well, we don't know what the population proportion is. But here we're assuming a population proportion. So we're assuming it is 0.08. And then we'll multiply that times one minus 0.08."}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Remember, all that is is, and sometimes we say, well, we don't know what the population proportion is. But here we're assuming a population proportion. So we're assuming it is 0.08. And then we'll multiply that times one minus 0.08. So we'll multiply that times 0.92. And this comes straight from, we've seen it in previous videos, the standard deviation of the sampling distribution of sample proportions. And then you divide that by n, which is 200 right over here."}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "And then we'll multiply that times one minus 0.08. So we'll multiply that times 0.92. And this comes straight from, we've seen it in previous videos, the standard deviation of the sampling distribution of sample proportions. And then you divide that by n, which is 200 right over here. And we could get a calculator out to figure this out. But this will give us some value, which just says how many standard deviations away from 0.08 is 0.11. And then we could use a z-table to figure out what's the probability of getting that far or further from the true proportion."}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "And then you divide that by n, which is 200 right over here. And we could get a calculator out to figure this out. But this will give us some value, which just says how many standard deviations away from 0.08 is 0.11. And then we could use a z-table to figure out what's the probability of getting that far or further from the true proportion. And then that will give us our p-value, which we can compare to significance level. Sometimes you will see a formula that looks something like this. And you say, hey, look, you have your sample proportion."}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "And then we could use a z-table to figure out what's the probability of getting that far or further from the true proportion. And then that will give us our p-value, which we can compare to significance level. Sometimes you will see a formula that looks something like this. And you say, hey, look, you have your sample proportion. You find the difference between that and the assumed proportion in the null hypothesis. That's what this little zero says. Now, this is the assumed population proportion from the null hypothesis."}, {"video_title": "Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "And you say, hey, look, you have your sample proportion. You find the difference between that and the assumed proportion in the null hypothesis. That's what this little zero says. Now, this is the assumed population proportion from the null hypothesis. And you divide that by the standard deviation, the assumed standard deviation of the sampling distribution of the sample proportions. So that would be our assumed population proportion times one minus our assumed population proportion divided by our sample size. And in future videos, we're gonna go all the way and calculate this and then look it up in a z-table and see what's the probability of getting that extreme or more extreme of a result and compare it to alpha."}, {"video_title": "Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "So what the stem and leaf plot does is it gives us the first digit in each number, and essentially you could say it called us the tens place, and then it gives us the ones place. So there was only one zoo that had four turtles, so you could view this as zero, zero, four, or four turtles. Then there's, let's see, so everything here, the tens place is a one, so this number right over here is really an 11, this is a 14, this right over here would be a 16, that's a 16, and so forth and so on, this would be a 17, 18. All of this, this is 23, this is 23, this is 26, because we have our tens place right over here, this is the first digit. So let's go ahead and answer the question, how many zoos had fewer than 46 turtles? So there are no zoos that had 40 anything turtles, and so all of these zoos here, so all of these had 30 something turtles, this, these had 20 something turtles, these have, in the teens, this has single digits. So it's literally as many zoos as we have listed here."}, {"video_title": "Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "All of this, this is 23, this is 23, this is 26, because we have our tens place right over here, this is the first digit. So let's go ahead and answer the question, how many zoos had fewer than 46 turtles? So there are no zoos that had 40 anything turtles, and so all of these zoos here, so all of these had 30 something turtles, this, these had 20 something turtles, these have, in the teens, this has single digits. So it's literally as many zoos as we have listed here. So one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17. 17 zoos have fewer than 46 turtles. Let's do another one."}, {"video_title": "Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "So it's literally as many zoos as we have listed here. So one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17. 17 zoos have fewer than 46 turtles. Let's do another one. The buyer for a chain of supermarkets created the following stem and leaf plot showing the number of coconuts at each of the stores. What was the smallest number of coconuts at any one grocery store? So the buyer for a chain of supermarkets created the following stem of leaf plots showing the number of coconuts at each of the stores."}, {"video_title": "Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "Let's do another one. The buyer for a chain of supermarkets created the following stem and leaf plot showing the number of coconuts at each of the stores. What was the smallest number of coconuts at any one grocery store? So the buyer for a chain of supermarkets created the following stem of leaf plots showing the number of coconuts at each of the stores. So at any one grocery store, the smallest number, well, that's this one right over here. And remember, it's not two, we have our tens places right over here, it's a one. So this right over here represents 12 coconuts at that store."}, {"video_title": "Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "So the buyer for a chain of supermarkets created the following stem of leaf plots showing the number of coconuts at each of the stores. So at any one grocery store, the smallest number, well, that's this one right over here. And remember, it's not two, we have our tens places right over here, it's a one. So this right over here represents 12 coconuts at that store. So we'll put 12 right over here. Let's try out another one. A statistician for a chain of department stores created the following stem and leaf plot showing the number of watches at each of the stores."}, {"video_title": "Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "So this right over here represents 12 coconuts at that store. So we'll put 12 right over here. Let's try out another one. A statistician for a chain of department stores created the following stem and leaf plot showing the number of watches at each of the stores. How many department stores have exactly seven watches? Well, that's only this one right over here, zero, seven, watches, this one right, this, this, and that one are not seven. This is representing 17 because it's in the row with one at the beginning."}, {"video_title": "Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "A statistician for a chain of department stores created the following stem and leaf plot showing the number of watches at each of the stores. How many department stores have exactly seven watches? Well, that's only this one right over here, zero, seven, watches, this one right, this, this, and that one are not seven. This is representing 17 because it's in the row with one at the beginning. This right over here represents 27 because it's in the row with the two at the beginning. So there's only one store that has exactly seven watches. Let's do one more, this is kind of fun."}, {"video_title": "Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "This is representing 17 because it's in the row with one at the beginning. This right over here represents 27 because it's in the row with the two at the beginning. So there's only one store that has exactly seven watches. Let's do one more, this is kind of fun. A zookeeper created the following stem and leaf plot showing the number of tigers at each major zoo in the country. How many zoos have more than 24 tigers? So we can ignore the zeros and the teens and we get into the 20s."}, {"video_title": "Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "Let's do one more, this is kind of fun. A zookeeper created the following stem and leaf plot showing the number of tigers at each major zoo in the country. How many zoos have more than 24 tigers? So we can ignore the zeros and the teens and we get into the 20s. This is 25, so that meets the criteria, and then you go to 28, 29. So all of these, all of these in the 30s, and all of these right over here, this three zero, this doesn't mean zero tigers, this is 30 tigers. This is 40 tigers."}, {"video_title": "Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3", "Sentence": "In the last video, we worked through, essentially, the probability distribution for this random variable defined as the number of free throws you make when taking six free throws, assuming you have a 70% free throw percentage. And I suggested that, hey, why don't you visualize this, draw, graph this probability distribution, this binomial probability distribution. And when I thought about it, I said, well, you know, I too would enjoy graphing it, and we might as well do it together, because whenever you graph these things, it makes it very visual, and kind of the shape of a binomial distribution like this. So let's do that. So let me maybe move over to the right a little bit. I really just need to be able to keep track of these things right over here. So let me draw some lines."}, {"video_title": "Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3", "Sentence": "So let's do that. So let me maybe move over to the right a little bit. I really just need to be able to keep track of these things right over here. So let me draw some lines. So let me, if I were to just draw one line there, and then another line here, and then we have the different percentages. So let's do that. See, the highest one is a little over 30%, 32%."}, {"video_title": "Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3", "Sentence": "So let me draw some lines. So let me, if I were to just draw one line there, and then another line here, and then we have the different percentages. So let's do that. See, the highest one is a little over 30%, 32%. So maybe we'll go as high as 40% here. 40%, and then this would be 20%. 20, that looks about halfway, 20%."}, {"video_title": "Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3", "Sentence": "See, the highest one is a little over 30%, 32%. So maybe we'll go as high as 40% here. 40%, and then this would be 20%. 20, that looks about halfway, 20%. This would be 10%. 10%, and this would be 30%. 30%."}, {"video_title": "Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3", "Sentence": "20, that looks about halfway, 20%. This would be 10%. 10%, and this would be 30%. 30%. And then in this axis, let's do the different values that the random variable could take on. So the random variable taking on the value zero. The random variable taking on the value one."}, {"video_title": "Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3", "Sentence": "30%. And then in this axis, let's do the different values that the random variable could take on. So the random variable taking on the value zero. The random variable taking on the value one. Zero, one. The random variable taking on two. Two, we're almost there."}, {"video_title": "Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3", "Sentence": "The random variable taking on the value one. Zero, one. The random variable taking on two. Two, we're almost there. Let's see, three. And then four. Four, and then five."}, {"video_title": "Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3", "Sentence": "Two, we're almost there. Let's see, three. And then four. Four, and then five. Five, and then finally six. Take x equals six. And then six, and now let's just graph all of these."}, {"video_title": "Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3", "Sentence": "Four, and then five. Five, and then finally six. Take x equals six. And then six, and now let's just graph all of these. So this first one, 0.1%, well that's barely gonna register on this graph right here, so I'll just kind of give it a little bit of a showing right over there. Actually, let me do it in that green color. So let me make sure, so in that green color, you're gonna have just a little bit of a showing."}, {"video_title": "Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3", "Sentence": "And then six, and now let's just graph all of these. So this first one, 0.1%, well that's barely gonna register on this graph right here, so I'll just kind of give it a little bit of a showing right over there. Actually, let me do it in that green color. So let me make sure, so in that green color, you're gonna have just a little bit of a showing. One as well is kind of barely a showing, so it shows up a little bit more. So let me draw it like that. That is 1% right over there."}, {"video_title": "Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3", "Sentence": "So let me make sure, so in that green color, you're gonna have just a little bit of a showing. One as well is kind of barely a showing, so it shows up a little bit more. So let me draw it like that. That is 1% right over there. Now two is 6%, which on this scale is gonna be about, it's about that high. So we draw it like that. So that is two, so that is 6%, right there."}, {"video_title": "Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3", "Sentence": "That is 1% right over there. Now two is 6%, which on this scale is gonna be about, it's about that high. So we draw it like that. So that is two, so that is 6%, right there. X equaling three, 18.5% shot of that happening. So 18.5, it gets us right about, it's a hand-drawn chart, or histogram, so you have to bear with me. So it's roughly there."}, {"video_title": "Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3", "Sentence": "So that is two, so that is 6%, right there. X equaling three, 18.5% shot of that happening. So 18.5, it gets us right about, it's a hand-drawn chart, or histogram, so you have to bear with me. So it's roughly there. And then four was 32.4%. So that is up here. So 32.4% is right, looks like that."}, {"video_title": "Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3", "Sentence": "So it's roughly there. And then four was 32.4%. So that is up here. So 32.4% is right, looks like that. So let me shade that in. 32.4%, and then five was 30.3%. So 30.3%, and slightly lower, just like that."}, {"video_title": "Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3", "Sentence": "So 32.4% is right, looks like that. So let me shade that in. 32.4%, and then five was 30.3%. So 30.3%, and slightly lower, just like that. And it looks like this. 30.3%, and finally six is 11.8%. So really this whole video was just an exercise in making a histogram, but it's useful because to actually visualize what the distribution looks like, and what's really interesting is to think about, well, how does this change as you change the free throw percentage, or as you change the number of shots you take, how does this change this binomial distribution?"}, {"video_title": "Interpreting two-way tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "They give us all of this data. They give us this, as they say, the two-way table of column relative frequencies. So for example, this column right over here is men. The column total is one, or you could say 100%. And we could see that 0.42 of the men, or 42% of the men, voted for Obama. We can see 52% of the men, or 0.52 of the men, voted for Romney. And we can see that the other, that the neither Obama, or 6% went for neither Obama or, nor Romney."}, {"video_title": "Interpreting two-way tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "The column total is one, or you could say 100%. And we could see that 0.42 of the men, or 42% of the men, voted for Obama. We can see 52% of the men, or 0.52 of the men, voted for Romney. And we can see that the other, that the neither Obama, or 6% went for neither Obama or, nor Romney. And for women, 52% went for Obama, 43% went for Romney, 5% went for other. And then these, this 52 plus 43 plus five will add up to 100% of the women. During the 2012 United States presidential election, were male voters more likely to vote for Romney than female voters?"}, {"video_title": "Interpreting two-way tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And we can see that the other, that the neither Obama, or 6% went for neither Obama or, nor Romney. And for women, 52% went for Obama, 43% went for Romney, 5% went for other. And then these, this 52 plus 43 plus five will add up to 100% of the women. During the 2012 United States presidential election, were male voters more likely to vote for Romney than female voters? So let's see. If we, there's a couple of ways you could think about it. Well, actually, let's do it this way."}, {"video_title": "Interpreting two-way tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "During the 2012 United States presidential election, were male voters more likely to vote for Romney than female voters? So let's see. If we, there's a couple of ways you could think about it. Well, actually, let's do it this way. Male voters, if you were a man, 52% of them voted for Romney, while for the women, 43% of them voted for Romney. So a man was more likely. There's a, if you randomly picked a man who voted, there was a 52% chance they voted for Romney, while if you randomly picked a woman, there was a 43%, a woman who voted, there was a 43% chance that she voted for Romney."}, {"video_title": "Interpreting two-way tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Well, actually, let's do it this way. Male voters, if you were a man, 52% of them voted for Romney, while for the women, 43% of them voted for Romney. So a man was more likely. There's a, if you randomly picked a man who voted, there was a 52% chance they voted for Romney, while if you randomly picked a woman, there was a 43%, a woman who voted, there was a 43% chance that she voted for Romney. So yes, male voters were more likely to vote for Romney than female voters. So the answer is yes. We're done."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "So let's define the random variable X. So let's say that X is equal to the number of made shots, number of made free throws when taking six free throws. So it's how many of the six do you make? And we're going to assume what we assumed in the first video in this series of these making free throws. So we're gonna assume the 70% free throw probability right over here. So assuming assumptions, assuming 70% free throw free throw percentage. All right, so let's figure out the probabilities of the different values that X could actually take on."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "And we're going to assume what we assumed in the first video in this series of these making free throws. So we're gonna assume the 70% free throw probability right over here. So assuming assumptions, assuming 70% free throw free throw percentage. All right, so let's figure out the probabilities of the different values that X could actually take on. So let's see, what is the probability, what is the probability that X is equal to zero? That even though you have a 70% free throw percentage, that you make none of the shots. And actually you could calculate this through probably some common sense without using all of these fancy things."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "All right, so let's figure out the probabilities of the different values that X could actually take on. So let's see, what is the probability, what is the probability that X is equal to zero? That even though you have a 70% free throw percentage, that you make none of the shots. And actually you could calculate this through probably some common sense without using all of these fancy things. But just to make things consistent, I'm gonna write it out. So this is going to be, it's going to be equal to six choose zero times 0.7 to the zeroth power times 0.3 to the sixth power. And this right over here is gonna end up being one."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "And actually you could calculate this through probably some common sense without using all of these fancy things. But just to make things consistent, I'm gonna write it out. So this is going to be, it's going to be equal to six choose zero times 0.7 to the zeroth power times 0.3 to the sixth power. And this right over here is gonna end up being one. This over here is going to end up being one. And so you're just gonna be left with 0.3 to the sixth power. And I calculated it ahead of time."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "And this right over here is gonna end up being one. This over here is going to end up being one. And so you're just gonna be left with 0.3 to the sixth power. And I calculated it ahead of time. So if we just round to the nearest, if we round our percentages to the nearest tenth, this is going to give you approximately, approximately, well if we round the decimal to the nearest thousandth, you're gonna get something like that, which is approximately equal to 0.1% chance of you missing all of them. So roughly, I'm speaking roughly here, one in a thousand, one in a thousand chance of that happening, of missing all six free throws. Now let's keep going."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "And I calculated it ahead of time. So if we just round to the nearest, if we round our percentages to the nearest tenth, this is going to give you approximately, approximately, well if we round the decimal to the nearest thousandth, you're gonna get something like that, which is approximately equal to 0.1% chance of you missing all of them. So roughly, I'm speaking roughly here, one in a thousand, one in a thousand chance of that happening, of missing all six free throws. Now let's keep going. This is fun. So what is the probability that our random variable is equal to one? Well this is going to be six choose one times 0.7 to the first power times 0.3 to the six minus first power, so that's the fifth power."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "Now let's keep going. This is fun. So what is the probability that our random variable is equal to one? Well this is going to be six choose one times 0.7 to the first power times 0.3 to the six minus first power, so that's the fifth power. And I calculated this, and this is approximately 0.01, or we could say 1%. So still a fairly low probability, 10 times more likely than this, roughly, but still a fairly low probability. Let's keep going."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "Well this is going to be six choose one times 0.7 to the first power times 0.3 to the six minus first power, so that's the fifth power. And I calculated this, and this is approximately 0.01, or we could say 1%. So still a fairly low probability, 10 times more likely than this, roughly, but still a fairly low probability. Let's keep going. So the probability that X is equal to two, well that's what our first video was, essentially. So this is going to be six choose two times 0.7 squared times 0.3 to the fourth power. And we saw that this is approximately going to be 0.06, or we could say 6%."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "Let's keep going. So the probability that X is equal to two, well that's what our first video was, essentially. So this is going to be six choose two times 0.7 squared times 0.3 to the fourth power. And we saw that this is approximately going to be 0.06, or we could say 6%. And obviously you could type these things in a calculator and get a much more precise answer, but just for the sake of just getting a sense of what these probabilities look like, that's why I'm giving these rough estimates. Kind of, I guess you could say to the closest, maybe tenth of a percent. And actually if you round to the closest tenth of a percent, you actually get to 6.0%, and this is 1.0%, because this we actually went to a tenth of a percent here."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "And we saw that this is approximately going to be 0.06, or we could say 6%. And obviously you could type these things in a calculator and get a much more precise answer, but just for the sake of just getting a sense of what these probabilities look like, that's why I'm giving these rough estimates. Kind of, I guess you could say to the closest, maybe tenth of a percent. And actually if you round to the closest tenth of a percent, you actually get to 6.0%, and this is 1.0%, because this we actually went to a tenth of a percent here. But let's keep going. We're obviously going to have to do a few more of these. So let me just make sure I have enough real estate."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "And actually if you round to the closest tenth of a percent, you actually get to 6.0%, and this is 1.0%, because this we actually went to a tenth of a percent here. But let's keep going. We're obviously going to have to do a few more of these. So let me just make sure I have enough real estate. All right, so the probability that our random variable is equal to three is going to be six choose three, and I'm sure you could probably fill this out on your own, but I'm going to do it. 0.7 to the third power times 0.3 to the six minus three, which is the third power, which is approximately equal to, well, it's going to be 0.185 or 18.5, 18.5%. So yeah, you know, that's definitely within the realm of possibility."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "So let me just make sure I have enough real estate. All right, so the probability that our random variable is equal to three is going to be six choose three, and I'm sure you could probably fill this out on your own, but I'm going to do it. 0.7 to the third power times 0.3 to the six minus three, which is the third power, which is approximately equal to, well, it's going to be 0.185 or 18.5, 18.5%. So yeah, you know, that's definitely within the realm of possibility. I mean, all of these are in the realm of possibility, but it's starting to be a non-insignificant probability. So now let's do the probability that our random variable is equal to four. Well, it's going to be six choose four times 0.7 to the fourth power times 0.3 to the six minus four, or second power, which is equal to, this is going to get equal to, or approximately, because I am taking away a little bit of the precision when I write these things down, 0.324, so approximately 32.4% chance of making exactly four out of the six free throws."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "So yeah, you know, that's definitely within the realm of possibility. I mean, all of these are in the realm of possibility, but it's starting to be a non-insignificant probability. So now let's do the probability that our random variable is equal to four. Well, it's going to be six choose four times 0.7 to the fourth power times 0.3 to the six minus four, or second power, which is equal to, this is going to get equal to, or approximately, because I am taking away a little bit of the precision when I write these things down, 0.324, so approximately 32.4% chance of making exactly four out of the six free throws. All right, two more to go. Let's see, I have not used purple as yet. So the probability that our random variable is equal to five, it's going to be six choose five, or times, I should say, 0.7 to the fifth power times 0.3 to the first power, and that is going to be roughly, roughly 0.303, which is 30.3%."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "Well, it's going to be six choose four times 0.7 to the fourth power times 0.3 to the six minus four, or second power, which is equal to, this is going to get equal to, or approximately, because I am taking away a little bit of the precision when I write these things down, 0.324, so approximately 32.4% chance of making exactly four out of the six free throws. All right, two more to go. Let's see, I have not used purple as yet. So the probability that our random variable is equal to five, it's going to be six choose five, or times, I should say, 0.7 to the fifth power times 0.3 to the first power, and that is going to be roughly, roughly 0.303, which is 30.3%. That's interesting, one more left. So the probability that I make all of them, of all six, is going to be equal to, is equal to six choose six, and 0.7 to the sixth power times 0.3 to the zeroth power, which is, this right over here is going to be one, this is going to be one, so it's really just 0.7 to the sixth power, and this is approximately 0.118, I calculated that ahead of time, which is 11.8%, and so there's something interesting that's going on here. The first time we looked at the binomial distribution, we said, hey, you know, there's this symmetry as we kind of got to some type of a peak and went down, but I don't see that symmetry here, and the reason why you're not seeing that symmetry is that you are more likely to make a free throw than not."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "So the probability that our random variable is equal to five, it's going to be six choose five, or times, I should say, 0.7 to the fifth power times 0.3 to the first power, and that is going to be roughly, roughly 0.303, which is 30.3%. That's interesting, one more left. So the probability that I make all of them, of all six, is going to be equal to, is equal to six choose six, and 0.7 to the sixth power times 0.3 to the zeroth power, which is, this right over here is going to be one, this is going to be one, so it's really just 0.7 to the sixth power, and this is approximately 0.118, I calculated that ahead of time, which is 11.8%, and so there's something interesting that's going on here. The first time we looked at the binomial distribution, we said, hey, you know, there's this symmetry as we kind of got to some type of a peak and went down, but I don't see that symmetry here, and the reason why you're not seeing that symmetry is that you are more likely to make a free throw than not. So you have a 70% free throw probability. This is no longer just flipping a fair coin. Where you will see the symmetry is in these coefficients."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "The first time we looked at the binomial distribution, we said, hey, you know, there's this symmetry as we kind of got to some type of a peak and went down, but I don't see that symmetry here, and the reason why you're not seeing that symmetry is that you are more likely to make a free throw than not. So you have a 70% free throw probability. This is no longer just flipping a fair coin. Where you will see the symmetry is in these coefficients. If you calculate these coefficients, six choose zero is one, six choose six is one. You would see that six choose one is six, and six choose five is six. You'd see six choose two is 15, and six choose four is also 15, and then six choose three is 20."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "Where you will see the symmetry is in these coefficients. If you calculate these coefficients, six choose zero is one, six choose six is one. You would see that six choose one is six, and six choose five is six. You'd see six choose two is 15, and six choose four is also 15, and then six choose three is 20. So you definitely see the symmetry in the coefficients, but then these things are weighted by the fact that you're more likely to make something than miss something. If these were both 0.5, then you would also see the symmetry right over here, and you can plot this to essentially visualize what the probability distribution looks like for this example, and I encourage you to do that, to take these different cases, just like we did in that first example with the fair coin, and plot these. But this essentially does give you the probability distribution for the random variable in question."}, {"video_title": "Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3", "Sentence": "You'd see six choose two is 15, and six choose four is also 15, and then six choose three is 20. So you definitely see the symmetry in the coefficients, but then these things are weighted by the fact that you're more likely to make something than miss something. If these were both 0.5, then you would also see the symmetry right over here, and you can plot this to essentially visualize what the probability distribution looks like for this example, and I encourage you to do that, to take these different cases, just like we did in that first example with the fair coin, and plot these. But this essentially does give you the probability distribution for the random variable in question. This is, I just wrote it out instead of just visualizing it, but it says, okay, well, so these are the different values that this random variable can take on. It can't take on negative one, or it can't be 15.5, or pi, or one million. These are the only seven values that this random variable can take on, and I've just given you the probabilities, or I guess you could say the rough probabilities, of the random variable taking on each of those seven values."}, {"video_title": "Sampling distribution of the difference in sample proportions -Probability example.mp3", "Sentence": "What we're gonna do in this video is build on that example and try to answer a little bit more about it. So in this situation, what we wanna do is find the probability, given what we already know about this sampling distribution's mean and standard deviation and shape. We wanna find the probability that the sample proportion of defects from plant B is greater than the sample proportion from plant A. So pause this video and see if you can figure this out. All right, now let's do this together. So first of all, let's just interpret what this is. The probability that the sample proportion of defects from plant B is greater than the sample proportion from plant A."}, {"video_title": "Sampling distribution of the difference in sample proportions -Probability example.mp3", "Sentence": "So pause this video and see if you can figure this out. All right, now let's do this together. So first of all, let's just interpret what this is. The probability that the sample proportion of defects from plant B is greater than the sample proportion from plant A. So if the sample proportion from plant B is greater than the proportion from plant A, then the difference between the sample proportions is going to be negative. So this is equivalent to the probability that the difference of the sample proportions, so the sample proportion from A minus the sample proportion from B is going to be less than zero. Or another way to think about it, that's going to be this area right over here."}, {"video_title": "Sampling distribution of the difference in sample proportions -Probability example.mp3", "Sentence": "The probability that the sample proportion of defects from plant B is greater than the sample proportion from plant A. So if the sample proportion from plant B is greater than the proportion from plant A, then the difference between the sample proportions is going to be negative. So this is equivalent to the probability that the difference of the sample proportions, so the sample proportion from A minus the sample proportion from B is going to be less than zero. Or another way to think about it, that's going to be this area right over here. Now, there's a bunch of ways that we can figure out this area, but the easiest, or one of the easiest, I guess there's many different ways to do it, is to figure out, well, how many, this is up to and including how many standard deviations below the mean, and then we could use a z-table. So what we just have to do is figure out what is the z-value here. And the z-value here, we just have to say, well, how many standard deviations below the mean is this?"}, {"video_title": "Sampling distribution of the difference in sample proportions -Probability example.mp3", "Sentence": "Or another way to think about it, that's going to be this area right over here. Now, there's a bunch of ways that we can figure out this area, but the easiest, or one of the easiest, I guess there's many different ways to do it, is to figure out, well, how many, this is up to and including how many standard deviations below the mean, and then we could use a z-table. So what we just have to do is figure out what is the z-value here. And the z-value here, we just have to say, well, how many standard deviations below the mean is this? And I'll do it up here. Let me square this off so I don't make it too messy. Z is going to be equal to, so we are negative 0.02 from the mean, or we're at 0.02 to the left of the mean, so I'll just do negative 0.02, and then over the standard deviation, which is 0.025, which is going to be equal to, get a calculator here, we get 0.02 divided by 0.025 is equal to that, and we are, of course, going to be to the left of the mean, so our z is going to be approximately negative 0.8, or 0.8, I'm saying approximately because this was approximate over here when we figured out the standard deviation."}, {"video_title": "Sampling distribution of the difference in sample proportions -Probability example.mp3", "Sentence": "And the z-value here, we just have to say, well, how many standard deviations below the mean is this? And I'll do it up here. Let me square this off so I don't make it too messy. Z is going to be equal to, so we are negative 0.02 from the mean, or we're at 0.02 to the left of the mean, so I'll just do negative 0.02, and then over the standard deviation, which is 0.025, which is going to be equal to, get a calculator here, we get 0.02 divided by 0.025 is equal to that, and we are, of course, going to be to the left of the mean, so our z is going to be approximately negative 0.8, or 0.8, I'm saying approximately because this was approximate over here when we figured out the standard deviation. So it is negative 0.8, and then we just have to use a z lookup table. And so if we look at a z lookup table, what we see here is if we're going to negative 0.8, negative 0.8 is right over here, so negative 0.8, and then we have zeros after that, so we're looking at this right over here. The area under the normal curve up to and including that z value, so we always have to make sure that we're looking at the right thing on this standard normal probabilities table right over here."}, {"video_title": "Sampling distribution of the difference in sample proportions -Probability example.mp3", "Sentence": "Z is going to be equal to, so we are negative 0.02 from the mean, or we're at 0.02 to the left of the mean, so I'll just do negative 0.02, and then over the standard deviation, which is 0.025, which is going to be equal to, get a calculator here, we get 0.02 divided by 0.025 is equal to that, and we are, of course, going to be to the left of the mean, so our z is going to be approximately negative 0.8, or 0.8, I'm saying approximately because this was approximate over here when we figured out the standard deviation. So it is negative 0.8, and then we just have to use a z lookup table. And so if we look at a z lookup table, what we see here is if we're going to negative 0.8, negative 0.8 is right over here, so negative 0.8, and then we have zeros after that, so we're looking at this right over here. The area under the normal curve up to and including that z value, so we always have to make sure that we're looking at the right thing on this standard normal probabilities table right over here. That gives us 0.21, or we could say this is approximately 21%. So let me get rid of this. And so we know that this right over here is approximately 21%, or we could say 0.21."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "The scores of the first four rounds and the lowest round are shown in the following dot plot. And we see it right over here, the lowest round she scores an 80. She also scores a 90 once, a 92 once, a 94 once, and a 96 once. It was discovered that Anna broke some rules when she scored 80. So that score, so I guess cheating didn't help her. So that score will be removed from the data set. So they removed that 80 right over there, or just left with the scores from the other four rounds."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "It was discovered that Anna broke some rules when she scored 80. So that score, so I guess cheating didn't help her. So that score will be removed from the data set. So they removed that 80 right over there, or just left with the scores from the other four rounds. How will the removal of the lowest round affect the mean and the median? So let's actually think about the median first. So the median is the middle number."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "So they removed that 80 right over there, or just left with the scores from the other four rounds. How will the removal of the lowest round affect the mean and the median? So let's actually think about the median first. So the median is the middle number. So over here, when you had five data points, the middle data point is gonna be the one that has two to the left and two to the right. So the median up here is going to be 92. The median up there is 92."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "So the median is the middle number. So over here, when you had five data points, the middle data point is gonna be the one that has two to the left and two to the right. So the median up here is going to be 92. The median up there is 92. And what's the median once you remove this? Now you only have four data points. When you're trying to find the median of an even number of numbers, you look at the middle two numbers."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "The median up there is 92. And what's the median once you remove this? Now you only have four data points. When you're trying to find the median of an even number of numbers, you look at the middle two numbers. So that's a 92 and a 94. And then you take the average of them. You go halfway between them to figure out the median."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "When you're trying to find the median of an even number of numbers, you look at the middle two numbers. So that's a 92 and a 94. And then you take the average of them. You go halfway between them to figure out the median. So the median here is going to be, let me do that a little bit clearer. The median over here is gonna be halfway between 92 and 94, which is 93. So the median, the median is 93."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "You go halfway between them to figure out the median. So the median here is going to be, let me do that a little bit clearer. The median over here is gonna be halfway between 92 and 94, which is 93. So the median, the median is 93. Median is 93. So removing the lowest data point, in this case, increased the median. So the median, let me write it down here."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "So the median, the median is 93. Median is 93. So removing the lowest data point, in this case, increased the median. So the median, let me write it down here. So the median increased by a little bit. The median increases. Now what's going to happen to the mean?"}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "So the median, let me write it down here. So the median increased by a little bit. The median increases. Now what's going to happen to the mean? What's going to happen to the mean? Well, one way to think about it, without even doing any calculations, is if you remove a number that is lower than the mean, lower than the existing mean, and I haven't calculated what the existing mean is, but if you remove that, the mean is going to go up. The mean is going to go up."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "Now what's going to happen to the mean? What's going to happen to the mean? Well, one way to think about it, without even doing any calculations, is if you remove a number that is lower than the mean, lower than the existing mean, and I haven't calculated what the existing mean is, but if you remove that, the mean is going to go up. The mean is going to go up. So hopefully that gives you some intuition. If you removed a number that's larger than the mean, your mean is going to go down, because you don't have that large number anymore. If you remove a number that's lower than the mean, well, you take that out, you don't have that small number bringing the average down, and so the mean will go up."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "The mean is going to go up. So hopefully that gives you some intuition. If you removed a number that's larger than the mean, your mean is going to go down, because you don't have that large number anymore. If you remove a number that's lower than the mean, well, you take that out, you don't have that small number bringing the average down, and so the mean will go up. But let's verify it mathematically. So let's calculate the mean over here. So we're going to add 80 plus 90 plus 92 plus 94 plus 96."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "If you remove a number that's lower than the mean, well, you take that out, you don't have that small number bringing the average down, and so the mean will go up. But let's verify it mathematically. So let's calculate the mean over here. So we're going to add 80 plus 90 plus 92 plus 94 plus 96. Those are our data points. And that gets us, two plus four is six, plus six is 12. And then we have one plus eight is nine."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "So we're going to add 80 plus 90 plus 92 plus 94 plus 96. Those are our data points. And that gets us, two plus four is six, plus six is 12. And then we have one plus eight is nine. And we essentially, this is, so these are nine, and you have another nine, another nine, another nine, another nine. You essentially have, this is five nines right over here. So this is going to be 450, 452."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "And then we have one plus eight is nine. And we essentially, this is, so these are nine, and you have another nine, another nine, another nine, another nine. You essentially have, this is five nines right over here. So this is going to be 450, 452. So that's the sum of the scores of these five rounds. And then you divide it by the number of rounds you have. So it'd be 452 divided by five."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "So this is going to be 450, 452. So that's the sum of the scores of these five rounds. And then you divide it by the number of rounds you have. So it'd be 452 divided by five. So 452 divided by five is going to give us, five goes into, doesn't go into four, it goes into 45 nine times. Nine times five is 45. You subtract, you get zero, bring down the two."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "So it'd be 452 divided by five. So 452 divided by five is going to give us, five goes into, doesn't go into four, it goes into 45 nine times. Nine times five is 45. You subtract, you get zero, bring down the two. Five goes into two zero times. Zero times five is, zero times five is zero. Subtract, you have two left over."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "You subtract, you get zero, bring down the two. Five goes into two zero times. Zero times five is, zero times five is zero. Subtract, you have two left over. So you can say that the mean here, the mean here is 90 and 2 5ths. Maybe, not nine and 2 5ths. 90 and 2 5ths."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "Subtract, you have two left over. So you can say that the mean here, the mean here is 90 and 2 5ths. Maybe, not nine and 2 5ths. 90 and 2 5ths. So the mean is right around here. So that's the mean of these data points right over there. And if you remove it, what is the mean going to be?"}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "90 and 2 5ths. So the mean is right around here. So that's the mean of these data points right over there. And if you remove it, what is the mean going to be? So here we're just going to take our 90 plus our 92 plus our 94 plus our 96. Add them together. So let's see, two plus four plus six is 12."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "And if you remove it, what is the mean going to be? So here we're just going to take our 90 plus our 92 plus our 94 plus our 96. Add them together. So let's see, two plus four plus six is 12. And you add these together, you're going to get 37. 372 divided by four. Because I have four data points now, not five."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "So let's see, two plus four plus six is 12. And you add these together, you're going to get 37. 372 divided by four. Because I have four data points now, not five. Four goes into three, let me do this in a place where you can see it. So four goes into 372. Goes into 37 nine times."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "Because I have four data points now, not five. Four goes into three, let me do this in a place where you can see it. So four goes into 372. Goes into 37 nine times. Nine times four is 36. Subtract, you get a one. Bring down the two."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "Goes into 37 nine times. Nine times four is 36. Subtract, you get a one. Bring down the two. It goes exactly three times. Three times four is 12. You have no remainder."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "Bring down the two. It goes exactly three times. Three times four is 12. You have no remainder. So the median and the mean here are both. So this is also the mean. The mean here is also 93."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "You have no remainder. So the median and the mean here are both. So this is also the mean. The mean here is also 93. So you see that the median, the median went from 92 to 93. It increased. The mean went from 90 and 2 5ths to 93."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "The mean here is also 93. So you see that the median, the median went from 92 to 93. It increased. The mean went from 90 and 2 5ths to 93. So the mean increased by more than the median. They both increased, but the mean increased by more. And it makes sense, because this number was way, way below all of these over here."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "The mean went from 90 and 2 5ths to 93. So the mean increased by more than the median. They both increased, but the mean increased by more. And it makes sense, because this number was way, way below all of these over here. So you can imagine, if you take this out, the mean should increase by a good amount. But let's see which of these choices are what we just described. But the mean and the median will decrease."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "And it makes sense, because this number was way, way below all of these over here. So you can imagine, if you take this out, the mean should increase by a good amount. But let's see which of these choices are what we just described. But the mean and the median will decrease. Nope. But the mean and the median will decrease. Nope."}, {"video_title": "Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3", "Sentence": "But the mean and the median will decrease. Nope. But the mean and the median will decrease. Nope. But the mean and the median will increase, but the mean will increase by more than the median. That's exactly, that's exactly what happened. The mean went from 90 and 2 5ths, or 90.4, went from 90.4, or 90 and 2 5ths, to 93."}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "As we go further in our statistical careers, it's going to be valuable to assume that certain distributions are normal distributions or sometimes to assume that they are binomial distributions because if we can do that, we can make all sorts of interesting inferences about them when we make that assumption. But one of the key things about normal distributions or binomial distributions is we assume that they're the sum or they can be viewed as the sum of a bunch of independent trials. So we have to assume that trials are independent. Now that is reasonable in a lot of situations, but sometimes, let's say you're conducting a survey of people exiting a mall, and in that case, and let's say you're saying whether they have done their taxes already, if they're exiting the mall, it's hard to do these samples with replacement. They're leaving the mall. You can't say, hey, hey, wait, I just asked you a question. Now you've answered it."}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "Now that is reasonable in a lot of situations, but sometimes, let's say you're conducting a survey of people exiting a mall, and in that case, and let's say you're saying whether they have done their taxes already, if they're exiting the mall, it's hard to do these samples with replacement. They're leaving the mall. You can't say, hey, hey, wait, I just asked you a question. Now you've answered it. Now go back into the mall because I want each trial to be truly independent. But we all know it feels intuitive that, hey, if there are 10,000 people in the mall and I'm going to sample 10 of them, does it really matter that it's truly independent? Doesn't it matter that we're just kind of close to being independent?"}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "Now you've answered it. Now go back into the mall because I want each trial to be truly independent. But we all know it feels intuitive that, hey, if there are 10,000 people in the mall and I'm going to sample 10 of them, does it really matter that it's truly independent? Doesn't it matter that we're just kind of close to being independent? And because of that idea and because we do wanna make inferences based on things being close to a binomial distribution or a normal distribution, we have something called the 10% rule. And the 10% rule says that if our sample, if our sample is less than or equal to 10% of the population, then it is okay to assume approximate independence, approximate independence. And there are some fairly sophisticated ways of coming up with this 10% threshold."}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "Doesn't it matter that we're just kind of close to being independent? And because of that idea and because we do wanna make inferences based on things being close to a binomial distribution or a normal distribution, we have something called the 10% rule. And the 10% rule says that if our sample, if our sample is less than or equal to 10% of the population, then it is okay to assume approximate independence, approximate independence. And there are some fairly sophisticated ways of coming up with this 10% threshold. People could have picked 9%. They could have picked 10.1%. But 10% is a nice round number."}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "And there are some fairly sophisticated ways of coming up with this 10% threshold. People could have picked 9%. They could have picked 10.1%. But 10% is a nice round number. And if we look at some tangible examples, it seems to do a pretty good job. So for example, right over here, let's let x be the number of boys from three trials of selecting from a classroom of n students where 50% of the class is a boy and 50% of the class is a girl. And so what we have over here is we have a bunch of different n's."}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "But 10% is a nice round number. And if we look at some tangible examples, it seems to do a pretty good job. So for example, right over here, let's let x be the number of boys from three trials of selecting from a classroom of n students where 50% of the class is a boy and 50% of the class is a girl. And so what we have over here is we have a bunch of different n's. What if we have 20 students in the class? What if we have 30? What if we have 100?"}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so what we have over here is we have a bunch of different n's. What if we have 20 students in the class? What if we have 30? What if we have 100? What if we have 10,000? And so we could find the probability that we select three boys with replacement in each of these scenarios. And we could also find the probability that we select three boys without replacement."}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "What if we have 100? What if we have 10,000? And so we could find the probability that we select three boys with replacement in each of these scenarios. And we could also find the probability that we select three boys without replacement. And then we could think about what proportion is our sample size of the entire population. And then we could say, does the 10% rule actually make sense? So this first column where we are picking three boys with replacement, in this case, because we are replacing, each of these trials are independent, are truly independent."}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "And we could also find the probability that we select three boys without replacement. And then we could think about what proportion is our sample size of the entire population. And then we could say, does the 10% rule actually make sense? So this first column where we are picking three boys with replacement, in this case, because we are replacing, each of these trials are independent, are truly independent. And if our trials are independent, then x would be truly a binomial variable. Here, we aren't independent because we are not replacing, so not independent. And so officially, in this column right over here when we're not replacing, x would not be considered a binomial random variable."}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "So this first column where we are picking three boys with replacement, in this case, because we are replacing, each of these trials are independent, are truly independent. And if our trials are independent, then x would be truly a binomial variable. Here, we aren't independent because we are not replacing, so not independent. And so officially, in this column right over here when we're not replacing, x would not be considered a binomial random variable. But let's see if there's a threshold where if our sample size is a small enough percentage of our entire population where we would feel not so bad about assuming x is close to being binomial. So in all of the cases where you have independent trials and 50% of the population is boys, 50% is girls, well, you're going to amount to 1 1\u20442 times 1 1\u20442 times 1 1\u20442. So in all of those situations, you have a 12.5% chance that x is going to be equal to three."}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so officially, in this column right over here when we're not replacing, x would not be considered a binomial random variable. But let's see if there's a threshold where if our sample size is a small enough percentage of our entire population where we would feel not so bad about assuming x is close to being binomial. So in all of the cases where you have independent trials and 50% of the population is boys, 50% is girls, well, you're going to amount to 1 1\u20442 times 1 1\u20442 times 1 1\u20442. So in all of those situations, you have a 12.5% chance that x is going to be equal to three. And in this case, x would be a binomial variable. But look over here. When three is a fairly large percentage of our population, in this case, it is 15%, the percent chance of getting three boys without replacement is 10.5%, which is reasonably different from 12 1\u20442%."}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "So in all of those situations, you have a 12.5% chance that x is going to be equal to three. And in this case, x would be a binomial variable. But look over here. When three is a fairly large percentage of our population, in this case, it is 15%, the percent chance of getting three boys without replacement is 10.5%, which is reasonably different from 12 1\u20442%. It is 2% different, but 2% relative to 12 1\u20442%. So that's someplace in between 10 and 20% difference in terms of the probability. So this is a reasonably big difference."}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "When three is a fairly large percentage of our population, in this case, it is 15%, the percent chance of getting three boys without replacement is 10.5%, which is reasonably different from 12 1\u20442%. It is 2% different, but 2% relative to 12 1\u20442%. So that's someplace in between 10 and 20% difference in terms of the probability. So this is a reasonably big difference. But as we increase the population size without increasing the sample size, we see that these numbers get closer and closer to each other, all the way so that if you have 10,000 people in your population and you're only doing three trials, that the numbers get very, very close. This is actually 12.49 something percent. But if you round to the nearest 10th of a percent, you see that they are close."}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "So this is a reasonably big difference. But as we increase the population size without increasing the sample size, we see that these numbers get closer and closer to each other, all the way so that if you have 10,000 people in your population and you're only doing three trials, that the numbers get very, very close. This is actually 12.49 something percent. But if you round to the nearest 10th of a percent, you see that they are close. So I think most people would say, all right, if your sample is 3 10,000th of the population, that you'd feel pretty good treating this column without replacement as being pretty close to being a binomial variable. And most people would say, all right, this first scenario where your sample size is 15% of your population, you wouldn't feel so good treating this without replacement column as a binomial random variable. But where do you draw the line?"}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "But if you round to the nearest 10th of a percent, you see that they are close. So I think most people would say, all right, if your sample is 3 10,000th of the population, that you'd feel pretty good treating this column without replacement as being pretty close to being a binomial variable. And most people would say, all right, this first scenario where your sample size is 15% of your population, you wouldn't feel so good treating this without replacement column as a binomial random variable. But where do you draw the line? And as we alluded to earlier in the video, the line is typically drawn at 10%. That if your sample size is less than or equal to 10% of your population, it's not unreasonable to treat your random variable, even though it's not officially binomial, to say, okay, maybe it is, maybe I can functionally treat it as binomial, and then from there I can make all of the powerful inferences that we tend to do in statistics. With that said, the lower the percentage the sample is of the population, the better."}, {"video_title": "10% Rule of assuming independence between trials Random variables AP Statistics Khan Academy.mp3", "Sentence": "But where do you draw the line? And as we alluded to earlier in the video, the line is typically drawn at 10%. That if your sample size is less than or equal to 10% of your population, it's not unreasonable to treat your random variable, even though it's not officially binomial, to say, okay, maybe it is, maybe I can functionally treat it as binomial, and then from there I can make all of the powerful inferences that we tend to do in statistics. With that said, the lower the percentage the sample is of the population, the better. Now to be clear, that's not saying that small sample sizes are better than large sample sizes. In statistics, large sample sizes tend to be a lot better than small sample sizes. But if you wanna make this independence assumption, so to speak, even when it's not exactly true, you want your sample to be a small percentage of the population."}, {"video_title": "Significance test for a proportion free response (part 2 with correction) Khan Academy.mp3", "Sentence": "In a previous video, I had worked through this AP Statistics example problem right over here, and then I had it looked over by folks who are familiar with how it is graded, including some people who had graded the AP Statistics exam themselves. And they pointed out a few problems with how I actually wrote things down. And so I thought in this video, I would correct those problems. And instead of just redoing the whole example again, the way that would get maximum points on the AP test, I thought it would be even more instructive to show where I went wrong. So the math in this problem was not incorrect, but if I were actually taking the AP Statistics exam, I was told that, say, for the conditions for inference, where we have this random condition here, in the example, I just pointed to the part of the problem where they tell us that we are dealing with a random sample. Assume that the 65 boxes purchased by the students are random samples. I am told that the AP graders do not like this."}, {"video_title": "Significance test for a proportion free response (part 2 with correction) Khan Academy.mp3", "Sentence": "And instead of just redoing the whole example again, the way that would get maximum points on the AP test, I thought it would be even more instructive to show where I went wrong. So the math in this problem was not incorrect, but if I were actually taking the AP Statistics exam, I was told that, say, for the conditions for inference, where we have this random condition here, in the example, I just pointed to the part of the problem where they tell us that we are dealing with a random sample. Assume that the 65 boxes purchased by the students are random samples. I am told that the AP graders do not like this. They do not want you to just point to a part of the passage that says that it's a random sample. Instead, what you could say, and they are functionally equivalent, but you wanna make sure that you're doing what people are looking for. Instead, what we could do is, it is stated, over here we could write, stated in problem, problem, that random sample that we're dealing with, that random sample taken."}, {"video_title": "Significance test for a proportion free response (part 2 with correction) Khan Academy.mp3", "Sentence": "I am told that the AP graders do not like this. They do not want you to just point to a part of the passage that says that it's a random sample. Instead, what you could say, and they are functionally equivalent, but you wanna make sure that you're doing what people are looking for. Instead, what we could do is, it is stated, over here we could write, stated in problem, problem, that random sample that we're dealing with, that random sample taken. And that would be sufficient, instead of drawing the arrow to that part over there. Then I could check it off. Now, the other thing that I didn't do, I talked about it in the previous video, but I didn't write it down."}, {"video_title": "Significance test for a proportion free response (part 2 with correction) Khan Academy.mp3", "Sentence": "Instead, what we could do is, it is stated, over here we could write, stated in problem, problem, that random sample that we're dealing with, that random sample taken. And that would be sufficient, instead of drawing the arrow to that part over there. Then I could check it off. Now, the other thing that I didn't do, I talked about it in the previous video, but I didn't write it down. And I really did wanna model what you would need to do on the actual AP exam is, I came, I said that we failed to reject the null hypothesis, and therefore, there's not enough evidence to suggest the alternative hypothesis. But if you're actually taking the AP exam, you wanna go one step further. You wanna really talk about the conclusion you're making."}, {"video_title": "Significance test for a proportion free response (part 2 with correction) Khan Academy.mp3", "Sentence": "Now, the other thing that I didn't do, I talked about it in the previous video, but I didn't write it down. And I really did wanna model what you would need to do on the actual AP exam is, I came, I said that we failed to reject the null hypothesis, and therefore, there's not enough evidence to suggest the alternative hypothesis. But if you're actually taking the AP exam, you wanna go one step further. You wanna really talk about the conclusion you're making. Because they asked this question, based on this sample, is there support for the student's belief that the proportion of boxes with vouchers is less than 0.2? And so now, I can just draw it back to that question. There is not, there is not support, support for student belief, student belief, student belief, that the proportion, proportion of boxes with vouchers, with vouchers, is less than, less than, is less than 0.2."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "We are told a large nationwide poll recently showed an unemployment rate of 9% in the United States. The mayor of a local town wonders if this national result holds true for her town. So she plans on taking a sample of her residents to see if the unemployment rate is significantly different than 9% in her town. Let P represent the unemployment rate in her town. Here are the hypotheses she'll use. So her null hypothesis is that, hey, the unemployment rate in her town is the same as for the country. And her alternative hypothesis is that it is not the same."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "Let P represent the unemployment rate in her town. Here are the hypotheses she'll use. So her null hypothesis is that, hey, the unemployment rate in her town is the same as for the country. And her alternative hypothesis is that it is not the same. Under which of the following conditions would the mayor commit a type one error? So pause this video and see if you can figure it out on your own. Now let's work through this together."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "And her alternative hypothesis is that it is not the same. Under which of the following conditions would the mayor commit a type one error? So pause this video and see if you can figure it out on your own. Now let's work through this together. So let's just remind ourselves what a type one error even is. This is a situation where we reject the null hypothesis even though it is true. Reject null hypothesis even though, even though our null hypothesis is true."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "Now let's work through this together. So let's just remind ourselves what a type one error even is. This is a situation where we reject the null hypothesis even though it is true. Reject null hypothesis even though, even though our null hypothesis is true. And in general, if you're committing either a type one or a type two error, you're doing the wrong thing. You're doing something that somehow contradicts reality even though you didn't intend to. And so in this case, that would be rejecting the hypothesis that the unemployment rate is 9% in this town even though it actually is 9% in this town."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "Reject null hypothesis even though, even though our null hypothesis is true. And in general, if you're committing either a type one or a type two error, you're doing the wrong thing. You're doing something that somehow contradicts reality even though you didn't intend to. And so in this case, that would be rejecting the hypothesis that the unemployment rate is 9% in this town even though it actually is 9% in this town. So let's see which of these choices match up to that. She concludes the town's unemployment rate is not 9% when it actually is. Yeah, in this situation, in order to conclude that the unemployment rate is not 9%, she would have to reject the null hypothesis even though the null hypothesis is actually true, even though the unemployment rate actually is 9%."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "And so in this case, that would be rejecting the hypothesis that the unemployment rate is 9% in this town even though it actually is 9% in this town. So let's see which of these choices match up to that. She concludes the town's unemployment rate is not 9% when it actually is. Yeah, in this situation, in order to conclude that the unemployment rate is not 9%, she would have to reject the null hypothesis even though the null hypothesis is actually true, even though the unemployment rate actually is 9%. So I'm liking this choice. But let's read the other ones just to make sure. She concludes the town's unemployment rate is not 9% when it actually is not."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "Yeah, in this situation, in order to conclude that the unemployment rate is not 9%, she would have to reject the null hypothesis even though the null hypothesis is actually true, even though the unemployment rate actually is 9%. So I'm liking this choice. But let's read the other ones just to make sure. She concludes the town's unemployment rate is not 9% when it actually is not. Well, this wouldn't be an error. If the null hypothesis isn't true, it's not a problem to reject it. So this one wouldn't be an error."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "She concludes the town's unemployment rate is not 9% when it actually is not. Well, this wouldn't be an error. If the null hypothesis isn't true, it's not a problem to reject it. So this one wouldn't be an error. She concludes the town's unemployment rate is 9% when it actually is. Well, once again, this would not be an error. This would be failing to reject the null hypothesis when the null hypothesis is actually true, not an error."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "So this one wouldn't be an error. She concludes the town's unemployment rate is 9% when it actually is. Well, once again, this would not be an error. This would be failing to reject the null hypothesis when the null hypothesis is actually true, not an error. Choice D, she concludes the town's unemployment rate is 9% when it actually is not. So this is a situation where she fails to reject the null hypothesis even though the null hypothesis is not true. So this one right over here, this one would actually be, this is an error, this is an error, but this is a type two error."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "This would be failing to reject the null hypothesis when the null hypothesis is actually true, not an error. Choice D, she concludes the town's unemployment rate is 9% when it actually is not. So this is a situation where she fails to reject the null hypothesis even though the null hypothesis is not true. So this one right over here, this one would actually be, this is an error, this is an error, but this is a type two error. So one way to think about it, first you say, okay, am I making an error? Am I rejecting something that's true or am I failing to reject something that's false? And the rejecting something that is true, that's type one, and failing to reject something that is false, that is type two."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "So this one right over here, this one would actually be, this is an error, this is an error, but this is a type two error. So one way to think about it, first you say, okay, am I making an error? Am I rejecting something that's true or am I failing to reject something that's false? And the rejecting something that is true, that's type one, and failing to reject something that is false, that is type two. And so with that in mind, let's do another example. A large university is curious if they should build another cafeteria. They plan to survey a sample of their students to see if there is strong evidence that the proportion interested in a meal plan is higher than 40%, in which case, they will consider building a new cafeteria."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "And the rejecting something that is true, that's type one, and failing to reject something that is false, that is type two. And so with that in mind, let's do another example. A large university is curious if they should build another cafeteria. They plan to survey a sample of their students to see if there is strong evidence that the proportion interested in a meal plan is higher than 40%, in which case, they will consider building a new cafeteria. Let P represent the proportion of students interested in a meal plan. Here are the hypotheses they'll use. So the null hypothesis is that 40% or fewer of the students are interested in a meal plan, while the alternative hypothesis is that more than 40% are interested."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "They plan to survey a sample of their students to see if there is strong evidence that the proportion interested in a meal plan is higher than 40%, in which case, they will consider building a new cafeteria. Let P represent the proportion of students interested in a meal plan. Here are the hypotheses they'll use. So the null hypothesis is that 40% or fewer of the students are interested in a meal plan, while the alternative hypothesis is that more than 40% are interested. What would be the consequence of a type two error in this context? So once again, pause this video and try to answer this for yourself. Okay, now let's do it together."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "So the null hypothesis is that 40% or fewer of the students are interested in a meal plan, while the alternative hypothesis is that more than 40% are interested. What would be the consequence of a type two error in this context? So once again, pause this video and try to answer this for yourself. Okay, now let's do it together. Let's just remind ourselves what a type two error is. We just talked about it. So failing, failing to reject, in this case, our null hypothesis, even, even though it is false."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "Okay, now let's do it together. Let's just remind ourselves what a type two error is. We just talked about it. So failing, failing to reject, in this case, our null hypothesis, even, even though it is false. So this would be a scenario where this is false, which would mean that more than 40% actually do want a meal plan, but you fail to reject this. So what would happen is is that you wouldn't build another cafeteria because you'd say, hey, no, there are not that many people who are interested in the meal plan, but you wouldn't, but actually, there are a lot of people who are interested in the meal plan, and so you probably won't have enough cafeteria space. And so this says, they don't consider building a new cafeteria when they should."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "So failing, failing to reject, in this case, our null hypothesis, even, even though it is false. So this would be a scenario where this is false, which would mean that more than 40% actually do want a meal plan, but you fail to reject this. So what would happen is is that you wouldn't build another cafeteria because you'd say, hey, no, there are not that many people who are interested in the meal plan, but you wouldn't, but actually, there are a lot of people who are interested in the meal plan, and so you probably won't have enough cafeteria space. And so this says, they don't consider building a new cafeteria when they should. Yeah, this is exactly right. They don't consider building a new cafeteria when they shouldn't. Well, this would just be a correct conclusion."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "And so this says, they don't consider building a new cafeteria when they should. Yeah, this is exactly right. They don't consider building a new cafeteria when they shouldn't. Well, this would just be a correct conclusion. They consider building a new cafeteria when they shouldn't. And so this is a scenario where they do reject the null hypothesis even though the null hypothesis is true. So this right over here would be a type one error."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "Well, this would just be a correct conclusion. They consider building a new cafeteria when they shouldn't. And so this is a scenario where they do reject the null hypothesis even though the null hypothesis is true. So this right over here would be a type one error. Type one error. Because if they're considering building a new cafeteria, that means they rejected the null hypothesis. Even when they shouldn't, that means that the null hypothesis was true, so type one."}, {"video_title": "Examples identifying Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "So this right over here would be a type one error. Type one error. Because if they're considering building a new cafeteria, that means they rejected the null hypothesis. Even when they shouldn't, that means that the null hypothesis was true, so type one. They consider building a new cafeteria when they should. Well, once again, this wouldn't be an error at all. This would be a correct conclusion."}, {"video_title": "Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "We have already seen a situation multiple times where there is some parameter associated with a population. Maybe it's the proportion of a population that supports a candidate. Maybe it's the mean of a population, the mean height of all the people in the city. And we've determined that it's unpractical, or there's no way for us to know the true population parameter, but we could try to estimate it by taking a sample size. So we take n samples, and then we calculate a statistic based on that. We've also seen that not only can we calculate the statistic which is trying to estimate this parameter, but we can construct a confidence interval about that statistic based on some confidence level. And so that confidence interval would look something like this."}, {"video_title": "Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And we've determined that it's unpractical, or there's no way for us to know the true population parameter, but we could try to estimate it by taking a sample size. So we take n samples, and then we calculate a statistic based on that. We've also seen that not only can we calculate the statistic which is trying to estimate this parameter, but we can construct a confidence interval about that statistic based on some confidence level. And so that confidence interval would look something like this. It would be the value of the statistic that we have just calculated, plus or minus some margin of error. And so we'll often say this critical value, z, and this will be based on the number of standard deviations we wanna go above and below that statistic. And so then we'll multiply that times the standard deviation of the sampling distribution for that statistic."}, {"video_title": "Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And so that confidence interval would look something like this. It would be the value of the statistic that we have just calculated, plus or minus some margin of error. And so we'll often say this critical value, z, and this will be based on the number of standard deviations we wanna go above and below that statistic. And so then we'll multiply that times the standard deviation of the sampling distribution for that statistic. Now what we'll see is we often don't know this. To know this, you oftentimes even need to know this parameter. For example, in the situation where the parameter that we're trying to estimate and construct confidence intervals for is say the population proportion, what percentage of the population supports a certain candidate?"}, {"video_title": "Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And so then we'll multiply that times the standard deviation of the sampling distribution for that statistic. Now what we'll see is we often don't know this. To know this, you oftentimes even need to know this parameter. For example, in the situation where the parameter that we're trying to estimate and construct confidence intervals for is say the population proportion, what percentage of the population supports a certain candidate? Well, in that world, the statistic is the sample proportion. So we would have the sample proportion, plus or minus z star, times, well, we can't calculate this unless we know the population proportion. So instead, we estimate this with the standard error of the statistic, which in this case is p hat times one minus p hat, the sample proportion times one minus the sample proportion over our sample size."}, {"video_title": "Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "For example, in the situation where the parameter that we're trying to estimate and construct confidence intervals for is say the population proportion, what percentage of the population supports a certain candidate? Well, in that world, the statistic is the sample proportion. So we would have the sample proportion, plus or minus z star, times, well, we can't calculate this unless we know the population proportion. So instead, we estimate this with the standard error of the statistic, which in this case is p hat times one minus p hat, the sample proportion times one minus the sample proportion over our sample size. If the parameter we're trying to estimate is the population mean, then our statistic is going to be the sample mean. So in that scenario, we are going to be looking at our statistic is our sample mean, plus or minus z star. Now if we knew the standard deviation of this population, we would know what the standard deviation of the sampling distribution of our statistic is."}, {"video_title": "Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So instead, we estimate this with the standard error of the statistic, which in this case is p hat times one minus p hat, the sample proportion times one minus the sample proportion over our sample size. If the parameter we're trying to estimate is the population mean, then our statistic is going to be the sample mean. So in that scenario, we are going to be looking at our statistic is our sample mean, plus or minus z star. Now if we knew the standard deviation of this population, we would know what the standard deviation of the sampling distribution of our statistic is. It would be equal to the standard deviation of our population times the square root of our sample size. But we often will not know this. In fact, it's very unusual to know this."}, {"video_title": "Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Now if we knew the standard deviation of this population, we would know what the standard deviation of the sampling distribution of our statistic is. It would be equal to the standard deviation of our population times the square root of our sample size. But we often will not know this. In fact, it's very unusual to know this. And so sometimes you will say, okay, if we don't know this, let's just figure out the sample standard deviation of our sample here. So instead, we'll say, okay, let's take our sample mean, plus or minus z star, times the sample standard deviation of our sample, which we can calculate, divided by the square root of n. Now this might seem pretty good if we're trying to construct a confidence interval for our sample, for our mean. But it turns out that this is not, not so good."}, {"video_title": "Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "In fact, it's very unusual to know this. And so sometimes you will say, okay, if we don't know this, let's just figure out the sample standard deviation of our sample here. So instead, we'll say, okay, let's take our sample mean, plus or minus z star, times the sample standard deviation of our sample, which we can calculate, divided by the square root of n. Now this might seem pretty good if we're trying to construct a confidence interval for our sample, for our mean. But it turns out that this is not, not so good. Because it turns out that this right over here is going to actually underestimate the actual interval, the true margin of error you need for your confidence level. And so that's why statisticians have invented another statistic. Instead of using z, they call it t. Instead of using a z table, they use a t table."}, {"video_title": "Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "But it turns out that this is not, not so good. Because it turns out that this right over here is going to actually underestimate the actual interval, the true margin of error you need for your confidence level. And so that's why statisticians have invented another statistic. Instead of using z, they call it t. Instead of using a z table, they use a t table. And we're going to see this in future videos. And so if you are actually trying to construct a confidence interval for a sample mean, and you don't know the true standard deviation of your population, which is normally the case, instead of doing this, what we're going to do is, we're gonna take our sample mean, plus or minus, and our critical value, we'll call that t star, times our sample standard deviation, which we can calculate, divided by the square root of n. And so the real functional difference is that this actually is going to give us the confidence interval that actually has the level of confidence that we want. If we have one at 95% level of confidence, if we keep computing this over and over again for multiple samples, that roughly 95% of the time, this interval will contain our true population mean."}, {"video_title": "Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "The sample mean was 350 kilograms and the sample standard deviation was 25 kilograms. The resulting 90% confidence interval for the mean amount of food was from 341 kilograms to 359 kilograms. Which of the following statements is a correct interpretation of the 90% confidence level? So like always, pause this video and see if you can answer this on your own. So before we even look at these choices, let's just make sure we're reading the statement or interpreting the statement correctly. A zookeeper is trying to figure out what the true expected amount of food an elephant would eat on a day. You could view that as the mean amount of food that an elephant would eat on a day."}, {"video_title": "Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So like always, pause this video and see if you can answer this on your own. So before we even look at these choices, let's just make sure we're reading the statement or interpreting the statement correctly. A zookeeper is trying to figure out what the true expected amount of food an elephant would eat on a day. You could view that as the mean amount of food that an elephant would eat on a day. If you view it as the number, all the possible days as the population, you could view this as the population mean for mean amount of food per day. Now the zookeeper doesn't know that and so instead, they're trying to estimate it by sampling 30 days. So let's think about it this way."}, {"video_title": "Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "You could view that as the mean amount of food that an elephant would eat on a day. If you view it as the number, all the possible days as the population, you could view this as the population mean for mean amount of food per day. Now the zookeeper doesn't know that and so instead, they're trying to estimate it by sampling 30 days. So let's think about it this way. Let's say that this is the true population mean, the true mean amount of food that an elephant will eat in a day. What the zookeeper can try to do is, well they take a sample and in this case, they took a sample of 30 days and they calculated a sample statistic, in this case the sample mean of 350 kilograms. I don't know if it's actually to the right of the true parameter but just for visualization purposes, let's say it is."}, {"video_title": "Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So let's think about it this way. Let's say that this is the true population mean, the true mean amount of food that an elephant will eat in a day. What the zookeeper can try to do is, well they take a sample and in this case, they took a sample of 30 days and they calculated a sample statistic, in this case the sample mean of 350 kilograms. I don't know if it's actually to the right of the true parameter but just for visualization purposes, let's say it is. So let's say sample mean and this is their first sample. It was 350 kilograms and then using this, using the sample, they were able to construct a confidence interval from 341 to 359 kilograms and so the confidence interval, I'll draw it like this. We actually aren't sure if it actually overlaps with the true mean like I'm drawing here but just for the sake of visualization purposes, let's say that this one happened to."}, {"video_title": "Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "I don't know if it's actually to the right of the true parameter but just for visualization purposes, let's say it is. So let's say sample mean and this is their first sample. It was 350 kilograms and then using this, using the sample, they were able to construct a confidence interval from 341 to 359 kilograms and so the confidence interval, I'll draw it like this. We actually aren't sure if it actually overlaps with the true mean like I'm drawing here but just for the sake of visualization purposes, let's say that this one happened to. The whole point of a 90% confidence level is if I kept doing this, so this is our first sample and the associated interval with that first sample and then if I did another sample, let's say this is the mean of that next sample so that's sample mean two and I have an associated confidence interval and that interval, not only the start and end points will change but the actual width of the interval might change depending on what my sample looks like. What a 90% confidence level means that if I keep doing this, that 90% of my confidence intervals should overlap with the true parameter, with the true population mean. So now with that out of the way, let's see which of these choices are consistent with that interpretation."}, {"video_title": "Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "We actually aren't sure if it actually overlaps with the true mean like I'm drawing here but just for the sake of visualization purposes, let's say that this one happened to. The whole point of a 90% confidence level is if I kept doing this, so this is our first sample and the associated interval with that first sample and then if I did another sample, let's say this is the mean of that next sample so that's sample mean two and I have an associated confidence interval and that interval, not only the start and end points will change but the actual width of the interval might change depending on what my sample looks like. What a 90% confidence level means that if I keep doing this, that 90% of my confidence intervals should overlap with the true parameter, with the true population mean. So now with that out of the way, let's see which of these choices are consistent with that interpretation. Choice A, the elephant ate between 341 kilograms and 359 kilograms on 90% of all of the days. No, that is definitely not what is going on here. We're not talking about what's happening on 90% of the days so let's rule this choice out."}, {"video_title": "Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So now with that out of the way, let's see which of these choices are consistent with that interpretation. Choice A, the elephant ate between 341 kilograms and 359 kilograms on 90% of all of the days. No, that is definitely not what is going on here. We're not talking about what's happening on 90% of the days so let's rule this choice out. There is a 0.9 probability that the true mean amount of food is between 341 kilograms and 359 kilograms. So this one is interesting and it is a tempting choice because when we do this one sample, you can kind of say, all right, if I did a bunch of these samples, 90% of them, if we have a 90% confidence interval or 90% confidence level should overlap with this true mean, with the population parameter. The reason why this is a little bit uncomfortable is it makes the true mean sound almost like a random variable that it could kind of jump around and it's the true mean that kind of is either gonna jump into this interval or not jump into this interval."}, {"video_title": "Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "We're not talking about what's happening on 90% of the days so let's rule this choice out. There is a 0.9 probability that the true mean amount of food is between 341 kilograms and 359 kilograms. So this one is interesting and it is a tempting choice because when we do this one sample, you can kind of say, all right, if I did a bunch of these samples, 90% of them, if we have a 90% confidence interval or 90% confidence level should overlap with this true mean, with the population parameter. The reason why this is a little bit uncomfortable is it makes the true mean sound almost like a random variable that it could kind of jump around and it's the true mean that kind of is either gonna jump into this interval or not jump into this interval. So it causes a little bit of unease. See choice, so I'm just gonna put a question mark here. In repeated sampling, okay, I like the way that this is starting."}, {"video_title": "Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "The reason why this is a little bit uncomfortable is it makes the true mean sound almost like a random variable that it could kind of jump around and it's the true mean that kind of is either gonna jump into this interval or not jump into this interval. So it causes a little bit of unease. See choice, so I'm just gonna put a question mark here. In repeated sampling, okay, I like the way that this is starting. In repeated sampling, this method produces intervals, yep, that's what it does, every time you sample, you produce an interval, that capture the population mean in about 90% of samples. Yeah, that's exactly what we're talking about. If we just kept doing this, that 90, that if we have well-constructed 90% confidence intervals, that if we kept doing this, 90% of these constructed sampled intervals should overlap with the true mean."}, {"video_title": "Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "In repeated sampling, okay, I like the way that this is starting. In repeated sampling, this method produces intervals, yep, that's what it does, every time you sample, you produce an interval, that capture the population mean in about 90% of samples. Yeah, that's exactly what we're talking about. If we just kept doing this, that 90, that if we have well-constructed 90% confidence intervals, that if we kept doing this, 90% of these constructed sampled intervals should overlap with the true mean. So I like this choice. But let's just read choice D to rule it out. In repeated sampling, this method produces a sample mean between 341 kilograms and 359 kilograms in about 90% of samples."}, {"video_title": "Interpreting confidence level example Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "If we just kept doing this, that 90, that if we have well-constructed 90% confidence intervals, that if we kept doing this, 90% of these constructed sampled intervals should overlap with the true mean. So I like this choice. But let's just read choice D to rule it out. In repeated sampling, this method produces a sample mean between 341 kilograms and 359 kilograms in about 90% of samples. No, the confidence interval does not put a constrain on that 90% of the time you will have a sample mean between these values. It is not trying to do that. It is definitely choice C."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "Let X represent the number of free desserts Kai gets in his two trips. Construct the theoretical probability distribution of X. All right, so pause this video and see if you can work through this before we do it together. All right, so first let's just think about the possible values that X could take on. This is the number of free desserts he gets, and he visits twice. So there's some world in which he doesn't get any free desserts, so that's zero in his two visits. Maybe on one of the visits, he gets a dessert, and the other one he doesn't."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "All right, so first let's just think about the possible values that X could take on. This is the number of free desserts he gets, and he visits twice. So there's some world in which he doesn't get any free desserts, so that's zero in his two visits. Maybe on one of the visits, he gets a dessert, and the other one he doesn't. And maybe in both of his visits, he actually is able to get a free dessert. So he's going to have some place from zero to two free desserts in a given week. So we just have to figure out the probability of each of these."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "Maybe on one of the visits, he gets a dessert, and the other one he doesn't. And maybe in both of his visits, he actually is able to get a free dessert. So he's going to have some place from zero to two free desserts in a given week. So we just have to figure out the probability of each of these. So let's first of all think about the probability, let me write it over here, the probability that capital X is equal to zero is going to be equal to what? Well, that's going to be the probability that he doesn't get a dessert on both days. And it's important to realize that these are independent events."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "So we just have to figure out the probability of each of these. So let's first of all think about the probability, let me write it over here, the probability that capital X is equal to zero is going to be equal to what? Well, that's going to be the probability that he doesn't get a dessert on both days. And it's important to realize that these are independent events. It's not like the restaurant's gonna say, oh, if you didn't get a dessert on one day, you're more likely to get it on the other day, or somehow if you got it on a previous day, you're less likely on another day, that they are independent events. So the probability of not getting it on any one day is four out of five. And the probability of not getting it on two of the days, I would just multiply them because they are independent events."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "And it's important to realize that these are independent events. It's not like the restaurant's gonna say, oh, if you didn't get a dessert on one day, you're more likely to get it on the other day, or somehow if you got it on a previous day, you're less likely on another day, that they are independent events. So the probability of not getting it on any one day is four out of five. And the probability of not getting it on two of the days, I would just multiply them because they are independent events. So four over five times four over five. So the probability that X is equal to zero is going to be 16 25ths, 16 over 25. Now, what about the probability that X is equal to one?"}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "And the probability of not getting it on two of the days, I would just multiply them because they are independent events. So four over five times four over five. So the probability that X is equal to zero is going to be 16 25ths, 16 over 25. Now, what about the probability that X is equal to one? What is this going to be? Well, there are two scenarios over here. There's one scenario where let's say on day one, he does not get the dessert, and on day two, he does get the dessert."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "Now, what about the probability that X is equal to one? What is this going to be? Well, there are two scenarios over here. There's one scenario where let's say on day one, he does not get the dessert, and on day two, he does get the dessert. But then of course, there's the other scenario where on day one, he gets the dessert, and then on day two, he doesn't get the dessert. These are the two scenarios where he's going to get X equals one. And so if we add these together, let's see, four fifths times one fifth, this is going to be four over 25."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "There's one scenario where let's say on day one, he does not get the dessert, and on day two, he does get the dessert. But then of course, there's the other scenario where on day one, he gets the dessert, and then on day two, he doesn't get the dessert. These are the two scenarios where he's going to get X equals one. And so if we add these together, let's see, four fifths times one fifth, this is going to be four over 25. And then this is going to be four over 25 again. And you add these two together, you're going to get eight 25ths. And then last but not least, and actually we could figure out this last one by subtracting 16 and eight from 25, which would actually give us one 25th."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "And so if we add these together, let's see, four fifths times one fifth, this is going to be four over 25. And then this is going to be four over 25 again. And you add these two together, you're going to get eight 25ths. And then last but not least, and actually we could figure out this last one by subtracting 16 and eight from 25, which would actually give us one 25th. But let's just write this out, the probability that X equals two, this is the probability he gets a dessert on both days. So one fifth chance on day one, and one fifth chance on the second day. So one fifth times one fifth is one 25th."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "And then last but not least, and actually we could figure out this last one by subtracting 16 and eight from 25, which would actually give us one 25th. But let's just write this out, the probability that X equals two, this is the probability he gets a dessert on both days. So one fifth chance on day one, and one fifth chance on the second day. So one fifth times one fifth is one 25th. And you can do a reality check here. These all need to add up to one, and they do indeed add up to one. 16 plus eight plus one is 25."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Let's say I have a bag, and in that bag I am going to put some green cubes in that bag. And in particular, I am going to put 8 green cubes. I'm also going to put some spheres in that bag. Let's say I'm going to put 9 spheres, and these are the green spheres. I'm also going to put some yellow cubes in that bag. Some yellow cubes. I'm going to put 5 of those."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Let's say I'm going to put 9 spheres, and these are the green spheres. I'm also going to put some yellow cubes in that bag. Some yellow cubes. I'm going to put 5 of those. And I'm also going to put some yellow spheres in this bag. Yellow spheres. And let's say I put 7 of those."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "I'm going to put 5 of those. And I'm also going to put some yellow spheres in this bag. Yellow spheres. And let's say I put 7 of those. I'm going to stick them all in this bag, and then I'm going to shake that bag, and then I'm going to pour it out, and I'm going to look at the first object that falls out of that bag. And what I want to think about in this video is what are the probabilities of getting different types of objects. So for example, what is the probability of getting a cube?"}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "And let's say I put 7 of those. I'm going to stick them all in this bag, and then I'm going to shake that bag, and then I'm going to pour it out, and I'm going to look at the first object that falls out of that bag. And what I want to think about in this video is what are the probabilities of getting different types of objects. So for example, what is the probability of getting a cube? A cube of any color. What is the probability of getting a cube? Well, to think about that, we should think about, or this is one way to think about it, what are all of the equally likely possibilities that might pop out of the bag?"}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So for example, what is the probability of getting a cube? A cube of any color. What is the probability of getting a cube? Well, to think about that, we should think about, or this is one way to think about it, what are all of the equally likely possibilities that might pop out of the bag? Well, we have 8 plus 9 is 17, 17 plus 5 is 22, 22 plus 7 is 29. So we have 29 objects. There are 29 objects in the bag."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Well, to think about that, we should think about, or this is one way to think about it, what are all of the equally likely possibilities that might pop out of the bag? Well, we have 8 plus 9 is 17, 17 plus 5 is 22, 22 plus 7 is 29. So we have 29 objects. There are 29 objects in the bag. Did I do that right? This is 14, yep, 29 objects. So let's draw all of the possible objects."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "There are 29 objects in the bag. Did I do that right? This is 14, yep, 29 objects. So let's draw all of the possible objects. And I'll represent it as this big area right over here. So these are all the possible objects. There are 29 possible objects."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So let's draw all of the possible objects. And I'll represent it as this big area right over here. So these are all the possible objects. There are 29 possible objects. So there's 29 equal possibilities for the outcome of my experiment of seeing what pops out of that bag, assuming that it's equally likely for a cube or a sphere to pop out first. And how many of them meet our constraint of being a cube? Well, I have 8 green cubes, and I have 5 yellow cubes."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "There are 29 possible objects. So there's 29 equal possibilities for the outcome of my experiment of seeing what pops out of that bag, assuming that it's equally likely for a cube or a sphere to pop out first. And how many of them meet our constraint of being a cube? Well, I have 8 green cubes, and I have 5 yellow cubes. So there are a total of 13 cubes. So let me draw that set of cubes. So there's 13 cubes."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Well, I have 8 green cubes, and I have 5 yellow cubes. So there are a total of 13 cubes. So let me draw that set of cubes. So there's 13 cubes. I could draw it like this. There are 13 cubes. This right here is the set of cubes."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So there's 13 cubes. I could draw it like this. There are 13 cubes. This right here is the set of cubes. This area, and I'm not drawing it exact, I'm approximating, represents the set of all the cubes. So the probability of getting a cube is the number of events that meet our criterion. So there's 13 possible cubes that have an equal likely chance of popping out over all of the possible equally likely events, which are 29."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "This right here is the set of cubes. This area, and I'm not drawing it exact, I'm approximating, represents the set of all the cubes. So the probability of getting a cube is the number of events that meet our criterion. So there's 13 possible cubes that have an equal likely chance of popping out over all of the possible equally likely events, which are 29. That includes the cubes and the spheres. Now let's ask a different question. What is the probability of getting a yellow object, either a cube or a sphere?"}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So there's 13 possible cubes that have an equal likely chance of popping out over all of the possible equally likely events, which are 29. That includes the cubes and the spheres. Now let's ask a different question. What is the probability of getting a yellow object, either a cube or a sphere? So once again, how many things meet our conditions here? We have 5 plus 7. There's 12 yellow objects in the bag."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "What is the probability of getting a yellow object, either a cube or a sphere? So once again, how many things meet our conditions here? We have 5 plus 7. There's 12 yellow objects in the bag. So we have 29 equally likely possibilities. I'll do that in the same color. We have 29 equally likely possibilities, and of those, 12 meet our criterion."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "There's 12 yellow objects in the bag. So we have 29 equally likely possibilities. I'll do that in the same color. We have 29 equally likely possibilities, and of those, 12 meet our criterion. So let me draw 12 right over here. I'll do my best attempt. It looks something like the set of yellow objects."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "We have 29 equally likely possibilities, and of those, 12 meet our criterion. So let me draw 12 right over here. I'll do my best attempt. It looks something like the set of yellow objects. There are 12 objects that are yellow. So the 12 that meet our conditions are 12 over all the possibilities, 29. So the probability of getting a cube, 13 29ths."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "It looks something like the set of yellow objects. There are 12 objects that are yellow. So the 12 that meet our conditions are 12 over all the possibilities, 29. So the probability of getting a cube, 13 29ths. Probability of getting a yellow, 12 29ths. Now let's ask something a little bit more interesting. What is the probability of getting a yellow cube?"}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So the probability of getting a cube, 13 29ths. Probability of getting a yellow, 12 29ths. Now let's ask something a little bit more interesting. What is the probability of getting a yellow cube? I'll put it in yellow, so we care about the color now. So this thing is yellow. What is the probability of getting a yellow cube?"}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "What is the probability of getting a yellow cube? I'll put it in yellow, so we care about the color now. So this thing is yellow. What is the probability of getting a yellow cube? Well, there's 29 equally likely possibilities. And of those 29 equally likely possibilities, 5 of those are yellow cubes, or yellow cubes. 5 of them."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "What is the probability of getting a yellow cube? Well, there's 29 equally likely possibilities. And of those 29 equally likely possibilities, 5 of those are yellow cubes, or yellow cubes. 5 of them. So the probability is 5 29ths. And where would we see that on this Venn diagram that I've drawn? This is just a way to visualize the different probabilities."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "5 of them. So the probability is 5 29ths. And where would we see that on this Venn diagram that I've drawn? This is just a way to visualize the different probabilities. And they become interesting when you start thinking about where sets overlap, or even where they don't overlap. So here we're thinking about things that are members of the set yellow. So they're in this set, and they are cubes."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "This is just a way to visualize the different probabilities. And they become interesting when you start thinking about where sets overlap, or even where they don't overlap. So here we're thinking about things that are members of the set yellow. So they're in this set, and they are cubes. So this area right over here, that's the overlap of these two sets. So this area right over here, this represents things that are both yellow and cubes, because they are inside both circles. So this right over here, let me write it over here."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So they're in this set, and they are cubes. So this area right over here, that's the overlap of these two sets. So this area right over here, this represents things that are both yellow and cubes, because they are inside both circles. So this right over here, let me write it over here. So there's 5 objects that are both yellow and cubes. Now let's ask, and this is probably the most interesting thing to ask, what is the probability of getting something that is yellow or a cube? A cube of any color."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So this right over here, let me write it over here. So there's 5 objects that are both yellow and cubes. Now let's ask, and this is probably the most interesting thing to ask, what is the probability of getting something that is yellow or a cube? A cube of any color. Well we still know that the denominator here is going to be 29. These are all of the equally likely possibilities that might jump out of the bag. But what are the possibilities that meet our conditions?"}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "A cube of any color. Well we still know that the denominator here is going to be 29. These are all of the equally likely possibilities that might jump out of the bag. But what are the possibilities that meet our conditions? Well one way to think about it is, well there's 12 things that would meet the yellow condition. So that would be this entire circle right over here. 12 things that meet the yellow condition."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "But what are the possibilities that meet our conditions? Well one way to think about it is, well there's 12 things that would meet the yellow condition. So that would be this entire circle right over here. 12 things that meet the yellow condition. So this right over here is 12. This is the number of yellow, that is 12. And then to that, we can't just add the number of cubes, because if we add the number of cubes, we've already counted these 5."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "12 things that meet the yellow condition. So this right over here is 12. This is the number of yellow, that is 12. And then to that, we can't just add the number of cubes, because if we add the number of cubes, we've already counted these 5. These 5 are counted as part of this 12. One way to think about it is, there are 7 yellow objects that are not cubes. Those are the spheres."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "And then to that, we can't just add the number of cubes, because if we add the number of cubes, we've already counted these 5. These 5 are counted as part of this 12. One way to think about it is, there are 7 yellow objects that are not cubes. Those are the spheres. There are 5 yellow objects that are cubes. And then there are 8 cubes that are not yellow. That's one way to think about it."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Those are the spheres. There are 5 yellow objects that are cubes. And then there are 8 cubes that are not yellow. That's one way to think about it. So when we counted this 12, the number of yellow, we counted all of this. So we can't just add the number of cubes to it, because then we would count this middle part again. So then we have to essentially count cubes, the number of cubes, which is 13."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "That's one way to think about it. So when we counted this 12, the number of yellow, we counted all of this. So we can't just add the number of cubes to it, because then we would count this middle part again. So then we have to essentially count cubes, the number of cubes, which is 13. So number of cubes. And we'll have to subtract out this middle section right over here. Let me do this."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So then we have to essentially count cubes, the number of cubes, which is 13. So number of cubes. And we'll have to subtract out this middle section right over here. Let me do this. So subtract out the middle section right over here. So minus 5. So this is the number of yellow cubes."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Let me do this. So subtract out the middle section right over here. So minus 5. So this is the number of yellow cubes. It feels weird to write the word yellow in green. The number of yellow cubes. Or another way to think about it, and you could just do this math right here, 12 plus 13 minus 5 is what?"}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So this is the number of yellow cubes. It feels weird to write the word yellow in green. The number of yellow cubes. Or another way to think about it, and you could just do this math right here, 12 plus 13 minus 5 is what? It's 20. Did I do that right? 12 minus, yep, it's 20."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Or another way to think about it, and you could just do this math right here, 12 plus 13 minus 5 is what? It's 20. Did I do that right? 12 minus, yep, it's 20. So that's one way you just get this is equal to 20 over 29. But the more interesting thing than even the answer of the probability of getting that is expressing this in terms of the other probabilities that we figured out earlier in the video. So let's think about this a little bit."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "12 minus, yep, it's 20. So that's one way you just get this is equal to 20 over 29. But the more interesting thing than even the answer of the probability of getting that is expressing this in terms of the other probabilities that we figured out earlier in the video. So let's think about this a little bit. We can rewrite this fraction right over here. We can rewrite this as 12 over 29 plus 13 over 29 minus 5 over 29. And this was the number of yellow over the total possibilities."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So let's think about this a little bit. We can rewrite this fraction right over here. We can rewrite this as 12 over 29 plus 13 over 29 minus 5 over 29. And this was the number of yellow over the total possibilities. So this right over here was the probability of getting a yellow. This right over here was the number of cubes over the total possibilities. So this is plus the probability of getting a cube."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "And this was the number of yellow over the total possibilities. So this right over here was the probability of getting a yellow. This right over here was the number of cubes over the total possibilities. So this is plus the probability of getting a cube. And this right over here was the number of yellow cubes over the total possibilities. So this right over here was minus the probability of yellow and a cube. Let me write it that way."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So this is plus the probability of getting a cube. And this right over here was the number of yellow cubes over the total possibilities. So this right over here was minus the probability of yellow and a cube. Let me write it that way. Minus the probability of yellow. I'll write yellow in yellow. Yellow and getting a cube."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Let me write it that way. Minus the probability of yellow. I'll write yellow in yellow. Yellow and getting a cube. And so what we've just done here, and you could play with the numbers, the numbers I just used as an example right here to make things a little bit concrete, but you can see this is a generalizable thing. If we have the probability of one condition or another condition, so let me rewrite it. The probability, and I'll just write a little bit more generally here."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Yellow and getting a cube. And so what we've just done here, and you could play with the numbers, the numbers I just used as an example right here to make things a little bit concrete, but you can see this is a generalizable thing. If we have the probability of one condition or another condition, so let me rewrite it. The probability, and I'll just write a little bit more generally here. This gives us an interesting idea. The probability of getting one condition of an object being a member of set A or a member of set B is equal to the probability that it is a member of set A plus the probability that it is a member of set B minus the probability that it is a member of both. And this is a really useful result."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "The probability, and I'll just write a little bit more generally here. This gives us an interesting idea. The probability of getting one condition of an object being a member of set A or a member of set B is equal to the probability that it is a member of set A plus the probability that it is a member of set B minus the probability that it is a member of both. And this is a really useful result. I think sometimes it's called the addition rule of probability, but I wanted to show you it's a completely common sense thing. The reason why you can't just add these two probabilities is because they might have some overlap. There's a probability of getting both."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "And this is a really useful result. I think sometimes it's called the addition rule of probability, but I wanted to show you it's a completely common sense thing. The reason why you can't just add these two probabilities is because they might have some overlap. There's a probability of getting both. And if you just added both of these, you would be double counting that overlap, which we've already seen earlier in this video. So you have to subtract one version of the overlap out so you are not double counting it. And I'll throw one other idea out."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "There's a probability of getting both. And if you just added both of these, you would be double counting that overlap, which we've already seen earlier in this video. So you have to subtract one version of the overlap out so you are not double counting it. And I'll throw one other idea out. Sometimes you have possibilities that have no overlap. So let's say this is a set of all possibilities. And let's say this is a set that meets condition A."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "And I'll throw one other idea out. Sometimes you have possibilities that have no overlap. So let's say this is a set of all possibilities. And let's say this is a set that meets condition A. Let's say that this is the set that meets condition B. So in this situation, there is no overlap. Nothing is a member of both set A and B."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "And let's say this is a set that meets condition A. Let's say that this is the set that meets condition B. So in this situation, there is no overlap. Nothing is a member of both set A and B. So in this situation, the probability of A and B is 0. There is no overlap. And these type of conditions, or these two events, are called mutually exclusive."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Nothing is a member of both set A and B. So in this situation, the probability of A and B is 0. There is no overlap. And these type of conditions, or these two events, are called mutually exclusive. So if events are mutually exclusive, that means that they both cannot happen at the same time. There's no event that meets both of these conditions. And if things are mutually exclusive, then you can say the probability of A or B is a probability of A plus B, because this thing is 0."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "And these type of conditions, or these two events, are called mutually exclusive. So if events are mutually exclusive, that means that they both cannot happen at the same time. There's no event that meets both of these conditions. And if things are mutually exclusive, then you can say the probability of A or B is a probability of A plus B, because this thing is 0. But if things are not mutually exclusive, you would have to subtract out the overlap. Probably the best way to think about it is to just always realize that you have to subtract out the overlap. And obviously, if something is mutually exclusive, the probability of getting A and B is going to be 0."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "So to make things concrete as quickly as possible, I'll start with a very tangible example of a binomial variable, and then we'll think a little bit more abstractly about what makes it binomial. So let's say that I have a coin. So this is my coin here. Doesn't even have to be a fair coin. Let me just draw this really fast. So that's my coin. And let's say on a given flip of that coin, the probability that I get heads is 0.6, and the probability that I get tails, well, it would be one minus 0.6, or 0.4."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Doesn't even have to be a fair coin. Let me just draw this really fast. So that's my coin. And let's say on a given flip of that coin, the probability that I get heads is 0.6, and the probability that I get tails, well, it would be one minus 0.6, or 0.4. And what I'm going to do is I'm going to define a random variable X as being equal to the number of heads after 10 flips of my coin. Now what makes this a binomial variable? Well, one of the first conditions that's often given for a binomial variable is that it's made up of a finite number of independent trials."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "And let's say on a given flip of that coin, the probability that I get heads is 0.6, and the probability that I get tails, well, it would be one minus 0.6, or 0.4. And what I'm going to do is I'm going to define a random variable X as being equal to the number of heads after 10 flips of my coin. Now what makes this a binomial variable? Well, one of the first conditions that's often given for a binomial variable is that it's made up of a finite number of independent trials. So it's made up, made up of independent, independent trials. Now what do I mean by independent trials? Well, a trial is each flip of my coin."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, one of the first conditions that's often given for a binomial variable is that it's made up of a finite number of independent trials. So it's made up, made up of independent, independent trials. Now what do I mean by independent trials? Well, a trial is each flip of my coin. So a flip is equal to a trial in the language of this statement that I just made. And what do I mean by each flip or each trial being independent? Well, the probability of whether I get heads or tails on each flip are independent of whether I just got heads or tails on some previous flip."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, a trial is each flip of my coin. So a flip is equal to a trial in the language of this statement that I just made. And what do I mean by each flip or each trial being independent? Well, the probability of whether I get heads or tails on each flip are independent of whether I just got heads or tails on some previous flip. So in this case, we are made up of independent trials. So another condition is each trial can be clearly classified as either a success or failure. Or another way of thinking about it, each trial clearly has one of two discrete outcomes."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, the probability of whether I get heads or tails on each flip are independent of whether I just got heads or tails on some previous flip. So in this case, we are made up of independent trials. So another condition is each trial can be clearly classified as either a success or failure. Or another way of thinking about it, each trial clearly has one of two discrete outcomes. So each trial, and in the example I'm giving, the flip is a trial, can be classified, can be classified as either success or failure. So in the context of this random variable X, we could define heads as a success because that's what we are happening to count up. And so you're either going to have success or failure."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Or another way of thinking about it, each trial clearly has one of two discrete outcomes. So each trial, and in the example I'm giving, the flip is a trial, can be classified, can be classified as either success or failure. So in the context of this random variable X, we could define heads as a success because that's what we are happening to count up. And so you're either going to have success or failure. You're either gonna have heads or tails on each of these trials. Now another condition for being a binomial variable is that you have a fixed number of trials. Fixed number of trials."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so you're either going to have success or failure. You're either gonna have heads or tails on each of these trials. Now another condition for being a binomial variable is that you have a fixed number of trials. Fixed number of trials. So in this case, we're saying that we have 10 trials, 10 flips of our coin. And then the last condition is the probability of success, and in this context, success is a heads, on each trial, each trial is constant. And we've already talked about it."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Fixed number of trials. So in this case, we're saying that we have 10 trials, 10 flips of our coin. And then the last condition is the probability of success, and in this context, success is a heads, on each trial, each trial is constant. And we've already talked about it. On each trial on each flip, the probability of heads is going to stay at 0.6. If for some reason that were to change from trial to trial, maybe if you were to swap the coin and each coin had a different probability, then this would no longer be a binomial variable. And so you might say, okay, that's reasonable."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "And we've already talked about it. On each trial on each flip, the probability of heads is going to stay at 0.6. If for some reason that were to change from trial to trial, maybe if you were to swap the coin and each coin had a different probability, then this would no longer be a binomial variable. And so you might say, okay, that's reasonable. I get why this is a binomial variable. Can you give me an example of something that is not a binomial variable? Well, let's say that I were to define the variable Y, and it's equal to the number of kings after taking two cards from a standard deck of cards, a standard deck, without replacement, without replacement."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so you might say, okay, that's reasonable. I get why this is a binomial variable. Can you give me an example of something that is not a binomial variable? Well, let's say that I were to define the variable Y, and it's equal to the number of kings after taking two cards from a standard deck of cards, a standard deck, without replacement, without replacement. So you might immediately say, well, this feels like it could be binomial. We have each trial can be classified as either a success or failure. For each trials, when I take a card out, if I get a king, that looks like that would be a success."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, let's say that I were to define the variable Y, and it's equal to the number of kings after taking two cards from a standard deck of cards, a standard deck, without replacement, without replacement. So you might immediately say, well, this feels like it could be binomial. We have each trial can be classified as either a success or failure. For each trials, when I take a card out, if I get a king, that looks like that would be a success. If I don't get a king, that would be a failure. So it seems to meet that right over there. It has a fixed number of trials."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "For each trials, when I take a card out, if I get a king, that looks like that would be a success. If I don't get a king, that would be a failure. So it seems to meet that right over there. It has a fixed number of trials. I'm taking two cards out of the deck, so it seems to meet that. But what about these conditions, that it's made up of independent trials, or that the probability of success on each trial is constant? Well, if I get a king, the probability of king on the first trial, probability, I say king, on first trial would be equal to, well, out of a deck of 52 cards, you're going to have four kings in it."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "It has a fixed number of trials. I'm taking two cards out of the deck, so it seems to meet that. But what about these conditions, that it's made up of independent trials, or that the probability of success on each trial is constant? Well, if I get a king, the probability of king on the first trial, probability, I say king, on first trial would be equal to, well, out of a deck of 52 cards, you're going to have four kings in it. So the probability of a king on the first trial would be four out of 52. But what about the probability of getting a king on the second, on the second trial? What would this be equal to?"}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, if I get a king, the probability of king on the first trial, probability, I say king, on first trial would be equal to, well, out of a deck of 52 cards, you're going to have four kings in it. So the probability of a king on the first trial would be four out of 52. But what about the probability of getting a king on the second, on the second trial? What would this be equal to? Well, it depends on what happened on the first trial. If the first trial, you had a king, well, then you would have, so let's see, this would be the situation given first trial, first king. Well, now, there would be three kings left in a deck of 51 cards."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "What would this be equal to? Well, it depends on what happened on the first trial. If the first trial, you had a king, well, then you would have, so let's see, this would be the situation given first trial, first king. Well, now, there would be three kings left in a deck of 51 cards. But if you did not get a king on the first trial, now you have four kings in a deck of 51 cards, because remember, we're doing it without replacement. You're just taking that first card, whatever you did, and you're taking it aside. So what's interesting here is this is not made up of independent trials."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, now, there would be three kings left in a deck of 51 cards. But if you did not get a king on the first trial, now you have four kings in a deck of 51 cards, because remember, we're doing it without replacement. You're just taking that first card, whatever you did, and you're taking it aside. So what's interesting here is this is not made up of independent trials. It does not meet this condition. The probability on your second trial is dependent on what happens on your first trial. And another way to think about it is because we aren't replacing each card that we're picking, the probability of success on each trial also is not constant."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "So what's interesting here is this is not made up of independent trials. It does not meet this condition. The probability on your second trial is dependent on what happens on your first trial. And another way to think about it is because we aren't replacing each card that we're picking, the probability of success on each trial also is not constant. And so that's why this right over here is not a binomial variable. Now, if y, if we got rid of, without replacement, and if we said we did replace every card after we picked it, then things would be different. Then we actually would be looking at a binomial variable."}, {"video_title": "Binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "And another way to think about it is because we aren't replacing each card that we're picking, the probability of success on each trial also is not constant. And so that's why this right over here is not a binomial variable. Now, if y, if we got rid of, without replacement, and if we said we did replace every card after we picked it, then things would be different. Then we actually would be looking at a binomial variable. So instead of without replacement, if I just said with replacement, well then, your probability of a king on each trial is going to be four out of 52. You have a finite number of trials. Your probability of success is going to stay constant, and they would be independent."}, {"video_title": "Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3", "Sentence": "And so the next question, given that we've been talking a lot about lines of regression or regression lines, is can we fit a regression line to this? Well, if we try to, we might get something that looks like this, or maybe something that looks like this. I'm just eyeballing it. Obviously, we could input it into a computer to try to develop a linear regression model to try to minimize the sum of the squared distances from the points to the line. But you can see it's pretty difficult. And some of you might be saying, well, this looks more like some type of an exponential. So maybe we could fit an exponential to it."}, {"video_title": "Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3", "Sentence": "Obviously, we could input it into a computer to try to develop a linear regression model to try to minimize the sum of the squared distances from the points to the line. But you can see it's pretty difficult. And some of you might be saying, well, this looks more like some type of an exponential. So maybe we could fit an exponential to it. So it could look something like that. And you wouldn't be wrong. But there is a way that we can apply our tools of linear regression to this data set."}, {"video_title": "Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3", "Sentence": "So maybe we could fit an exponential to it. So it could look something like that. And you wouldn't be wrong. But there is a way that we can apply our tools of linear regression to this data set. And the way we can is instead of plotting x versus y, we can think about x versus the logarithm of y. So this is this exact same data set. You see the x values are the same."}, {"video_title": "Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3", "Sentence": "But there is a way that we can apply our tools of linear regression to this data set. And the way we can is instead of plotting x versus y, we can think about x versus the logarithm of y. So this is this exact same data set. You see the x values are the same. But for the y values, I just took the log base 10 of all of these. So 10 to the what power is equal to 2,307.23. 10 to the 3.36 power is equal to 2,307.23."}, {"video_title": "Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3", "Sentence": "You see the x values are the same. But for the y values, I just took the log base 10 of all of these. So 10 to the what power is equal to 2,307.23. 10 to the 3.36 power is equal to 2,307.23. I did that for all of these data points. I did it on a spreadsheet. And if you were to plot all of these, something neat happens."}, {"video_title": "Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3", "Sentence": "10 to the 3.36 power is equal to 2,307.23. I did that for all of these data points. I did it on a spreadsheet. And if you were to plot all of these, something neat happens. All of a sudden, when we're plotting x versus the log of y or the log of y versus x, all of a sudden it looks linear. Now, be clear. The true relationship between x and y is not linear."}, {"video_title": "Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3", "Sentence": "And if you were to plot all of these, something neat happens. All of a sudden, when we're plotting x versus the log of y or the log of y versus x, all of a sudden it looks linear. Now, be clear. The true relationship between x and y is not linear. It looks like some type of an exponential relationship. But the value of transforming the data, and there's different ways you can do it. In this case, the value of taking the log of y and thinking about it that way is now we can use our tools of linear regression."}, {"video_title": "Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3", "Sentence": "The true relationship between x and y is not linear. It looks like some type of an exponential relationship. But the value of transforming the data, and there's different ways you can do it. In this case, the value of taking the log of y and thinking about it that way is now we can use our tools of linear regression. Because this data set, you could actually fit a linear regression line to this quite well. You could imagine a line that looks something like this. It would fit the data quite well."}, {"video_title": "Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3", "Sentence": "In this case, the value of taking the log of y and thinking about it that way is now we can use our tools of linear regression. Because this data set, you could actually fit a linear regression line to this quite well. You could imagine a line that looks something like this. It would fit the data quite well. And the reason why you might wanna do this versus trying to fit an exponential is because we've already developed so many tools around linear regression and hypothesis testing around the slope and confidence intervals. And so this might be the direction you wanna go at. And what's neat is once you fit a linear regression, it's not difficult to mathematically unwind from your linear model back to an exponential one."}, {"video_title": "Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3", "Sentence": "It would fit the data quite well. And the reason why you might wanna do this versus trying to fit an exponential is because we've already developed so many tools around linear regression and hypothesis testing around the slope and confidence intervals. And so this might be the direction you wanna go at. And what's neat is once you fit a linear regression, it's not difficult to mathematically unwind from your linear model back to an exponential one. So the big takeaway here is is that the tools of linear regression can be useful even when the underlying relationship between x and y are nonlinear. And the way that we do that is by transforming the data. Here, we took a logarithm of the y's, and that helped us see a more linear relationship of log y versus x."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And as we begin our journey into the world of statistics, we will be doing a lot of what we can call descriptive statistics. So if we have a bunch of data, and if we want to tell something about all of that data without giving them all of the data, can we somehow describe it with a smaller set of numbers? So that's what we're going to focus on. And then once we build our toolkit on the descriptive statistics, then we can start to make inferences about that data, start to make conclusions, start to make judgments, and we'll start to do a lot of inferential, inferential statistics, make inferences. So with that out of the way, let's think about how we can describe the data. So let's say we have a set of numbers. We can consider this to be data."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then once we build our toolkit on the descriptive statistics, then we can start to make inferences about that data, start to make conclusions, start to make judgments, and we'll start to do a lot of inferential, inferential statistics, make inferences. So with that out of the way, let's think about how we can describe the data. So let's say we have a set of numbers. We can consider this to be data. Maybe we're measuring the heights of our plants in our garden. And let's say we have six plants, and the heights are four inches, three inches, one inch, six inches, and another one's one inch, and then another one is seven inches. And let's say someone just said, in another room, not looking at your plants, just said, well, you know, how tall are your plants?"}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We can consider this to be data. Maybe we're measuring the heights of our plants in our garden. And let's say we have six plants, and the heights are four inches, three inches, one inch, six inches, and another one's one inch, and then another one is seven inches. And let's say someone just said, in another room, not looking at your plants, just said, well, you know, how tall are your plants? And they only want to hear one number. They want to somehow have one number that represents all of these different heights of plants. How would you do that?"}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And let's say someone just said, in another room, not looking at your plants, just said, well, you know, how tall are your plants? And they only want to hear one number. They want to somehow have one number that represents all of these different heights of plants. How would you do that? Well, you'd say, well, how can I find something that maybe I want a typical number, maybe I want some number that somehow represents the middle, maybe I want the most frequent number, maybe I want the number that somehow represents the center of all of these numbers. And if you said any of those things, you would actually have done the same things that the people who first came up with descriptive statistics said. They said, well, how can we do it?"}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "How would you do that? Well, you'd say, well, how can I find something that maybe I want a typical number, maybe I want some number that somehow represents the middle, maybe I want the most frequent number, maybe I want the number that somehow represents the center of all of these numbers. And if you said any of those things, you would actually have done the same things that the people who first came up with descriptive statistics said. They said, well, how can we do it? And we'll start by thinking of the idea of average. Average. And in everyday terminology, average has a very particular meaning."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "They said, well, how can we do it? And we'll start by thinking of the idea of average. Average. And in everyday terminology, average has a very particular meaning. As we'll see, when many people talk about average, they're talking about the arithmetic mean, which we'll see shortly. But in statistics, average means something more general. It really means give me a typical, give me a typical, or give me a middle, give me a middle number."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And in everyday terminology, average has a very particular meaning. As we'll see, when many people talk about average, they're talking about the arithmetic mean, which we'll see shortly. But in statistics, average means something more general. It really means give me a typical, give me a typical, or give me a middle, give me a middle number. Or, and these are ors, and really, it's an attempt to find a measure of central tendency. Central, central tendency. So once again, you have a bunch of numbers."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It really means give me a typical, give me a typical, or give me a middle, give me a middle number. Or, and these are ors, and really, it's an attempt to find a measure of central tendency. Central, central tendency. So once again, you have a bunch of numbers. You're somehow trying to represent these with one number. We'll call it the average. That's somehow typical or a middle or the center somehow of these numbers."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So once again, you have a bunch of numbers. You're somehow trying to represent these with one number. We'll call it the average. That's somehow typical or a middle or the center somehow of these numbers. And as we'll see, there's many types of averages. The first is the one that you're probably most familiar with. It's the one that people talk about, hey, the average on this exam or the average height."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "That's somehow typical or a middle or the center somehow of these numbers. And as we'll see, there's many types of averages. The first is the one that you're probably most familiar with. It's the one that people talk about, hey, the average on this exam or the average height. And that's the arithmetic mean. So let me write it in, I'll write it in yellow. Arith, arithmetic, arithmetic mean."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It's the one that people talk about, hey, the average on this exam or the average height. And that's the arithmetic mean. So let me write it in, I'll write it in yellow. Arith, arithmetic, arithmetic mean. When arithmetic is a noun, we call it arithmetic. When it's an adjective like this, we call it arithmetic. Arithmetic mean."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Arith, arithmetic, arithmetic mean. When arithmetic is a noun, we call it arithmetic. When it's an adjective like this, we call it arithmetic. Arithmetic mean. And this is really just the sum of all the numbers divided by, and this is a human constructed definition that we've found useful, the sum of all of these numbers divided by the number of numbers we have. So given that, what is the arithmetic mean of this data set? Well, let's just compute it."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Arithmetic mean. And this is really just the sum of all the numbers divided by, and this is a human constructed definition that we've found useful, the sum of all of these numbers divided by the number of numbers we have. So given that, what is the arithmetic mean of this data set? Well, let's just compute it. It's going to be four plus three plus one plus six plus one plus seven over the number of data points we have. So we have six data points, so we're gonna divide by six. And we get four plus three is seven, plus one is eight, plus six is 14, plus one is 15, plus seven, 15 plus seven is 22."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, let's just compute it. It's going to be four plus three plus one plus six plus one plus seven over the number of data points we have. So we have six data points, so we're gonna divide by six. And we get four plus three is seven, plus one is eight, plus six is 14, plus one is 15, plus seven, 15 plus seven is 22. We'll do that one more time. You have seven, eight, 14, 15, 22. All of that over six."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And we get four plus three is seven, plus one is eight, plus six is 14, plus one is 15, plus seven, 15 plus seven is 22. We'll do that one more time. You have seven, eight, 14, 15, 22. All of that over six. And we could write this as a mixed number. Six goes into 22 three times with the remainder of four. So it's three and 4 6, which is the same thing as three and 2 3rds."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "All of that over six. And we could write this as a mixed number. Six goes into 22 three times with the remainder of four. So it's three and 4 6, which is the same thing as three and 2 3rds. We could write this as a decimal with 3.6 repeating. So this is also 3.6 repeating. We could write it any one of those ways."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So it's three and 4 6, which is the same thing as three and 2 3rds. We could write this as a decimal with 3.6 repeating. So this is also 3.6 repeating. We could write it any one of those ways. But this is kind of a representative number. This is trying to get at a central tendency. Once again, these are human constructed."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We could write it any one of those ways. But this is kind of a representative number. This is trying to get at a central tendency. Once again, these are human constructed. No one ever, it's not like someone just found some religious document that said, this is the way that the arithmetic mean must be defined. It's not as pure of a computation as say finding the circumference of the circle, which there really is. That was kind of, we studied the universe and that just fell out of our study of the universe."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Once again, these are human constructed. No one ever, it's not like someone just found some religious document that said, this is the way that the arithmetic mean must be defined. It's not as pure of a computation as say finding the circumference of the circle, which there really is. That was kind of, we studied the universe and that just fell out of our study of the universe. It's a human constructed definition that we found useful. Now, there are other ways to measure the average or find a typical or middle value. The other very typical way is the median."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "That was kind of, we studied the universe and that just fell out of our study of the universe. It's a human constructed definition that we found useful. Now, there are other ways to measure the average or find a typical or middle value. The other very typical way is the median. And I will write median, I'm running out of colors. I will write median in pink. So there is the median."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "The other very typical way is the median. And I will write median, I'm running out of colors. I will write median in pink. So there is the median. And the median is literally looking for the middle number. So if you were to order all the numbers in your set and find the middle one, then that is your median. So given that, what's the median of this set of numbers going to be?"}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So there is the median. And the median is literally looking for the middle number. So if you were to order all the numbers in your set and find the middle one, then that is your median. So given that, what's the median of this set of numbers going to be? Let's try to figure it out. Let's try to order it. So we have one, then we have another one, then we have a three, then we have a four, a six and a seven."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So given that, what's the median of this set of numbers going to be? Let's try to figure it out. Let's try to order it. So we have one, then we have another one, then we have a three, then we have a four, a six and a seven. So all I did is I reordered this. And so what's the middle number? Well, you look here, since we have an even number of numbers we have six numbers."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So we have one, then we have another one, then we have a three, then we have a four, a six and a seven. So all I did is I reordered this. And so what's the middle number? Well, you look here, since we have an even number of numbers we have six numbers. There's not one middle number. You actually have two middle numbers here. You have two middle numbers right over here."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, you look here, since we have an even number of numbers we have six numbers. There's not one middle number. You actually have two middle numbers here. You have two middle numbers right over here. You have the three and the four. And in this case, when you have two middle numbers, you actually go halfway between these two numbers. Or essentially, you're taking the arithmetic mean of these two numbers to find the median."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "You have two middle numbers right over here. You have the three and the four. And in this case, when you have two middle numbers, you actually go halfway between these two numbers. Or essentially, you're taking the arithmetic mean of these two numbers to find the median. So the median is going to be halfway in between three and four which is going to be 3.5. So the median in this case is 3.5. So if you have an even number of numbers, the median or the middle two, essentially the arithmetic mean of the middle two are halfway between the middle two."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Or essentially, you're taking the arithmetic mean of these two numbers to find the median. So the median is going to be halfway in between three and four which is going to be 3.5. So the median in this case is 3.5. So if you have an even number of numbers, the median or the middle two, essentially the arithmetic mean of the middle two are halfway between the middle two. If you have an odd number of numbers, it's a little bit easier to compute. And just so that we see that, let me give you another data set. Let's say our data set, and I'll order it for us."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So if you have an even number of numbers, the median or the middle two, essentially the arithmetic mean of the middle two are halfway between the middle two. If you have an odd number of numbers, it's a little bit easier to compute. And just so that we see that, let me give you another data set. Let's say our data set, and I'll order it for us. Let's say our data set was 0,750, I don't know, 10,000, 10,000, and 1,000,000. 1,000,000. Let's say that that is our data set."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let's say our data set, and I'll order it for us. Let's say our data set was 0,750, I don't know, 10,000, 10,000, and 1,000,000. 1,000,000. Let's say that that is our data set. Kind of a crazy data set. But in this situation, what is our median? Well, here we have five numbers."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let's say that that is our data set. Kind of a crazy data set. But in this situation, what is our median? Well, here we have five numbers. We have an odd number of numbers. So it's easier to pick out a middle. The middle is a number that is greater than two of the numbers and is less than two of the numbers."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, here we have five numbers. We have an odd number of numbers. So it's easier to pick out a middle. The middle is a number that is greater than two of the numbers and is less than two of the numbers. It's exactly in the middle. So in this case, our median is 50. Now, the third measure of central tendency, and this is the one that's probably used least often in life, is the mode."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "The middle is a number that is greater than two of the numbers and is less than two of the numbers. It's exactly in the middle. So in this case, our median is 50. Now, the third measure of central tendency, and this is the one that's probably used least often in life, is the mode. And people often forget about it and it sounds like something very complex. But what we'll see is it's actually a very straightforward idea. And in some ways, it is the most basic idea."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Now, the third measure of central tendency, and this is the one that's probably used least often in life, is the mode. And people often forget about it and it sounds like something very complex. But what we'll see is it's actually a very straightforward idea. And in some ways, it is the most basic idea. So the mode is actually the most common number in a data set, if there is a most common number. If all the numbers are represented equally, if there's no one single most common number, then you have no mode. But given that definition of the mode, what is the single most common number in our original data set, in this data set right over here?"}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And in some ways, it is the most basic idea. So the mode is actually the most common number in a data set, if there is a most common number. If all the numbers are represented equally, if there's no one single most common number, then you have no mode. But given that definition of the mode, what is the single most common number in our original data set, in this data set right over here? Well, we only have one four. We only have one three. But we have two ones."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "But given that definition of the mode, what is the single most common number in our original data set, in this data set right over here? Well, we only have one four. We only have one three. But we have two ones. We have one six and one seven. So the number that shows up the most number of times here is our one. So the mode, the most typical number, the most common number here is a one."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "But we have two ones. We have one six and one seven. So the number that shows up the most number of times here is our one. So the mode, the most typical number, the most common number here is a one. So you see, these are all different ways of trying to get at a typical or middle or central tendency. But you do it in very, very different ways. And as we study more and more statistics, we'll see that they're good for different things."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So the mode, the most typical number, the most common number here is a one. So you see, these are all different ways of trying to get at a typical or middle or central tendency. But you do it in very, very different ways. And as we study more and more statistics, we'll see that they're good for different things. This is used very frequently. The median is really good if you have some kind of crazy number out here that could have otherwise skewed the arithmetic mean. The mode could also be useful in situations like that, especially if you do have one number that's showing up a lot more frequently."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "Each dot plot below represents a different set of data. We see that here. Order the dot plots from largest standard deviation top to smallest standard deviation bottom. So pause this video and see if you can do that, or at least if you could rank these from largest standard deviation to smallest standard deviation. All right, now let's work through this together. And I'm doing this on Khan Academy where I can move these around to order them. But let's just remind ourselves what the standard deviation is or how we can perceive it."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "So pause this video and see if you can do that, or at least if you could rank these from largest standard deviation to smallest standard deviation. All right, now let's work through this together. And I'm doing this on Khan Academy where I can move these around to order them. But let's just remind ourselves what the standard deviation is or how we can perceive it. You could view the standard deviation as a measure of the typical distance from each of the data points to the mean. So the largest standard deviation, which we wanna put on top, would be the one where typically our data points are further from the mean and our smallest standard deviation would be the ones where it feels like, on average, our data points are closer to the mean. When all of these examples, our mean looks to be right in the center, right between 50 and 100, so right around 75."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "But let's just remind ourselves what the standard deviation is or how we can perceive it. You could view the standard deviation as a measure of the typical distance from each of the data points to the mean. So the largest standard deviation, which we wanna put on top, would be the one where typically our data points are further from the mean and our smallest standard deviation would be the ones where it feels like, on average, our data points are closer to the mean. When all of these examples, our mean looks to be right in the center, right between 50 and 100, so right around 75. So it's really about how spread apart they are from that. And if you look at this first one, it has these two data points, the one on the left and one on the right, that are pretty far, and then you have these two that are a little bit closer and then these two that are inside. This one right over here, to get from this top one to this middle one, you essentially are taking this data point and making it go further, and taking this data point and making it go further."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "When all of these examples, our mean looks to be right in the center, right between 50 and 100, so right around 75. So it's really about how spread apart they are from that. And if you look at this first one, it has these two data points, the one on the left and one on the right, that are pretty far, and then you have these two that are a little bit closer and then these two that are inside. This one right over here, to get from this top one to this middle one, you essentially are taking this data point and making it go further, and taking this data point and making it go further. And so this one is going to have a higher standard deviation than that one. So let me put it just like that. And I just wanna make it very clear."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "This one right over here, to get from this top one to this middle one, you essentially are taking this data point and making it go further, and taking this data point and making it go further. And so this one is going to have a higher standard deviation than that one. So let me put it just like that. And I just wanna make it very clear. Keep track of what's the difference between these two things. Here you have this data point and this data point that was closer in, and then if you move it further, that's going to make your typical distance from the middle more, which is exactly what happened there. Now what about this one?"}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "And I just wanna make it very clear. Keep track of what's the difference between these two things. Here you have this data point and this data point that was closer in, and then if you move it further, that's going to make your typical distance from the middle more, which is exactly what happened there. Now what about this one? Well, this one is starting here and then taking this point and taking this point and moving it closer. And so that would make our typical distance from the middle, from the mean, shorter. So this would have the smallest standard deviation and this would have the largest."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "Now what about this one? Well, this one is starting here and then taking this point and taking this point and moving it closer. And so that would make our typical distance from the middle, from the mean, shorter. So this would have the smallest standard deviation and this would have the largest. Let's do another example. So same idea, order the dot plots from largest standard deviation on the top to smallest standard deviation on the bottom. Pause this video and see if you can figure that out."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "So this would have the smallest standard deviation and this would have the largest. Let's do another example. So same idea, order the dot plots from largest standard deviation on the top to smallest standard deviation on the bottom. Pause this video and see if you can figure that out. So this is interesting because these all have different means. Just eyeballing it, the mean for this first one is right around here. The mean for the second one is right around here at around 10."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "Pause this video and see if you can figure that out. So this is interesting because these all have different means. Just eyeballing it, the mean for this first one is right around here. The mean for the second one is right around here at around 10. And the mean for the third one, it looks like the same mean as this top one. And so pause this video. How would you order them?"}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "The mean for the second one is right around here at around 10. And the mean for the third one, it looks like the same mean as this top one. And so pause this video. How would you order them? All right, so just eyeballing it, these, this middle one right over here, your typical data point seems furthest from the mean. You definitely have, if the mean is here, you have these, this data point and this data point that are quite far from that mean. And even this data point and this data point are at least as far as any of the data points that we have in the top or the bottom one."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "How would you order them? All right, so just eyeballing it, these, this middle one right over here, your typical data point seems furthest from the mean. You definitely have, if the mean is here, you have these, this data point and this data point that are quite far from that mean. And even this data point and this data point are at least as far as any of the data points that we have in the top or the bottom one. So I would say this has the largest standard deviation. And if I were to compare between these two, if you think about how you would get the difference between these two is if you took this data point and moved it at, and you moved it to the mean, and if you took this data point and you moved it to the mean, you would get this third situation. And so this third situation, you have the fewest data points that are sitting away from the mean relative to this one."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "Probability, a word that you've probably heard a lot of and you are probably a little bit familiar with it, but hopefully this will give you a little deeper understanding. So let's say that I have a fair coin over here. So when I talk about a fair coin, I mean that it has an equal chance of landing on one side or another. So you can maybe view it as the sides are equal, the weight is the same on either side. If I flip it in the air, it's not more likely to land on one side or the other. It's equally likely. And so you have one side of this coin, so this would be the heads, I guess."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "So you can maybe view it as the sides are equal, the weight is the same on either side. If I flip it in the air, it's not more likely to land on one side or the other. It's equally likely. And so you have one side of this coin, so this would be the heads, I guess. Try to draw George Washington. I'll assume it's a quarter of some kind. And then the other side, of course, is the tails."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "And so you have one side of this coin, so this would be the heads, I guess. Try to draw George Washington. I'll assume it's a quarter of some kind. And then the other side, of course, is the tails. So that is heads. The other side right over there is tails. And so if I were to ask you, what is the probability?"}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "And then the other side, of course, is the tails. So that is heads. The other side right over there is tails. And so if I were to ask you, what is the probability? I'm going to flip a coin, and I want to know what is the probability of getting heads. And I could write that like this. The probability of getting heads."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "And so if I were to ask you, what is the probability? I'm going to flip a coin, and I want to know what is the probability of getting heads. And I could write that like this. The probability of getting heads. And you probably, just based on that question, have a sense of what probability is asking. It's asking for some type of way of getting your hands around an event that's fundamentally random. We don't know whether it's heads or tails, but we can start to describe the chances of it being heads or tails."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "The probability of getting heads. And you probably, just based on that question, have a sense of what probability is asking. It's asking for some type of way of getting your hands around an event that's fundamentally random. We don't know whether it's heads or tails, but we can start to describe the chances of it being heads or tails. And we'll talk about different ways of describing that. So one way to think about it, and this is the way that probability tends to be introduced in textbooks, is you say, well look, how many different equally likely possibilities are there? So how many equally likely possibilities?"}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "We don't know whether it's heads or tails, but we can start to describe the chances of it being heads or tails. And we'll talk about different ways of describing that. So one way to think about it, and this is the way that probability tends to be introduced in textbooks, is you say, well look, how many different equally likely possibilities are there? So how many equally likely possibilities? So number of equally likely possibilities. And of the number of equally likely possibilities, I care about the number that contain my event right here. So the number of possibilities that meet my constraint."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "So how many equally likely possibilities? So number of equally likely possibilities. And of the number of equally likely possibilities, I care about the number that contain my event right here. So the number of possibilities that meet my constraint. That meet my conditions. So in the case of the probability of figuring out heads, what is the number of equally likely possibilities? Well there's only two possibilities."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "So the number of possibilities that meet my constraint. That meet my conditions. So in the case of the probability of figuring out heads, what is the number of equally likely possibilities? Well there's only two possibilities. We're assuming that the coin can't land on its corner and just stand straight up. We're assuming that it lands flat. So there's two possibilities here."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "Well there's only two possibilities. We're assuming that the coin can't land on its corner and just stand straight up. We're assuming that it lands flat. So there's two possibilities here. Two equally likely possibilities. You could either get heads or you could get tails. And what's the number of possibilities that meet my conditions?"}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "So there's two possibilities here. Two equally likely possibilities. You could either get heads or you could get tails. And what's the number of possibilities that meet my conditions? Well there's only one. The condition of heads. So it'll be 1 over 2."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "And what's the number of possibilities that meet my conditions? Well there's only one. The condition of heads. So it'll be 1 over 2. So the one way to think about it is the probability of getting heads is equal to 1 over 2. Is equal to 1 half. If I wanted to write that as a percentage, we know that 1 half is the same thing as 50%."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "So it'll be 1 over 2. So the one way to think about it is the probability of getting heads is equal to 1 over 2. Is equal to 1 half. If I wanted to write that as a percentage, we know that 1 half is the same thing as 50%. Now another way to think about or conceptualize probability that will give you this exact same answer, is to say, well if I were to run the experiment of flipping a coin. So this flip, you view this as an experiment. I know this isn't the kind of experiment that you're used to."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "If I wanted to write that as a percentage, we know that 1 half is the same thing as 50%. Now another way to think about or conceptualize probability that will give you this exact same answer, is to say, well if I were to run the experiment of flipping a coin. So this flip, you view this as an experiment. I know this isn't the kind of experiment that you're used to. You know, you normally think an experiment is doing something in chemistry or physics or all the rest. But an experiment is every time you run this random event. So one way to think about probability is if I were to do this experiment, an experiment many, many, many times."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "I know this isn't the kind of experiment that you're used to. You know, you normally think an experiment is doing something in chemistry or physics or all the rest. But an experiment is every time you run this random event. So one way to think about probability is if I were to do this experiment, an experiment many, many, many times. If I were to do it a thousand times or a million times or a billion times or a trillion times, and the more the better. What percentage of those would give me what I care about? What percentage of those would give me heads?"}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "So one way to think about probability is if I were to do this experiment, an experiment many, many, many times. If I were to do it a thousand times or a million times or a billion times or a trillion times, and the more the better. What percentage of those would give me what I care about? What percentage of those would give me heads? And so another way to think about this 50% probability of getting heads, is if I were to run this experiment tons of times. If I were to run this forever and closer or an infinite number of times, what percentage of those would be heads? You would get this 50%."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "What percentage of those would give me heads? And so another way to think about this 50% probability of getting heads, is if I were to run this experiment tons of times. If I were to run this forever and closer or an infinite number of times, what percentage of those would be heads? You would get this 50%. And you can run that simulation. You can flip a coin. And it's actually a fun thing to do."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "You would get this 50%. And you can run that simulation. You can flip a coin. And it's actually a fun thing to do. I encourage you to do it. If you take 100 or 200 quarters or pennies, stick them in a big box, shake the box. So you're kind of simultaneously flipping all of the coins."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "And it's actually a fun thing to do. I encourage you to do it. If you take 100 or 200 quarters or pennies, stick them in a big box, shake the box. So you're kind of simultaneously flipping all of the coins. And then count how many of those are going to be heads. And you're going to see that the larger the number that you are doing, the more likely you're going to get something really close to 50%. There's always some chance, even if you flip the coin a million times, there's some super duper small chance that you get all tails."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "So you're kind of simultaneously flipping all of the coins. And then count how many of those are going to be heads. And you're going to see that the larger the number that you are doing, the more likely you're going to get something really close to 50%. There's always some chance, even if you flip the coin a million times, there's some super duper small chance that you get all tails. But the more you do, the more likely that things are going to trend towards 50% of them are going to be heads. Now let's just apply these same ideas. And while we're starting with probability, at least kind of the basic, this is probably an easier thing to conceptualize."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "There's always some chance, even if you flip the coin a million times, there's some super duper small chance that you get all tails. But the more you do, the more likely that things are going to trend towards 50% of them are going to be heads. Now let's just apply these same ideas. And while we're starting with probability, at least kind of the basic, this is probably an easier thing to conceptualize. But a lot of times this is actually a helpful one too, this idea that if you run the experiment many, many, many, many times, what percentage of those trials are going to give you what you're asking for? In this case, it was heads. Now let's do another very typical example when you first learn probability."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "And while we're starting with probability, at least kind of the basic, this is probably an easier thing to conceptualize. But a lot of times this is actually a helpful one too, this idea that if you run the experiment many, many, many, many times, what percentage of those trials are going to give you what you're asking for? In this case, it was heads. Now let's do another very typical example when you first learn probability. And this is the idea of rolling a die. So here's my die right over here. And of course you have the different sides of the die."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "Now let's do another very typical example when you first learn probability. And this is the idea of rolling a die. So here's my die right over here. And of course you have the different sides of the die. So that's the 1, that's the 2, that's the 3. And what I want to do, and we know of course that there are, and I'm assuming this is a fair die, and so there are 6 equally likely possibilities. When you roll this, you could get a 1, a 2, a 3, a 4, a 5, or a 6."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "And of course you have the different sides of the die. So that's the 1, that's the 2, that's the 3. And what I want to do, and we know of course that there are, and I'm assuming this is a fair die, and so there are 6 equally likely possibilities. When you roll this, you could get a 1, a 2, a 3, a 4, a 5, or a 6. And they are all equally likely. So if I were to ask you, what is the probability, given that I'm rolling a fair die, so the experiment is rolling this fair die, what is the probability of getting a 1? Well, what are the number of equally likely possibilities?"}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "When you roll this, you could get a 1, a 2, a 3, a 4, a 5, or a 6. And they are all equally likely. So if I were to ask you, what is the probability, given that I'm rolling a fair die, so the experiment is rolling this fair die, what is the probability of getting a 1? Well, what are the number of equally likely possibilities? Well, I have 6 equally likely possibilities. And how many of those meet my conditions? Well, only one of them meets my condition, that right there."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "Well, what are the number of equally likely possibilities? Well, I have 6 equally likely possibilities. And how many of those meet my conditions? Well, only one of them meets my condition, that right there. So there is a 1 6 probability of rolling a 1. What is the probability of rolling a 1 or a 6? Well, once again, there are 6 equally likely possibilities for what I can get."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "Well, only one of them meets my condition, that right there. So there is a 1 6 probability of rolling a 1. What is the probability of rolling a 1 or a 6? Well, once again, there are 6 equally likely possibilities for what I can get. And there are now 2 possibilities that meet my conditions. I could roll a 1 or I could roll a 6. So now there are 2 possibilities that meet my constraints, my conditions."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "Well, once again, there are 6 equally likely possibilities for what I can get. And there are now 2 possibilities that meet my conditions. I could roll a 1 or I could roll a 6. So now there are 2 possibilities that meet my constraints, my conditions. So there is a 1 3 probability of rolling a 1 or a 6. Now what is the probability, this might seem a little silly to even ask this question, but I'll ask it just to make it clear. What is the probability of rolling a 2 and a 3?"}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "So now there are 2 possibilities that meet my constraints, my conditions. So there is a 1 3 probability of rolling a 1 or a 6. Now what is the probability, this might seem a little silly to even ask this question, but I'll ask it just to make it clear. What is the probability of rolling a 2 and a 3? And I'm just talking about one roll of the die. Well, in any roll of the die, I can only get a 2 or a 3. I'm not talking about taking 2 rolls of this die."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "What is the probability of rolling a 2 and a 3? And I'm just talking about one roll of the die. Well, in any roll of the die, I can only get a 2 or a 3. I'm not talking about taking 2 rolls of this die. So in this situation, there are 6 possibilities, but none of these possibilities are 2 and a 3. 2 and a 3 cannot exist on one trial. You cannot get a 2 and a 3 in the same experiment."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "I'm not talking about taking 2 rolls of this die. So in this situation, there are 6 possibilities, but none of these possibilities are 2 and a 3. 2 and a 3 cannot exist on one trial. You cannot get a 2 and a 3 in the same experiment. Getting a 2 and a 3 are mutually exclusive events. They cannot happen at the same time. So the probability of this is actually 0."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "You cannot get a 2 and a 3 in the same experiment. Getting a 2 and a 3 are mutually exclusive events. They cannot happen at the same time. So the probability of this is actually 0. There's no way to roll this normal die and all of a sudden you get a 2 and a 3. In fact, I don't want to confuse you with that because it's kind of abstract and impossible. So let's cross this out right over here."}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "So the probability of this is actually 0. There's no way to roll this normal die and all of a sudden you get a 2 and a 3. In fact, I don't want to confuse you with that because it's kind of abstract and impossible. So let's cross this out right over here. Now what is the probability of getting an even number? So once again, you have 6 equally likely possibilities when I roll that die. And which of these possibilities meet my conditions, the condition of being even?"}, {"video_title": "Probability explained Independent and dependent events Probability and Statistics Khan Academy.mp3", "Sentence": "So let's cross this out right over here. Now what is the probability of getting an even number? So once again, you have 6 equally likely possibilities when I roll that die. And which of these possibilities meet my conditions, the condition of being even? Well, 2 is even, 4 is even, and 6 is even. So 3 of the possibilities meet my conditions, meet my constraints. So this is 1 half."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And that's the central limit theorem. And what it tells us is we can start off with any distribution that has a well-defined mean and variance, and if it has a well-defined variance, it has a well-defined standard deviation. And it could be a continuous distribution or a discrete one. I'll draw a discrete one just because it's easier to imagine, at least for the purposes of this video. So let's say I have a discrete probability distribution function, and I want to be very careful not to make it look anything close to a normal distribution, because I want to show you the power of the central limit theorem. So let's say I have a distribution, let's say I can take on values 1 through 6. 1, 2, 3, 4, 5, 6, it's some kind of crazy dice that's very likely to get a 1."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "I'll draw a discrete one just because it's easier to imagine, at least for the purposes of this video. So let's say I have a discrete probability distribution function, and I want to be very careful not to make it look anything close to a normal distribution, because I want to show you the power of the central limit theorem. So let's say I have a distribution, let's say I can take on values 1 through 6. 1, 2, 3, 4, 5, 6, it's some kind of crazy dice that's very likely to get a 1. Let's say it's impossible, let me make that a straight line. You're very high likelihood of getting a 1. Let's say it's impossible to get a 2."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "1, 2, 3, 4, 5, 6, it's some kind of crazy dice that's very likely to get a 1. Let's say it's impossible, let me make that a straight line. You're very high likelihood of getting a 1. Let's say it's impossible to get a 2. Let's say it's an okay likelihood of getting a 3 or a 4. Let's say it's impossible to get a 5. Let's say it's very likely to get a 6 like that."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Let's say it's impossible to get a 2. Let's say it's an okay likelihood of getting a 3 or a 4. Let's say it's impossible to get a 5. Let's say it's very likely to get a 6 like that. So that's my probability distribution function. If I were to draw a mean, this is symmetric, so maybe the mean would be something like that. It would be halfway, so that would be my mean right there, the standard deviation."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Let's say it's very likely to get a 6 like that. So that's my probability distribution function. If I were to draw a mean, this is symmetric, so maybe the mean would be something like that. It would be halfway, so that would be my mean right there, the standard deviation. Maybe it would be that far and that far above and below the mean. But that's my discrete probability distribution function. Now what I'm going to do here, instead of just taking samples of this random variable that's described by this probability distribution function, I'm going to take samples of it, but I'm going to average the samples, and then look at those samples and see the frequency of the averages that I get."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "It would be halfway, so that would be my mean right there, the standard deviation. Maybe it would be that far and that far above and below the mean. But that's my discrete probability distribution function. Now what I'm going to do here, instead of just taking samples of this random variable that's described by this probability distribution function, I'm going to take samples of it, but I'm going to average the samples, and then look at those samples and see the frequency of the averages that I get. And when I say average, I mean the mean. Let me define something. Let's say my sample size, and I could put any number here, but let's say first off we try a sample size of n is equal to 4."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Now what I'm going to do here, instead of just taking samples of this random variable that's described by this probability distribution function, I'm going to take samples of it, but I'm going to average the samples, and then look at those samples and see the frequency of the averages that I get. And when I say average, I mean the mean. Let me define something. Let's say my sample size, and I could put any number here, but let's say first off we try a sample size of n is equal to 4. And what that means is I'm going to take 4 samples from this. So let's say the first time I take 4 samples, so my sample size is 4, let's say I get a 1, let's say I get another 1, and let's say I get a 3, and I get a 6. So that right there is my first sample of sample size 4."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Let's say my sample size, and I could put any number here, but let's say first off we try a sample size of n is equal to 4. And what that means is I'm going to take 4 samples from this. So let's say the first time I take 4 samples, so my sample size is 4, let's say I get a 1, let's say I get another 1, and let's say I get a 3, and I get a 6. So that right there is my first sample of sample size 4. I know the terminology can get confusing because this is a sample that's made up of 4 samples. But when we talk about the sample mean and the sampling distribution of the sample mean, which we're going to talk more and more about over the next few videos, normally the sample refers to the set of samples from your distribution, and the sample size tells you how many you actually took from your distribution. But the terminology can be very confusing because you could easily view one of these as a sample."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So that right there is my first sample of sample size 4. I know the terminology can get confusing because this is a sample that's made up of 4 samples. But when we talk about the sample mean and the sampling distribution of the sample mean, which we're going to talk more and more about over the next few videos, normally the sample refers to the set of samples from your distribution, and the sample size tells you how many you actually took from your distribution. But the terminology can be very confusing because you could easily view one of these as a sample. But we're taking 4 samples from here. We have a sample size of 4. And what I'm going to do is I'm going to average them."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But the terminology can be very confusing because you could easily view one of these as a sample. But we're taking 4 samples from here. We have a sample size of 4. And what I'm going to do is I'm going to average them. So let's say the mean, I want to be very careful when I say average, the mean of this first sample of size 4 is what? 1 plus 1 is 2, 2 plus 3 is 5, 5 plus 6 is 11, 11 divided by 4 is 2.75. That is my first sample mean for my first sample of size 4."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And what I'm going to do is I'm going to average them. So let's say the mean, I want to be very careful when I say average, the mean of this first sample of size 4 is what? 1 plus 1 is 2, 2 plus 3 is 5, 5 plus 6 is 11, 11 divided by 4 is 2.75. That is my first sample mean for my first sample of size 4. Let me do another one. My second sample of size 4, let's say that I get a 3, a 4, let's say I get another 3, and let's say I get a 1. I just didn't happen to get a 6 that time."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "That is my first sample mean for my first sample of size 4. Let me do another one. My second sample of size 4, let's say that I get a 3, a 4, let's say I get another 3, and let's say I get a 1. I just didn't happen to get a 6 that time. And notice, I can't get a 2 or a 5. It's impossible for this distribution. The chance of getting a 2 or a 5 is 0, so I can't have any 2s or 5s over here."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "I just didn't happen to get a 6 that time. And notice, I can't get a 2 or a 5. It's impossible for this distribution. The chance of getting a 2 or a 5 is 0, so I can't have any 2s or 5s over here. So for the second sample of sample size 4, my sample mean, so my second sample mean is going to be 3 plus 4 is 7, 7 plus 3 is 10, plus 1 is 11, 11 divided by 4, once again is 2.75. Let me do one more, because I really want to make it clear what we're doing here. So I do one more."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "The chance of getting a 2 or a 5 is 0, so I can't have any 2s or 5s over here. So for the second sample of sample size 4, my sample mean, so my second sample mean is going to be 3 plus 4 is 7, 7 plus 3 is 10, plus 1 is 11, 11 divided by 4, once again is 2.75. Let me do one more, because I really want to make it clear what we're doing here. So I do one more. Actually, we're going to do a gazillion more, but let me just do one more in detail. So let's say my third sample of sample size 4, I get, so I'm going to literally take 4 samples. So my sample is made up of 4 samples from this original crazy distribution."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So I do one more. Actually, we're going to do a gazillion more, but let me just do one more in detail. So let's say my third sample of sample size 4, I get, so I'm going to literally take 4 samples. So my sample is made up of 4 samples from this original crazy distribution. Let's say I get a 1, a 1, and a 6, and a 6. And so my third sample mean is going to be 1 plus 1 is 2, 2 plus 6 is 8, 8 plus 6 is 14, 14 divided by 4, 14 divided by 4 is what? 3.5."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So my sample is made up of 4 samples from this original crazy distribution. Let's say I get a 1, a 1, and a 6, and a 6. And so my third sample mean is going to be 1 plus 1 is 2, 2 plus 6 is 8, 8 plus 6 is 14, 14 divided by 4, 14 divided by 4 is what? 3.5. And as I find each of these sample means, so for each of my samples of sample size 4, I figure out a mean. And as I do each of them, I'm going to plot it on a frequency distribution. And this is all going to amaze you in a few seconds."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "3.5. And as I find each of these sample means, so for each of my samples of sample size 4, I figure out a mean. And as I do each of them, I'm going to plot it on a frequency distribution. And this is all going to amaze you in a few seconds. So I plot this all on a frequency distribution. So I say, okay, on my first sample, my first sample mean was 2.75. So I'm plotting the actual frequency of the sample means I get for each sample."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And this is all going to amaze you in a few seconds. So I plot this all on a frequency distribution. So I say, okay, on my first sample, my first sample mean was 2.75. So I'm plotting the actual frequency of the sample means I get for each sample. So 2.75, I got it one time, so I'll put a little plot there. So that's from that one right there. And I got, the next time, I also got a 2.75."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So I'm plotting the actual frequency of the sample means I get for each sample. So 2.75, I got it one time, so I'll put a little plot there. So that's from that one right there. And I got, the next time, I also got a 2.75. That's a 2.75 there, so I got twice. So I'll plot the frequency right there. Then I got a 3.5, so all the possible values, I could have a 3, I could have a 3.25, I could have a 3.5."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And I got, the next time, I also got a 2.75. That's a 2.75 there, so I got twice. So I'll plot the frequency right there. Then I got a 3.5, so all the possible values, I could have a 3, I could have a 3.25, I could have a 3.5. So then I have the 3.5, so I'll plot it right there. And what I'm going to do is I'm going to keep taking these samples. Maybe I'll take 10,000 of them."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Then I got a 3.5, so all the possible values, I could have a 3, I could have a 3.25, I could have a 3.5. So then I have the 3.5, so I'll plot it right there. And what I'm going to do is I'm going to keep taking these samples. Maybe I'll take 10,000 of them. So I'm going to keep taking these samples. So I go all the way to S, you know, 10,000. I just do a bunch of these."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Maybe I'll take 10,000 of them. So I'm going to keep taking these samples. So I go all the way to S, you know, 10,000. I just do a bunch of these. And what it's going to look like over time is each of these, I'm going to make it a dot because I'm going to have to zoom out. So if I look at it like this, over time, still it's all the values that it might be able to take on. You know, 2.75 might be here."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "I just do a bunch of these. And what it's going to look like over time is each of these, I'm going to make it a dot because I'm going to have to zoom out. So if I look at it like this, over time, still it's all the values that it might be able to take on. You know, 2.75 might be here. So this first dot is going to be right there. And that second one is going to be right there. And that one at 3.5 is going to look right there."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "You know, 2.75 might be here. So this first dot is going to be right there. And that second one is going to be right there. And that one at 3.5 is going to look right there. But I'm going to do it 10,000 times because I'm going to have 10,000 dots. And let's say as I do it, I'm going to just keep plotting them. I'm just going to keep plotting the frequencies."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And that one at 3.5 is going to look right there. But I'm going to do it 10,000 times because I'm going to have 10,000 dots. And let's say as I do it, I'm going to just keep plotting them. I'm just going to keep plotting the frequencies. I'm just going to keep plotting them over and over and over again. And what you're going to see is as I take many, many samples of size 4, I'm going to have something that's going to start kind of approximating a normal distribution. So each of these dots represent an incidence of a sample mean."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "I'm just going to keep plotting the frequencies. I'm just going to keep plotting them over and over and over again. And what you're going to see is as I take many, many samples of size 4, I'm going to have something that's going to start kind of approximating a normal distribution. So each of these dots represent an incidence of a sample mean. So as I keep adding on this column right here, that means I kept getting the sample mean 2.75. So over time, I'm going to have something that's starting to approximate a normal distribution. And that is the neat thing about the central limit theorem."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So each of these dots represent an incidence of a sample mean. So as I keep adding on this column right here, that means I kept getting the sample mean 2.75. So over time, I'm going to have something that's starting to approximate a normal distribution. And that is the neat thing about the central limit theorem. So the central limit to N, this was the case for, so in orange, that's the case for N is equal to 4. This was for sample size of 4. Now, if I did the same thing with the sample size of maybe 20."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And that is the neat thing about the central limit theorem. So the central limit to N, this was the case for, so in orange, that's the case for N is equal to 4. This was for sample size of 4. Now, if I did the same thing with the sample size of maybe 20. So in this case, instead of just taking 4 samples from my original crazy distribution, every sample, I take 20 instances of my random variable and I average those 20, and then I plot the sample mean on here. So in that case, I'm going to have a distribution that looks like this. And we'll discuss this in more videos."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Now, if I did the same thing with the sample size of maybe 20. So in this case, instead of just taking 4 samples from my original crazy distribution, every sample, I take 20 instances of my random variable and I average those 20, and then I plot the sample mean on here. So in that case, I'm going to have a distribution that looks like this. And we'll discuss this in more videos. But it turns out if I were to plot the sample, 10,000 of the sample means here, I'm going to have something that, two things, it's going to even more closely approximate a normal distribution, and we're going to see in future videos it's actually going to have a smaller, well, let me be clear, it's going to have the same mean. So that's the mean, this is going to have the same mean. But it's going to have a smaller standard deviation."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And we'll discuss this in more videos. But it turns out if I were to plot the sample, 10,000 of the sample means here, I'm going to have something that, two things, it's going to even more closely approximate a normal distribution, and we're going to see in future videos it's actually going to have a smaller, well, let me be clear, it's going to have the same mean. So that's the mean, this is going to have the same mean. But it's going to have a smaller standard deviation. So I want to, well, I should plot these from the bottom because you kind of stack it. One, you get one, then another instance, then another instance. But this is going to more and more approach a normal distribution."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But it's going to have a smaller standard deviation. So I want to, well, I should plot these from the bottom because you kind of stack it. One, you get one, then another instance, then another instance. But this is going to more and more approach a normal distribution. So the reality is, and this is what's super cool about the central limit theorem, as your sample size becomes larger, or you can even say as it approaches infinity, but you really don't have to get that close to infinity to really get close to a normal distribution. So if you have a sample size of 10 or 20, you're already getting very close to a normal distribution. In fact, about as good an approximation as we see in our everyday life."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But this is going to more and more approach a normal distribution. So the reality is, and this is what's super cool about the central limit theorem, as your sample size becomes larger, or you can even say as it approaches infinity, but you really don't have to get that close to infinity to really get close to a normal distribution. So if you have a sample size of 10 or 20, you're already getting very close to a normal distribution. In fact, about as good an approximation as we see in our everyday life. What's cool is we can start with some crazy distribution. This has nothing to do with a normal distribution. But if we have a sample size, this was n equals 4, but if we have a sample size of n equals 10 or n equals 100, and we were to take 100 of these instead of 4 here and average them and then plot that average, the frequency of it, then we would take 100 again, average them, take the mean, plot that again."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "In fact, about as good an approximation as we see in our everyday life. What's cool is we can start with some crazy distribution. This has nothing to do with a normal distribution. But if we have a sample size, this was n equals 4, but if we have a sample size of n equals 10 or n equals 100, and we were to take 100 of these instead of 4 here and average them and then plot that average, the frequency of it, then we would take 100 again, average them, take the mean, plot that again. And if we were to do that a bunch of times, in fact, if we were to do that an infinite time, we would find that, especially if we had an infinite sample size, we would find a perfect normal distribution. That's the crazy thing. And it doesn't apply just to taking the sample mean."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But if we have a sample size, this was n equals 4, but if we have a sample size of n equals 10 or n equals 100, and we were to take 100 of these instead of 4 here and average them and then plot that average, the frequency of it, then we would take 100 again, average them, take the mean, plot that again. And if we were to do that a bunch of times, in fact, if we were to do that an infinite time, we would find that, especially if we had an infinite sample size, we would find a perfect normal distribution. That's the crazy thing. And it doesn't apply just to taking the sample mean. Here we took the sample mean every time, but you could have also taken the sample sum. The central limit theorem would have still applied. But that's what's so super useful about it."}, {"video_title": "Central limit theorem Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And it doesn't apply just to taking the sample mean. Here we took the sample mean every time, but you could have also taken the sample sum. The central limit theorem would have still applied. But that's what's so super useful about it. Because in life, there's all sorts of processes out there, proteins bumping into each other, people doing crazy things, humans interacting in weird ways, and you don't know the probability distribution functions for any of those things. But what the central limit theorem tells us is that if we add a bunch of those actions together, assuming that they all have the same distribution, or if we were to take the mean of all of those actions together, and if we were to plot the frequency of those means, we do get a normal distribution. And that's frankly why the normal distribution shows up so much in statistics, and why, frankly, it's a very good approximation for the sum or the means of a lot of processes."}, {"video_title": "Identifying individuals, variables and categorical variables in a data set Khan Academy.mp3", "Sentence": "We're told that millions of Americans rely on caffeine to get them up in the morning, which is true, although if I drank caffeine in the morning, I'm very sensitive, I wouldn't be able to sleep at night. Here's nutritional data on some popular drinks at Ben's Beans Coffee Shop. All right, so here we have the different names of the drinks, and then here we have the type of the drink, and it looks like they're either hot or cold. Here we have the calories for each of those drinks. Here we have the sugar content in grams for each of those drinks, and here we have the caffeine in milligrams for each of those drinks. And then we are asked, the individuals in this data set are, and we have three choices, Ben's Beans customers, Ben's Beans drinks, or the caffeine contents. Now, we have to be careful."}, {"video_title": "Identifying individuals, variables and categorical variables in a data set Khan Academy.mp3", "Sentence": "Here we have the calories for each of those drinks. Here we have the sugar content in grams for each of those drinks, and here we have the caffeine in milligrams for each of those drinks. And then we are asked, the individuals in this data set are, and we have three choices, Ben's Beans customers, Ben's Beans drinks, or the caffeine contents. Now, we have to be careful. When someone says the individuals in a data set, they don't necessarily mean that they have to be people. They could be things, and the individuals in this data set, each of these rows, they're referring to a certain type of drink at Ben's Beans Coffee Shops. So the different types of drinks that Ben's Beans offers, those are the individuals in this data set."}, {"video_title": "Identifying individuals, variables and categorical variables in a data set Khan Academy.mp3", "Sentence": "Now, we have to be careful. When someone says the individuals in a data set, they don't necessarily mean that they have to be people. They could be things, and the individuals in this data set, each of these rows, they're referring to a certain type of drink at Ben's Beans Coffee Shops. So the different types of drinks that Ben's Beans offers, those are the individuals in this data set. So they're Ben's Beans drinks. Next, they ask us, the data set contains, and they say how many variables and how many of those variables are categorical. So if we look up here, let's look at the variables."}, {"video_title": "Identifying individuals, variables and categorical variables in a data set Khan Academy.mp3", "Sentence": "So the different types of drinks that Ben's Beans offers, those are the individuals in this data set. So they're Ben's Beans drinks. Next, they ask us, the data set contains, and they say how many variables and how many of those variables are categorical. So if we look up here, let's look at the variables. So this first column that is essentially giving us the type of drink, this wouldn't be a variable. This would be more of an identifier, but all of these other columns are representing variables. So for example, type is a variable."}, {"video_title": "Identifying individuals, variables and categorical variables in a data set Khan Academy.mp3", "Sentence": "So if we look up here, let's look at the variables. So this first column that is essentially giving us the type of drink, this wouldn't be a variable. This would be more of an identifier, but all of these other columns are representing variables. So for example, type is a variable. It can either be hot or cold. And because it can only take on one of, kind of a number of buckets, it's either going to be hot or cold, it's going to fit in one category or another. And you don't just have two categories."}, {"video_title": "Identifying individuals, variables and categorical variables in a data set Khan Academy.mp3", "Sentence": "So for example, type is a variable. It can either be hot or cold. And because it can only take on one of, kind of a number of buckets, it's either going to be hot or cold, it's going to fit in one category or another. And you don't just have two categories. You could have more than two categories. But it isn't just some type of variable number that can take on a bunch of different values. So this right over here is a categorical variable."}, {"video_title": "Identifying individuals, variables and categorical variables in a data set Khan Academy.mp3", "Sentence": "And you don't just have two categories. You could have more than two categories. But it isn't just some type of variable number that can take on a bunch of different values. So this right over here is a categorical variable. Calories is not a categorical variable. You could have something with 4.1 calories. You could have something with 178."}, {"video_title": "Identifying individuals, variables and categorical variables in a data set Khan Academy.mp3", "Sentence": "So this right over here is a categorical variable. Calories is not a categorical variable. You could have something with 4.1 calories. You could have something with 178. Things aren't fitting into nice buckets. Same thing for sugars and for the caffeine, that these are quantitative variables that don't just fit into a category. And so here, I would say that we have four variables, one, two, three, four, one of which is categorical."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "And power is an idea that you might encounter in a first-year statistics course. It turns out that it's fairly difficult to calculate, but it's interesting to know what it means and what are the levers that might increase the power or decrease the power in a significance test. So just to cut to the chase, power is a probability. You can view it as the probability that you're doing the right thing when the null hypothesis is not true. And the right thing is you should reject the null hypothesis if it's not true. So it's the probability of rejecting, rejecting your null hypothesis given that the null hypothesis is false. So you could view it as a conditional probability like that."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "You can view it as the probability that you're doing the right thing when the null hypothesis is not true. And the right thing is you should reject the null hypothesis if it's not true. So it's the probability of rejecting, rejecting your null hypothesis given that the null hypothesis is false. So you could view it as a conditional probability like that. But there's other ways to conceptualize it. We can connect it to type two errors. For example, you could say this is equal to one minus the probability of not rejecting, one minus the probability of not rejecting, not rejecting the null hypothesis given that the null hypothesis is false."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So you could view it as a conditional probability like that. But there's other ways to conceptualize it. We can connect it to type two errors. For example, you could say this is equal to one minus the probability of not rejecting, one minus the probability of not rejecting, not rejecting the null hypothesis given that the null hypothesis is false. And this thing that I just described, not rejecting the null hypothesis given that the null hypothesis is false, this is, that's the definition of a type, type two error. So you could view it as just the probability of not making a type two error or one minus the probability of making a type two error. Hopefully that's not confusing."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "For example, you could say this is equal to one minus the probability of not rejecting, one minus the probability of not rejecting, not rejecting the null hypothesis given that the null hypothesis is false. And this thing that I just described, not rejecting the null hypothesis given that the null hypothesis is false, this is, that's the definition of a type, type two error. So you could view it as just the probability of not making a type two error or one minus the probability of making a type two error. Hopefully that's not confusing. So let me just write it the other way. So you could say it's the probability of not making, not making a type, type two error. So what are the things that would actually drive power?"}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "Hopefully that's not confusing. So let me just write it the other way. So you could say it's the probability of not making, not making a type, type two error. So what are the things that would actually drive power? And to help us conceptualize that, I'll draw two sampling distributions. One, if we assume that the null hypothesis is true, and one where we assume that the null hypothesis is false and the true population parameter is something different than the null hypothesis is saying. So for example, let's say that we have a null hypothesis that our population mean is equal to, let's just call it mu one, and we have an alternative hypothesis, so H sub A, that says, hey, no, the population mean is not equal to mu one."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So what are the things that would actually drive power? And to help us conceptualize that, I'll draw two sampling distributions. One, if we assume that the null hypothesis is true, and one where we assume that the null hypothesis is false and the true population parameter is something different than the null hypothesis is saying. So for example, let's say that we have a null hypothesis that our population mean is equal to, let's just call it mu one, and we have an alternative hypothesis, so H sub A, that says, hey, no, the population mean is not equal to mu one. So if you assumed a world where the null hypothesis is true, so I'll do that in blue. So if we assume the null hypothesis is true, what would be our sampling distribution? Remember, what we do in significance tests is we have some form of a population, let me draw that."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So for example, let's say that we have a null hypothesis that our population mean is equal to, let's just call it mu one, and we have an alternative hypothesis, so H sub A, that says, hey, no, the population mean is not equal to mu one. So if you assumed a world where the null hypothesis is true, so I'll do that in blue. So if we assume the null hypothesis is true, what would be our sampling distribution? Remember, what we do in significance tests is we have some form of a population, let me draw that. You have a population right over here, and our hypotheses are making some statement about a parameter in that population, and to test it, we take a sample of a certain size, we calculate a statistic, in this case, we would be the sample mean, and we say if we assume that our null hypothesis is true, what is the probability of getting that sample statistic? And if that's below a threshold, which we call a significance level, we reject the null hypothesis. And so that world that we have been living in, one way to think about it, in a world where you assume the null hypothesis is true, you might have a sampling distribution that looks something like this."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "Remember, what we do in significance tests is we have some form of a population, let me draw that. You have a population right over here, and our hypotheses are making some statement about a parameter in that population, and to test it, we take a sample of a certain size, we calculate a statistic, in this case, we would be the sample mean, and we say if we assume that our null hypothesis is true, what is the probability of getting that sample statistic? And if that's below a threshold, which we call a significance level, we reject the null hypothesis. And so that world that we have been living in, one way to think about it, in a world where you assume the null hypothesis is true, you might have a sampling distribution that looks something like this. The null hypothesis is true, then the center of your sampling distribution would be right over here at mu one, and given your sample size, you would get a certain sampling distribution for the sample means. If your sample size increases, this will be narrow, or if it decreases, this thing is going to be wider, and you set a significance level, which is essentially your probability of rejecting the null hypothesis, even if it is true. You could even view it as, and we've talked about it, you can view your significance level as a probability of making a type one error."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "And so that world that we have been living in, one way to think about it, in a world where you assume the null hypothesis is true, you might have a sampling distribution that looks something like this. The null hypothesis is true, then the center of your sampling distribution would be right over here at mu one, and given your sample size, you would get a certain sampling distribution for the sample means. If your sample size increases, this will be narrow, or if it decreases, this thing is going to be wider, and you set a significance level, which is essentially your probability of rejecting the null hypothesis, even if it is true. You could even view it as, and we've talked about it, you can view your significance level as a probability of making a type one error. So your significance level is some area, and so let's say it's this area that I'm shading in orange right over here, that would be your significance level. So if you took a sample right over here, and you calculated its sample mean, and you happen to fall in this area, or this area, or this area right over here, then you would reject your null hypothesis. Now, if the null hypothesis actually was true, you would be committing a type one error without knowing about it."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "You could even view it as, and we've talked about it, you can view your significance level as a probability of making a type one error. So your significance level is some area, and so let's say it's this area that I'm shading in orange right over here, that would be your significance level. So if you took a sample right over here, and you calculated its sample mean, and you happen to fall in this area, or this area, or this area right over here, then you would reject your null hypothesis. Now, if the null hypothesis actually was true, you would be committing a type one error without knowing about it. But for power, we are concerned with a type two error. So in this one, it's a conditional probability that our null hypothesis is false. And so let's construct another sampling distribution in the case where our null hypothesis is false."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "Now, if the null hypothesis actually was true, you would be committing a type one error without knowing about it. But for power, we are concerned with a type two error. So in this one, it's a conditional probability that our null hypothesis is false. And so let's construct another sampling distribution in the case where our null hypothesis is false. So let me just continue this line right over here, and I'll do that. So let's imagine a world where our null hypothesis is false, and it's actually the case that our mean is mu two. And let's say that mu two is right over here."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "And so let's construct another sampling distribution in the case where our null hypothesis is false. So let me just continue this line right over here, and I'll do that. So let's imagine a world where our null hypothesis is false, and it's actually the case that our mean is mu two. And let's say that mu two is right over here. And in this reality, our sampling distribution might look something like this. Once again, it'll be for a given sample size. The larger the sample size, the narrower this bell curve would be."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "And let's say that mu two is right over here. And in this reality, our sampling distribution might look something like this. Once again, it'll be for a given sample size. The larger the sample size, the narrower this bell curve would be. And so it might look something like this. So in which situation? So in this world, we should be rejecting the null hypothesis."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "The larger the sample size, the narrower this bell curve would be. And so it might look something like this. So in which situation? So in this world, we should be rejecting the null hypothesis. But what are the samples in which case we are not rejecting the null hypothesis even though we should? Well, we're not going to reject the null hypothesis if we get samples in, if we get a sample here or a sample here or a sample here. A sample where if you assume the null hypothesis is true, the probability isn't that unlikely."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So in this world, we should be rejecting the null hypothesis. But what are the samples in which case we are not rejecting the null hypothesis even though we should? Well, we're not going to reject the null hypothesis if we get samples in, if we get a sample here or a sample here or a sample here. A sample where if you assume the null hypothesis is true, the probability isn't that unlikely. And so the probability of making a type two error when we should reject the null hypothesis but we don't is actually this area right over here. And the power, the probability of rejecting the null hypothesis given that it's false, so given that it's false would be this red distribution, that would be the rest of this area right over here. So how can we increase the power?"}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "A sample where if you assume the null hypothesis is true, the probability isn't that unlikely. And so the probability of making a type two error when we should reject the null hypothesis but we don't is actually this area right over here. And the power, the probability of rejecting the null hypothesis given that it's false, so given that it's false would be this red distribution, that would be the rest of this area right over here. So how can we increase the power? Well, one way is to increase our alpha, increase our significance level. If we increase our significance level, say from that, remember the significance level is an area. So if we want it to go up, if we increase the area, and it looked something like that, now by expanding that significance area, we have increased the power because now this yellow area is larger, we've pushed this boundary to the left of it."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So how can we increase the power? Well, one way is to increase our alpha, increase our significance level. If we increase our significance level, say from that, remember the significance level is an area. So if we want it to go up, if we increase the area, and it looked something like that, now by expanding that significance area, we have increased the power because now this yellow area is larger, we've pushed this boundary to the left of it. Now you might say, oh, well, hey, if we wanna increase the power, power sounds like a good thing. Why don't we just always increase alpha? Well, the problem with that is if you increase alpha, so let me write this down."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So if we want it to go up, if we increase the area, and it looked something like that, now by expanding that significance area, we have increased the power because now this yellow area is larger, we've pushed this boundary to the left of it. Now you might say, oh, well, hey, if we wanna increase the power, power sounds like a good thing. Why don't we just always increase alpha? Well, the problem with that is if you increase alpha, so let me write this down. So if you take alpha, your significance level, and you increase it, that will increase the power, that will increase the power, but it's also going to increase your probability of a type one error. Because remember, that's one way to conceptualize what alpha is, what your significance level is. It's a probability of a type one error."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "Well, the problem with that is if you increase alpha, so let me write this down. So if you take alpha, your significance level, and you increase it, that will increase the power, that will increase the power, but it's also going to increase your probability of a type one error. Because remember, that's one way to conceptualize what alpha is, what your significance level is. It's a probability of a type one error. Now what are other ways to increase your power? Well, if you increase your sample size, then both of these distributions will, these sampling distributions are going to get narrower. And so if these sampling distributions, if both of these sampling distributions get narrower, then that situation where you are not rejecting your null hypothesis, even though you should, is going to have a lot less area."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "It's a probability of a type one error. Now what are other ways to increase your power? Well, if you increase your sample size, then both of these distributions will, these sampling distributions are going to get narrower. And so if these sampling distributions, if both of these sampling distributions get narrower, then that situation where you are not rejecting your null hypothesis, even though you should, is going to have a lot less area. There's gonna be, one way to think about it, there's going to be a lot less overlap between these two sampling distributions. And so let me write that down. So another way is to, if you increase n, your sample size, that's going to increase your power."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "And so if these sampling distributions, if both of these sampling distributions get narrower, then that situation where you are not rejecting your null hypothesis, even though you should, is going to have a lot less area. There's gonna be, one way to think about it, there's going to be a lot less overlap between these two sampling distributions. And so let me write that down. So another way is to, if you increase n, your sample size, that's going to increase your power. And this, in general, is always a good thing if you can do it. Now other things that may or may not be under your control is, well, the less variability there is in the data set, that would also make these sampling distributions narrower, and that would also increase the power. So less variability, and you could measure that as by variance or standard deviation of your underlying data set, that would increase your power."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So another way is to, if you increase n, your sample size, that's going to increase your power. And this, in general, is always a good thing if you can do it. Now other things that may or may not be under your control is, well, the less variability there is in the data set, that would also make these sampling distributions narrower, and that would also increase the power. So less variability, and you could measure that as by variance or standard deviation of your underlying data set, that would increase your power. Another thing that would increase the power is if the true parameter is further away than what the null hypothesis is saying. So if you say true parameter far from null hypothesis, what it's saying, that also will increase the power. So these two are not typically under your control, but the sample size is, and the significance level is."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So less variability, and you could measure that as by variance or standard deviation of your underlying data set, that would increase your power. Another thing that would increase the power is if the true parameter is further away than what the null hypothesis is saying. So if you say true parameter far from null hypothesis, what it's saying, that also will increase the power. So these two are not typically under your control, but the sample size is, and the significance level is. Significance level, there's a trade-off, though. If you increase the power through that, you're also increasing the probability of a type one error. So for a lot of researchers, they might say, hey, if a type two error is worse, I'm willing to make this trade-off."}, {"video_title": "Introduction to power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So these two are not typically under your control, but the sample size is, and the significance level is. Significance level, there's a trade-off, though. If you increase the power through that, you're also increasing the probability of a type one error. So for a lot of researchers, they might say, hey, if a type two error is worse, I'm willing to make this trade-off. I'll increase the significance level. But if a type one error is actually what I'm afraid of, then I wouldn't wanna use this lever. But in any case, increasing your sample size, if you can do it, is going to be a good thing."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "Each problem has only one correct answer. What is the probability of randomly guessing the correct answer on both problems? Now, the probability of guessing the correct answer on each problem, these are independent events. So let's write this down. The probability of correct on problem number one is independent. Or let me write it this way. Probability of correct on number one and probability of correct on problem two are independent."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "So let's write this down. The probability of correct on problem number one is independent. Or let me write it this way. Probability of correct on number one and probability of correct on problem two are independent. Independent. Which means that the outcome of one of the events of guessing on the first problem isn't going to affect the probability of guessing correctly on the second problem, independent events. So the probability of guessing on both of them, so that means that the probability of being correct on one and number two is going to be equal to the product of these probabilities."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "Probability of correct on number one and probability of correct on problem two are independent. Independent. Which means that the outcome of one of the events of guessing on the first problem isn't going to affect the probability of guessing correctly on the second problem, independent events. So the probability of guessing on both of them, so that means that the probability of being correct on one and number two is going to be equal to the product of these probabilities. And we're going to see why that is visually in a second. But it's going to be the probability of correct on number one times the probability of being correct on number two. Now, what are each of these probabilities?"}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "So the probability of guessing on both of them, so that means that the probability of being correct on one and number two is going to be equal to the product of these probabilities. And we're going to see why that is visually in a second. But it's going to be the probability of correct on number one times the probability of being correct on number two. Now, what are each of these probabilities? On number one, there are four choices. There are four possible outcomes. And only one of them is going to be correct."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "Now, what are each of these probabilities? On number one, there are four choices. There are four possible outcomes. And only one of them is going to be correct. Each one only has one correct answer. So the probability of being correct on problem one is 1 fourth, and then the probability of being correct on problem number two has three choices. So there's three possible outcomes."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "And only one of them is going to be correct. Each one only has one correct answer. So the probability of being correct on problem one is 1 fourth, and then the probability of being correct on problem number two has three choices. So there's three possible outcomes. And there's only one correct one. So only one of them are correct. So probability of correct on number two is 1 third."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "So there's three possible outcomes. And there's only one correct one. So only one of them are correct. So probability of correct on number two is 1 third. Probability of guessing correct on number one is 1 fourth. The probability of doing on both of them is going to be its product. So it's going to be equal to 1 fourth times 1 third is 1 twelfth."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "So probability of correct on number two is 1 third. Probability of guessing correct on number one is 1 fourth. The probability of doing on both of them is going to be its product. So it's going to be equal to 1 fourth times 1 third is 1 twelfth. Now, to see kind of visually why this makes sense, let's draw a little chart here. And we did a similar thing when we thought about rolling two separate dice. So let's think about problem number one has four choices, only one of which is correct."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "So it's going to be equal to 1 fourth times 1 third is 1 twelfth. Now, to see kind of visually why this makes sense, let's draw a little chart here. And we did a similar thing when we thought about rolling two separate dice. So let's think about problem number one has four choices, only one of which is correct. So let's write incorrect choice one, incorrect choice two, incorrect choice three, and then it has the correct choice over there. So those are the four choices. They're not going to necessarily be in that order on the exam, but we can just list them in this order."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "So let's think about problem number one has four choices, only one of which is correct. So let's write incorrect choice one, incorrect choice two, incorrect choice three, and then it has the correct choice over there. So those are the four choices. They're not going to necessarily be in that order on the exam, but we can just list them in this order. Now, problem number two has three choices, only one of which is correct. So problem number two has incorrect choice one, incorrect choice two, and then let's say the third choice is correct. It's not necessarily in that order, but we know it has two incorrect and one correct choices."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "They're not going to necessarily be in that order on the exam, but we can just list them in this order. Now, problem number two has three choices, only one of which is correct. So problem number two has incorrect choice one, incorrect choice two, and then let's say the third choice is correct. It's not necessarily in that order, but we know it has two incorrect and one correct choices. Now, what are all of the different possible outcomes? We can draw a grid here, all of these possible outcomes. Let's draw all of the outcomes."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "It's not necessarily in that order, but we know it has two incorrect and one correct choices. Now, what are all of the different possible outcomes? We can draw a grid here, all of these possible outcomes. Let's draw all of the outcomes. Each of these cells or each of these boxes in a grid are a possible outcome. You're just guessing. You're randomly choosing one of these four."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "Let's draw all of the outcomes. Each of these cells or each of these boxes in a grid are a possible outcome. You're just guessing. You're randomly choosing one of these four. You're randomly choosing one of these four. So you might get incorrect choice one and incorrect choice one, incorrect choice in problem number one, and then incorrect choice in problem number two. That would be that cell right there."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "You're randomly choosing one of these four. You're randomly choosing one of these four. So you might get incorrect choice one and incorrect choice one, incorrect choice in problem number one, and then incorrect choice in problem number two. That would be that cell right there. Maybe you get problem number one correct, but you get incorrect choice number two in problem number two. So these would represent all of the possible outcomes when you guess on each problem. And which of these outcomes represent getting correct on both?"}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "That would be that cell right there. Maybe you get problem number one correct, but you get incorrect choice number two in problem number two. So these would represent all of the possible outcomes when you guess on each problem. And which of these outcomes represent getting correct on both? Well, getting correct on both is only this one. Correct on choice one and correct on choice on problem number two. And so that's one of the possible outcomes."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "And which of these outcomes represent getting correct on both? Well, getting correct on both is only this one. Correct on choice one and correct on choice on problem number two. And so that's one of the possible outcomes. And how many total outcomes are there? There's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Out of 12 possible outcomes."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "And so that's one of the possible outcomes. And how many total outcomes are there? There's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Out of 12 possible outcomes. Or, since these are independent events, you can multiply. You see that there are 12 outcomes because there's 12 possible outcomes. So there's four possible outcomes for problem number one times the three possible outcomes for problem number two, and that's also where you get a 12."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "We're trying to fit four points. And in previous videos, we actually came up with the equation of this least squares regression line. What I'm going to do now is plot the residuals for each of these points. So what is a residual? Well, just as a reminder, your residual for a given point is equal to the actual minus the expected. So how do I make that tangible? Well, what's the residual for this point right over here?"}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "So what is a residual? Well, just as a reminder, your residual for a given point is equal to the actual minus the expected. So how do I make that tangible? Well, what's the residual for this point right over here? For this point here, the actual y, when x equals one, is one, but the expected, when x equals one for this least squares regression line, 2.5 times one minus two, well, that's going to be.5. And so our residual is one minus.5, so we have a positive, we have a positive 0.5 residual. Over for this point, you have zero residual."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "Well, what's the residual for this point right over here? For this point here, the actual y, when x equals one, is one, but the expected, when x equals one for this least squares regression line, 2.5 times one minus two, well, that's going to be.5. And so our residual is one minus.5, so we have a positive, we have a positive 0.5 residual. Over for this point, you have zero residual. The actual is the expected. For this point right over here, the actual, when x equals two, for y is two, but the expected is three. So our residual over here, once again, the actual is y equals two when x equals two."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "Over for this point, you have zero residual. The actual is the expected. For this point right over here, the actual, when x equals two, for y is two, but the expected is three. So our residual over here, once again, the actual is y equals two when x equals two. The expected, two times 2.5 minus two is three, so this is going to be two minus three, which equals a residual of negative one. And then over here, our residual, our actual, when x equals three, is six. Our expected, when x equals three, is 5.5."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "So our residual over here, once again, the actual is y equals two when x equals two. The expected, two times 2.5 minus two is three, so this is going to be two minus three, which equals a residual of negative one. And then over here, our residual, our actual, when x equals three, is six. Our expected, when x equals three, is 5.5. So six minus 5.5, that is a positive 0.5. So those are the residuals, but how do we plot it? Well, we would set up our axes."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "Our expected, when x equals three, is 5.5. So six minus 5.5, that is a positive 0.5. So those are the residuals, but how do we plot it? Well, we would set up our axes. Let me do it right over here. One, two, and three. And let's see, the maximum residual here is positive 0.5, and then the minimum one here is negative one."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "Well, we would set up our axes. Let me do it right over here. One, two, and three. And let's see, the maximum residual here is positive 0.5, and then the minimum one here is negative one. So let's see, this could be 0.51, negative 0.5, negative one. So this is negative one, this is positive one here. And so when x equals one, what was the residual?"}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "And let's see, the maximum residual here is positive 0.5, and then the minimum one here is negative one. So let's see, this could be 0.51, negative 0.5, negative one. So this is negative one, this is positive one here. And so when x equals one, what was the residual? Well, the actual was one, expected was 0.5, one minus 0.5 is 0.5. So this right over here, we can plot right over here, the residual is 0.5. When x equals two, we actually have two data points."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "And so when x equals one, what was the residual? Well, the actual was one, expected was 0.5, one minus 0.5 is 0.5. So this right over here, we can plot right over here, the residual is 0.5. When x equals two, we actually have two data points. First, I'll do this one. When we have the point two comma three, the residual there is zero. So for one of them, the residual is zero."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "When x equals two, we actually have two data points. First, I'll do this one. When we have the point two comma three, the residual there is zero. So for one of them, the residual is zero. Now for the other one, the residual is negative one. Let me do that in a different color. For the other one, the residual is negative one, so we would plot it right over here."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "So for one of them, the residual is zero. Now for the other one, the residual is negative one. Let me do that in a different color. For the other one, the residual is negative one, so we would plot it right over here. And then this last point, the residual is positive 0.5. So it is just like that. And so this thing that I have just created, where we're just seeing for each x where we have a corresponding point, we plot the point above or below the line based on the residual."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "For the other one, the residual is negative one, so we would plot it right over here. And then this last point, the residual is positive 0.5. So it is just like that. And so this thing that I have just created, where we're just seeing for each x where we have a corresponding point, we plot the point above or below the line based on the residual. This is called a residual plot. Now one question is, why do people even go through the trouble of creating a residual plot like this? The answer is, regardless of whether the regression line is upward sloping or downward sloping, this gives you a sense of how good a fit it is and whether a line is good at explaining the relationship between the variables."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "And so this thing that I have just created, where we're just seeing for each x where we have a corresponding point, we plot the point above or below the line based on the residual. This is called a residual plot. Now one question is, why do people even go through the trouble of creating a residual plot like this? The answer is, regardless of whether the regression line is upward sloping or downward sloping, this gives you a sense of how good a fit it is and whether a line is good at explaining the relationship between the variables. The general idea is, if you see the points pretty evenly scattered or randomly scattered above and below this line, you don't really discern any trend here, then a line is probably a good model for the data. But if you do see some type of trend, if the residuals had an upward trend like this, or if they were curving up and then curving down, or they had a downward trend, then you might say, hey, this line isn't a good fit, and maybe we would have to do a nonlinear model. What are some examples of other residual plots?"}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "The answer is, regardless of whether the regression line is upward sloping or downward sloping, this gives you a sense of how good a fit it is and whether a line is good at explaining the relationship between the variables. The general idea is, if you see the points pretty evenly scattered or randomly scattered above and below this line, you don't really discern any trend here, then a line is probably a good model for the data. But if you do see some type of trend, if the residuals had an upward trend like this, or if they were curving up and then curving down, or they had a downward trend, then you might say, hey, this line isn't a good fit, and maybe we would have to do a nonlinear model. What are some examples of other residual plots? And let's try to analyze them a bit. So right here you have a regression line and its corresponding residual plot. And once again, you see here, the residual is slightly positive."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "What are some examples of other residual plots? And let's try to analyze them a bit. So right here you have a regression line and its corresponding residual plot. And once again, you see here, the residual is slightly positive. The actual is slightly above the line, and you see it right over there, it's slightly positive. This one's even more positive, you see it there. But like the example we just looked at, it looks like these residuals are pretty evenly scattered above and below the line."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "And once again, you see here, the residual is slightly positive. The actual is slightly above the line, and you see it right over there, it's slightly positive. This one's even more positive, you see it there. But like the example we just looked at, it looks like these residuals are pretty evenly scattered above and below the line. There isn't any discernible trend. And so I would say that a linear model here, and in particular this regression line, is a good model for this data. But if we see something like this, a different picture emerges."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "But like the example we just looked at, it looks like these residuals are pretty evenly scattered above and below the line. There isn't any discernible trend. And so I would say that a linear model here, and in particular this regression line, is a good model for this data. But if we see something like this, a different picture emerges. When I look at just the residual plot, it doesn't look like they're evenly scattered. It looks like there's some type of trend here. I'm going down here, but then I'm going back up."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "But if we see something like this, a different picture emerges. When I look at just the residual plot, it doesn't look like they're evenly scattered. It looks like there's some type of trend here. I'm going down here, but then I'm going back up. When you see something like this, where on the residual plot you're going below the x-axis and then above, then it might say, hey, a linear model might not be appropriate. Maybe some type of nonlinear model, some type of nonlinear curve might better fit the data, or the relationship between the y and the x is nonlinear. Another way you could think about it is when you have a lot of residuals that are pretty far away from the x-axis in the residual plot, you would also say this line isn't such a good fit."}, {"video_title": "Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3", "Sentence": "And what you see here is a simulation that allows us to keep sampling from our gumball machine and start approximating the sampling distribution of the sample proportion. So her simulation focuses on green gumballs, but we talked about yellow before. In the yellow gumballs, we said 60% were yellow, so let's make 60% here green. And then let's take samples of 10, just like we did before. And then let's just start with one sample. So we're gonna draw one sample, and what we wanna show is we wanna show the percentages, which is the proportion of each sample that are green. So if we draw that first sample, notice, out of the 10, five ended up being green, and then it plotted that right over here under 50%."}, {"video_title": "Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3", "Sentence": "And then let's take samples of 10, just like we did before. And then let's just start with one sample. So we're gonna draw one sample, and what we wanna show is we wanna show the percentages, which is the proportion of each sample that are green. So if we draw that first sample, notice, out of the 10, five ended up being green, and then it plotted that right over here under 50%. We have one situation where 50% were green. Now let's do another sample. So this sample, 60% are green, and so let's keep going."}, {"video_title": "Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3", "Sentence": "So if we draw that first sample, notice, out of the 10, five ended up being green, and then it plotted that right over here under 50%. We have one situation where 50% were green. Now let's do another sample. So this sample, 60% are green, and so let's keep going. Let's draw another sample. And now that one, we have 50% are green, and so notice now we see here on this distribution, two of them had 50% green, and we could keep drawing samples. And let's just really increase."}, {"video_title": "Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3", "Sentence": "So this sample, 60% are green, and so let's keep going. Let's draw another sample. And now that one, we have 50% are green, and so notice now we see here on this distribution, two of them had 50% green, and we could keep drawing samples. And let's just really increase. So we're gonna do 50 samples of 10 at a time. And so here, we can quickly get to a fairly large number of samples. So here, we're over 1,000 samples."}, {"video_title": "Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3", "Sentence": "And let's just really increase. So we're gonna do 50 samples of 10 at a time. And so here, we can quickly get to a fairly large number of samples. So here, we're over 1,000 samples. And what's interesting here is we're seeing experimentally that our sample, the mean of our sample proportion here is 0.62. What we calculated a few minutes ago was that it should be 0.6. We also see that the standard deviation of our sample proportion is 0.16, and what we calculated was approximately 0.15."}, {"video_title": "Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3", "Sentence": "So here, we're over 1,000 samples. And what's interesting here is we're seeing experimentally that our sample, the mean of our sample proportion here is 0.62. What we calculated a few minutes ago was that it should be 0.6. We also see that the standard deviation of our sample proportion is 0.16, and what we calculated was approximately 0.15. And as we draw more and more samples, we should get even closer and closer to those values. And we see that for the most part, we are getting closer and closer. In fact, now that it's rounded, we're at exactly those values that we had calculated before."}, {"video_title": "Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3", "Sentence": "We also see that the standard deviation of our sample proportion is 0.16, and what we calculated was approximately 0.15. And as we draw more and more samples, we should get even closer and closer to those values. And we see that for the most part, we are getting closer and closer. In fact, now that it's rounded, we're at exactly those values that we had calculated before. Now, one interesting thing to observe is when your population proportion is not too close to zero and not too close to one, this looks pretty close to a normal distribution. And that makes sense because we saw the relation between the sampling distribution of the sample proportion and a binomial random variable. But what if our population proportion is closer to zero?"}, {"video_title": "Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3", "Sentence": "In fact, now that it's rounded, we're at exactly those values that we had calculated before. Now, one interesting thing to observe is when your population proportion is not too close to zero and not too close to one, this looks pretty close to a normal distribution. And that makes sense because we saw the relation between the sampling distribution of the sample proportion and a binomial random variable. But what if our population proportion is closer to zero? So let's say our population proportion is 10%, 0.1. What do you think the distribution is going to look like then? Well, we know that the mean of our sampling distribution is going to be 10%, and so you can imagine that the distribution is going to be right skewed."}, {"video_title": "Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3", "Sentence": "But what if our population proportion is closer to zero? So let's say our population proportion is 10%, 0.1. What do you think the distribution is going to look like then? Well, we know that the mean of our sampling distribution is going to be 10%, and so you can imagine that the distribution is going to be right skewed. But let's actually see that. So here we see that our distribution is indeed right skewed. And that makes sense because you can only get values from zero to one, and if your mean is closer to zero, then you're gonna see the meat of your distribution here, and then you're gonna see a long tail to the right, which creates that right skew."}, {"video_title": "Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3", "Sentence": "Well, we know that the mean of our sampling distribution is going to be 10%, and so you can imagine that the distribution is going to be right skewed. But let's actually see that. So here we see that our distribution is indeed right skewed. And that makes sense because you can only get values from zero to one, and if your mean is closer to zero, then you're gonna see the meat of your distribution here, and then you're gonna see a long tail to the right, which creates that right skew. And if your population proportion was close to one, well, you can imagine the opposite's going to happen. You're going to end up with a left skew. And we indeed see right over here a left skew."}, {"video_title": "Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3", "Sentence": "And that makes sense because you can only get values from zero to one, and if your mean is closer to zero, then you're gonna see the meat of your distribution here, and then you're gonna see a long tail to the right, which creates that right skew. And if your population proportion was close to one, well, you can imagine the opposite's going to happen. You're going to end up with a left skew. And we indeed see right over here a left skew. Now, the other interesting thing to appreciate is the larger your samples, the smaller the standard deviation. And so let's do a population proportion that is right in between. And so here, this is similar to what we saw before."}, {"video_title": "Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3", "Sentence": "And we indeed see right over here a left skew. Now, the other interesting thing to appreciate is the larger your samples, the smaller the standard deviation. And so let's do a population proportion that is right in between. And so here, this is similar to what we saw before. This is looking roughly normal. But now, and that's when we had sample size of 10. But what if we have a sample size of 50 every time?"}, {"video_title": "Sampling distribution of sample proportion part 2 AP Statistics Khan Academy.mp3", "Sentence": "And so here, this is similar to what we saw before. This is looking roughly normal. But now, and that's when we had sample size of 10. But what if we have a sample size of 50 every time? Well, notice, now it looks like a much tighter distribution. This isn't even going all the way to one yet, but it is a much tighter distribution. And the reason why that made sense, the standard deviation of your sample proportion, it is inversely proportional to the square root of n. And so that makes sense."}, {"video_title": "Sample statistic bias worked example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "We're told Alejandro was curious if sample median was an unbiased estimator of population median. He placed ping pong balls numbered from zero to 32, so I guess that would be, what, 33 ping pong balls, in a drum and mixed them well. Note that the median of the population is 16, right? The median number, of course, yes, and that population is 16. He then took a random sample of five balls and calculated the median of the sample. So we have this population of balls. He takes a, we know the population parameter."}, {"video_title": "Sample statistic bias worked example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "The median number, of course, yes, and that population is 16. He then took a random sample of five balls and calculated the median of the sample. So we have this population of balls. He takes a, we know the population parameter. We know that the population median is 16, but then he starts taking a sample of five balls, so n equals five, and he calculates a sample median. Sample median. And then he replaced the balls and repeated this process for a total of 50 trials."}, {"video_title": "Sample statistic bias worked example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "He takes a, we know the population parameter. We know that the population median is 16, but then he starts taking a sample of five balls, so n equals five, and he calculates a sample median. Sample median. And then he replaced the balls and repeated this process for a total of 50 trials. His results are summarized in the dot plot below, where each dot represents the sample median from a sample of five balls. So he does this. He takes these five balls, puts them back in, then he does it again, then he does it again, and every time, he calculates the sample median for that sample, and he plots that on the dot plot."}, {"video_title": "Sample statistic bias worked example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And then he replaced the balls and repeated this process for a total of 50 trials. His results are summarized in the dot plot below, where each dot represents the sample median from a sample of five balls. So he does this. He takes these five balls, puts them back in, then he does it again, then he does it again, and every time, he calculates the sample median for that sample, and he plots that on the dot plot. So, and he'll do this for 50 samples. And each dot here represents that sample statistic, so it shows that four times we got a sample median. In four of those 50 samples, we got a sample median of 20."}, {"video_title": "Sample statistic bias worked example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "He takes these five balls, puts them back in, then he does it again, then he does it again, and every time, he calculates the sample median for that sample, and he plots that on the dot plot. So, and he'll do this for 50 samples. And each dot here represents that sample statistic, so it shows that four times we got a sample median. In four of those 50 samples, we got a sample median of 20. In five of those sample medians, we got a sample median of 10. And so what he ends up creating with these dots is really an approximation of the sampling distribution of the sample medians. Now, to judge whether it is a biased or unbiased estimator for the population median, well, actually, pause the video."}, {"video_title": "Sample statistic bias worked example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "In four of those 50 samples, we got a sample median of 20. In five of those sample medians, we got a sample median of 10. And so what he ends up creating with these dots is really an approximation of the sampling distribution of the sample medians. Now, to judge whether it is a biased or unbiased estimator for the population median, well, actually, pause the video. See if you can figure that out. All right, now let's do this together. Now, to judge it, let's think about where the true population parameter is, the population median."}, {"video_title": "Sample statistic bias worked example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "Now, to judge whether it is a biased or unbiased estimator for the population median, well, actually, pause the video. See if you can figure that out. All right, now let's do this together. Now, to judge it, let's think about where the true population parameter is, the population median. It's 16, we know that. And so that is right over here, the true population parameter. So if we were dealing with a biased, a biased estimator for the population parameter, then as we get our approximation of the sampling distribution, we would expect it to be somewhat skewed."}, {"video_title": "Sample statistic bias worked example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "Now, to judge it, let's think about where the true population parameter is, the population median. It's 16, we know that. And so that is right over here, the true population parameter. So if we were dealing with a biased, a biased estimator for the population parameter, then as we get our approximation of the sampling distribution, we would expect it to be somewhat skewed. So for example, if the sampling, if this approximation of the sampling distribution looks something like that, then we say, okay, that looks like a biased estimator. Or if it was looking something like that, we'd say, okay, that looks like a biased estimator. But if this approximation for our sampling distribution that Alejandro is constructing, where we see that roughly the same proportion of the sample statistics came out below as came out above the true parameter, and it's not, it doesn't have to be exact, but it seems roughly the case, this seems pretty unbiased."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And I've taken some exercises from the Khan Academy exercises here, and I'm just gonna solve it on my scratch pad. The following data points represent the number of animal crackers in each kid's lunchbox. Sort the data from least to greatest, and then find the interquartile range of the data set. And I encourage you to do this before I take a shot at it. All right, so let's first sort it. And if we were actually doing this on the Khan Academy exercise, you could just drag these, you could just click and drag these numbers around to sort them, but I'll just do it by hand. So let's see, the lowest number here looks like it's a four."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And I encourage you to do this before I take a shot at it. All right, so let's first sort it. And if we were actually doing this on the Khan Academy exercise, you could just drag these, you could just click and drag these numbers around to sort them, but I'll just do it by hand. So let's see, the lowest number here looks like it's a four. So I have that four, and then I have another four. And then I have another four. And let's see, are there any fives?"}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So let's see, the lowest number here looks like it's a four. So I have that four, and then I have another four. And then I have another four. And let's see, are there any fives? No fives, but there is a six. So then there's a six, and then there's a seven. There doesn't seem to be an eight or a nine, but then we get to a 10."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And let's see, are there any fives? No fives, but there is a six. So then there's a six, and then there's a seven. There doesn't seem to be an eight or a nine, but then we get to a 10. And then we get to 11, 12, no 13, but then we get 14. And then finally we have a 15. So the first thing we want to do is figure out the median here."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "There doesn't seem to be an eight or a nine, but then we get to a 10. And then we get to 11, 12, no 13, but then we get 14. And then finally we have a 15. So the first thing we want to do is figure out the median here. So the median's the middle number. I have one, two, three, four, five, six, seven, eight, nine numbers, so there's going to be just one middle number. I have an odd number of numbers here."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So the first thing we want to do is figure out the median here. So the median's the middle number. I have one, two, three, four, five, six, seven, eight, nine numbers, so there's going to be just one middle number. I have an odd number of numbers here. And it's going to be the number that has four to the left and four to the right, and that middle number, the median, is going to be 10. Notice I have four to the left and four to the right. And the interquartile range is all about figuring out the difference between the middle of the first half and the middle of the second half."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "I have an odd number of numbers here. And it's going to be the number that has four to the left and four to the right, and that middle number, the median, is going to be 10. Notice I have four to the left and four to the right. And the interquartile range is all about figuring out the difference between the middle of the first half and the middle of the second half. It's a measure of spread, how far apart all of these data points are. And so let's figure out the middle of the first half. So we're going to ignore the median here and just look at these first four numbers."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And the interquartile range is all about figuring out the difference between the middle of the first half and the middle of the second half. It's a measure of spread, how far apart all of these data points are. And so let's figure out the middle of the first half. So we're going to ignore the median here and just look at these first four numbers. And so out of these first four numbers, I have, since I have an even number of numbers, I'm going to calculate the median using the middle two numbers. So I'm going to look at the middle two numbers here, and I'm going to take their average. So the average of four and six, halfway between four and six is five."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So we're going to ignore the median here and just look at these first four numbers. And so out of these first four numbers, I have, since I have an even number of numbers, I'm going to calculate the median using the middle two numbers. So I'm going to look at the middle two numbers here, and I'm going to take their average. So the average of four and six, halfway between four and six is five. Or you could say four plus six is, four plus six is equal to 10. But then I want to divide that by two. So this is going to be equal to five."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So the average of four and six, halfway between four and six is five. Or you could say four plus six is, four plus six is equal to 10. But then I want to divide that by two. So this is going to be equal to five. So the middle of the first half is five. You can imagine it right over there. And in the middle of the second half, I'm going to have to do the same thing."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So this is going to be equal to five. So the middle of the first half is five. You can imagine it right over there. And in the middle of the second half, I'm going to have to do the same thing. I have four numbers, so I'm going to look at the middle two numbers. The middle two numbers are 12 and 14. The average of 12 and 14 is going to be 13."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And in the middle of the second half, I'm going to have to do the same thing. I have four numbers, so I'm going to look at the middle two numbers. The middle two numbers are 12 and 14. The average of 12 and 14 is going to be 13. Is going to be 13. If you took 12 plus 14 over two, that's going to be 26 over two, which is equal to 13. But an easier way for numbers like this, you say, hey, 13 is right exactly halfway between 12 and 14."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "The average of 12 and 14 is going to be 13. Is going to be 13. If you took 12 plus 14 over two, that's going to be 26 over two, which is equal to 13. But an easier way for numbers like this, you say, hey, 13 is right exactly halfway between 12 and 14. So there you have it. I have the middle of the first half, this five. I have the middle of the second half, 13."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "But an easier way for numbers like this, you say, hey, 13 is right exactly halfway between 12 and 14. So there you have it. I have the middle of the first half, this five. I have the middle of the second half, 13. To calculate the interquartile range, I just have to find the difference between these two things. So the interquartile range for this first example is going to be 13 minus five. The middle of the second half minus the middle of the first half, which is going to be equal to eight."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "I have the middle of the second half, 13. To calculate the interquartile range, I just have to find the difference between these two things. So the interquartile range for this first example is going to be 13 minus five. The middle of the second half minus the middle of the first half, which is going to be equal to eight. Let's do some more of these. This is strangely fun. Find the interquartile range of the data in the dot plot below."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "The middle of the second half minus the middle of the first half, which is going to be equal to eight. Let's do some more of these. This is strangely fun. Find the interquartile range of the data in the dot plot below. Songs on each album in Shane's collection. And so let's see what's going on here. And like always, I encourage you to take a shot at it."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Find the interquartile range of the data in the dot plot below. Songs on each album in Shane's collection. And so let's see what's going on here. And like always, I encourage you to take a shot at it. So this is just representing the data in a different way, but we could write this again as an ordered list. So let's do that. We have one song, or we have one album with seven songs, I guess you could say."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And like always, I encourage you to take a shot at it. So this is just representing the data in a different way, but we could write this again as an ordered list. So let's do that. We have one song, or we have one album with seven songs, I guess you could say. So we have a seven. We have two albums with nine songs. So we have two nines."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We have one song, or we have one album with seven songs, I guess you could say. So we have a seven. We have two albums with nine songs. So we have two nines. Let me write those. We have two nines. Then we have three tens."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So we have two nines. Let me write those. We have two nines. Then we have three tens. Let's cross those out. So 10, 10, 10. Then we have an 11."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Then we have three tens. Let's cross those out. So 10, 10, 10. Then we have an 11. We have an 11. We have two 12s. Two 12s."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Then we have an 11. We have an 11. We have two 12s. Two 12s. And then finally we have, so I used those already, and then we have an album with 14 songs. 14. So all I did here is I wrote this data like this."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Two 12s. And then finally we have, so I used those already, and then we have an album with 14 songs. 14. So all I did here is I wrote this data like this. So we could see, okay, this album has seven songs, this album has nine, this album has nine. And the way I wrote it, it's already in order. So I can immediately start calculating the median."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So all I did here is I wrote this data like this. So we could see, okay, this album has seven songs, this album has nine, this album has nine. And the way I wrote it, it's already in order. So I can immediately start calculating the median. Let's see, I have one, two, three, four, five, six, seven, eight, nine, 10 numbers. I have an even number of numbers. So to calculate the median, I'm gonna have to look at the middle two numbers."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So I can immediately start calculating the median. Let's see, I have one, two, three, four, five, six, seven, eight, nine, 10 numbers. I have an even number of numbers. So to calculate the median, I'm gonna have to look at the middle two numbers. So the middle two numbers look like it's these two tens here because I have four to the left of them and then four to the right of them. And so since I'm calculating the median using two numbers, it's going to be halfway between them. It's going to be the average of these two numbers."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So to calculate the median, I'm gonna have to look at the middle two numbers. So the middle two numbers look like it's these two tens here because I have four to the left of them and then four to the right of them. And so since I'm calculating the median using two numbers, it's going to be halfway between them. It's going to be the average of these two numbers. Well, the average of 10 and 10 is just going to be 10. So the median is going to be 10. Median is going to be 10."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It's going to be the average of these two numbers. Well, the average of 10 and 10 is just going to be 10. So the median is going to be 10. Median is going to be 10. And in a case like this where I calculated the median using the middle two numbers, I can now include this left 10 in the first half and I can include this right 10 in the second half. So let's do that. So the first half is going to be those five numbers and then the second half is going to be these five numbers."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Median is going to be 10. And in a case like this where I calculated the median using the middle two numbers, I can now include this left 10 in the first half and I can include this right 10 in the second half. So let's do that. So the first half is going to be those five numbers and then the second half is going to be these five numbers. And it makes sense because we're literally just looking at first half, it's going to be five numbers, second half is going to be five numbers. If I had a true middle number like the previous example, then we ignore that when we look at the first and second half or at least that's the way that we're doing it in these examples. But what's the median of this first half if we look at these five numbers?"}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So the first half is going to be those five numbers and then the second half is going to be these five numbers. And it makes sense because we're literally just looking at first half, it's going to be five numbers, second half is going to be five numbers. If I had a true middle number like the previous example, then we ignore that when we look at the first and second half or at least that's the way that we're doing it in these examples. But what's the median of this first half if we look at these five numbers? Well, if you have five numbers, if you have an odd number of numbers, you're going to have one middle number and it's going to be the one that has two on either sides. This has two to the left and it has two to the right. So the median of the first half, the middle of the first half is nine right over here."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "But what's the median of this first half if we look at these five numbers? Well, if you have five numbers, if you have an odd number of numbers, you're going to have one middle number and it's going to be the one that has two on either sides. This has two to the left and it has two to the right. So the median of the first half, the middle of the first half is nine right over here. And the middle of the second half, I have one, two, three, four, five numbers and this 12 is right in the middle. You have two to the left and two to the right. So the median of the second half is 12."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "Anthony Dune is analyzing his basketball statistics. The following table shows a probability model for the results from his next two free throws. And so it has various outcomes of those two free throws and then the corresponding probability, missing both free throws, 0.2, making exactly one free throw, 0.5, and making both free throws, 0.1. Is this a valid probability model? Pause this video and see if you can make a conclusion there. So let's think about what makes a valid probability model. One, the sum of the probabilities of all the scenarios need to add up to 100%, so we would definitely want to check that."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "Is this a valid probability model? Pause this video and see if you can make a conclusion there. So let's think about what makes a valid probability model. One, the sum of the probabilities of all the scenarios need to add up to 100%, so we would definitely want to check that. And also, they would all have to be positive values, or I guess I should say, none of them can be negative values. You could have a scenario that has a 0% probability. And so all of these look like positive probabilities, so we meet that second test, that all the probabilities are non-negative, but do they add up to 100%?"}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "One, the sum of the probabilities of all the scenarios need to add up to 100%, so we would definitely want to check that. And also, they would all have to be positive values, or I guess I should say, none of them can be negative values. You could have a scenario that has a 0% probability. And so all of these look like positive probabilities, so we meet that second test, that all the probabilities are non-negative, but do they add up to 100%? So if I had 0.2 to 0.5, that is 0.7 plus 0.1, they add up to 0.8, or they add up to 80%. So this is not a valid probability model. In order for it to be valid, they would all, all the various scenarios need to add up exactly to 100%."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so all of these look like positive probabilities, so we meet that second test, that all the probabilities are non-negative, but do they add up to 100%? So if I had 0.2 to 0.5, that is 0.7 plus 0.1, they add up to 0.8, or they add up to 80%. So this is not a valid probability model. In order for it to be valid, they would all, all the various scenarios need to add up exactly to 100%. In this case, we only add up to 80%. If we added up to 1.1 or 110%, then we would also have a problem, but we can just write no. Let's do another example."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "In order for it to be valid, they would all, all the various scenarios need to add up exactly to 100%. In this case, we only add up to 80%. If we added up to 1.1 or 110%, then we would also have a problem, but we can just write no. Let's do another example. So here, we are told, you are a space alien. You visit planet Earth and abduct 97 chickens, 47 cows, and 77 humans. Then, you randomly select one Earth creature from your sample to experiment on."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "Let's do another example. So here, we are told, you are a space alien. You visit planet Earth and abduct 97 chickens, 47 cows, and 77 humans. Then, you randomly select one Earth creature from your sample to experiment on. Each creature has an equal probability of getting selected. Create a probability model to show how likely you are to select each type of Earth creature. Input your answers as fractions or as decimals rounded to the nearest hundredth."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "Then, you randomly select one Earth creature from your sample to experiment on. Each creature has an equal probability of getting selected. Create a probability model to show how likely you are to select each type of Earth creature. Input your answers as fractions or as decimals rounded to the nearest hundredth. So in the last example, we wanted to see whether a probability model was valid, whether it was legitimate. Here, we wanna construct a legitimate probability model. Well, how would we do that?"}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "Input your answers as fractions or as decimals rounded to the nearest hundredth. So in the last example, we wanted to see whether a probability model was valid, whether it was legitimate. Here, we wanna construct a legitimate probability model. Well, how would we do that? Well, the estimated probability of getting a chicken is going to be the fraction that you're sampling from that is, are chickens, because any one of the animals are equally likely to be selected. 97 of the 97 plus 47 plus 77 animals are chickens. And so, what is this going to be?"}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, how would we do that? Well, the estimated probability of getting a chicken is going to be the fraction that you're sampling from that is, are chickens, because any one of the animals are equally likely to be selected. 97 of the 97 plus 47 plus 77 animals are chickens. And so, what is this going to be? It's going to be 97 over 97, 47, and 77. You add them up. Three sevens is 21."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so, what is this going to be? It's going to be 97 over 97, 47, and 77. You add them up. Three sevens is 21. And then, let's see, two plus nine is 11, plus four is 15, plus seven is 22, so 221. So 97 of the 221 animals are chickens. And so, I'll just write 97, two 21s."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "Three sevens is 21. And then, let's see, two plus nine is 11, plus four is 15, plus seven is 22, so 221. So 97 of the 221 animals are chickens. And so, I'll just write 97, two 21s. They say that we can answer as fractions, so the problem's gonna go that way. What about cows? Well, 47 of the 221 are cows."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so, I'll just write 97, two 21s. They say that we can answer as fractions, so the problem's gonna go that way. What about cows? Well, 47 of the 221 are cows. So there's a 47, two 21st probability of getting a cow. And then, last but not least, you have 77 of the 221s are human. Is this a legitimate probability distribution?"}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, 47 of the 221 are cows. So there's a 47, two 21st probability of getting a cow. And then, last but not least, you have 77 of the 221s are human. Is this a legitimate probability distribution? Well, add these up. If you add these three fractions up, the denominator's gonna be 221, and we already know that 97 plus 47 plus 77 is 221, so it definitely adds up to one, and none of these are negative. So this is a legitimate probability distribution."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "In the United States, we're talking about on the order of 300 million people. So ideally, if you could somehow magically do it, you would survey or somehow observe all 300 million people and take the mean of how many hours of TV they watch on a given day. And then that will give you the parameter, the population mean. But we've already talked about, in a case like this, that's very impractical. Even if you tried to do it, by the time you did it, your data might be stale because some people might have passed away, other people might have been born. Who knows what might have happened? And so this is a truth that is out there."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But we've already talked about, in a case like this, that's very impractical. Even if you tried to do it, by the time you did it, your data might be stale because some people might have passed away, other people might have been born. Who knows what might have happened? And so this is a truth that is out there. There is a theoretical population mean for the amount of the average or the mean hours of TV watched per day by Americans. There is a truth here at any given point in time. It's just pretty much impossible to come up with the exact answer, to come up with this exact truth."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And so this is a truth that is out there. There is a theoretical population mean for the amount of the average or the mean hours of TV watched per day by Americans. There is a truth here at any given point in time. It's just pretty much impossible to come up with the exact answer, to come up with this exact truth. But you don't give up. You say, well, maybe I don't have to survey all 300 million or observe all 300 million. Instead, I'm just going to observe a sample."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "It's just pretty much impossible to come up with the exact answer, to come up with this exact truth. But you don't give up. You say, well, maybe I don't have to survey all 300 million or observe all 300 million. Instead, I'm just going to observe a sample. I'm just going to observe a sample right over here. And let's say, for the sake, make the computation simple. You do a sample of six."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Instead, I'm just going to observe a sample. I'm just going to observe a sample right over here. And let's say, for the sake, make the computation simple. You do a sample of six. And we'll talk about later why six might not be as large of a sample as you would like. But you survey how much TV these folks watch. And you get that you find one person who watched 1 and 1 half hours, another person watched 2 and 1 half hours, another person watched 4 hours."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "You do a sample of six. And we'll talk about later why six might not be as large of a sample as you would like. But you survey how much TV these folks watch. And you get that you find one person who watched 1 and 1 half hours, another person watched 2 and 1 half hours, another person watched 4 hours. And then you get one person who watched 2 hours. And then you get two people who watched 1 hour each. So given this data from your sample, what do you get as your sample mean?"}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And you get that you find one person who watched 1 and 1 half hours, another person watched 2 and 1 half hours, another person watched 4 hours. And then you get one person who watched 2 hours. And then you get two people who watched 1 hour each. So given this data from your sample, what do you get as your sample mean? Well, the sample mean, which we would denote by lowercase x with a bar over it, it's just the sum of all of these divided by the number of data points we have. So let's see, we have 1.5 plus 2.5 plus 4 plus 2 plus 1 plus 1. And all of that divided by 6, which gives us, see, the numerator 1.5 plus 2.5 is 4."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So given this data from your sample, what do you get as your sample mean? Well, the sample mean, which we would denote by lowercase x with a bar over it, it's just the sum of all of these divided by the number of data points we have. So let's see, we have 1.5 plus 2.5 plus 4 plus 2 plus 1 plus 1. And all of that divided by 6, which gives us, see, the numerator 1.5 plus 2.5 is 4. Plus 4 is 8. Plus 2 is 10. Plus 2 more is 12."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And all of that divided by 6, which gives us, see, the numerator 1.5 plus 2.5 is 4. Plus 4 is 8. Plus 2 is 10. Plus 2 more is 12. So this is going to be 12 over 6, which is equal to 2 hours of television. So at least for your sample, you say my sample mean is 2 hours of television. It's an estimate."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Plus 2 more is 12. So this is going to be 12 over 6, which is equal to 2 hours of television. So at least for your sample, you say my sample mean is 2 hours of television. It's an estimate. It's a statistic that is trying to estimate this parameter, this thing that's very hard to know. But it's our best shot. Maybe we'll get a better answer if we get more data points."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "It's an estimate. It's a statistic that is trying to estimate this parameter, this thing that's very hard to know. But it's our best shot. Maybe we'll get a better answer if we get more data points. But this is what we have so far. Now, the next question you ask yourself is, well, I don't want to just estimate my population mean. I also want to estimate another parameter."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Maybe we'll get a better answer if we get more data points. But this is what we have so far. Now, the next question you ask yourself is, well, I don't want to just estimate my population mean. I also want to estimate another parameter. I also am interested in estimating my population variance. So once again, since we can't survey everyone in the population, this is pretty much impossible to know. But we're going to attempt to estimate this parameter."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "I also want to estimate another parameter. I also am interested in estimating my population variance. So once again, since we can't survey everyone in the population, this is pretty much impossible to know. But we're going to attempt to estimate this parameter. We attempted to estimate the mean. Now we will also attempt to estimate this parameter, this variance parameter. So how would you do it?"}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But we're going to attempt to estimate this parameter. We attempted to estimate the mean. Now we will also attempt to estimate this parameter, this variance parameter. So how would you do it? Well, reasonable logic would say, well, maybe we'll do the same thing with the sample as we would have done with the population. When you're doing the population variance, you would take each data point in the population, find the distance between that and the population mean, take the square of that difference, and then add up all those squares of those difference, and then divide by the number of data points you have. So let's try that over here."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So how would you do it? Well, reasonable logic would say, well, maybe we'll do the same thing with the sample as we would have done with the population. When you're doing the population variance, you would take each data point in the population, find the distance between that and the population mean, take the square of that difference, and then add up all those squares of those difference, and then divide by the number of data points you have. So let's try that over here. So let's try to take each of these data points and find the difference. Let me do that in a different color. Each of these data points and find the difference between that data point and our sample mean, not the population mean."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So let's try that over here. So let's try to take each of these data points and find the difference. Let me do that in a different color. Each of these data points and find the difference between that data point and our sample mean, not the population mean. We don't know what the population mean. The sample mean, so that's that first data point plus the second data point plus the second data point. So it's 4 minus 2 squared plus 1 minus 2 squared."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Each of these data points and find the difference between that data point and our sample mean, not the population mean. We don't know what the population mean. The sample mean, so that's that first data point plus the second data point plus the second data point. So it's 4 minus 2 squared plus 1 minus 2 squared. And this is what you would have done if you were taking a population variance. If this was your entire population, this is how you would find a population mean here if this was your entire population. And you would find the squared distances from each of those data points and then divide by the number of data points."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So it's 4 minus 2 squared plus 1 minus 2 squared. And this is what you would have done if you were taking a population variance. If this was your entire population, this is how you would find a population mean here if this was your entire population. And you would find the squared distances from each of those data points and then divide by the number of data points. So let's just think about this a little bit. 1 minus 2 squared, then you have 2.5 minus 2 squared. 2.5 minus 2, 2 being the sample mean, squared plus, let me see this green color, plus 2 minus 2 squared, plus 1 minus 2 squared."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And you would find the squared distances from each of those data points and then divide by the number of data points. So let's just think about this a little bit. 1 minus 2 squared, then you have 2.5 minus 2 squared. 2.5 minus 2, 2 being the sample mean, squared plus, let me see this green color, plus 2 minus 2 squared, plus 1 minus 2 squared. And then maybe you would divide by the number of data points that you have, where you have the number of data points. So in this case, we're dividing by 6. And what would we get in this circumstance?"}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "2.5 minus 2, 2 being the sample mean, squared plus, let me see this green color, plus 2 minus 2 squared, plus 1 minus 2 squared. And then maybe you would divide by the number of data points that you have, where you have the number of data points. So in this case, we're dividing by 6. And what would we get in this circumstance? Well, if we just do the computation, 1.5 minus 2 is negative 0.5. We square that. This becomes a positive 0.25."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And what would we get in this circumstance? Well, if we just do the computation, 1.5 minus 2 is negative 0.5. We square that. This becomes a positive 0.25. 4 minus 2 squared is going to be 2 squared, which is 4. 1 minus 2 squared, well, that's negative 1 squared, which is just 1. 2.5 minus 2 is 0.5 squared is 0.25."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "This becomes a positive 0.25. 4 minus 2 squared is going to be 2 squared, which is 4. 1 minus 2 squared, well, that's negative 1 squared, which is just 1. 2.5 minus 2 is 0.5 squared is 0.25. 2 minus 2 squared, well, this is 0. And then 1 minus 2 squared is 1. It's negative 1 squared, so we just get 1."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "2.5 minus 2 is 0.5 squared is 0.25. 2 minus 2 squared, well, this is 0. And then 1 minus 2 squared is 1. It's negative 1 squared, so we just get 1. And if we add all of this up, let's see. Let me add the whole numbers first. 4 plus 1 is 5, plus 1 is 6."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "It's negative 1 squared, so we just get 1. And if we add all of this up, let's see. Let me add the whole numbers first. 4 plus 1 is 5, plus 1 is 6. And then we have 0.25. So this is going to be equal to 6.5. Let me write this in a neutral color."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "4 plus 1 is 5, plus 1 is 6. And then we have 0.25. So this is going to be equal to 6.5. Let me write this in a neutral color. So this is going to be 6.5 over this 6 right over here. And we could write this as, well, there's a couple of ways that we could write this, but I'll just get the calculator out and we can just calculate it. So 6.5 divided by 6 gets us, if we round, it's approximately 1.08."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Let me write this in a neutral color. So this is going to be 6.5 over this 6 right over here. And we could write this as, well, there's a couple of ways that we could write this, but I'll just get the calculator out and we can just calculate it. So 6.5 divided by 6 gets us, if we round, it's approximately 1.08. So it's approximately 1.08 is this calculation. Now, what we have to think about is whether this is the best calculation, whether this is the best estimate for the population variance, given the data that we have. You can always argue that we could have more data."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So 6.5 divided by 6 gets us, if we round, it's approximately 1.08. So it's approximately 1.08 is this calculation. Now, what we have to think about is whether this is the best calculation, whether this is the best estimate for the population variance, given the data that we have. You can always argue that we could have more data. But given the data we have, is this the best calculation that we can make to estimate the population variance? And I'll have you think about that for a second. Well, it turns out that this is close."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "You can always argue that we could have more data. But given the data we have, is this the best calculation that we can make to estimate the population variance? And I'll have you think about that for a second. Well, it turns out that this is close. This is close to the best calculation, the best estimate that we can make, given the data we have. And sometimes this will be called the sample variance. But it's a particular type of sample variance where we just divide by the number of data points we have."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Well, it turns out that this is close. This is close to the best calculation, the best estimate that we can make, given the data we have. And sometimes this will be called the sample variance. But it's a particular type of sample variance where we just divide by the number of data points we have. And so people will write just an n over here. So this is one way to define a sample variance. In an attempt to estimate our population variance."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But it's a particular type of sample variance where we just divide by the number of data points we have. And so people will write just an n over here. So this is one way to define a sample variance. In an attempt to estimate our population variance. But it turns out, and in the next video I'll give you an intuitive explanation of why it turns out this way. And then I would also like to write a computer simulation that at least experimentally makes you feel a little bit better. But it turns out you're going to get a better estimate."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "In an attempt to estimate our population variance. But it turns out, and in the next video I'll give you an intuitive explanation of why it turns out this way. And then I would also like to write a computer simulation that at least experimentally makes you feel a little bit better. But it turns out you're going to get a better estimate. And it's a little bit weird and voodooish at first when you first think about it. You're going to get a better estimate for your population variance. If you don't divide by 6, if you don't divide by the number of data points you have, but you divide by 1 less than the number of data points you have."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But it turns out you're going to get a better estimate. And it's a little bit weird and voodooish at first when you first think about it. You're going to get a better estimate for your population variance. If you don't divide by 6, if you don't divide by the number of data points you have, but you divide by 1 less than the number of data points you have. So how would we do that? And we can denote that as sample variance. So when most people talk about the sample variance, they're talking about a sample variance where you do this calculation."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "If you don't divide by 6, if you don't divide by the number of data points you have, but you divide by 1 less than the number of data points you have. So how would we do that? And we can denote that as sample variance. So when most people talk about the sample variance, they're talking about a sample variance where you do this calculation. But instead of dividing by 6, you were to divide by 5. So they'd say you divide by n minus 1. So what would we get in those circumstances?"}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So when most people talk about the sample variance, they're talking about a sample variance where you do this calculation. But instead of dividing by 6, you were to divide by 5. So they'd say you divide by n minus 1. So what would we get in those circumstances? Well, the top part's going to be the exact same thing. We're going to get 6.5. But then our denominator, our n is 6."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So what would we get in those circumstances? Well, the top part's going to be the exact same thing. We're going to get 6.5. But then our denominator, our n is 6. We have 6 data points. But we're going to divide by 1 less than 6. We're going to divide by 5."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But then our denominator, our n is 6. We have 6 data points. But we're going to divide by 1 less than 6. We're going to divide by 5. And 6.5 divided by 5 is equal to 1.3. So when we calculate our sample variance, this technique, which is the more mainstream technique. And I know it seems voodoo."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "We're going to divide by 5. And 6.5 divided by 5 is equal to 1.3. So when we calculate our sample variance, this technique, which is the more mainstream technique. And I know it seems voodoo. Why are we dividing by n minus 1? Well, for population variance, we divide by n. But remember, we're trying to estimate the population variance. And it turns out that this is a better estimate."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And I know it seems voodoo. Why are we dividing by n minus 1? Well, for population variance, we divide by n. But remember, we're trying to estimate the population variance. And it turns out that this is a better estimate. Because this calculation is underestimating what the population variance is. This is a better estimate. We don't know for sure what it is."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And it turns out that this is a better estimate. Because this calculation is underestimating what the population variance is. This is a better estimate. We don't know for sure what it is. These both could be way off. It could be just by chance what we happen to sample. But over many samples, and there's many ways to think about it, this is going to be a better calculation."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "We don't know for sure what it is. These both could be way off. It could be just by chance what we happen to sample. But over many samples, and there's many ways to think about it, this is going to be a better calculation. It's going to give you a better estimate. And so how would we write this down? How would we write this down with mathematical notation?"}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But over many samples, and there's many ways to think about it, this is going to be a better calculation. It's going to give you a better estimate. And so how would we write this down? How would we write this down with mathematical notation? Well, we could, remember, we're taking the sum. And we're taking each of the data points. So we'll start with the first data point, all the way to the nth data point."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "How would we write this down with mathematical notation? Well, we could, remember, we're taking the sum. And we're taking each of the data points. So we'll start with the first data point, all the way to the nth data point. This lowercase n says that, hey, we're looking at the sample. If I said uppercase N, that usually denotes that we're trying to sum up everything in the population. Here we're looking at a sample of size lowercase n. And we're taking each data point."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So we'll start with the first data point, all the way to the nth data point. This lowercase n says that, hey, we're looking at the sample. If I said uppercase N, that usually denotes that we're trying to sum up everything in the population. Here we're looking at a sample of size lowercase n. And we're taking each data point. So each x sub i. And from it, we're subtracting the sample mean. And then we're squaring it."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Here we're looking at a sample of size lowercase n. And we're taking each data point. So each x sub i. And from it, we're subtracting the sample mean. And then we're squaring it. We're taking the sum of the squared distances. And then we're dividing not by the number of data points we have, but by 1 less than the number of data points we have. So this calculation, where we summed up all of this, and then we divided by 5, not by 6, this is the standard definition of sample variance."}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "A nutritionist wants to estimate the average caloric content of the burritos at a popular restaurant. They obtain a random sample of 14 burritos and measure their caloric content. Their sample data are roughly symmetric with a mean of 700 calories and a standard deviation of 50 calories. Based on this sample, which of the following is a 95% confidence interval for the mean caloric content of these burritos? So pause this video and see if you can figure it out. All right, what's going on here? So there's a population of burritos here."}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Based on this sample, which of the following is a 95% confidence interval for the mean caloric content of these burritos? So pause this video and see if you can figure it out. All right, what's going on here? So there's a population of burritos here. There is a mean caloric content that the nutritionist wants to figure out but doesn't know the true population parameter here, the population mean. And so they take a sample of 14 burritos and they calculate some sample statistics. They calculate the sample mean, which is 700."}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So there's a population of burritos here. There is a mean caloric content that the nutritionist wants to figure out but doesn't know the true population parameter here, the population mean. And so they take a sample of 14 burritos and they calculate some sample statistics. They calculate the sample mean, which is 700. They also calculate the sample standard deviation, which is equal to 50. And they wanna use this data to construct a 95% confidence interval. And so our confidence interval is going to take the form, and we've seen this before, our sample mean plus or minus our critical value times the sample standard deviation divided by the square root of n. The reason why we're using a t-statistic is because we don't know the actual standard deviation for the population."}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "They calculate the sample mean, which is 700. They also calculate the sample standard deviation, which is equal to 50. And they wanna use this data to construct a 95% confidence interval. And so our confidence interval is going to take the form, and we've seen this before, our sample mean plus or minus our critical value times the sample standard deviation divided by the square root of n. The reason why we're using a t-statistic is because we don't know the actual standard deviation for the population. If we knew the standard deviation for the population, we would use that instead of our sample standard deviation. And if we use that, if we used sigma, which is a population parameter, then we could use a z-statistic right over here. We would use a z-distribution."}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And so our confidence interval is going to take the form, and we've seen this before, our sample mean plus or minus our critical value times the sample standard deviation divided by the square root of n. The reason why we're using a t-statistic is because we don't know the actual standard deviation for the population. If we knew the standard deviation for the population, we would use that instead of our sample standard deviation. And if we use that, if we used sigma, which is a population parameter, then we could use a z-statistic right over here. We would use a z-distribution. But since we're using this sample standard deviation, that's why we're using a t-statistic. But now let's do that. So what is this going to be?"}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "We would use a z-distribution. But since we're using this sample standard deviation, that's why we're using a t-statistic. But now let's do that. So what is this going to be? So our sample mean is 700, they tell us that. So it's going to be 700 plus or minus, plus or minus. So what would be our critical value for a 95% confidence interval?"}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So what is this going to be? So our sample mean is 700, they tell us that. So it's going to be 700 plus or minus, plus or minus. So what would be our critical value for a 95% confidence interval? Well, we will just get out our t-table. And with a t-table, remember, you have to care about degrees of freedom. And if our sample size is 14, then that means you take 14 minus one, so degrees of freedom is n minus one."}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So what would be our critical value for a 95% confidence interval? Well, we will just get out our t-table. And with a t-table, remember, you have to care about degrees of freedom. And if our sample size is 14, then that means you take 14 minus one, so degrees of freedom is n minus one. So that's gonna be 14 minus one is equal to 13. So we have 13 degrees of freedom that we have to keep in mind when we look at our t-table. So let's look at our t-table."}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And if our sample size is 14, then that means you take 14 minus one, so degrees of freedom is n minus one. So that's gonna be 14 minus one is equal to 13. So we have 13 degrees of freedom that we have to keep in mind when we look at our t-table. So let's look at our t-table. So 95% confidence interval and 13 degrees of freedom. So degrees of freedom right over here. So we have 13 degrees of freedom."}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So let's look at our t-table. So 95% confidence interval and 13 degrees of freedom. So degrees of freedom right over here. So we have 13 degrees of freedom. So that is this row right over here. And if we want a 95% confidence level, then that means our tail probability, remember, if our distribution, let me see if I'll draw it really small, little small distribution right over here. So if you want 95% of the area in the middle, that means you have 5% not shaded in, and that's evenly divided on each side, so that means you have 2 1\u20442% at the tails, 2 1\u20442%."}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So we have 13 degrees of freedom. So that is this row right over here. And if we want a 95% confidence level, then that means our tail probability, remember, if our distribution, let me see if I'll draw it really small, little small distribution right over here. So if you want 95% of the area in the middle, that means you have 5% not shaded in, and that's evenly divided on each side, so that means you have 2 1\u20442% at the tails, 2 1\u20442%. So what you wanna look for is a tail probability of 2 1\u20442%. So that is this right over here,.025, that's 2 1\u20442%. And so there you go, that is our critical value, 2.160."}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So if you want 95% of the area in the middle, that means you have 5% not shaded in, and that's evenly divided on each side, so that means you have 2 1\u20442% at the tails, 2 1\u20442%. So what you wanna look for is a tail probability of 2 1\u20442%. So that is this right over here,.025, that's 2 1\u20442%. And so there you go, that is our critical value, 2.160. So this, so this part right over here, so this is going to be two, let me do that in a darker color. This is going to be 2.160 times, what's our sample standard deviation? It's 50 over the square root of n, square root of 14."}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And so there you go, that is our critical value, 2.160. So this, so this part right over here, so this is going to be two, let me do that in a darker color. This is going to be 2.160 times, what's our sample standard deviation? It's 50 over the square root of n, square root of 14. So all of our choices have the 700 there. So we just need to figure out what our margin of error, this part of it, and we could use a calculator for that. Okay, 2.16, I could write a zero there, it doesn't really matter, times 50 divided by the square root of 14, square root of 14."}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "It's 50 over the square root of n, square root of 14. So all of our choices have the 700 there. So we just need to figure out what our margin of error, this part of it, and we could use a calculator for that. Okay, 2.16, I could write a zero there, it doesn't really matter, times 50 divided by the square root of 14, square root of 14. We get a little bit of a drum roll here, I think. 28.86, so this part right over here is approximately 28.86. That's our margin of error."}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Okay, 2.16, I could write a zero there, it doesn't really matter, times 50 divided by the square root of 14, square root of 14. We get a little bit of a drum roll here, I think. 28.86, so this part right over here is approximately 28.86. That's our margin of error. And we see out of all of these choices here, if we round to the nearest tenth, that'd be 28.9. So this is approximately 28.9, which is this choice right over here. This was an awfully close one, I guess they're trying to make sure that we're looking at enough digits."}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "That's our margin of error. And we see out of all of these choices here, if we round to the nearest tenth, that'd be 28.9. So this is approximately 28.9, which is this choice right over here. This was an awfully close one, I guess they're trying to make sure that we're looking at enough digits. So there we have it. We have established our 95% confidence interval. Now one thing that we should keep in mind is, is this a valid confidence interval?"}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "This was an awfully close one, I guess they're trying to make sure that we're looking at enough digits. So there we have it. We have established our 95% confidence interval. Now one thing that we should keep in mind is, is this a valid confidence interval? Did we meet our conditions for a valid confidence interval? And here we have to think, well, did we take a random sample and they tell us that they obtained a random sample of 14 burritos, so we check that one. Is the sampling distribution roughly normal?"}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Now one thing that we should keep in mind is, is this a valid confidence interval? Did we meet our conditions for a valid confidence interval? And here we have to think, well, did we take a random sample and they tell us that they obtained a random sample of 14 burritos, so we check that one. Is the sampling distribution roughly normal? Well, either you take, if you take over 30 samples, then it would be, but here we only took 14. But they do tell us that the sample data is roughly symmetric. And so if it's roughly symmetric and it has no significant outliers, then this is reasonable that you can assume that it is roughly normal."}, {"video_title": "Example constructing a t interval for a mean Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Is the sampling distribution roughly normal? Well, either you take, if you take over 30 samples, then it would be, but here we only took 14. But they do tell us that the sample data is roughly symmetric. And so if it's roughly symmetric and it has no significant outliers, then this is reasonable that you can assume that it is roughly normal. And then the last condition is the independence condition. And here, if we aren't sampling with replacement, and it doesn't look like we are, if we're not sampling with replacement, this has to be less than 10% of, this has to be less than 10% of the population of burritos. And we're assuming that there's going to be more than 140 burritos that the universe, that the population, that this popular restaurant makes."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "A set of laptop prices are normally distributed with a mean of $750 and a standard deviation of $60. What proportion of laptop prices are between $624 and $768? So let's think about what they are asking. So we have a normal distribution for the prices. So it would look something like this. And this is just my hand-drawn sketch of a normal distribution. So it would look something like this."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So we have a normal distribution for the prices. So it would look something like this. And this is just my hand-drawn sketch of a normal distribution. So it would look something like this. It should be symmetric. So I'm making it as symmetric as I can hand-draw it. And we have the mean right in the center."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So it would look something like this. It should be symmetric. So I'm making it as symmetric as I can hand-draw it. And we have the mean right in the center. So the mean would be right there. And that is $750. They also tell us that we have a standard deviation of $60."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "And we have the mean right in the center. So the mean would be right there. And that is $750. They also tell us that we have a standard deviation of $60. So that means one standard deviation above the mean would be roughly right over here. And that'd be 750 plus 60. So that would be $810."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "They also tell us that we have a standard deviation of $60. So that means one standard deviation above the mean would be roughly right over here. And that'd be 750 plus 60. So that would be $810. One standard deviation below the mean would put us right about there. And that would be 750 minus $60, which would be $690. And then they tell us what proportion of laptop prices are between $624 and $768?"}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So that would be $810. One standard deviation below the mean would put us right about there. And that would be 750 minus $60, which would be $690. And then they tell us what proportion of laptop prices are between $624 and $768? So the lower bound, $624, that's going to actually be more than another standard deviation less. So that's going to be right around here. So that is $624."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "And then they tell us what proportion of laptop prices are between $624 and $768? So the lower bound, $624, that's going to actually be more than another standard deviation less. So that's going to be right around here. So that is $624. And $768 would put us right at about there. And once again, this is just a hand-drawn sketch. But that is 768."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So that is $624. And $768 would put us right at about there. And once again, this is just a hand-drawn sketch. But that is 768. And so what proportion are between those two values? We want to find essentially the area under this distribution between these two values. The way we are going to approach it, we're going to figure out the z-score for 768."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "But that is 768. And so what proportion are between those two values? We want to find essentially the area under this distribution between these two values. The way we are going to approach it, we're going to figure out the z-score for 768. It's going to be positive because it's above the mean. And then we're going to use a z-table to figure out what proportion is below 768. So essentially we're going to figure out this entire area."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "The way we are going to approach it, we're going to figure out the z-score for 768. It's going to be positive because it's above the mean. And then we're going to use a z-table to figure out what proportion is below 768. So essentially we're going to figure out this entire area. We're even gonna figure out the stuff that's below 624. That's what that z-table will give us. And then we'll figure out the z-score for 624."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So essentially we're going to figure out this entire area. We're even gonna figure out the stuff that's below 624. That's what that z-table will give us. And then we'll figure out the z-score for 624. That will be negative two point something. And we will use a z-table again to figure out the proportion that is less than that. And so then we can subtract this red area from the proportion that is less than 768 to get this area in between."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "And then we'll figure out the z-score for 624. That will be negative two point something. And we will use a z-table again to figure out the proportion that is less than that. And so then we can subtract this red area from the proportion that is less than 768 to get this area in between. So let's do that. Let's figure out first the z-score for 768. And then we'll do it for 624."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "And so then we can subtract this red area from the proportion that is less than 768 to get this area in between. So let's do that. Let's figure out first the z-score for 768. And then we'll do it for 624. The z-score for 768, I'll write it like that, is going to be 768 minus 750 over the standard deviation, over 60. So this is going to be equal to 18 over 60, which is the same thing as six over, let's see if we divide the numerator and denominator by three, 6 20ths, and this is the same thing as 0.30. So that is the z-score for this upper bound."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "And then we'll do it for 624. The z-score for 768, I'll write it like that, is going to be 768 minus 750 over the standard deviation, over 60. So this is going to be equal to 18 over 60, which is the same thing as six over, let's see if we divide the numerator and denominator by three, 6 20ths, and this is the same thing as 0.30. So that is the z-score for this upper bound. Let's figure out what proportion is less than that. For that, we take out a z-table. Get our z-table."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So that is the z-score for this upper bound. Let's figure out what proportion is less than that. For that, we take out a z-table. Get our z-table. And let's see, we want to get 0.30. So this is zero. And so this is 0.30, this first column, and we've done this in other videos, this goes up until the 10th place for our z-score, and then if we want to get to our 100th place, that's what these other columns give us."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "Get our z-table. And let's see, we want to get 0.30. So this is zero. And so this is 0.30, this first column, and we've done this in other videos, this goes up until the 10th place for our z-score, and then if we want to get to our 100th place, that's what these other columns give us. But we're at 0.30, so we're going to be in this row, and our 100th place is right over here, it's a zero. So this is the proportion that is less than $768. So 0.6179."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "And so this is 0.30, this first column, and we've done this in other videos, this goes up until the 10th place for our z-score, and then if we want to get to our 100th place, that's what these other columns give us. But we're at 0.30, so we're going to be in this row, and our 100th place is right over here, it's a zero. So this is the proportion that is less than $768. So 0.6179. So 0.6179. So now let's do the same exercise, but do it for the proportion that's below $624. The z-score for 624 is going to be equal to 624 minus the mean of 750, all of that over 60."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So 0.6179. So 0.6179. So now let's do the same exercise, but do it for the proportion that's below $624. The z-score for 624 is going to be equal to 624 minus the mean of 750, all of that over 60. And so what is that going to be? I'll get my calculator out for this one, don't want to make a careless error. 624 minus 750 is equal to, 624 minus 750 is equal to, and then divide by 60 is equal to negative 2.1."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "The z-score for 624 is going to be equal to 624 minus the mean of 750, all of that over 60. And so what is that going to be? I'll get my calculator out for this one, don't want to make a careless error. 624 minus 750 is equal to, 624 minus 750 is equal to, and then divide by 60 is equal to negative 2.1. So that lower bound is 2.1 standard deviations below the mean, or you could say it has a z-score of negative 2.1, is equal to negative 2.1. And so to figure out the proportion that is less than that, this red area right over here, we go back to our z-table. And we'd actually go to the first part of the z-table."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "624 minus 750 is equal to, 624 minus 750 is equal to, and then divide by 60 is equal to negative 2.1. So that lower bound is 2.1 standard deviations below the mean, or you could say it has a z-score of negative 2.1, is equal to negative 2.1. And so to figure out the proportion that is less than that, this red area right over here, we go back to our z-table. And we'd actually go to the first part of the z-table. So same idea, but this starts at negative, a z-score of negative 3.4, 3.4 standard deviations below the mean. But just like we saw before, this is our zero hundredths, one hundredth, two hundredth, so on and so forth. And we want to go to negative 2.1, we could say negative 2.10, just to be precise."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "And we'd actually go to the first part of the z-table. So same idea, but this starts at negative, a z-score of negative 3.4, 3.4 standard deviations below the mean. But just like we saw before, this is our zero hundredths, one hundredth, two hundredth, so on and so forth. And we want to go to negative 2.1, we could say negative 2.10, just to be precise. So this is going to get us, let's see, negative 2.1, there we go. And so we are negative 2.1, and it's negative 2.10, so we have zero hundredths. So we're gonna be right here on our table."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "And we want to go to negative 2.1, we could say negative 2.10, just to be precise. So this is going to get us, let's see, negative 2.1, there we go. And so we are negative 2.1, and it's negative 2.10, so we have zero hundredths. So we're gonna be right here on our table. So we see the proportion that is less than 624 is.0179, or 0.0179. So 0.0179. So 0.0179."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So we're gonna be right here on our table. So we see the proportion that is less than 624 is.0179, or 0.0179. So 0.0179. So 0.0179. And so if we want to figure out the proportion that's in between the two, we just subtract this red area from this entire area, the entire proportion that's less than 768, to get what's in between. 0.6179, once again, I know I keep repeating it, that's this entire area right over here, and we're gonna subtract out what we have in red. So minus 0.0179, so we're gonna subtract this out, to get 0.6."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "It has yellow and green and pink and blue gumballs. Let me throw a few blue ones in there. And what we're going to concern ourselves in this video are the yellow gumballs. And let's say that we know that the proportion of yellow gumballs over here is p. This right over here is a population, population parameter. Parameter. And for the sake of argument, just to make things concrete, let's just say that 60% of the gumballs are yellow, or 0.6 of them. Now let's review some things that we have seen before."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "And let's say that we know that the proportion of yellow gumballs over here is p. This right over here is a population, population parameter. Parameter. And for the sake of argument, just to make things concrete, let's just say that 60% of the gumballs are yellow, or 0.6 of them. Now let's review some things that we have seen before. I'm gonna define our Bernoulli random variable. Let's call this capital Y, which is equal to one if when we take one random gumball out of that machine, we get a, we pick a yellow gumball. And it's equal to zero if when we pick one random gumball out of that machine, we don't pick yellow."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "Now let's review some things that we have seen before. I'm gonna define our Bernoulli random variable. Let's call this capital Y, which is equal to one if when we take one random gumball out of that machine, we get a, we pick a yellow gumball. And it's equal to zero if when we pick one random gumball out of that machine, we don't pick yellow. So not yellow. From previous videos, we know some interesting things about this Bernoulli random variable. We know its mean."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "And it's equal to zero if when we pick one random gumball out of that machine, we don't pick yellow. So not yellow. From previous videos, we know some interesting things about this Bernoulli random variable. We know its mean. We know the mean of our Bernoulli random variable is going to be the proportion of yellow balls in this population. So it's going to be equal to P, which in this particular case we know is 0.6. And we know what the standard deviation of this Bernoulli random variable is."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "We know its mean. We know the mean of our Bernoulli random variable is going to be the proportion of yellow balls in this population. So it's going to be equal to P, which in this particular case we know is 0.6. And we know what the standard deviation of this Bernoulli random variable is. It is going to be P times one minus P. Actually, that's the variance. We want to take the square root of that to get the standard deviation. So in this particular scenario, that's going to be the square root of 0.6 times 0.4."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "And we know what the standard deviation of this Bernoulli random variable is. It is going to be P times one minus P. Actually, that's the variance. We want to take the square root of that to get the standard deviation. So in this particular scenario, that's going to be the square root of 0.6 times 0.4. Fair enough. This is all review so far. But now let me define, let me define another random variable, X, which is equal to the sum of 10 independent, independent trials, trials, trials of Y."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "So in this particular scenario, that's going to be the square root of 0.6 times 0.4. Fair enough. This is all review so far. But now let me define, let me define another random variable, X, which is equal to the sum of 10 independent, independent trials, trials, trials of Y. Now, we have seen random variables like this before. This is a binomial random variable. Now, what do we know about its mean and standard deviation?"}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "But now let me define, let me define another random variable, X, which is equal to the sum of 10 independent, independent trials, trials, trials of Y. Now, we have seen random variables like this before. This is a binomial random variable. Now, what do we know about its mean and standard deviation? Well, in previous videos, we know that the mean of this binomial random variable is just going to be equal to N times the mean of each of the Bernoulli trials right over here. So N times P, which in this particular situation is going to be, N is 10, we're doing 10 trials, and P is 0.6, which is equal to six. And that makes sense."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "Now, what do we know about its mean and standard deviation? Well, in previous videos, we know that the mean of this binomial random variable is just going to be equal to N times the mean of each of the Bernoulli trials right over here. So N times P, which in this particular situation is going to be, N is 10, we're doing 10 trials, and P is 0.6, which is equal to six. And that makes sense. If 60% of the balls here are yellow, and if you were to take a sample, or if you were to take 10 trials, so if you were to take 10 balls one at a time, they have to be independent, so you keep looking at them and then replacing them, but if you took 10, then you would expect that six of them would be yellow. They're not always going to be six yellow, but that would be maybe what you would expect. All right, now what's the standard deviation here?"}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "And that makes sense. If 60% of the balls here are yellow, and if you were to take a sample, or if you were to take 10 trials, so if you were to take 10 balls one at a time, they have to be independent, so you keep looking at them and then replacing them, but if you took 10, then you would expect that six of them would be yellow. They're not always going to be six yellow, but that would be maybe what you would expect. All right, now what's the standard deviation here? So our standard deviation is equal to, and we have proved this in other videos, it's equal to the square root of N times P times one minus P. Notice, you just put an N right over here under the radical sign, and so this is going to get us, in this particular situation, to 10 times 0.6 times 0.4, and then the square root of everything. So all of this is review. If it's unfamiliar, I encourage you to review some of the videos on Bernoulli random variables and on binomial random variables."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "All right, now what's the standard deviation here? So our standard deviation is equal to, and we have proved this in other videos, it's equal to the square root of N times P times one minus P. Notice, you just put an N right over here under the radical sign, and so this is going to get us, in this particular situation, to 10 times 0.6 times 0.4, and then the square root of everything. So all of this is review. If it's unfamiliar, I encourage you to review some of the videos on Bernoulli random variables and on binomial random variables. But what we're going to do in this video is think about a sampling distribution, and it's going to be the sampling distribution for a sample statistic known as the sample proportion, which we actually talked about when we first introduced sampling distributions. So let's say, so let's just park all of this. This is background right over here."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "If it's unfamiliar, I encourage you to review some of the videos on Bernoulli random variables and on binomial random variables. But what we're going to do in this video is think about a sampling distribution, and it's going to be the sampling distribution for a sample statistic known as the sample proportion, which we actually talked about when we first introduced sampling distributions. So let's say, so let's just park all of this. This is background right over here. Let's start taking samples of 10, and I didn't pick that randomly. I want to make it reconcile with what we did with our random variable here. And so let's take a sample of 10 gumballs, and let's calculate the proportion that are yellow."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "This is background right over here. Let's start taking samples of 10, and I didn't pick that randomly. I want to make it reconcile with what we did with our random variable here. And so let's take a sample of 10 gumballs, and let's calculate the proportion that are yellow. And so we will call that our sample proportion, and I might as well just do that in yellow. So we want to calculate the sample proportion that are yellow. And what is this equivalent to?"}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "And so let's take a sample of 10 gumballs, and let's calculate the proportion that are yellow. And so we will call that our sample proportion, and I might as well just do that in yellow. So we want to calculate the sample proportion that are yellow. And what is this equivalent to? Well, you could say, well, this is just equivalent to my random variable X. I want to count the number of gumballs that are yellow, and then I'm gonna divide it by the sample size. So I'm gonna divide it by N. And in this case, it would be X divided by 10. I know what some of you are thinking."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "And what is this equivalent to? Well, you could say, well, this is just equivalent to my random variable X. I want to count the number of gumballs that are yellow, and then I'm gonna divide it by the sample size. So I'm gonna divide it by N. And in this case, it would be X divided by 10. I know what some of you are thinking. Wait, wait, wait, hold on for a second. X is sum of 10 independent, independent trials right over here. To be independent, you can't just take 10 gumballs."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "I know what some of you are thinking. Wait, wait, wait, hold on for a second. X is sum of 10 independent, independent trials right over here. To be independent, you can't just take 10 gumballs. You have to take them one at a time and then replace them back in order for them to be truly independent. But remember, we have our 10% rule. We have our 10% rule, which tells us that if a sample is less than or equal to 10% of the population, then you can treat each of the gumballs in this situation as being independent."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "To be independent, you can't just take 10 gumballs. You have to take them one at a time and then replace them back in order for them to be truly independent. But remember, we have our 10% rule. We have our 10% rule, which tells us that if a sample is less than or equal to 10% of the population, then you can treat each of the gumballs in this situation as being independent. So let's just say for the sake of argument that there are 10,000 gumballs in here. And so we can feel pretty good that these samples, that each of the things in the sample are independent of each other by our 10% rule. And so each of these are, each of these 10 gumballs, what we see, they are going to be independent."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "We have our 10% rule, which tells us that if a sample is less than or equal to 10% of the population, then you can treat each of the gumballs in this situation as being independent. So let's just say for the sake of argument that there are 10,000 gumballs in here. And so we can feel pretty good that these samples, that each of the things in the sample are independent of each other by our 10% rule. And so each of these are, each of these 10 gumballs, what we see, they are going to be independent. I'm gonna put them in quotes by the 10% rule. And so in that situation, we can make this claim or we can feel good that this claim is roughly true. And so let's say for that first sample that we do, our sample proportion is equal to 0.3."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "And so each of these are, each of these 10 gumballs, what we see, they are going to be independent. I'm gonna put them in quotes by the 10% rule. And so in that situation, we can make this claim or we can feel good that this claim is roughly true. And so let's say for that first sample that we do, our sample proportion is equal to 0.3. So three of our gumballs just happened, three of our 10 gumballs just happened to be yellow. Then we do it again. So we take another sample."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "And so let's say for that first sample that we do, our sample proportion is equal to 0.3. So three of our gumballs just happened, three of our 10 gumballs just happened to be yellow. Then we do it again. So we take another sample. We calculate our sample proportion, the statistic again. Remember, it's trying to estimate our population parameter. And let's say that time it happens to be seven out of 10."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "So we take another sample. We calculate our sample proportion, the statistic again. Remember, it's trying to estimate our population parameter. And let's say that time it happens to be seven out of 10. And we just keep doing that. And if we keep doing that and we plot it on a dot chart or dot distribution, I guess we could say, where our possible outcomes, you could have zero out of 10, one out of 10, two, three, four, five, that's so half of them, six, seven, eight, nine, 10. So that would be all of them."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "And let's say that time it happens to be seven out of 10. And we just keep doing that. And if we keep doing that and we plot it on a dot chart or dot distribution, I guess we could say, where our possible outcomes, you could have zero out of 10, one out of 10, two, three, four, five, that's so half of them, six, seven, eight, nine, 10. So that would be all of them. And so you could plot, okay, 0.3, one, two, three. That's one scenario where I got zero, where my sample proportion is 0.3. Then 0.7, that's one situation where I got a 0.7."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "So that would be all of them. And so you could plot, okay, 0.3, one, two, three. That's one scenario where I got zero, where my sample proportion is 0.3. Then 0.7, that's one situation where I got a 0.7. And let's say I were to take another sample of 10 and I were to get 0.7, then you would plot that over here. And if you kept taking samples and kept calculating these sample proportions and you kept plotting it here, you would get a better and better and better approximation for the sampling distribution of the sampling proportion. But how can we actually characterize the true sampling distribution for the sample proportion?"}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "Then 0.7, that's one situation where I got a 0.7. And let's say I were to take another sample of 10 and I were to get 0.7, then you would plot that over here. And if you kept taking samples and kept calculating these sample proportions and you kept plotting it here, you would get a better and better and better approximation for the sampling distribution of the sampling proportion. But how can we actually characterize the true sampling distribution for the sample proportion? What is going to be the mean of this sampling distribution? And what is going to be the standard deviation? Well, we can derive that from what we see right over here."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "But how can we actually characterize the true sampling distribution for the sample proportion? What is going to be the mean of this sampling distribution? And what is going to be the standard deviation? Well, we can derive that from what we see right over here. The mean of our sampling distribution of our sample proportion is just going to be equal to the mean of our random variable X divided by N. It's just going to be the mean of X divided by N, which is equal to what? Well, the mean of X is N times P. This is N times P. You divided it by N, you're going to get P. And that makes sense. One way to think about it, the expected value for your sample proportion is going to be the proportion of gumballs that you actually see."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "Well, we can derive that from what we see right over here. The mean of our sampling distribution of our sample proportion is just going to be equal to the mean of our random variable X divided by N. It's just going to be the mean of X divided by N, which is equal to what? Well, the mean of X is N times P. This is N times P. You divided it by N, you're going to get P. And that makes sense. One way to think about it, the expected value for your sample proportion is going to be the proportion of gumballs that you actually see. And so this is also a good indicator that this is going to be a reasonably unbiased estimator. Now let's think about the standard deviation for our sample proportion. Well, we can just view that as our standard deviation of our binomial random variable X divided by N. So this is going to be equal to the square root of N times P times one minus P, all of that over N, which is the same thing as if we put this, if we divide by N inside the radical, it'd be the same thing as the square root of N, P times one minus P over N squared."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "One way to think about it, the expected value for your sample proportion is going to be the proportion of gumballs that you actually see. And so this is also a good indicator that this is going to be a reasonably unbiased estimator. Now let's think about the standard deviation for our sample proportion. Well, we can just view that as our standard deviation of our binomial random variable X divided by N. So this is going to be equal to the square root of N times P times one minus P, all of that over N, which is the same thing as if we put this, if we divide by N inside the radical, it'd be the same thing as the square root of N, P times one minus P over N squared. Divide the numerator and denominator by N, you will get the square root of P times one minus P, all of that over N. And so in this particular situation where our parameter is 0.6, where our population parameter is 0.6, so it's going to be 0.6, that's the true proportion for our population. And then what would our standard deviation be for our sample proportion? Well, it's going to be equal to the square root of 0.6 times 0.4, all of that over 10."}, {"video_title": "Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3", "Sentence": "Well, we can just view that as our standard deviation of our binomial random variable X divided by N. So this is going to be equal to the square root of N times P times one minus P, all of that over N, which is the same thing as if we put this, if we divide by N inside the radical, it'd be the same thing as the square root of N, P times one minus P over N squared. Divide the numerator and denominator by N, you will get the square root of P times one minus P, all of that over N. And so in this particular situation where our parameter is 0.6, where our population parameter is 0.6, so it's going to be 0.6, that's the true proportion for our population. And then what would our standard deviation be for our sample proportion? Well, it's going to be equal to the square root of 0.6 times 0.4, all of that over 10. And we can get a calculator out to calculate that. So if we take 0.6 times 0.4 equals, divided by 10 equals, and then we take the square root of that, and we get it's approximately 0.15. Approximately 0.15."}, {"video_title": "Using a confidence interval to test slope More on regression AP Statistics Khan Academy.mp3", "Sentence": "Hashim wants to use this interval to test the null hypothesis that the true slope of the population regression line, so this is a population parameter right here for the slope of the population regression line, is equal to zero, versus the alternative hypothesis is that the true slope of the population regression line is not equal to zero, at the alpha is equal to 0.05 level of significance. Assume that all conditions for inference have been met. So given the information that we just have about what Hashim is doing, what would be his conclusion? Would he reject the null hypothesis, which would suggest the alternative, or would he be unable to reject the null hypothesis? Well, let's just think about this a little bit. We have a 95% confidence interval. Let me write this down."}, {"video_title": "Using a confidence interval to test slope More on regression AP Statistics Khan Academy.mp3", "Sentence": "Would he reject the null hypothesis, which would suggest the alternative, or would he be unable to reject the null hypothesis? Well, let's just think about this a little bit. We have a 95% confidence interval. Let me write this down. So our 95% confidence, confidence interval, could write it like this, or you could say that it goes from 0.39 minus 0.23, so that'd be 0.16, so it goes from 0.16 until 0.39 plus 0.23 is going to be what? 0.62. Now what a 95% confidence interval tells us is that 95% of the time that we take a sample and we construct a 95% confidence interval, that 95% of the time we do this, it should overlap with the true population parameter that we are trying to estimate."}, {"video_title": "Using a confidence interval to test slope More on regression AP Statistics Khan Academy.mp3", "Sentence": "Let me write this down. So our 95% confidence, confidence interval, could write it like this, or you could say that it goes from 0.39 minus 0.23, so that'd be 0.16, so it goes from 0.16 until 0.39 plus 0.23 is going to be what? 0.62. Now what a 95% confidence interval tells us is that 95% of the time that we take a sample and we construct a 95% confidence interval, that 95% of the time we do this, it should overlap with the true population parameter that we are trying to estimate. But in this hypothesis test, remember, we are assuming that the true population parameter is equal to zero, and that does not overlap with this confidence interval. So assuming, let me write this down, assuming null hypothesis is true, null hypothesis is true, we are in the less than or equal to 5% chance of situations, situations, where, where beta not overlap, overlap, with 95% intervals. And the whole notion of hypothesis testing is you assume the null hypothesis, you take your sample, and then if you get statistics, and if the probability of getting those statistics or something even more extreme than those statistics is less than your significance level, then you reject the null hypothesis."}, {"video_title": "Using a confidence interval to test slope More on regression AP Statistics Khan Academy.mp3", "Sentence": "Now what a 95% confidence interval tells us is that 95% of the time that we take a sample and we construct a 95% confidence interval, that 95% of the time we do this, it should overlap with the true population parameter that we are trying to estimate. But in this hypothesis test, remember, we are assuming that the true population parameter is equal to zero, and that does not overlap with this confidence interval. So assuming, let me write this down, assuming null hypothesis is true, null hypothesis is true, we are in the less than or equal to 5% chance of situations, situations, where, where beta not overlap, overlap, with 95% intervals. And the whole notion of hypothesis testing is you assume the null hypothesis, you take your sample, and then if you get statistics, and if the probability of getting those statistics or something even more extreme than those statistics is less than your significance level, then you reject the null hypothesis. And that's exactly what's happening here. And this null hypothesis value is nowhere even close to overlapping, it's over 16 hundredths below the low end of this bound. And so because of that, we would reject the null hypothesis."}, {"video_title": "Using a confidence interval to test slope More on regression AP Statistics Khan Academy.mp3", "Sentence": "And the whole notion of hypothesis testing is you assume the null hypothesis, you take your sample, and then if you get statistics, and if the probability of getting those statistics or something even more extreme than those statistics is less than your significance level, then you reject the null hypothesis. And that's exactly what's happening here. And this null hypothesis value is nowhere even close to overlapping, it's over 16 hundredths below the low end of this bound. And so because of that, we would reject the null hypothesis. Reject the null hypothesis, which suggests the alternative, which suggests the alternative hypothesis. And one way to interpret this alternative hypothesis, that beta is not equal to zero, is that there is, there is a non-zero linear relationship, relationship, between, between ages and backpack weights. Ages and backpack weights."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "Women have a mean height of 170 centimeters with a standard deviation of six centimeters. The male and female heights are each normally distributed. We independently, randomly select a man and a woman. What is the probability that the woman is taller than the man? So I encourage you to pause this video and think through it and I'll give you a hint. What if we were to define the random variable m as equal to the height of a randomly selected man, height of random man. What if we defined the random variable w to be equal to the height of a random woman, woman, and we defined a third random variable in terms of these first two?"}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "What is the probability that the woman is taller than the man? So I encourage you to pause this video and think through it and I'll give you a hint. What if we were to define the random variable m as equal to the height of a randomly selected man, height of random man. What if we defined the random variable w to be equal to the height of a random woman, woman, and we defined a third random variable in terms of these first two? So let me call this d for difference. And it is equal to the difference in height between a randomly selected man and a randomly selected woman. So d, the random variable d, is equal to the random variable m minus the random variable w. So the first two are clearly normally distributed."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "What if we defined the random variable w to be equal to the height of a random woman, woman, and we defined a third random variable in terms of these first two? So let me call this d for difference. And it is equal to the difference in height between a randomly selected man and a randomly selected woman. So d, the random variable d, is equal to the random variable m minus the random variable w. So the first two are clearly normally distributed. They tell us that right over here. The male and female heights are each normally distributed. And we also know, or you're about to know, that the difference of random variables that are each normally distributed is also going to be normally distributed."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So d, the random variable d, is equal to the random variable m minus the random variable w. So the first two are clearly normally distributed. They tell us that right over here. The male and female heights are each normally distributed. And we also know, or you're about to know, that the difference of random variables that are each normally distributed is also going to be normally distributed. So given this, can you think about how to tackle this question? The probability that the woman is taller than the man. All right, now let's work through this together and to help us visualize, I'll draw the normal distribution curves for these three random variables."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "And we also know, or you're about to know, that the difference of random variables that are each normally distributed is also going to be normally distributed. So given this, can you think about how to tackle this question? The probability that the woman is taller than the man. All right, now let's work through this together and to help us visualize, I'll draw the normal distribution curves for these three random variables. So this first one is for the variable m. And so right here in the middle, that is the mean of m. And we know that this is going to be equal to 178 centimeters. We'll assume everything is in centimeters. We also know that it has a standard deviation of eight centimeters."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "All right, now let's work through this together and to help us visualize, I'll draw the normal distribution curves for these three random variables. So this first one is for the variable m. And so right here in the middle, that is the mean of m. And we know that this is going to be equal to 178 centimeters. We'll assume everything is in centimeters. We also know that it has a standard deviation of eight centimeters. So for example, if this is one standard deviation above, this is one standard deviation below, this point right over here would be eight centimeters more than 178, so that would be 186. And this would be eight centimeters below that, so this would be 170 centimeters. So this is for the random variable m. Now let's think about the random variable w. The random variable w, the mean of w, they tell us, is 170."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "We also know that it has a standard deviation of eight centimeters. So for example, if this is one standard deviation above, this is one standard deviation below, this point right over here would be eight centimeters more than 178, so that would be 186. And this would be eight centimeters below that, so this would be 170 centimeters. So this is for the random variable m. Now let's think about the random variable w. The random variable w, the mean of w, they tell us, is 170. And one standard deviation above the mean is going to be six centimeters above the mean. The standard deviation is six, six centimeters. So this would be minus six, is to go to one standard deviation below the mean."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So this is for the random variable m. Now let's think about the random variable w. The random variable w, the mean of w, they tell us, is 170. And one standard deviation above the mean is going to be six centimeters above the mean. The standard deviation is six, six centimeters. So this would be minus six, is to go to one standard deviation below the mean. Now let's think about the difference between the two, the random variable d. So let me think about this a little bit. The random variable d, the mean of d is going to be equal to the differences in the means of these random variables. So it's going to be equal to the mean of m, the mean of m, minus the mean of w, minus the mean of w. Well we know both of these, this is gonna be 178 minus 170."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So this would be minus six, is to go to one standard deviation below the mean. Now let's think about the difference between the two, the random variable d. So let me think about this a little bit. The random variable d, the mean of d is going to be equal to the differences in the means of these random variables. So it's going to be equal to the mean of m, the mean of m, minus the mean of w, minus the mean of w. Well we know both of these, this is gonna be 178 minus 170. So let me write that down. This is equal to 178 centimeters minus 170 centimeters, which is going to be equal to, I'll do it in this color, this is going to be equal to eight centimeters. So this is eight right over here."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So it's going to be equal to the mean of m, the mean of m, minus the mean of w, minus the mean of w. Well we know both of these, this is gonna be 178 minus 170. So let me write that down. This is equal to 178 centimeters minus 170 centimeters, which is going to be equal to, I'll do it in this color, this is going to be equal to eight centimeters. So this is eight right over here. Now what about the standard deviation? Assuming these two random variables are independent, and they tell us that we are independently, randomly selecting a man and a woman, the height of the man shouldn't affect the height of the woman or vice versa. Assuming that these two are independent variables, if you take the sum or the difference of these, then the spread will increase."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So this is eight right over here. Now what about the standard deviation? Assuming these two random variables are independent, and they tell us that we are independently, randomly selecting a man and a woman, the height of the man shouldn't affect the height of the woman or vice versa. Assuming that these two are independent variables, if you take the sum or the difference of these, then the spread will increase. But you won't just add the standard deviations. What you would actually do is say, the variance of the difference is going to be the sum of these two variances. So let me write that down."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "Assuming that these two are independent variables, if you take the sum or the difference of these, then the spread will increase. But you won't just add the standard deviations. What you would actually do is say, the variance of the difference is going to be the sum of these two variances. So let me write that down. So I could write variance with VAR, or I could write it as the standard deviation squared. So let me write that. The standard deviation of D, of our difference, squared, which is the variance, is going to be equal to the variance of our variable M, plus the variance of our variable W. Now this might be a little bit counterintuitive."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So let me write that down. So I could write variance with VAR, or I could write it as the standard deviation squared. So let me write that. The standard deviation of D, of our difference, squared, which is the variance, is going to be equal to the variance of our variable M, plus the variance of our variable W. Now this might be a little bit counterintuitive. This might have made sense to you if this was plus right over here, but it doesn't matter if we are adding or subtracting, and these are truly independent variables, then regardless of whether we're adding or subtracting, you would add the variances. And so we can figure this out. This is going to be equal to, the standard deviation of variable M is eight."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "The standard deviation of D, of our difference, squared, which is the variance, is going to be equal to the variance of our variable M, plus the variance of our variable W. Now this might be a little bit counterintuitive. This might have made sense to you if this was plus right over here, but it doesn't matter if we are adding or subtracting, and these are truly independent variables, then regardless of whether we're adding or subtracting, you would add the variances. And so we can figure this out. This is going to be equal to, the standard deviation of variable M is eight. So eight squared is going to be 64. And then we have six squared. This right over here is six."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "This is going to be equal to, the standard deviation of variable M is eight. So eight squared is going to be 64. And then we have six squared. This right over here is six. Six squared is going to be 36. You add these two together, this is going to be equal to 100. And so the variance of this distribution right over here is going to be equal to 100."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "This right over here is six. Six squared is going to be 36. You add these two together, this is going to be equal to 100. And so the variance of this distribution right over here is going to be equal to 100. Well what's the standard deviation of that distribution? Well it's going to be equal to the square root of the variance. So the square root of 100, which is equal to 10."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so the variance of this distribution right over here is going to be equal to 100. Well what's the standard deviation of that distribution? Well it's going to be equal to the square root of the variance. So the square root of 100, which is equal to 10. So for example, one standard deviation above the mean is going to be 18. One standard deviation below the mean is going to be equal to negative two. And so now using this distribution, we can actually answer this question."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So the square root of 100, which is equal to 10. So for example, one standard deviation above the mean is going to be 18. One standard deviation below the mean is going to be equal to negative two. And so now using this distribution, we can actually answer this question. What is the probability that the woman is taller than the man? Well we can rewrite that question as saying, what is the probability that the random variable D is, what conditions would it be? Pause the video and think about it."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so now using this distribution, we can actually answer this question. What is the probability that the woman is taller than the man? Well we can rewrite that question as saying, what is the probability that the random variable D is, what conditions would it be? Pause the video and think about it. Well the situations where the woman is taller than the man, if the woman is taller than the man, then this is going to be a negative value. Then D is going to be less than zero. So what we really want to do is figure out the probability that D is less than zero."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "Pause the video and think about it. Well the situations where the woman is taller than the man, if the woman is taller than the man, then this is going to be a negative value. Then D is going to be less than zero. So what we really want to do is figure out the probability that D is less than zero. And so what we want to do, if we say zero is right over, if we said that zero is right over here on our distribution, so that is D is equal to zero, we want to figure out, well what is the area under the curve less than that? So we want to figure out this entire area. There's a couple of ways you could do this."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So what we really want to do is figure out the probability that D is less than zero. And so what we want to do, if we say zero is right over, if we said that zero is right over here on our distribution, so that is D is equal to zero, we want to figure out, well what is the area under the curve less than that? So we want to figure out this entire area. There's a couple of ways you could do this. You could figure out the z-score for D equaling zero, and that's pretty straightforward. You could just say this z is equal to zero minus our mean of eight divided by our standard deviation of 10. So it's negative eight over 10, which is equal to negative 8 1\u204410."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "There's a couple of ways you could do this. You could figure out the z-score for D equaling zero, and that's pretty straightforward. You could just say this z is equal to zero minus our mean of eight divided by our standard deviation of 10. So it's negative eight over 10, which is equal to negative 8 1\u204410. So you could look up a z-table and say, what is the total area under the curve below z is equal to negative 0.8? Another way you could do this is you could use a graphing calculator. I have a TI-84 here, where you have a normal cumulative distribution function."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So it's negative eight over 10, which is equal to negative 8 1\u204410. So you could look up a z-table and say, what is the total area under the curve below z is equal to negative 0.8? Another way you could do this is you could use a graphing calculator. I have a TI-84 here, where you have a normal cumulative distribution function. I'm gonna press second, vars, and that gets me to distribution. And so I have these various functions. I want normal cumulative distribution function."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "I have a TI-84 here, where you have a normal cumulative distribution function. I'm gonna press second, vars, and that gets me to distribution. And so I have these various functions. I want normal cumulative distribution function. So that is choice two. And then the lower bound. Well, I want to go to negative infinity."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "I want normal cumulative distribution function. So that is choice two. And then the lower bound. Well, I want to go to negative infinity. Well, calculators don't have a negative infinity button, but you could put in a very, very, very, very negative number that, for our purposes, is equivalent to negative infinity. So we could say negative one times 10 to the 99th power. And the way we do that is second, this two capital Es are saying, essentially times 10 to the, and I'll say 99th power."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, I want to go to negative infinity. Well, calculators don't have a negative infinity button, but you could put in a very, very, very, very negative number that, for our purposes, is equivalent to negative infinity. So we could say negative one times 10 to the 99th power. And the way we do that is second, this two capital Es are saying, essentially times 10 to the, and I'll say 99th power. So this is a very, very, very negative number. The upper bound here, we want to go, let me delete this. The upper bound is going to be zero."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "And the way we do that is second, this two capital Es are saying, essentially times 10 to the, and I'll say 99th power. So this is a very, very, very negative number. The upper bound here, we want to go, let me delete this. The upper bound is going to be zero. We're finding the area from negative infinity all the way to zero. The mean here, well, we've already figured that out. The mean is eight."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "The upper bound is going to be zero. We're finding the area from negative infinity all the way to zero. The mean here, well, we've already figured that out. The mean is eight. And then the standard deviation here, we figured this out too, this is equal to 10. And so when we pick this, we're gonna go back to the main screen, enter. So this is, we could have just typed this in directly on the main screen."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "The mean is eight. And then the standard deviation here, we figured this out too, this is equal to 10. And so when we pick this, we're gonna go back to the main screen, enter. So this is, we could have just typed this in directly on the main screen. This says, look, we're looking at a normal distribution. We want to find the cumulative area between two bounds. In this case, it's from negative infinity to zero."}, {"video_title": "Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So this is, we could have just typed this in directly on the main screen. This says, look, we're looking at a normal distribution. We want to find the cumulative area between two bounds. In this case, it's from negative infinity to zero. From negative infinity to zero, where the mean is eight and the standard deviation is 10. We press enter, and we get approximately 0.212, is approximately 0.212. Or you could say, what is the probability that the woman is taller than the man?"}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "A study from the year 2000 showed 132 cases of myopia in 400 randomly selected people. A separate study from 2015 showed 228 cases in 600 randomly selected people. So what we're going to do in this video is do a hypothesis test to see if we have evidence to suggest the researcher's suspicion that myopia is becoming more common over time. If at any point you are inspired, I encourage you to pause the video and try to work through things on your own, but here I go, I'm going to do it with you. So let's just start off by setting our null and alternative hypothesis. So remember, our null hypothesis, this would be the no news here. So that would be that contrary to their suspicions, that myopia is not becoming more common over time."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "If at any point you are inspired, I encourage you to pause the video and try to work through things on your own, but here I go, I'm going to do it with you. So let's just start off by setting our null and alternative hypothesis. So remember, our null hypothesis, this would be the no news here. So that would be that contrary to their suspicions, that myopia is not becoming more common over time. And so the way that we're measuring more common over time is we could look at the proportion of folks who have myopia in 2015 and compare that to the proportion in 2000. So our null hypothesis is that there's no difference, is that the true proportion of folks who have myopia in 2015 is equal to the proportion of folks who have myopia in 2000. And then our alternative hypothesis, remember, they suspect that it's becoming more common over time."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "So that would be that contrary to their suspicions, that myopia is not becoming more common over time. And so the way that we're measuring more common over time is we could look at the proportion of folks who have myopia in 2015 and compare that to the proportion in 2000. So our null hypothesis is that there's no difference, is that the true proportion of folks who have myopia in 2015 is equal to the proportion of folks who have myopia in 2000. And then our alternative hypothesis, remember, they suspect that it's becoming more common over time. So that would be a situation where our true proportion in 2015 is greater than the true proportion in 2000. In this scenario, myopia would be becoming more common over time because 2015 happens after 2000. So before we even go about testing our null hypothesis, seeing if we can reject it or not, which would suggest our alternative, you have to look at your conditions for inference."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "And then our alternative hypothesis, remember, they suspect that it's becoming more common over time. So that would be a situation where our true proportion in 2015 is greater than the true proportion in 2000. In this scenario, myopia would be becoming more common over time because 2015 happens after 2000. So before we even go about testing our null hypothesis, seeing if we can reject it or not, which would suggest our alternative, you have to look at your conditions for inference. And we've done this many times before. You have your random condition, and it looks like we meet that because in both of the samples, we have 400 randomly selected people, randomly selected people, so that looks good. Then you have your normal condition."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "So before we even go about testing our null hypothesis, seeing if we can reject it or not, which would suggest our alternative, you have to look at your conditions for inference. And we've done this many times before. You have your random condition, and it looks like we meet that because in both of the samples, we have 400 randomly selected people, randomly selected people, so that looks good. Then you have your normal condition. And to meet your normal condition, your number of successes and failures in each of the samples have to be at least 10. And we see that that is the case. We have 132 successes, so to speak, not that it's a success for someone to have myopia, but the way this is being constructed, that would be a success."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "Then you have your normal condition. And to meet your normal condition, your number of successes and failures in each of the samples have to be at least 10. And we see that that is the case. We have 132 successes, so to speak, not that it's a success for someone to have myopia, but the way this is being constructed, that would be a success. And then 400 minus 132 failures. In each case, either of those numbers would be greater than 10. And same thing for the sample from 2015."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "We have 132 successes, so to speak, not that it's a success for someone to have myopia, but the way this is being constructed, that would be a success. And then 400 minus 132 failures. In each case, either of those numbers would be greater than 10. And same thing for the sample from 2015. So we're meeting both of those. And then the last condition that we always talk about is the independence condition. And two ways to get there."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "And same thing for the sample from 2015. So we're meeting both of those. And then the last condition that we always talk about is the independence condition. And two ways to get there. Either you are sampling with replacement, or you feel good that your sample size is no more than 10% of the population. And I think it is safe to say that even this larger sample of 600, that there's more than 6,000 people out there. And so I think it's reasonable to say that we're meeting that independence condition, even though they're not making it explicit here."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "And two ways to get there. Either you are sampling with replacement, or you feel good that your sample size is no more than 10% of the population. And I think it is safe to say that even this larger sample of 600, that there's more than 6,000 people out there. And so I think it's reasonable to say that we're meeting that independence condition, even though they're not making it explicit here. But it's good to always think about this. Now the next thing you wanna do in a hypothesis test is set your significance level, your alpha. And I'll set my significance level to 0.05."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "And so I think it's reasonable to say that we're meeting that independence condition, even though they're not making it explicit here. But it's good to always think about this. Now the next thing you wanna do in a hypothesis test is set your significance level, your alpha. And I'll set my significance level to 0.05. So we're now going to assume the null hypothesis and say, well, what is the probability of getting a difference between 2015 and 2000 that is at least as large as the one that we got? And if that probability is less than our significance level, then we would reject our null hypothesis, and that would suggest the alternative. If that probability is greater than our significance level, then we fail to reject the null hypothesis, and we fail to have evidence for the researcher's suspicion."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "And I'll set my significance level to 0.05. So we're now going to assume the null hypothesis and say, well, what is the probability of getting a difference between 2015 and 2000 that is at least as large as the one that we got? And if that probability is less than our significance level, then we would reject our null hypothesis, and that would suggest the alternative. If that probability is greater than our significance level, then we fail to reject the null hypothesis, and we fail to have evidence for the researcher's suspicion. So let's move ahead with that. So what we wanna do is let's come up with a z value or a z score. So our z is going to be equal to the sample proportion in 2015 minus our sample proportion in 2000, all of that over the standard deviation of the sampling distribution of the difference between the sample proportions in 2015 and 2000."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "If that probability is greater than our significance level, then we fail to reject the null hypothesis, and we fail to have evidence for the researcher's suspicion. So let's move ahead with that. So what we wanna do is let's come up with a z value or a z score. So our z is going to be equal to the sample proportion in 2015 minus our sample proportion in 2000, all of that over the standard deviation of the sampling distribution of the difference between the sample proportions in 2015 and 2000. Now, this is going to be, and I will say approximately equal to, we can calculate this numerator exactly, but this denominator we are going to estimate. So this numerator is going to be, let's see, in 2015, I'll use some different colors, 2015, we have 228 cases out of 600. So it's 228 out of 600."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "So our z is going to be equal to the sample proportion in 2015 minus our sample proportion in 2000, all of that over the standard deviation of the sampling distribution of the difference between the sample proportions in 2015 and 2000. Now, this is going to be, and I will say approximately equal to, we can calculate this numerator exactly, but this denominator we are going to estimate. So this numerator is going to be, let's see, in 2015, I'll use some different colors, 2015, we have 228 cases out of 600. So it's 228 out of 600. And then in 2000, we have 132 cases out of 400. So minus 132 over 400. And then all of that over the square root."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "So it's 228 out of 600. And then in 2000, we have 132 cases out of 400. So minus 132 over 400. And then all of that over the square root. And what we use in the denominator here under the radical sign is we use the combined proportion. And we could write that as P hat sub C. And the reason why we use the combined proportion, we've talked about this in previous videos, is remember, when we do a hypothesis test, we assume that our null hypothesis is true. And if our null hypothesis is true, there's no difference between the proportions in 2015 and 2000."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "And then all of that over the square root. And what we use in the denominator here under the radical sign is we use the combined proportion. And we could write that as P hat sub C. And the reason why we use the combined proportion, we've talked about this in previous videos, is remember, when we do a hypothesis test, we assume that our null hypothesis is true. And if our null hypothesis is true, there's no difference between the proportions in 2015 and 2000. And so to get a better estimate of the true proportion, well, we should just add up our samples. So our sample size would be 600 plus 400. 600 plus 400."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "And if our null hypothesis is true, there's no difference between the proportions in 2015 and 2000. And so to get a better estimate of the true proportion, well, we should just add up our samples. So our sample size would be 600 plus 400. 600 plus 400. And the number of cases of myopia would be 228 plus 132. Plus 132, which would get us to, what is this? 360 over 1000, which is equal to 0.36."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "600 plus 400. And the number of cases of myopia would be 228 plus 132. Plus 132, which would get us to, what is this? 360 over 1000, which is equal to 0.36. And there, and we can use that inside the expression when we're trying to estimate our standard deviation of this sampling distribution. So this is going to be 0.36 times one minus 0.36, which would be 0.64, over the sample size in 2015, which is 600, plus 0.36 times 0.64, over the sample size in 2000, which is equal to 400. And let's see, before I even get my calculator out, I think I can simplify this a little bit."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "360 over 1000, which is equal to 0.36. And there, and we can use that inside the expression when we're trying to estimate our standard deviation of this sampling distribution. So this is going to be 0.36 times one minus 0.36, which would be 0.64, over the sample size in 2015, which is 600, plus 0.36 times 0.64, over the sample size in 2000, which is equal to 400. And let's see, before I even get my calculator out, I think I can simplify this a little bit. 228 over 600, 228 divided by six is going to be equal to 38. So this would be 0.38. Let's see, 132 divided by four would be 33."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "And let's see, before I even get my calculator out, I think I can simplify this a little bit. 228 over 600, 228 divided by six is going to be equal to 38. So this would be 0.38. Let's see, 132 divided by four would be 33. So this would be 0.33. And so our entire numerator is going to be 0.05. So 0.05."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "Let's see, 132 divided by four would be 33. So this would be 0.33. And so our entire numerator is going to be 0.05. So 0.05. And so now I could put this into my calculator and I will get 0.05 divided by the square root of, let's see, I'm gonna have 0.36 times 0.64 divided by 600 plus 0.36 times 0.64 divided by 400 is going to get me approximately 1.61. So this is going to be approximately 1.61. And so one way to think about it is the difference that we got between our sample proportions between 2015 and 2000 of 0.05, that that is 1.61 standard deviations above our mean of our sampling distribution if we assume that the null hypothesis is true."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "So 0.05. And so now I could put this into my calculator and I will get 0.05 divided by the square root of, let's see, I'm gonna have 0.36 times 0.64 divided by 600 plus 0.36 times 0.64 divided by 400 is going to get me approximately 1.61. So this is going to be approximately 1.61. And so one way to think about it is the difference that we got between our sample proportions between 2015 and 2000 of 0.05, that that is 1.61 standard deviations above our mean of our sampling distribution if we assume that the null hypothesis is true. And so from this, we can calculate our p-value. Remember, our p-value, our p-value is equal to the probability that our z-score is at least that big, is greater than or equal to 1.61. And one way to think about it, if we look at the sampling distribution, or really we could just look at any normal distribution now since we have normalized for z."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "And so one way to think about it is the difference that we got between our sample proportions between 2015 and 2000 of 0.05, that that is 1.61 standard deviations above our mean of our sampling distribution if we assume that the null hypothesis is true. And so from this, we can calculate our p-value. Remember, our p-value, our p-value is equal to the probability that our z-score is at least that big, is greater than or equal to 1.61. And one way to think about it, if we look at the sampling distribution, or really we could just look at any normal distribution now since we have normalized for z. So we're looking at 1.61 standard deviations above the mean. So z is equal to 1.61. So we're thinking about this area right over here."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "And one way to think about it, if we look at the sampling distribution, or really we could just look at any normal distribution now since we have normalized for z. So we're looking at 1.61 standard deviations above the mean. So z is equal to 1.61. So we're thinking about this area right over here. That would be our p-value. And to help us with that, we can get out a z-table. And we see this z-table gives us the cumulative area up to some z-score."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "So we're thinking about this area right over here. That would be our p-value. And to help us with that, we can get out a z-table. And we see this z-table gives us the cumulative area up to some z-score. And so we would just have to, whatever this gives us, we would just have to do one minus that. So if we go to 1.61, we get.9463. So it'd be one minus.9463 is equal to one minus 0.9463, which is equal to, let's see, it's 0.0537, 0.0537."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "And we see this z-table gives us the cumulative area up to some z-score. And so we would just have to, whatever this gives us, we would just have to do one minus that. So if we go to 1.61, we get.9463. So it'd be one minus.9463 is equal to one minus 0.9463, which is equal to, let's see, it's 0.0537, 0.0537. And notice, this p-value is ever so slightly higher than our significance level. But this is why we wanna set our significance level ahead of time. We don't wanna get tempted to say, oh, I'm so close, let me just raise my significance level a little bit more so that I can reject my null hypothesis and then I can have something that I can tell my friends about."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "So it'd be one minus.9463 is equal to one minus 0.9463, which is equal to, let's see, it's 0.0537, 0.0537. And notice, this p-value is ever so slightly higher than our significance level. But this is why we wanna set our significance level ahead of time. We don't wanna get tempted to say, oh, I'm so close, let me just raise my significance level a little bit more so that I can reject my null hypothesis and then I can have something that I can tell my friends about. No, that would not be good science. That would not be good statistics. We have to be disciplined."}, {"video_title": "Hypothesis test for difference in proportions example AP Statistics Khan Academy.mp3", "Sentence": "We don't wanna get tempted to say, oh, I'm so close, let me just raise my significance level a little bit more so that I can reject my null hypothesis and then I can have something that I can tell my friends about. No, that would not be good science. That would not be good statistics. We have to be disciplined. So here, because our p-value, our p-value is greater than our significance level, even though it's by a very small amount, we fail to reject our null hypothesis. And another way to think about it in terms of the context of the question, we can say that there is not enough evidence to suggest that myopia becoming more common over time. Myopia becoming more common over time."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "So let me draw it. I'll assume it's a quarter or something. Let's see. So this is a quarter. Let me draw my best attempt at a profile of George Washington. Well, that'll do. So it's a fair coin, and we're going to flip it a bunch of times and figure out the different probabilities."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "So this is a quarter. Let me draw my best attempt at a profile of George Washington. Well, that'll do. So it's a fair coin, and we're going to flip it a bunch of times and figure out the different probabilities. So let's start with a straightforward one. Let's just flip it once. So with one flip of the coin, what's the probability of getting heads?"}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "So it's a fair coin, and we're going to flip it a bunch of times and figure out the different probabilities. So let's start with a straightforward one. Let's just flip it once. So with one flip of the coin, what's the probability of getting heads? Well, there's two equally likely possibilities, and the one with heads is one of those two equally likely possibilities, so there's a 1 half chance. Same thing if we were to ask what is the probability of getting tails. There are two equally likely possibilities, and one of those gives us tails, so 1 half."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "So with one flip of the coin, what's the probability of getting heads? Well, there's two equally likely possibilities, and the one with heads is one of those two equally likely possibilities, so there's a 1 half chance. Same thing if we were to ask what is the probability of getting tails. There are two equally likely possibilities, and one of those gives us tails, so 1 half. And this is one thing to realize. If you take the probabilities of heads plus the probabilities of tails, you get 1 half plus 1 half, which is 1. And this is generally 2."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "There are two equally likely possibilities, and one of those gives us tails, so 1 half. And this is one thing to realize. If you take the probabilities of heads plus the probabilities of tails, you get 1 half plus 1 half, which is 1. And this is generally 2. The sum of the probabilities of all of the possible events should be equal to 1, and that makes sense because you're adding up all of these fractions, and the numerator will then add up to all of the possible events. The denominator is always all of the possible events, so you have all of the possible events over all of the possible events when you add all of these things up. Now let's take it up a notch."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "And this is generally 2. The sum of the probabilities of all of the possible events should be equal to 1, and that makes sense because you're adding up all of these fractions, and the numerator will then add up to all of the possible events. The denominator is always all of the possible events, so you have all of the possible events over all of the possible events when you add all of these things up. Now let's take it up a notch. Let's figure out the probability of... I'm going to take this coin, and I'm going to flip it twice. The probability of getting heads and then getting another heads."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "Now let's take it up a notch. Let's figure out the probability of... I'm going to take this coin, and I'm going to flip it twice. The probability of getting heads and then getting another heads. The probability of getting a head and then another head. So there's two ways to think about it. One way is to just think about all of the different possibilities."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "The probability of getting heads and then getting another heads. The probability of getting a head and then another head. So there's two ways to think about it. One way is to just think about all of the different possibilities. I could get a head on the first flip and a head on the second flip, head on the first flip, tail on the second flip. I could get tails on the first flip, heads on the second flip, or I could get tails on both flips. So there's four distinct, equally likely possibilities."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "One way is to just think about all of the different possibilities. I could get a head on the first flip and a head on the second flip, head on the first flip, tail on the second flip. I could get tails on the first flip, heads on the second flip, or I could get tails on both flips. So there's four distinct, equally likely possibilities. Four distinct, equally likely outcomes here. One way to think about it is on the first flip, I have two possibilities. On the second flip, I have another two possibilities."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "So there's four distinct, equally likely possibilities. Four distinct, equally likely outcomes here. One way to think about it is on the first flip, I have two possibilities. On the second flip, I have another two possibilities. I could have heads or tails, heads or tails. So I have four possibilities. For each of these possibilities, for each of these two, I have two possibilities here."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "On the second flip, I have another two possibilities. I could have heads or tails, heads or tails. So I have four possibilities. For each of these possibilities, for each of these two, I have two possibilities here. So either way, I have four equally likely possibilities. How many of those meet our constraints? Well, we have it right over here."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "For each of these possibilities, for each of these two, I have two possibilities here. So either way, I have four equally likely possibilities. How many of those meet our constraints? Well, we have it right over here. This one right over here, having two heads meets our constraints. So this is, and there's only one of those possibilities. I've only circled one of the four scenarios."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "Well, we have it right over here. This one right over here, having two heads meets our constraints. So this is, and there's only one of those possibilities. I've only circled one of the four scenarios. So there's a 1 4th chance of that happening. Another way you could think about this, and this is because these are independent events, and this is a very important idea to understand in probability, and we'll also study scenarios that are not independent, but these are independent events. What happens in the first flip in no way affects what happens in the second flip."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "I've only circled one of the four scenarios. So there's a 1 4th chance of that happening. Another way you could think about this, and this is because these are independent events, and this is a very important idea to understand in probability, and we'll also study scenarios that are not independent, but these are independent events. What happens in the first flip in no way affects what happens in the second flip. This is actually one thing that many people don't realize. There's something called the gambler's fallacy, where someone thinks if I got a bunch of heads in a row, then all of a sudden it becomes more likely on the next flip to get a tail. That is not the case."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "What happens in the first flip in no way affects what happens in the second flip. This is actually one thing that many people don't realize. There's something called the gambler's fallacy, where someone thinks if I got a bunch of heads in a row, then all of a sudden it becomes more likely on the next flip to get a tail. That is not the case. Every flip is an independent event. What happened in the past in these flips does not affect the probabilities going forward. So the probability of getting heads on the first flip in no way, or the fact that you got heads on the first flip, in no way affects that you got a heads on the second flip."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "That is not the case. Every flip is an independent event. What happened in the past in these flips does not affect the probabilities going forward. So the probability of getting heads on the first flip in no way, or the fact that you got heads on the first flip, in no way affects that you got a heads on the second flip. So if you can make that assumption, you can say that the probability of getting heads and heads, or heads and then heads is going to be the same thing as the probability of getting heads on the first flip times the probability of getting heads on the second flip. And we know the probability of getting heads on the first flip is 1 half, and the probability of getting heads on the second flip is 1 half. So we have 1 half times 1 half which is equal to 1."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "So the probability of getting heads on the first flip in no way, or the fact that you got heads on the first flip, in no way affects that you got a heads on the second flip. So if you can make that assumption, you can say that the probability of getting heads and heads, or heads and then heads is going to be the same thing as the probability of getting heads on the first flip times the probability of getting heads on the second flip. And we know the probability of getting heads on the first flip is 1 half, and the probability of getting heads on the second flip is 1 half. So we have 1 half times 1 half which is equal to 1. 1 4th, which is exactly what we got when we tried out all of the different scenarios, all of the equally likely possibilities. Let's take it up another notch. Let's figure out the probability, and we've kind of been ignoring tails, so let's pay some attention to tails."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "So we have 1 half times 1 half which is equal to 1. 1 4th, which is exactly what we got when we tried out all of the different scenarios, all of the equally likely possibilities. Let's take it up another notch. Let's figure out the probability, and we've kind of been ignoring tails, so let's pay some attention to tails. The probability of getting tails and then heads and then tails. So this exact series of events. So I'm not saying in any order, two tails and a head."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "Let's figure out the probability, and we've kind of been ignoring tails, so let's pay some attention to tails. The probability of getting tails and then heads and then tails. So this exact series of events. So I'm not saying in any order, two tails and a head. I'm saying this exact order. The first flip is a tails, second flip is a heads, and then third flip is a tail. So once again, these are all independent events."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "So I'm not saying in any order, two tails and a head. I'm saying this exact order. The first flip is a tails, second flip is a heads, and then third flip is a tail. So once again, these are all independent events. The fact that I get tails on the first flip in no way affects the probability of getting heads on the second flip and that in no way affects the probability of getting a tails on the third flip. So because these are independent events, we can say this is the same thing as the probability of getting tails on the first flip, times the probability of getting heads on the second flip, times the probability of getting tails on the third flip. And we know these are all independent events."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "So once again, these are all independent events. The fact that I get tails on the first flip in no way affects the probability of getting heads on the second flip and that in no way affects the probability of getting a tails on the third flip. So because these are independent events, we can say this is the same thing as the probability of getting tails on the first flip, times the probability of getting heads on the second flip, times the probability of getting tails on the third flip. And we know these are all independent events. So this right over here is 1 half times 1 half times 1 half. 1 half times 1 half is 1 fourth. 1 fourth times 1 half is equal to 1 eighth."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "And we know these are all independent events. So this right over here is 1 half times 1 half times 1 half. 1 half times 1 half is 1 fourth. 1 fourth times 1 half is equal to 1 eighth. So this is equal to 1 eighth. And we can verify it. Let's try it all of the different scenarios again."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "1 fourth times 1 half is equal to 1 eighth. So this is equal to 1 eighth. And we can verify it. Let's try it all of the different scenarios again. So you could get heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "Let's try it all of the different scenarios again. So you could get heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails. You can get tails, heads, heads. This is a little tricky sometimes."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "You could get heads, tails, heads. You could get heads, tails, tails. You can get tails, heads, heads. This is a little tricky sometimes. You want to make sure you're being exhaustive in all of the different possibilities here. You could get tails, heads, tails. You could get tails, tails, heads."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "This is a little tricky sometimes. You want to make sure you're being exhaustive in all of the different possibilities here. You could get tails, heads, tails. You could get tails, tails, heads. Or you could get tails, tails, tails. And what we see here is that we got exactly 8 equally likely possibilities. We have 8 equally likely possibilities."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "You could get tails, tails, heads. Or you could get tails, tails, tails. And what we see here is that we got exactly 8 equally likely possibilities. We have 8 equally likely possibilities. And the tail, heads, tails is exactly one of them. It is this possibility right over here. So it is one of 8 equally likely possibilities."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "So it tells us a large bakery makes thousands of cupcakes daily in two shifts, shift A and shift B. Suppose that on average, cupcakes from shift A weigh 130 grams with a standard deviation of four grams. For shift B, the mean and standard deviation are 125 grams and three grams respectively. Assume independence between shifts. Every day, the bakery takes a simple random sample of 40 cupcakes from each shift. They calculate the mean weight for each sample, then look at the difference A minus B between the sample means. Find the probability that the mean weights from the samples are more than six grams apart from each other."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "Assume independence between shifts. Every day, the bakery takes a simple random sample of 40 cupcakes from each shift. They calculate the mean weight for each sample, then look at the difference A minus B between the sample means. Find the probability that the mean weights from the samples are more than six grams apart from each other. So I'm actually not gonna tell you immediately to pause this video and try to work through this on your own. First, I'm gonna think about how we could break this down. And then I'll ask you to pause and try to tackle each of those parts by itself."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "Find the probability that the mean weights from the samples are more than six grams apart from each other. So I'm actually not gonna tell you immediately to pause this video and try to work through this on your own. First, I'm gonna think about how we could break this down. And then I'll ask you to pause and try to tackle each of those parts by itself. So in order to tackle this eventual question, we're going to have to think about the mean of the sampling distribution for the difference in sample means. So sample mean from group A minus sample mean for group B. We're gonna have to think about the standard deviation of the sampling distribution for the difference in sample means."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "And then I'll ask you to pause and try to tackle each of those parts by itself. So in order to tackle this eventual question, we're going to have to think about the mean of the sampling distribution for the difference in sample means. So sample mean from group A minus sample mean for group B. We're gonna have to think about the standard deviation of the sampling distribution for the difference in sample means. And we're going to have to think about is this distribution normal? If we're able to figure out these three things, then we just have to figure out, well, how many standard deviations away from the mean is this? And we could use your standard Z table to figure out the probability."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "We're gonna have to think about the standard deviation of the sampling distribution for the difference in sample means. And we're going to have to think about is this distribution normal? If we're able to figure out these three things, then we just have to figure out, well, how many standard deviations away from the mean is this? And we could use your standard Z table to figure out the probability. So now I encourage you to pause this video and try to tackle this first part. What is the mean of the sampling distribution for the difference in sample means? All right, now let's work through this together."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "And we could use your standard Z table to figure out the probability. So now I encourage you to pause this video and try to tackle this first part. What is the mean of the sampling distribution for the difference in sample means? All right, now let's work through this together. So the mean of the sampling distribution for the difference in sample means, and we have seen this before, this is going to be equal to the difference between the means of the sampling distribution for each of the sample means. So that mean minus this mean. And we also know that the mean of the sampling distribution for each of these sample means, that's just going to be the mean of the population that we are sampling from."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "All right, now let's work through this together. So the mean of the sampling distribution for the difference in sample means, and we have seen this before, this is going to be equal to the difference between the means of the sampling distribution for each of the sample means. So that mean minus this mean. And we also know that the mean of the sampling distribution for each of these sample means, that's just going to be the mean of the population that we are sampling from. So this mean right over here is just going to be the mean, the population mean for shift A, which is going to be 130 grams. I'll just write that there. And then the mean of the sampling distribution for the sample means from shift B, we can see that that's just going to be the population mean for shift B, which is right over here."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "And we also know that the mean of the sampling distribution for each of these sample means, that's just going to be the mean of the population that we are sampling from. So this mean right over here is just going to be the mean, the population mean for shift A, which is going to be 130 grams. I'll just write that there. And then the mean of the sampling distribution for the sample means from shift B, we can see that that's just going to be the population mean for shift B, which is right over here. So minus 125 grams. And of course, this is just going to be equal to five grams. So we have answered the first part."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "And then the mean of the sampling distribution for the sample means from shift B, we can see that that's just going to be the population mean for shift B, which is right over here. So minus 125 grams. And of course, this is just going to be equal to five grams. So we have answered the first part. We know the mean of the sampling distribution of the difference in sample means. Now, what about the standard deviation? So for that, let's think actually about variances because the math's a little bit easier with variances."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "So we have answered the first part. We know the mean of the sampling distribution of the difference in sample means. Now, what about the standard deviation? So for that, let's think actually about variances because the math's a little bit easier with variances. And then from that, we can derive standard deviations. So we know that the variance of the sampling distribution for the difference in sample means, assuming that your two samples are independent and you're sampling with replacement, if you're sampling with replacement, it's actually going to be the sum of the variances of the sampling distribution for each of the sample means. So it's going to be that plus this right over here."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "So for that, let's think actually about variances because the math's a little bit easier with variances. And then from that, we can derive standard deviations. So we know that the variance of the sampling distribution for the difference in sample means, assuming that your two samples are independent and you're sampling with replacement, if you're sampling with replacement, it's actually going to be the sum of the variances of the sampling distribution for each of the sample means. So it's going to be that plus this right over here. Now, you might be saying, wait, we're not sampling with replacement. Well, we also know that if each of the sample sizes are less than 10% of the population, then the difference is negligible. And so we could still use this formula."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "So it's going to be that plus this right over here. Now, you might be saying, wait, we're not sampling with replacement. Well, we also know that if each of the sample sizes are less than 10% of the population, then the difference is negligible. And so we could still use this formula. And so you could see that the simple random sample here is 40 from each shift. And they say that a large bakery makes thousands of cupcakes daily in two shifts. So even if it was a thousand, 10% of that would be a hundred."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "And so we could still use this formula. And so you could see that the simple random sample here is 40 from each shift. And they say that a large bakery makes thousands of cupcakes daily in two shifts. So even if it was a thousand, 10% of that would be a hundred. This is less than 10%. So we meet that condition. So we can use the same formula that you would use if you were sampling with replacement."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "So even if it was a thousand, 10% of that would be a hundred. This is less than 10%. So we meet that condition. So we can use the same formula that you would use if you were sampling with replacement. So this first variance right over here of the sampling distribution for the sample means from shift A, this is going to be equal to the variance of shift A, the population variance of shift A divided by your sample size. And then this over here, it's gonna be the same thing for shift B. It's going to be the variance of shift B divided by your sample size."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "So we can use the same formula that you would use if you were sampling with replacement. So this first variance right over here of the sampling distribution for the sample means from shift A, this is going to be equal to the variance of shift A, the population variance of shift A divided by your sample size. And then this over here, it's gonna be the same thing for shift B. It's going to be the variance of shift B divided by your sample size. And so this is going to be equal to what? Well, the variance from shift A is going to be the square of the standard deviation from shift A. The standard deviation's right over there."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "It's going to be the variance of shift B divided by your sample size. And so this is going to be equal to what? Well, the variance from shift A is going to be the square of the standard deviation from shift A. The standard deviation's right over there. And so that's going to be 16. We could write grams squared if we wanna keep the units there. And then we're going to divide by the sample size."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "The standard deviation's right over there. And so that's going to be 16. We could write grams squared if we wanna keep the units there. And then we're going to divide by the sample size. We know that the sample size in each case, 40 cupcakes at a time for each sample. And then for shift B, we know that the standard deviation, the population standard deviation for shift B is three grams. You square that and you get nine grams squared."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "And then we're going to divide by the sample size. We know that the sample size in each case, 40 cupcakes at a time for each sample. And then for shift B, we know that the standard deviation, the population standard deviation for shift B is three grams. You square that and you get nine grams squared. A gram squared is kind of an interesting idea, but that's what the units are working out to be right now. And our sample size is still equal to 40. And so this is going to be equal to, let's see, 16 plus nine is 25, common denominator of 40."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "You square that and you get nine grams squared. A gram squared is kind of an interesting idea, but that's what the units are working out to be right now. And our sample size is still equal to 40. And so this is going to be equal to, let's see, 16 plus nine is 25, common denominator of 40. So it's 25 over 40, which is the same thing as 5 8ths, 5 8ths of a gram squared, which is a little bit strange units, but this now tells us what the standard deviation is going to be because it's just gonna be the square root of all of this business. So the standard deviation of the sampling distribution for the difference in sample means over here is going to be the square root of 5 8ths. And now of course, the units are back to grams, which makes sense."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "And so this is going to be equal to, let's see, 16 plus nine is 25, common denominator of 40. So it's 25 over 40, which is the same thing as 5 8ths, 5 8ths of a gram squared, which is a little bit strange units, but this now tells us what the standard deviation is going to be because it's just gonna be the square root of all of this business. So the standard deviation of the sampling distribution for the difference in sample means over here is going to be the square root of 5 8ths. And now of course, the units are back to grams, which makes sense. And this is approximately going to be equal to, get my calculator out, five divided by eight equals, and then we take the square root of that. And it's going to be approximately 0.79, 0.79. So the next question, before we try to figure out the probability is, is are we dealing with a normal distribution here when we think about the sampling distribution for the difference in sample means?"}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "And now of course, the units are back to grams, which makes sense. And this is approximately going to be equal to, get my calculator out, five divided by eight equals, and then we take the square root of that. And it's going to be approximately 0.79, 0.79. So the next question, before we try to figure out the probability is, is are we dealing with a normal distribution here when we think about the sampling distribution for the difference in sample means? And so I encourage you to pause the video again and think about that. So there's two ways that we can assume that the sampling distribution for the difference in sampling means is normal. If the original populations that each of the sample means are being calculated from are normal, then that means that the sampling distribution for each of the sample means is going to be normal."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "So the next question, before we try to figure out the probability is, is are we dealing with a normal distribution here when we think about the sampling distribution for the difference in sample means? And so I encourage you to pause the video again and think about that. So there's two ways that we can assume that the sampling distribution for the difference in sampling means is normal. If the original populations that each of the sample means are being calculated from are normal, then that means that the sampling distribution for each of the sample means is going to be normal. And that means that the difference of the sampling distributions are going to be normal. Now, we don't know for a fact that the weights of the cupcakes from each shift are normal distributions, but we also know that the sampling distribution of the sampling means can be modeled as being approximately normal if the two sample sizes are greater than or equal to 30. And we know that each of these samples are definitely greater than or equal to 30, they are 40."}, {"video_title": "Sampling distribution of the difference in sample means AP Statistics Khan Academy.mp3", "Sentence": "If the original populations that each of the sample means are being calculated from are normal, then that means that the sampling distribution for each of the sample means is going to be normal. And that means that the difference of the sampling distributions are going to be normal. Now, we don't know for a fact that the weights of the cupcakes from each shift are normal distributions, but we also know that the sampling distribution of the sampling means can be modeled as being approximately normal if the two sample sizes are greater than or equal to 30. And we know that each of these samples are definitely greater than or equal to 30, they are 40. So that tells us that the sampling distribution of the difference in sample means is also normal. So we've established the things that we need to then calculate the probability. So I encourage you, pause the video and see if you can use that information to calculate that probability."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "Each person snapped their fingers with their dominant hand for 10 seconds and their non-dominant hand for 10 seconds, where if you're right-handed, right hand would be your dominant hand. If you were left-handed, left hand would be your dominant hand. Each participant flipped a coin to determine which hand they would use first, because if you always used your dominant hand first, maybe you're tired by the time you're doing your non-dominant hand or there's something else. So here, it's random which one you use first. Here are the data for how many snaps they performed with each hand, the difference for each participant, and summary statistics. And this is actually real data from the Khan Academy content team. And so you see for each of the participants, for Jeff right over here, he was able to do 44 snaps in 10 seconds on his dominant hand, which is impressive, more than I think I could do."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "So here, it's random which one you use first. Here are the data for how many snaps they performed with each hand, the difference for each participant, and summary statistics. And this is actually real data from the Khan Academy content team. And so you see for each of the participants, for Jeff right over here, he was able to do 44 snaps in 10 seconds on his dominant hand, which is impressive, more than I think I could do. And he was even able to do 35 on his non-dominant hand. And so the difference here, the dominant hand minus the non-dominant was nine. And then they tabulated this data for all five members."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "And so you see for each of the participants, for Jeff right over here, he was able to do 44 snaps in 10 seconds on his dominant hand, which is impressive, more than I think I could do. And he was even able to do 35 on his non-dominant hand. And so the difference here, the dominant hand minus the non-dominant was nine. And then they tabulated this data for all five members. Now, they also calculated summary statistics for them. But this is the really interesting thing right over here. This is the difference between the dominant and the non-dominant hand."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "And then they tabulated this data for all five members. Now, they also calculated summary statistics for them. But this is the really interesting thing right over here. This is the difference between the dominant and the non-dominant hand. And so what they did here, the mean difference, what they did is they took this row right over here, and they calculated the mean, which they got to be 6.8. And then they calculated the standard deviation of these differences right over here, which they got to be approximately 1.64. And then we are asked, create and interpret a 95% confidence interval for mean difference in number of snaps for these participants."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "This is the difference between the dominant and the non-dominant hand. And so what they did here, the mean difference, what they did is they took this row right over here, and they calculated the mean, which they got to be 6.8. And then they calculated the standard deviation of these differences right over here, which they got to be approximately 1.64. And then we are asked, create and interpret a 95% confidence interval for mean difference in number of snaps for these participants. So pause this video, see if you can make some headway here. See if you can think about how to approach this. So what's interesting here is we're not trying to construct a confidence interval for just the mean number of snaps for the dominant hand or the mean number of snaps for the non-dominant hand."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "And then we are asked, create and interpret a 95% confidence interval for mean difference in number of snaps for these participants. So pause this video, see if you can make some headway here. See if you can think about how to approach this. So what's interesting here is we're not trying to construct a confidence interval for just the mean number of snaps for the dominant hand or the mean number of snaps for the non-dominant hand. We're constructing a 95% confidence interval for a mean difference. Now, you might say, wait, wait, wait. You know, I have two different samples here, and then this third sample is, or this third data is somehow constructed from these other two."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "So what's interesting here is we're not trying to construct a confidence interval for just the mean number of snaps for the dominant hand or the mean number of snaps for the non-dominant hand. We're constructing a 95% confidence interval for a mean difference. Now, you might say, wait, wait, wait. You know, I have two different samples here, and then this third sample is, or this third data is somehow constructed from these other two. But one way to think about it, this is matched pairs design. So in a matched pairs design, what you do is, for each participant, for each member in your sample, you will make them do the control and the treatment. So for example, you could do the control as how many they can do in the dominant hand in 10 seconds, and the treatment is how many they can do in the non-dominant hand."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "You know, I have two different samples here, and then this third sample is, or this third data is somehow constructed from these other two. But one way to think about it, this is matched pairs design. So in a matched pairs design, what you do is, for each participant, for each member in your sample, you will make them do the control and the treatment. So for example, you could do the control as how many they can do in the dominant hand in 10 seconds, and the treatment is how many they can do in the non-dominant hand. And in matched pairs design, you're really concerned about the difference. And so what you can really view this as is you just have one sample size of five for which you are calculating the difference for each member of that sample and the standard deviation across that entire sample. Now, before we calculate the confidence interval, let's just remind ourselves some of our conditions that we like to think about when we are constructing confidence intervals."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "So for example, you could do the control as how many they can do in the dominant hand in 10 seconds, and the treatment is how many they can do in the non-dominant hand. And in matched pairs design, you're really concerned about the difference. And so what you can really view this as is you just have one sample size of five for which you are calculating the difference for each member of that sample and the standard deviation across that entire sample. Now, before we calculate the confidence interval, let's just remind ourselves some of our conditions that we like to think about when we are constructing confidence intervals. The first condition we think about is whether our sample is random. Now, if we were trying to make some type of judgment about all human beings and their snapping ability, this would not be a random sample. These people all work at Khan Academy."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "Now, before we calculate the confidence interval, let's just remind ourselves some of our conditions that we like to think about when we are constructing confidence intervals. The first condition we think about is whether our sample is random. Now, if we were trying to make some type of judgment about all human beings and their snapping ability, this would not be a random sample. These people all work at Khan Academy. Maybe somehow in our interview process, we select for people who snap particularly well. But whatever inferences we make, we can say, hey, this is roughly true about this group of friends. Now, the next condition we wanna think about is the normal condition."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "These people all work at Khan Academy. Maybe somehow in our interview process, we select for people who snap particularly well. But whatever inferences we make, we can say, hey, this is roughly true about this group of friends. Now, the next condition we wanna think about is the normal condition. Now, there's a couple of ways to think about it. If we had sample size of 30 or larger, the central limit theorem says, okay, the distribution, the sampling distribution would be roughly normal, the sampling distribution of the sample means. But obviously, our sample size is much smaller than that."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "Now, the next condition we wanna think about is the normal condition. Now, there's a couple of ways to think about it. If we had sample size of 30 or larger, the central limit theorem says, okay, the distribution, the sampling distribution would be roughly normal, the sampling distribution of the sample means. But obviously, our sample size is much smaller than that. One way to think about it, we could just plot our data points and see whether they seem to be skewed in any way. And if we just do a little dot plot right over here, we could say, let's say make this zero, one, two, three, four, five, six, seven, eight, and nine. So we have one data point where the difference was nine, one data point where the difference is five, one data point where the difference is eight, one data point where the difference is six, and another data point where the difference is six."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "But obviously, our sample size is much smaller than that. One way to think about it, we could just plot our data points and see whether they seem to be skewed in any way. And if we just do a little dot plot right over here, we could say, let's say make this zero, one, two, three, four, five, six, seven, eight, and nine. So we have one data point where the difference was nine, one data point where the difference is five, one data point where the difference is eight, one data point where the difference is six, and another data point where the difference is six. And so this doesn't look massively skewed in any way. Our mean difference was right over here, it was about 6.8. It looks roughly symmetric."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "So we have one data point where the difference was nine, one data point where the difference is five, one data point where the difference is eight, one data point where the difference is six, and another data point where the difference is six. And so this doesn't look massively skewed in any way. Our mean difference was right over here, it was about 6.8. It looks roughly symmetric. So we can feel okay about this normal distribution. This isn't the best study that one could conduct. This is obviously a small sample size."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "It looks roughly symmetric. So we can feel okay about this normal distribution. This isn't the best study that one could conduct. This is obviously a small sample size. It's not random of the entire population. But maybe we could go with it. Also, when you think about biological processes, like how well someone snaps, which is a product of a lot of things happening in a human body, and it's the sum of many, many processes, those things also tend to have a roughly normal distribution but I won't go into too much depth there."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "This is obviously a small sample size. It's not random of the entire population. But maybe we could go with it. Also, when you think about biological processes, like how well someone snaps, which is a product of a lot of things happening in a human body, and it's the sum of many, many processes, those things also tend to have a roughly normal distribution but I won't go into too much depth there. But all of these things, once again, this isn't a super robust study, but this is a fun thing for friends to do if they have nothing else to do. All right. Now the third one is independence."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "Also, when you think about biological processes, like how well someone snaps, which is a product of a lot of things happening in a human body, and it's the sum of many, many processes, those things also tend to have a roughly normal distribution but I won't go into too much depth there. But all of these things, once again, this isn't a super robust study, but this is a fun thing for friends to do if they have nothing else to do. All right. Now the third one is independence. And this one actually we can feel pretty good about because Jeff's difference right over here really shouldn't impact David's difference or David's difference really shouldn't impact Kim's difference, especially if they're not observing each other. And let's just say for the sake of argument that they did it all independently in a closed room with a independent observer so they weren't trying to get competitive or something like that. But needless to say, this isn't super robust study, but we can still calculate a 95% confidence interval."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "Now the third one is independence. And this one actually we can feel pretty good about because Jeff's difference right over here really shouldn't impact David's difference or David's difference really shouldn't impact Kim's difference, especially if they're not observing each other. And let's just say for the sake of argument that they did it all independently in a closed room with a independent observer so they weren't trying to get competitive or something like that. But needless to say, this isn't super robust study, but we can still calculate a 95% confidence interval. So how do we do that? Well, we've done this so many times. Our confidence interval would be our sample mean."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "But needless to say, this isn't super robust study, but we can still calculate a 95% confidence interval. So how do we do that? Well, we've done this so many times. Our confidence interval would be our sample mean. So it would be the mean of our difference, the mean of our difference plus or minus. Now we don't know the population standard deviation, so we're going to use our sample standard deviation. And if you're using a sample standard deviation and this confidence interval is all about the mean, and so our critical value here is going to be based on a t-table, on a t-statistic, and then we're gonna multiply that times the sample standard deviation of the differences divided by the square root of our sample size, divided by the square root of five."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "Our confidence interval would be our sample mean. So it would be the mean of our difference, the mean of our difference plus or minus. Now we don't know the population standard deviation, so we're going to use our sample standard deviation. And if you're using a sample standard deviation and this confidence interval is all about the mean, and so our critical value here is going to be based on a t-table, on a t-statistic, and then we're gonna multiply that times the sample standard deviation of the differences divided by the square root of our sample size, divided by the square root of five. Now we know most of this data here, and let me just write it down over here. We know the mean, the sample mean right over here is 6.8, so it's going to be 6.8 plus or minus. And now what will be our critical value here?"}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "And if you're using a sample standard deviation and this confidence interval is all about the mean, and so our critical value here is going to be based on a t-table, on a t-statistic, and then we're gonna multiply that times the sample standard deviation of the differences divided by the square root of our sample size, divided by the square root of five. Now we know most of this data here, and let me just write it down over here. We know the mean, the sample mean right over here is 6.8, so it's going to be 6.8 plus or minus. And now what will be our critical value here? Well, we want to have a 95% confidence interval. And what's our degrees of freedom? Well, it's one less than our sample size, so our degrees of freedom right over here is equal to four."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "And now what will be our critical value here? Well, we want to have a 95% confidence interval. And what's our degrees of freedom? Well, it's one less than our sample size, so our degrees of freedom right over here is equal to four. And so we're ready to use a t-table. So this is a truncated t-table that I could fit on my screen here. And so there's a couple of ways to think about it."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "Well, it's one less than our sample size, so our degrees of freedom right over here is equal to four. And so we're ready to use a t-table. So this is a truncated t-table that I could fit on my screen here. And so there's a couple of ways to think about it. Here they actually give us the confidence level, and the reason why that corresponds to a tail probability of.025 is if you take the middle 95% of a distribution, you're going to have 2.5% on either end. That's going to be your tail probability. So that's all that's going on over there."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "And so there's a couple of ways to think about it. Here they actually give us the confidence level, and the reason why that corresponds to a tail probability of.025 is if you take the middle 95% of a distribution, you're going to have 2.5% on either end. That's going to be your tail probability. So that's all that's going on over there. So we're going to be in this column right over here. And which degree of freedom do we use, or degrees of freedom? Well, it's gonna be four degrees of freedom."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "So that's all that's going on over there. So we're going to be in this column right over here. And which degree of freedom do we use, or degrees of freedom? Well, it's gonna be four degrees of freedom. Our sample size is five, five minus one is four. So this is going to be our critical value, 2.776. So we have 2.776 as our critical value, and then times our sample standard deviation, well, the sample standard deviation for our difference is right over here is 1.64."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "Well, it's gonna be four degrees of freedom. Our sample size is five, five minus one is four. So this is going to be our critical value, 2.776. So we have 2.776 as our critical value, and then times our sample standard deviation, well, the sample standard deviation for our difference is right over here is 1.64. And then we're going to divide that by the square root of our sample size. So the square root of our sample size, we already wrote a five in there. Sometimes I just write an N there."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "So we have 2.776 as our critical value, and then times our sample standard deviation, well, the sample standard deviation for our difference is right over here is 1.64. And then we're going to divide that by the square root of our sample size. So the square root of our sample size, we already wrote a five in there. Sometimes I just write an N there. And so what is this going to be equal to? First, let's calculate just the margin of error right over here. So this is going to be 2.776 times 1.64 divided by the square root of five."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "Sometimes I just write an N there. And so what is this going to be equal to? First, let's calculate just the margin of error right over here. So this is going to be 2.776 times 1.64 divided by the square root of five. And we get a margin of error of approximately 2.036. So this is going to be 6.8 plus or minus 2.036. It's approximately equal to that, where this is our margin of error."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "So this is going to be 2.776 times 1.64 divided by the square root of five. And we get a margin of error of approximately 2.036. So this is going to be 6.8 plus or minus 2.036. It's approximately equal to that, where this is our margin of error. And if we actually wanted to write out the interval, we could just take 6.8 minus this and 6.8 plus that. So let's do that again with the calculator. So 6.8 minus 2.036 is equal to 4.764."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "It's approximately equal to that, where this is our margin of error. And if we actually wanted to write out the interval, we could just take 6.8 minus this and 6.8 plus that. So let's do that again with the calculator. So 6.8 minus 2.036 is equal to 4.764. So our confidence interval starts at 4.764 approximately. And it goes to, let's see, I could actually do this one in my head. If I add 2.036 to 6.8, that is going to be 8.836."}, {"video_title": "Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3", "Sentence": "So 6.8 minus 2.036 is equal to 4.764. So our confidence interval starts at 4.764 approximately. And it goes to, let's see, I could actually do this one in my head. If I add 2.036 to 6.8, that is going to be 8.836. Now, how would we interpret this confidence interval right over here? One way to interpret it is to say that we are 95% confident that this interval captures the true mean difference in snaps for these friends. We could also say that there appears to be a difference in the mean number of snaps, since zero is not captured in this interval."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And I'm using theirs because it's open source. It's actually quite a good book. The problems are, I think, good practice for us. So let's see, number three. You could go to their site, and I think you can download the book. Assume that the mean weight of one-year-old girls in the US is normally distributed with a mean of about 9.5 grams. That's got to be kilograms."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "So let's see, number three. You could go to their site, and I think you can download the book. Assume that the mean weight of one-year-old girls in the US is normally distributed with a mean of about 9.5 grams. That's got to be kilograms. I have a 10-month-old son, and he weighs about 20 pounds, which is about 9 kilograms. Not 9.5. 9.5 grams is nothing."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "That's got to be kilograms. I have a 10-month-old son, and he weighs about 20 pounds, which is about 9 kilograms. Not 9.5. 9.5 grams is nothing. This would be, you know, we're talking about like mice or something. This has got to be kilograms. But anyway, it's about 9.5 kilograms with a standard deviation of approximately 1.1 grams."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "9.5 grams is nothing. This would be, you know, we're talking about like mice or something. This has got to be kilograms. But anyway, it's about 9.5 kilograms with a standard deviation of approximately 1.1 grams. So the mean is equal to 9.5 kilograms, I'm assuming. And the standard deviation is equal to 1.1 grams. Without using a calculator."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "But anyway, it's about 9.5 kilograms with a standard deviation of approximately 1.1 grams. So the mean is equal to 9.5 kilograms, I'm assuming. And the standard deviation is equal to 1.1 grams. Without using a calculator. So that's an interesting clue. Estimate the percentage of one-year-old girls in the US that meet the following conditions. So when they say that without a calculator estimate, that's a big clue or a big giveaway that we're supposed to use the empirical rule."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Without using a calculator. So that's an interesting clue. Estimate the percentage of one-year-old girls in the US that meet the following conditions. So when they say that without a calculator estimate, that's a big clue or a big giveaway that we're supposed to use the empirical rule. The empirical rule. Sometimes called the 68-95-99.7 rule. And this is actually, if you remember, this is the name of the rule."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "So when they say that without a calculator estimate, that's a big clue or a big giveaway that we're supposed to use the empirical rule. The empirical rule. Sometimes called the 68-95-99.7 rule. And this is actually, if you remember, this is the name of the rule. You've essentially remembered the rule. What that tells us is if we have a normal distribution, I'll do a bit of a review here before we jump into this problem. If we have a normal distribution, let me draw a normal distribution."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And this is actually, if you remember, this is the name of the rule. You've essentially remembered the rule. What that tells us is if we have a normal distribution, I'll do a bit of a review here before we jump into this problem. If we have a normal distribution, let me draw a normal distribution. Say it looks like that. That's my normal distribution. I didn't draw it perfectly, but you get the idea."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "If we have a normal distribution, let me draw a normal distribution. Say it looks like that. That's my normal distribution. I didn't draw it perfectly, but you get the idea. It should be symmetrical. This is our mean right there. That's our mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "I didn't draw it perfectly, but you get the idea. It should be symmetrical. This is our mean right there. That's our mean. If we go one standard deviation above the mean and one standard deviation below the mean. So this is our mean plus one standard deviation. This is our mean minus one standard deviation."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "That's our mean. If we go one standard deviation above the mean and one standard deviation below the mean. So this is our mean plus one standard deviation. This is our mean minus one standard deviation. The probability of finding a result, if we're dealing with a perfect normal distribution, that's between one standard deviation below the mean and one standard deviation above the mean, that would be this area. And it would be, you could guess, 68%. 68% chance you're going to get something within one standard deviation of the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "This is our mean minus one standard deviation. The probability of finding a result, if we're dealing with a perfect normal distribution, that's between one standard deviation below the mean and one standard deviation above the mean, that would be this area. And it would be, you could guess, 68%. 68% chance you're going to get something within one standard deviation of the mean. Either a standard deviation below or above or anywhere in between. Now, if we're talking about two standard deviations around the mean, so if we go down another standard deviation, so we go down another standard deviation in that direction and another standard deviation above the mean, and we were to ask ourselves, what's the probability of finding something within those two or within that range, then it's, you could guess it, 95%. And that includes this middle area right here."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "68% chance you're going to get something within one standard deviation of the mean. Either a standard deviation below or above or anywhere in between. Now, if we're talking about two standard deviations around the mean, so if we go down another standard deviation, so we go down another standard deviation in that direction and another standard deviation above the mean, and we were to ask ourselves, what's the probability of finding something within those two or within that range, then it's, you could guess it, 95%. And that includes this middle area right here. So the 68% is a subset of that 95%. And I think you know where this is going. If we go three standard deviations below the mean and above the mean, the empirical rule, or the 68, 95, 99.7 rule tells us that there is a 99.7% chance of finding a result in a normal distribution that is within three standard deviations of the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And that includes this middle area right here. So the 68% is a subset of that 95%. And I think you know where this is going. If we go three standard deviations below the mean and above the mean, the empirical rule, or the 68, 95, 99.7 rule tells us that there is a 99.7% chance of finding a result in a normal distribution that is within three standard deviations of the mean. So above three standard deviations below the mean and below three standard deviations above the mean. That's what the empirical rule tells us. Now let's see if we can apply it to this problem."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "If we go three standard deviations below the mean and above the mean, the empirical rule, or the 68, 95, 99.7 rule tells us that there is a 99.7% chance of finding a result in a normal distribution that is within three standard deviations of the mean. So above three standard deviations below the mean and below three standard deviations above the mean. That's what the empirical rule tells us. Now let's see if we can apply it to this problem. So they gave us the mean and the standard deviation. Let me draw that out. Let me draw my axis first as best as I can."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Now let's see if we can apply it to this problem. So they gave us the mean and the standard deviation. Let me draw that out. Let me draw my axis first as best as I can. That's my axis. Let me draw my bell curve. That's about as good as a bell curve is you can expect a freehand drawer to do."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Let me draw my axis first as best as I can. That's my axis. Let me draw my bell curve. That's about as good as a bell curve is you can expect a freehand drawer to do. And the mean here is 9 point, and this should be symmetric. This height should be the same as that height there. I think you get the idea."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "That's about as good as a bell curve is you can expect a freehand drawer to do. And the mean here is 9 point, and this should be symmetric. This height should be the same as that height there. I think you get the idea. I'm not a computer. 9.5 is the mean. I won't write the units."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "I think you get the idea. I'm not a computer. 9.5 is the mean. I won't write the units. It's all in kilograms. One standard deviation above the mean. So one standard deviation above the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "I won't write the units. It's all in kilograms. One standard deviation above the mean. So one standard deviation above the mean. We should add 1.1 to that. They told us the standard deviation is 1.1. That's going to be 10.6."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "So one standard deviation above the mean. We should add 1.1 to that. They told us the standard deviation is 1.1. That's going to be 10.6. If we go, let me just draw a little dotted line there. One standard deviation below the mean. One standard deviation below the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "That's going to be 10.6. If we go, let me just draw a little dotted line there. One standard deviation below the mean. One standard deviation below the mean. We're going to subtract 1.1 from 9.5. And so that would be 8.4. If we go two standard deviations above the mean, we would add another standard deviation here."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "One standard deviation below the mean. We're going to subtract 1.1 from 9.5. And so that would be 8.4. If we go two standard deviations above the mean, we would add another standard deviation here. We went one standard deviation, two standard deviations. That would get us to 11.7. And if we were to go three standard deviations, we'd add 1.1 again."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "If we go two standard deviations above the mean, we would add another standard deviation here. We went one standard deviation, two standard deviations. That would get us to 11.7. And if we were to go three standard deviations, we'd add 1.1 again. That would get us to 12.8. Doing it on the other side. One standard deviation below the mean is 8.4."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And if we were to go three standard deviations, we'd add 1.1 again. That would get us to 12.8. Doing it on the other side. One standard deviation below the mean is 8.4. Two standard deviations below the mean. Subtract 1.1 again would be 7.3. And then three standard deviations below the mean, maybe right there, would be 6.2 kilograms."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "One standard deviation below the mean is 8.4. Two standard deviations below the mean. Subtract 1.1 again would be 7.3. And then three standard deviations below the mean, maybe right there, would be 6.2 kilograms. So that's our setup for the problem. So what's the probability that we would find a one-year-old girl in the US that weighs less than 8.4 kilograms? Or maybe I should say whose mass is less than 8.4 kilograms."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And then three standard deviations below the mean, maybe right there, would be 6.2 kilograms. So that's our setup for the problem. So what's the probability that we would find a one-year-old girl in the US that weighs less than 8.4 kilograms? Or maybe I should say whose mass is less than 8.4 kilograms. So if we look here, the probability of finding a baby, or a female baby, that's one year old, with a mass or a weight of less than 8.4 kilograms, that's this area right here. I said mass because kilograms is actually a unit of mass. But most people use it as weight as well."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Or maybe I should say whose mass is less than 8.4 kilograms. So if we look here, the probability of finding a baby, or a female baby, that's one year old, with a mass or a weight of less than 8.4 kilograms, that's this area right here. I said mass because kilograms is actually a unit of mass. But most people use it as weight as well. So that's that area right there. So how can we figure out that area under this normal distribution using the empirical rule? Well, we know what this area is."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "But most people use it as weight as well. So that's that area right there. So how can we figure out that area under this normal distribution using the empirical rule? Well, we know what this area is. We know what this area between minus one standard deviation and plus one standard deviation is. We know that that is 68%. And if that's 68%, then that means in the parts that aren't in that middle region, you have 32%."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Well, we know what this area is. We know what this area between minus one standard deviation and plus one standard deviation is. We know that that is 68%. And if that's 68%, then that means in the parts that aren't in that middle region, you have 32%. Because the area under the entire normal distribution is 100% or 100%, or 1%, depending on how you want to think about it, because you can't have all of the possibilities combined, it can only add up to 1. You can't have it more than 100% there. So if you add up this leg and this leg, so this plus that leg, is going to be the remainder."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And if that's 68%, then that means in the parts that aren't in that middle region, you have 32%. Because the area under the entire normal distribution is 100% or 100%, or 1%, depending on how you want to think about it, because you can't have all of the possibilities combined, it can only add up to 1. You can't have it more than 100% there. So if you add up this leg and this leg, so this plus that leg, is going to be the remainder. So 100 minus 68, that's 32%. 32% is if you add up this left leg and this right leg over here. And this is a perfect normal distribution."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "So if you add up this leg and this leg, so this plus that leg, is going to be the remainder. So 100 minus 68, that's 32%. 32% is if you add up this left leg and this right leg over here. And this is a perfect normal distribution. They told us it's normally distributed, so it's going to be perfectly symmetrical. So if this side and that side add up to 32, but they're both symmetrical, meaning they have the exact same area, then this side right here, do it in pink, this side right here, I ended up looking more like purple, would be 16%. And this side right here would be 16%."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And this is a perfect normal distribution. They told us it's normally distributed, so it's going to be perfectly symmetrical. So if this side and that side add up to 32, but they're both symmetrical, meaning they have the exact same area, then this side right here, do it in pink, this side right here, I ended up looking more like purple, would be 16%. And this side right here would be 16%. So your probability of getting a result more than one standard deviation above the mean, so this right-hand side would be 16%, or the probability of having a result less than one standard deviation below the mean, that's this right here, 16%. So they want to know the probability of having a baby, or at one year's old, less than 8.4 kilograms. Less than 8.4 kilograms is this area right here, and that's 16%."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And this side right here would be 16%. So your probability of getting a result more than one standard deviation above the mean, so this right-hand side would be 16%, or the probability of having a result less than one standard deviation below the mean, that's this right here, 16%. So they want to know the probability of having a baby, or at one year's old, less than 8.4 kilograms. Less than 8.4 kilograms is this area right here, and that's 16%. So that's 16% for part A. Let's do part B. Between 7.3 and 11.7 kilograms."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Less than 8.4 kilograms is this area right here, and that's 16%. So that's 16% for part A. Let's do part B. Between 7.3 and 11.7 kilograms. So between 7.3, that's right there, that's two standard deviations below the mean, and 11.7. It's one, two standard deviations above the mean. So they're essentially asking us, what's the probability of getting a result within two standard deviations of the mean?"}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Between 7.3 and 11.7 kilograms. So between 7.3, that's right there, that's two standard deviations below the mean, and 11.7. It's one, two standard deviations above the mean. So they're essentially asking us, what's the probability of getting a result within two standard deviations of the mean? This is the mean right here. This is two standard deviations below. This is two standard deviations above."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "So they're essentially asking us, what's the probability of getting a result within two standard deviations of the mean? This is the mean right here. This is two standard deviations below. This is two standard deviations above. Well, that's pretty straightforward. The empirical rule tells us, between two standard deviations, you have a 95% chance of getting that result. Or a 95% chance of getting a result that is within two standard deviations."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "This is two standard deviations above. Well, that's pretty straightforward. The empirical rule tells us, between two standard deviations, you have a 95% chance of getting that result. Or a 95% chance of getting a result that is within two standard deviations. So the empirical rule just gives us that answer. And then finally, part C. The probability of having a one-year-old US baby girl more than 12.8 kilograms. So 12.8 kilograms is three standard deviations above the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Or a 95% chance of getting a result that is within two standard deviations. So the empirical rule just gives us that answer. And then finally, part C. The probability of having a one-year-old US baby girl more than 12.8 kilograms. So 12.8 kilograms is three standard deviations above the mean. So we want to know the probability of having more than three standard deviations above the mean. So that is this area way out there. I drew an orange."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "So 12.8 kilograms is three standard deviations above the mean. So we want to know the probability of having more than three standard deviations above the mean. So that is this area way out there. I drew an orange. Maybe I should do it in a different color to really contrast it. So it's this long tail out here. This little small area."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "I drew an orange. Maybe I should do it in a different color to really contrast it. So it's this long tail out here. This little small area. So what is that probability? So let's turn back to our empirical rule. Well, we know the probability."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "This little small area. So what is that probability? So let's turn back to our empirical rule. Well, we know the probability. We know this area. We know the area between minus three standard deviations and plus three standard deviations. We know this."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Well, we know the probability. We know this area. We know the area between minus three standard deviations and plus three standard deviations. We know this. I can, since this is the last problem, I can color the whole thing in. We know this area right here. Between minus three and plus three, that is 99.7%."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "We know this. I can, since this is the last problem, I can color the whole thing in. We know this area right here. Between minus three and plus three, that is 99.7%. The bulk of the results fall under there. I mean, almost all of them. So what do we have left over for the two tails?"}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Between minus three and plus three, that is 99.7%. The bulk of the results fall under there. I mean, almost all of them. So what do we have left over for the two tails? Remember, there are two tails. This is one of them. And then you have the results that are less than three standard deviations below the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "So what do we have left over for the two tails? Remember, there are two tails. This is one of them. And then you have the results that are less than three standard deviations below the mean. This tail right there. So that tells us that less than three standard deviations below the mean and more than three standard deviations above the mean combined have to be the rest. Well, the rest, it's only 0.3% for the rest."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And then you have the results that are less than three standard deviations below the mean. This tail right there. So that tells us that less than three standard deviations below the mean and more than three standard deviations above the mean combined have to be the rest. Well, the rest, it's only 0.3% for the rest. And these two things are symmetrical. They're going to be equal. So this right here has to be half of this, or 0.15%."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Well, the rest, it's only 0.3% for the rest. And these two things are symmetrical. They're going to be equal. So this right here has to be half of this, or 0.15%. And this right here is going to be 0.15%. So the probability of having a one-year-old baby girl in the US that is more than 12.8 kilograms, if you assume a perfect normal distribution, is the area under this curve, the area that is more than three standard deviations above the mean. And that is 0.15%."}, {"video_title": "Classifying shapes of distributions AP Statistics Khan Academy.mp3", "Sentence": "And what we're gonna do in this video is think about how to classify them, or use the words that people typically use to classify distributions. So let's first look at this distribution right over here. That's the distribution of the lengths of houseflies. So someone went out there and measured a bunch of houseflies, and then said, hey, look, there's many houseflies that are between 6 1\u204410 of a centimeter and 6 1\u20442 1\u204410 of a centimeter. Looks like there's about 40 houseflies there. And then if you say between 6 1\u20442 and 7 1\u204410, there's about 30 houseflies. And if you were to say between 5 1\u20442 1\u204410 and 6 1\u204410, it looks like it's about the same amount."}, {"video_title": "Classifying shapes of distributions AP Statistics Khan Academy.mp3", "Sentence": "So someone went out there and measured a bunch of houseflies, and then said, hey, look, there's many houseflies that are between 6 1\u204410 of a centimeter and 6 1\u20442 1\u204410 of a centimeter. Looks like there's about 40 houseflies there. And then if you say between 6 1\u20442 and 7 1\u204410, there's about 30 houseflies. And if you were to say between 5 1\u20442 1\u204410 and 6 1\u204410, it looks like it's about the same amount. This type of distribution is usually described as being symmetric. Why is it called that? Because if you were to draw a line down the middle of this distribution, both sides look like mirror images of each other."}, {"video_title": "Classifying shapes of distributions AP Statistics Khan Academy.mp3", "Sentence": "And if you were to say between 5 1\u20442 1\u204410 and 6 1\u204410, it looks like it's about the same amount. This type of distribution is usually described as being symmetric. Why is it called that? Because if you were to draw a line down the middle of this distribution, both sides look like mirror images of each other. This one looks pretty exactly symmetric, but more typically, when you're collecting data, you'll see roughly symmetric distributions. Now here we have a distribution that gives us the dates on pennies. So someone went out there, observed a bunch of pennies, looked at the dates on them."}, {"video_title": "Classifying shapes of distributions AP Statistics Khan Academy.mp3", "Sentence": "Because if you were to draw a line down the middle of this distribution, both sides look like mirror images of each other. This one looks pretty exactly symmetric, but more typically, when you're collecting data, you'll see roughly symmetric distributions. Now here we have a distribution that gives us the dates on pennies. So someone went out there, observed a bunch of pennies, looked at the dates on them. They saw many pennies, looks like a little bit more than 55 pennies, had a date between 2010 and 2020, while very few pennies had a date older than 1980 on them. And this type of distribution, when you have a tail to the left, you can see it right over here, you have a long tail to the left, this is known as a left-skewed distribution, left-skewed. Now in future videos, we'll come up with more technical definitions of what makes it left-skewed, but the way that you can recognize it is you have the high points of your distribution on the right, but then you have this long tail that skews it to the left."}, {"video_title": "Classifying shapes of distributions AP Statistics Khan Academy.mp3", "Sentence": "So someone went out there, observed a bunch of pennies, looked at the dates on them. They saw many pennies, looks like a little bit more than 55 pennies, had a date between 2010 and 2020, while very few pennies had a date older than 1980 on them. And this type of distribution, when you have a tail to the left, you can see it right over here, you have a long tail to the left, this is known as a left-skewed distribution, left-skewed. Now in future videos, we'll come up with more technical definitions of what makes it left-skewed, but the way that you can recognize it is you have the high points of your distribution on the right, but then you have this long tail that skews it to the left. Now the other side of a left-skewed, you might say, well, that would be a right-skewed distribution, and that's exactly what we see right over here. This is a distribution of state representatives, and as you can see, most of the states in the United States have between zero and 10 representatives. It looks like it's a little over 35."}, {"video_title": "Classifying shapes of distributions AP Statistics Khan Academy.mp3", "Sentence": "Now in future videos, we'll come up with more technical definitions of what makes it left-skewed, but the way that you can recognize it is you have the high points of your distribution on the right, but then you have this long tail that skews it to the left. Now the other side of a left-skewed, you might say, well, that would be a right-skewed distribution, and that's exactly what we see right over here. This is a distribution of state representatives, and as you can see, most of the states in the United States have between zero and 10 representatives. It looks like it's a little over 35. None of them actually have zero. They all have at least one representative, but they would fall into this bucket, while very few have more than 50 representatives. So here, where the bulk of our distribution is to the left, but we have this tail that skews us to the right, this is known as a right-skewed distribution."}, {"video_title": "Classifying shapes of distributions AP Statistics Khan Academy.mp3", "Sentence": "It looks like it's a little over 35. None of them actually have zero. They all have at least one representative, but they would fall into this bucket, while very few have more than 50 representatives. So here, where the bulk of our distribution is to the left, but we have this tail that skews us to the right, this is known as a right-skewed distribution. Now if we look at this next distribution, what would this be? Pause this video and think about it. Well, this could be a distribution of maybe someone went around the office and surveyed how many cups of coffee each person drank, and if they found someone who drank one cup of coffee per day, maybe this would be them, and then they found another person who drinks one cup of coffee, that's them."}, {"video_title": "Classifying shapes of distributions AP Statistics Khan Academy.mp3", "Sentence": "So here, where the bulk of our distribution is to the left, but we have this tail that skews us to the right, this is known as a right-skewed distribution. Now if we look at this next distribution, what would this be? Pause this video and think about it. Well, this could be a distribution of maybe someone went around the office and surveyed how many cups of coffee each person drank, and if they found someone who drank one cup of coffee per day, maybe this would be them, and then they found another person who drinks one cup of coffee, that's them. Then they found three people who drank two cups of coffee. Well, this is a very similar situation to what we saw on the dates on pennies. A large amount of our data fell into this right bucket of three cups of coffee, but then we had this tail to the left."}, {"video_title": "Classifying shapes of distributions AP Statistics Khan Academy.mp3", "Sentence": "Well, this could be a distribution of maybe someone went around the office and surveyed how many cups of coffee each person drank, and if they found someone who drank one cup of coffee per day, maybe this would be them, and then they found another person who drinks one cup of coffee, that's them. Then they found three people who drank two cups of coffee. Well, this is a very similar situation to what we saw on the dates on pennies. A large amount of our data fell into this right bucket of three cups of coffee, but then we had this tail to the left. So this would be left-skewed. Now these right two distributions are interesting. One could argue that this distribution here, which is telling us the number of days that we had different high temperatures, that this looks roughly symmetric, or actually even looks exactly symmetric, because if you did that little exercise of drawing a dotted line down the middle, it looks like the two sides are mirror images of each other."}, {"video_title": "Classifying shapes of distributions AP Statistics Khan Academy.mp3", "Sentence": "A large amount of our data fell into this right bucket of three cups of coffee, but then we had this tail to the left. So this would be left-skewed. Now these right two distributions are interesting. One could argue that this distribution here, which is telling us the number of days that we had different high temperatures, that this looks roughly symmetric, or actually even looks exactly symmetric, because if you did that little exercise of drawing a dotted line down the middle, it looks like the two sides are mirror images of each other. Now that would not be technically incorrect, but typically when you see these two peaks, this would typically be called a bimodal distribution. So even though bimodal distributions can sometimes be symmetric or roughly symmetric, you wanna be more precise. And here, when you have these two peaks, that's where the bi comes from, you would call it bimodal, and this makes sense because you have a lot of days that are warm that might happen during the summer, and you might have a lot of days that are cold that are happening during the winter."}, {"video_title": "Classifying shapes of distributions AP Statistics Khan Academy.mp3", "Sentence": "One could argue that this distribution here, which is telling us the number of days that we had different high temperatures, that this looks roughly symmetric, or actually even looks exactly symmetric, because if you did that little exercise of drawing a dotted line down the middle, it looks like the two sides are mirror images of each other. Now that would not be technically incorrect, but typically when you see these two peaks, this would typically be called a bimodal distribution. So even though bimodal distributions can sometimes be symmetric or roughly symmetric, you wanna be more precise. And here, when you have these two peaks, that's where the bi comes from, you would call it bimodal, and this makes sense because you have a lot of days that are warm that might happen during the summer, and you might have a lot of days that are cold that are happening during the winter. Now this last distribution here, the results from dye rolls, one could argue as well that this is roughly symmetric. It's not exact, it's not perfectly symmetric, but when you look at this dotted line here on the left and the right sides, it looks roughly symmetric. But a more exact classification here would be that it looks approximately uniform."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "And so what we could do is we could set up some buckets of time studied and some buckets of percent correct, and then we could survey the students and or look at the data from the scores on the test. And then we can place students in these buckets. So what you see right over here, this is a two-way table, and you can also view this as a joint distribution along these two dimensions. So one way to read this is that 20 out of the 200 total students got between 60 and 79% on the test and studied between 21 and 40 minutes. So there's all sorts of interesting things that we could try to glean from this, but what we're going to focus on this video is two more types of distributions other than the joint distribution that we see in this data. One type is a marginal distribution. And a marginal distribution is just focusing on one of these dimensions."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "So one way to read this is that 20 out of the 200 total students got between 60 and 79% on the test and studied between 21 and 40 minutes. So there's all sorts of interesting things that we could try to glean from this, but what we're going to focus on this video is two more types of distributions other than the joint distribution that we see in this data. One type is a marginal distribution. And a marginal distribution is just focusing on one of these dimensions. And one way to think about it is you can determine it by looking at the margin. So for example, if you wanted to figure out the marginal distribution of the percent correct, what you could do is look at the total of these rows. So these counts right over here give you the marginal distribution of the percent correct."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "And a marginal distribution is just focusing on one of these dimensions. And one way to think about it is you can determine it by looking at the margin. So for example, if you wanted to figure out the marginal distribution of the percent correct, what you could do is look at the total of these rows. So these counts right over here give you the marginal distribution of the percent correct. 40 out of the 200 got between 80 and 100. 60 out of the 200 got between 60 and 79, so on and so forth. Now a marginal distribution could be represented as counts or as percentages."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "So these counts right over here give you the marginal distribution of the percent correct. 40 out of the 200 got between 80 and 100. 60 out of the 200 got between 60 and 79, so on and so forth. Now a marginal distribution could be represented as counts or as percentages. So if you represent it as percentages, you would divide each of these counts by the total, which is 200. So 40 over 200, that would be 20%. 60 out of 200, that'd be 30%."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "Now a marginal distribution could be represented as counts or as percentages. So if you represent it as percentages, you would divide each of these counts by the total, which is 200. So 40 over 200, that would be 20%. 60 out of 200, that'd be 30%. 70 out of 200, that would be 35%. 20 out of 200 is 10%. And 10 out of 200 is 5%."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "60 out of 200, that'd be 30%. 70 out of 200, that would be 35%. 20 out of 200 is 10%. And 10 out of 200 is 5%. So this right over here in terms of percentages gives you the marginal distribution of the percent correct based on these buckets. So you could say 10% got between a 20 and a 39. Now you could also think about marginal distributions the other way."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "And 10 out of 200 is 5%. So this right over here in terms of percentages gives you the marginal distribution of the percent correct based on these buckets. So you could say 10% got between a 20 and a 39. Now you could also think about marginal distributions the other way. You could think about the marginal distribution for the time studied in the class. And so then you would look at these counts right over here. You'd say a total of 14 students studied between zero and 20 minutes."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "Now you could also think about marginal distributions the other way. You could think about the marginal distribution for the time studied in the class. And so then you would look at these counts right over here. You'd say a total of 14 students studied between zero and 20 minutes. You're not thinking about the percent correct anymore. A total of 30 studied between 21 and 40 minutes. And likewise, you could write these as percentages."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "You'd say a total of 14 students studied between zero and 20 minutes. You're not thinking about the percent correct anymore. A total of 30 studied between 21 and 40 minutes. And likewise, you could write these as percentages. This would be 7%. This would be 15%. This would be 43%."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "And likewise, you could write these as percentages. This would be 7%. This would be 15%. This would be 43%. And this would be 35% right over there. Now another idea that you might sometimes see when people are trying to interpret a joint distribution like this or get more information or more realizations from it is to think about something known as a conditional distribution. Conditional distribution."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "This would be 43%. And this would be 35% right over there. Now another idea that you might sometimes see when people are trying to interpret a joint distribution like this or get more information or more realizations from it is to think about something known as a conditional distribution. Conditional distribution. And this is the distribution of one variable given something true about the other variable. So for example, an example of a conditional distribution would be the distribution, distribution of percent correct, correct, given that students, students, students study between, let's say, 41 and 60 minutes. Between 41 and 60 minutes."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "Conditional distribution. And this is the distribution of one variable given something true about the other variable. So for example, an example of a conditional distribution would be the distribution, distribution of percent correct, correct, given that students, students, students study between, let's say, 41 and 60 minutes. Between 41 and 60 minutes. Well, to think about that, you would first look at your condition. Okay, let's look at the students who have studied between 41 and 60 minutes. And so that would be this column right over here."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "Between 41 and 60 minutes. Well, to think about that, you would first look at your condition. Okay, let's look at the students who have studied between 41 and 60 minutes. And so that would be this column right over here. And then that column, the information in it, can give you your conditional distribution. Now an important thing to realize is a marginal distribution can be represented as counts for the various buckets or percentages, while the standard practice for conditional distribution is to think in terms of percentages. So the conditional distribution of the percent correct given that students study between 41 and 60 minutes, it would look something like this."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "And so that would be this column right over here. And then that column, the information in it, can give you your conditional distribution. Now an important thing to realize is a marginal distribution can be represented as counts for the various buckets or percentages, while the standard practice for conditional distribution is to think in terms of percentages. So the conditional distribution of the percent correct given that students study between 41 and 60 minutes, it would look something like this. Let me get a little bit more space. So if we set up the various categories, 80 to 100, 60 to 79, 40 to 59, continue it over here, 20 to 39, and zero to 19, what we'd wanna do is calculate the percentage that fall into each of these buckets, given that we're studying between 41 and 60 minutes. So this first one, 80 to 100, it would be 16 out of the 86 students."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "So the conditional distribution of the percent correct given that students study between 41 and 60 minutes, it would look something like this. Let me get a little bit more space. So if we set up the various categories, 80 to 100, 60 to 79, 40 to 59, continue it over here, 20 to 39, and zero to 19, what we'd wanna do is calculate the percentage that fall into each of these buckets, given that we're studying between 41 and 60 minutes. So this first one, 80 to 100, it would be 16 out of the 86 students. So we would write 16 out of 86, which is equal to, 16 divided by 86 is equal to, I'll just round to one decimal place, it's roughly 18.6%. 18.6, approximately equal to 18.6%. And then to get the full conditional distribution, we would keep doing that."}, {"video_title": "Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3", "Sentence": "We are told a restaurant owner installed a new automated drink machine. The machine is designed to dispense 530 milliliters of liquid on the medium-sized setting. The owner suspects that the machine may be dispensing too much in medium drinks. They decide to take a sample of 30 medium drinks to see if the average amount is significantly greater than 500 milliliters. What are appropriate hypotheses for their significance test? And they actually give us four choices here. I'll scroll down a little bit so that you can see all of the choices."}, {"video_title": "Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3", "Sentence": "They decide to take a sample of 30 medium drinks to see if the average amount is significantly greater than 500 milliliters. What are appropriate hypotheses for their significance test? And they actually give us four choices here. I'll scroll down a little bit so that you can see all of the choices. So like always, pause this video and see if you can have a go at it. Okay, now let's do this together. So let's just remind ourselves what a null hypothesis is and what an alternative hypothesis is."}, {"video_title": "Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3", "Sentence": "I'll scroll down a little bit so that you can see all of the choices. So like always, pause this video and see if you can have a go at it. Okay, now let's do this together. So let's just remind ourselves what a null hypothesis is and what an alternative hypothesis is. One way to view a null hypothesis, this is the hypothesis where things are happening as expected. Sometimes people will describe this as the no difference hypothesis. It'll often have a statement of equality where the population parameter is equal to the value where the value is what people were kind of assuming all along."}, {"video_title": "Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3", "Sentence": "So let's just remind ourselves what a null hypothesis is and what an alternative hypothesis is. One way to view a null hypothesis, this is the hypothesis where things are happening as expected. Sometimes people will describe this as the no difference hypothesis. It'll often have a statement of equality where the population parameter is equal to the value where the value is what people were kind of assuming all along. The alternative hypothesis, this is a claim where if you have evidence to back up that claim, that would be new news. You are saying, hey, there's something interesting going on here. There is a difference."}, {"video_title": "Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3", "Sentence": "It'll often have a statement of equality where the population parameter is equal to the value where the value is what people were kind of assuming all along. The alternative hypothesis, this is a claim where if you have evidence to back up that claim, that would be new news. You are saying, hey, there's something interesting going on here. There is a difference. And so in this context, the no difference, we would say the null hypothesis would be we would care about the population parameter, and here we care about the average amount of drink dispensed in the medium setting. So the population parameter there would be the mean, and that the mean would be equal to 530 milliliters because that's what the drink machine is supposed to do. And then the alternative hypothesis, this is what the owner fears, is that the mean actually might be larger than that, larger than 530 milliliters."}, {"video_title": "Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3", "Sentence": "There is a difference. And so in this context, the no difference, we would say the null hypothesis would be we would care about the population parameter, and here we care about the average amount of drink dispensed in the medium setting. So the population parameter there would be the mean, and that the mean would be equal to 530 milliliters because that's what the drink machine is supposed to do. And then the alternative hypothesis, this is what the owner fears, is that the mean actually might be larger than that, larger than 530 milliliters. And so let's see, which of these choices is this? Well, these first two choices are talking about proportion, but it's really the average amount that we're talking about. We see it up here."}, {"video_title": "Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3", "Sentence": "And then the alternative hypothesis, this is what the owner fears, is that the mean actually might be larger than that, larger than 530 milliliters. And so let's see, which of these choices is this? Well, these first two choices are talking about proportion, but it's really the average amount that we're talking about. We see it up here. They decided to take a sample of 30 medium drinks to see if the average amount, they're not talking about proportions here, they're talking about averages, and in this case, we're talking about estimating the population parameter, the population mean for how much drink is dispensed on that setting. And so this one is looking like this right over here. Only these two are even dealing with the mean."}, {"video_title": "Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3", "Sentence": "We see it up here. They decided to take a sample of 30 medium drinks to see if the average amount, they're not talking about proportions here, they're talking about averages, and in this case, we're talking about estimating the population parameter, the population mean for how much drink is dispensed on that setting. And so this one is looking like this right over here. Only these two are even dealing with the mean. And the difference between this one and this one is this says the mean is greater than 530 milliliters, and that indeed is the owner's fear. And this over here, this alternative hypothesis is that it's dispensing, on average, less than 530 milliliters, but that's not what the owner is afraid of, and so that's not the kind of the news that we're trying to find some evidence for. So I would definitely pick choice C. Let's do another example."}, {"video_title": "Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3", "Sentence": "Only these two are even dealing with the mean. And the difference between this one and this one is this says the mean is greater than 530 milliliters, and that indeed is the owner's fear. And this over here, this alternative hypothesis is that it's dispensing, on average, less than 530 milliliters, but that's not what the owner is afraid of, and so that's not the kind of the news that we're trying to find some evidence for. So I would definitely pick choice C. Let's do another example. The National Sleep Foundation recommends that teenagers aged 14 to 17 years old get at least eight hours of sleep per night for proper health and wellness. A statistics class at a large high school suspects that students at their school are getting less than eight hours of sleep on average. To test their theory, they randomly sample 42 of these students and ask them how many hours of sleep they get per night."}, {"video_title": "Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3", "Sentence": "So I would definitely pick choice C. Let's do another example. The National Sleep Foundation recommends that teenagers aged 14 to 17 years old get at least eight hours of sleep per night for proper health and wellness. A statistics class at a large high school suspects that students at their school are getting less than eight hours of sleep on average. To test their theory, they randomly sample 42 of these students and ask them how many hours of sleep they get per night. The mean from this sample, the mean from the sample, is 7.5 hours. Here's their alternative hypothesis. The average amount of sleep students at their school get per night is, what is an appropriate ending to their alternative hypothesis?"}, {"video_title": "Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3", "Sentence": "To test their theory, they randomly sample 42 of these students and ask them how many hours of sleep they get per night. The mean from this sample, the mean from the sample, is 7.5 hours. Here's their alternative hypothesis. The average amount of sleep students at their school get per night is, what is an appropriate ending to their alternative hypothesis? So pause this video and see if you can think about that. So let's just first think about a good null hypothesis. So the null hypothesis is, hey, there's actually no news here that everything is what people were always assuming."}, {"video_title": "Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3", "Sentence": "The average amount of sleep students at their school get per night is, what is an appropriate ending to their alternative hypothesis? So pause this video and see if you can think about that. So let's just first think about a good null hypothesis. So the null hypothesis is, hey, there's actually no news here that everything is what people were always assuming. And so the null hypothesis here is that, no, the students are getting at least eight hours of sleep per night. And so that would be that, and remember, we care about the population of students, and we care about the population of students at the school, and so we would say, well, the null hypothesis is that the parameter for the students at that school, the mean amount of sleep that they're getting, is indeed greater than or equal to eight hours. And a good clue for the alternative hypothesis is when you see something like this, where they say a statistics class at a large high school suspects, so they suspect that things might be different than what people have always been assuming or actually what's good for students."}, {"video_title": "Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3", "Sentence": "So the null hypothesis is, hey, there's actually no news here that everything is what people were always assuming. And so the null hypothesis here is that, no, the students are getting at least eight hours of sleep per night. And so that would be that, and remember, we care about the population of students, and we care about the population of students at the school, and so we would say, well, the null hypothesis is that the parameter for the students at that school, the mean amount of sleep that they're getting, is indeed greater than or equal to eight hours. And a good clue for the alternative hypothesis is when you see something like this, where they say a statistics class at a large high school suspects, so they suspect that things might be different than what people have always been assuming or actually what's good for students. And so they suspect that students at their school are getting less than eight hours of sleep on average. And so they suspect that the population parameter, the population mean for their school is actually less than eight hours. And so if you wanted to write this out in words, the average amount of sleep students at their school get per night is less than eight hours."}, {"video_title": "Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3", "Sentence": "And a good clue for the alternative hypothesis is when you see something like this, where they say a statistics class at a large high school suspects, so they suspect that things might be different than what people have always been assuming or actually what's good for students. And so they suspect that students at their school are getting less than eight hours of sleep on average. And so they suspect that the population parameter, the population mean for their school is actually less than eight hours. And so if you wanted to write this out in words, the average amount of sleep students at their school get per night is less than eight hours. Now, one thing to watch out for is, one, you wanna make sure you're getting the right parameter. Sometimes it's often a population mean, sometimes it's a population proportion. But the other thing that sometimes folks get stuck up on, but the other thing that sometimes confuses folks is, well, we are measuring, is that we are calculating a statistic from a sample."}, {"video_title": "Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3", "Sentence": "And so if you wanted to write this out in words, the average amount of sleep students at their school get per night is less than eight hours. Now, one thing to watch out for is, one, you wanna make sure you're getting the right parameter. Sometimes it's often a population mean, sometimes it's a population proportion. But the other thing that sometimes folks get stuck up on, but the other thing that sometimes confuses folks is, well, we are measuring, is that we are calculating a statistic from a sample. Here we're calculating the sample mean, but the sample statistics are not what should be involved in your hypotheses. Your hypotheses are claims about your population that you care about. Here, the population is the students at the high school."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "So they all agree to put in their salaries into a computer, and so these are their salaries, they're measured in thousands, so one makes 35,000, 50,000, 50,000, 50,000, 56,000, two make 60,000, one make 75,000, and one makes 250,000, so she's doing very well for herself. And the computer spits out a bunch of parameters based on this data here. So it spits out two typical measures of central tendency. The mean is roughly 76.2, the computer would calculate it by adding up all of these numbers, these nine numbers, and then dividing by nine. And the median is 56. And median is quite easy to calculate, you just order the numbers and you take the middle number here, which is 56. Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure?"}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "The mean is roughly 76.2, the computer would calculate it by adding up all of these numbers, these nine numbers, and then dividing by nine. And the median is 56. And median is quite easy to calculate, you just order the numbers and you take the middle number here, which is 56. Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure? All right, so let's think about this a little bit. I'm gonna plot it on a line here, I'm gonna plot my data so we get a better sense, so we just don't see them, so we just don't see things as numbers, but we see where those numbers sit relative to each other. So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure? All right, so let's think about this a little bit. I'm gonna plot it on a line here, I'm gonna plot my data so we get a better sense, so we just don't see them, so we just don't see things as numbers, but we see where those numbers sit relative to each other. So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough. I could draw this a little bit neater, but 60, 70, 80, 90, actually let me just clean this up a little bit more too, this one right over here would be a little bit closer. So this one, let me just put it right around here, so that's 40, and then this would be 30, 20, 10. Okay, that's pretty good."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough. I could draw this a little bit neater, but 60, 70, 80, 90, actually let me just clean this up a little bit more too, this one right over here would be a little bit closer. So this one, let me just put it right around here, so that's 40, and then this would be 30, 20, 10. Okay, that's pretty good. So let's plot this data. So one student makes 35,000, so that is right over there. Two make 50, or three make 50,000, so one, two, and three, I'll put it like that."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "Okay, that's pretty good. So let's plot this data. So one student makes 35,000, so that is right over there. Two make 50, or three make 50,000, so one, two, and three, I'll put it like that. One makes 56,000, which would put them right over here. One makes 60,000, or actually two make 60,000, so it's like that. One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "Two make 50, or three make 50,000, so one, two, and three, I'll put it like that. One makes 56,000, which would put them right over here. One makes 60,000, or actually two make 60,000, so it's like that. One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there. And then when we calculate the mean, as 76.2 is our measure of central tendency, 76.2 is right over there. So is this a good measure of central tendency? Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there. And then when we calculate the mean, as 76.2 is our measure of central tendency, 76.2 is right over there. So is this a good measure of central tendency? Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one. And the reason is is that you have this one, that our data is skewed significantly by this data point at $250,000. It is so far from the rest of the distribution, from the rest of the data, that it has skewed the mean. And this is something that you see in general."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one. And the reason is is that you have this one, that our data is skewed significantly by this data point at $250,000. It is so far from the rest of the distribution, from the rest of the data, that it has skewed the mean. And this is something that you see in general. If you have data that is skewed, and especially things like salary data, where someone might make, most people are making 50, 60, $70,000, but someone might make $2 million, and so that will skew the average, or skew the mean, I should say, when you add them all up and divide by the number of data points you have. In this case, especially when you have data points that would skew the mean, median is much more robust. The median at 56 sits right over here, which seems to be much more indicative for central tendency."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "And this is something that you see in general. If you have data that is skewed, and especially things like salary data, where someone might make, most people are making 50, 60, $70,000, but someone might make $2 million, and so that will skew the average, or skew the mean, I should say, when you add them all up and divide by the number of data points you have. In this case, especially when you have data points that would skew the mean, median is much more robust. The median at 56 sits right over here, which seems to be much more indicative for central tendency. And think about it. Even if you made this, instead of 250,000, if you made this 250,000,000, which would be $250 million, which is a ginormous amount of money to make, it would skew the mean incredibly, but it actually would not even change the median, because the median, it doesn't matter how high this number gets. This could be a trillion dollars."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "The median at 56 sits right over here, which seems to be much more indicative for central tendency. And think about it. Even if you made this, instead of 250,000, if you made this 250,000,000, which would be $250 million, which is a ginormous amount of money to make, it would skew the mean incredibly, but it actually would not even change the median, because the median, it doesn't matter how high this number gets. This could be a trillion dollars. This could be a quadrillion dollars. The median is going to stay the same. So the median is much more robust if you have a skewed data set."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "This could be a trillion dollars. This could be a quadrillion dollars. The median is going to stay the same. So the median is much more robust if you have a skewed data set. Mean makes a little bit more sense if you have a symmetric data set, or if you have things that are, where things are roughly above and below the mean, or things aren't skewed incredibly in one direction, especially by a handful of data points like we have right over here. So in this example, the median is a much better measure of central tendency. And so what about spread?"}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "So the median is much more robust if you have a skewed data set. Mean makes a little bit more sense if you have a symmetric data set, or if you have things that are, where things are roughly above and below the mean, or things aren't skewed incredibly in one direction, especially by a handful of data points like we have right over here. So in this example, the median is a much better measure of central tendency. And so what about spread? Well, you might immediately say, well, Sal, you already told us that the mean is not so good, and the standard deviation is based on the mean. You take each of these data points, find their distance from the mean, square that number, add up those squared distances, divide by the number of data points if we're taking the population standard deviation, and then you take the square root of the whole thing. And so since this is based on the mean, which isn't a good measure of central tendency in this situation, and this is also going to skew that standard deviation, this is going to be, this is a lot larger than if you look at the actual, you want an indication of the spread."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "And so what about spread? Well, you might immediately say, well, Sal, you already told us that the mean is not so good, and the standard deviation is based on the mean. You take each of these data points, find their distance from the mean, square that number, add up those squared distances, divide by the number of data points if we're taking the population standard deviation, and then you take the square root of the whole thing. And so since this is based on the mean, which isn't a good measure of central tendency in this situation, and this is also going to skew that standard deviation, this is going to be, this is a lot larger than if you look at the actual, you want an indication of the spread. Yes, you have this one data point that's way far away from either the mean or the median, depending on how you want to think about it, but most of the data points seem much closer. And so for that situation, not only are we using the median, but the interquartile range is once again more robust. How do we calculate the interquartile range?"}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "And so since this is based on the mean, which isn't a good measure of central tendency in this situation, and this is also going to skew that standard deviation, this is going to be, this is a lot larger than if you look at the actual, you want an indication of the spread. Yes, you have this one data point that's way far away from either the mean or the median, depending on how you want to think about it, but most of the data points seem much closer. And so for that situation, not only are we using the median, but the interquartile range is once again more robust. How do we calculate the interquartile range? Well, you take the median, and then you take the bottom group of numbers and calculate the median of those. So that's 50 right over here. And then you take the top group of numbers, the upper group of numbers, and the median there, 60 and 75, is 67.5."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "How do we calculate the interquartile range? Well, you take the median, and then you take the bottom group of numbers and calculate the median of those. So that's 50 right over here. And then you take the top group of numbers, the upper group of numbers, and the median there, 60 and 75, is 67.5. If this looks unfamiliar, we have many videos on interquartile range and calculating standard deviation and median and mean. This is just a little bit of a review. And then the difference between these two is 17.5."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "And then you take the top group of numbers, the upper group of numbers, and the median there, 60 and 75, is 67.5. If this looks unfamiliar, we have many videos on interquartile range and calculating standard deviation and median and mean. This is just a little bit of a review. And then the difference between these two is 17.5. And notice, this distance between these two, this 17.5, this isn't going to change even if this is $250 billion. So once again, it is both of these measures are more robust when you have a skewed data set. So the big takeaway here is mean and standard deviation, they're not bad."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "And then the difference between these two is 17.5. And notice, this distance between these two, this 17.5, this isn't going to change even if this is $250 billion. So once again, it is both of these measures are more robust when you have a skewed data set. So the big takeaway here is mean and standard deviation, they're not bad. If you have a roughly symmetric data set, if you don't have any significant outliers, things that really skew the data set, mean and standard deviation can be quite solid. But if you're looking at something that could get really skewed by a handful of data points, median might be a median in interquartile range. Median for central tendency, interquartile range for spread around that central tendency."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "So the big takeaway here is mean and standard deviation, they're not bad. If you have a roughly symmetric data set, if you don't have any significant outliers, things that really skew the data set, mean and standard deviation can be quite solid. But if you're looking at something that could get really skewed by a handful of data points, median might be a median in interquartile range. Median for central tendency, interquartile range for spread around that central tendency. And that's why you'll see when people talk about salaries, they'll often talk about median because you could have some skewed salaries, especially on the upside. When you talk about things like home prices, you'll see median often measured more typically than mean because home prices in a neighborhood or in a city, a lot of the houses might be in the $200,000, $300,000 range but maybe there's one ginormous mansion that is $100 million. And if you calculated mean, that would skew and give a false impression of the average or the central tendency of prices in that city."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "Let's say it's a bunch of balls, each of them have a number written on it. For that population, we could calculate parameters. So a parameter you could view as a truth about that population. We've covered this in other videos. So for example, you could have the population mean, the mean of the numbers written on top of that ball. You could have the population standard deviation. You could have the proportion of balls that are even, whatever, these are all population parameters."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "We've covered this in other videos. So for example, you could have the population mean, the mean of the numbers written on top of that ball. You could have the population standard deviation. You could have the proportion of balls that are even, whatever, these are all population parameters. Now, we know from many other videos that you might not know the population parameter or it might not even be easy to find. And so the way that we try to estimate a population parameter is by taking a sample. So this right over here is a sample of size n, sample of size n. And then we can calculate a statistic from that sample, based on that sample, maybe we picked n balls from there."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "You could have the proportion of balls that are even, whatever, these are all population parameters. Now, we know from many other videos that you might not know the population parameter or it might not even be easy to find. And so the way that we try to estimate a population parameter is by taking a sample. So this right over here is a sample of size n, sample of size n. And then we can calculate a statistic from that sample, based on that sample, maybe we picked n balls from there. And so from that, we can calculate a statistic that is used to estimate this parameter. But we know that this is a random sample right over here. So every time we take a sample, the statistic that we calculate for that sample is not necessarily going to be the same as the population parameter."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "So this right over here is a sample of size n, sample of size n. And then we can calculate a statistic from that sample, based on that sample, maybe we picked n balls from there. And so from that, we can calculate a statistic that is used to estimate this parameter. But we know that this is a random sample right over here. So every time we take a sample, the statistic that we calculate for that sample is not necessarily going to be the same as the population parameter. In fact, if we were to take a random sample of size n again and then we were to calculate the statistic again, we could very well get a different value. So these are all going to be estimates of this parameter. And so an interesting question is, is what is the distribution of the values that I could get for these statistics?"}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "So every time we take a sample, the statistic that we calculate for that sample is not necessarily going to be the same as the population parameter. In fact, if we were to take a random sample of size n again and then we were to calculate the statistic again, we could very well get a different value. So these are all going to be estimates of this parameter. And so an interesting question is, is what is the distribution of the values that I could get for these statistics? What is the frequency with which I could get different values for the statistic that is trying to estimate this parameter? And that distribution is what a sampling distribution is. So let's make this even a little bit more concrete."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And so an interesting question is, is what is the distribution of the values that I could get for these statistics? What is the frequency with which I could get different values for the statistic that is trying to estimate this parameter? And that distribution is what a sampling distribution is. So let's make this even a little bit more concrete. Let's imagine where our population, I'm gonna make this a very simple example. Let's say our population has three balls in it, one, two, three, and they're numbered one, two, and three. And it's very easy to calculate, let's say the parameter that we care about right over here is the population mean."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "So let's make this even a little bit more concrete. Let's imagine where our population, I'm gonna make this a very simple example. Let's say our population has three balls in it, one, two, three, and they're numbered one, two, and three. And it's very easy to calculate, let's say the parameter that we care about right over here is the population mean. And that of course is going to be one plus two plus three, all of that over three, which is six divided by three, which is two. So that is our population parameter. But let's say that we wanted to take samples, let's say samples of two balls at a time, and every time we take a ball, we'll replace it."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And it's very easy to calculate, let's say the parameter that we care about right over here is the population mean. And that of course is going to be one plus two plus three, all of that over three, which is six divided by three, which is two. So that is our population parameter. But let's say that we wanted to take samples, let's say samples of two balls at a time, and every time we take a ball, we'll replace it. So each ball we take, it is an independent pick. And we're gonna use those samples of two balls at a time in order to estimate the population mean. So for example, this could be our first sample of size two."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "But let's say that we wanted to take samples, let's say samples of two balls at a time, and every time we take a ball, we'll replace it. So each ball we take, it is an independent pick. And we're gonna use those samples of two balls at a time in order to estimate the population mean. So for example, this could be our first sample of size two. And let's say in that first sample, I pick a one and let's say I pick a two. Well, then I can calculate the sample statistic here. In this case, it would be the sample mean, which is used to estimate the population mean."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "So for example, this could be our first sample of size two. And let's say in that first sample, I pick a one and let's say I pick a two. Well, then I can calculate the sample statistic here. In this case, it would be the sample mean, which is used to estimate the population mean. And in this, for this sample of two, it's going to be 1.5. Then I can do it again. And let's say I get a one and I get a three."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "In this case, it would be the sample mean, which is used to estimate the population mean. And in this, for this sample of two, it's going to be 1.5. Then I can do it again. And let's say I get a one and I get a three. Well, now when I calculate the sample mean, the average of one and three or the mean of one and three is going to be equal to two. Let's think about all of the different scenarios of samples we can get and what the associated sample means are going to be. And then we can get see the frequency of getting those sample means."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And let's say I get a one and I get a three. Well, now when I calculate the sample mean, the average of one and three or the mean of one and three is going to be equal to two. Let's think about all of the different scenarios of samples we can get and what the associated sample means are going to be. And then we can get see the frequency of getting those sample means. And so let me draw a little bit of a table here. So make a table right over here. And let's see, these are the numbers that we pick."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And then we can get see the frequency of getting those sample means. And so let me draw a little bit of a table here. So make a table right over here. And let's see, these are the numbers that we pick. And remember, when we pick one ball, we'll record that number, then we'll put it back in, and then we'll pick another ball. So these are going to be independent events and it's going to be with replacement. And so let's say we could pick a one and then a one."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And let's see, these are the numbers that we pick. And remember, when we pick one ball, we'll record that number, then we'll put it back in, and then we'll pick another ball. So these are going to be independent events and it's going to be with replacement. And so let's say we could pick a one and then a one. We could pick a one, then a two, a one, and a three. We could pick a two and then a one. We could pick a two and a two, a two and a three."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And so let's say we could pick a one and then a one. We could pick a one, then a two, a one, and a three. We could pick a two and then a one. We could pick a two and a two, a two and a three. We could pick a three and a one, a three and a two, or a three and a three. There's three possible balls for the first pick and three possible balls for the second because we're doing it with replacement. And so what is the sample mean in each of these for all of these combinations?"}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "We could pick a two and a two, a two and a three. We could pick a three and a one, a three and a two, or a three and a three. There's three possible balls for the first pick and three possible balls for the second because we're doing it with replacement. And so what is the sample mean in each of these for all of these combinations? So for this one, the sample mean is one. Here it is 1.5. Here it is two."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And so what is the sample mean in each of these for all of these combinations? So for this one, the sample mean is one. Here it is 1.5. Here it is two. Here it is 1.5. Here it is two. Here it is 2.5."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "Here it is two. Here it is 1.5. Here it is two. Here it is 2.5. Here it is two. Here it is 2.5. And then here it is three."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "Here it is 2.5. Here it is two. Here it is 2.5. And then here it is three. And so we can now plot the frequencies of these possible sample means that we can get. And that plot will be a sampling distribution of the sample means. So let's do that."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And then here it is three. And so we can now plot the frequencies of these possible sample means that we can get. And that plot will be a sampling distribution of the sample means. So let's do that. So let me make a little chart right over here, a little graph right over here. So these are the possible, possible sample means. We can get a one, we could get a 1.5."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "So let's do that. So let me make a little chart right over here, a little graph right over here. So these are the possible, possible sample means. We can get a one, we could get a 1.5. We could get a two, we could get a 2.5, or we can get a three. And now let's see the frequency of it. And I will put that over here."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "We can get a one, we could get a 1.5. We could get a two, we could get a 2.5, or we can get a three. And now let's see the frequency of it. And I will put that over here. And so let's see, how many ones out of our nine possibilities we have, how many were one? Well, only one of the sample means was one. And so the relative frequency, if we just said the number, we could make this line go up one, or we could just say, hey, this is going to be one out of the nine possibilities."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And I will put that over here. And so let's see, how many ones out of our nine possibilities we have, how many were one? Well, only one of the sample means was one. And so the relative frequency, if we just said the number, we could make this line go up one, or we could just say, hey, this is going to be one out of the nine possibilities. And so let me just make that, I'll call this right over here. This is 1 9th. Now what about 1.5?"}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And so the relative frequency, if we just said the number, we could make this line go up one, or we could just say, hey, this is going to be one out of the nine possibilities. And so let me just make that, I'll call this right over here. This is 1 9th. Now what about 1.5? Well, let's see, there's one, two of these possibilities out of nine. So 1.5, it would look like this. This right over here is two over nine."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "Now what about 1.5? Well, let's see, there's one, two of these possibilities out of nine. So 1.5, it would look like this. This right over here is two over nine. And now what about two? Well, we can see there's one, two, three. So three out of the nine possibilities, we got a two."}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "This right over here is two over nine. And now what about two? Well, we can see there's one, two, three. So three out of the nine possibilities, we got a two. So we could say this is two, or we could say this is 3 9th, which is the same thing, of course, as 1 3rd. So this right over here is three over nine. And then what about 2.5?"}, {"video_title": "Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "So three out of the nine possibilities, we got a two. So we could say this is two, or we could say this is 3 9th, which is the same thing, of course, as 1 3rd. So this right over here is three over nine. And then what about 2.5? Well, there's two 2.5s, so two out of the nine times. Another way you could interpret this is when you take a random sample with replacement of two balls, you have a 2 9th chance of having a sample mean of 2.5. And then last but not least, right over here, there's one scenario out of the nine where you get 2 3, so 1 9th."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "A high school newspaper doesn't know this figure, but they are curious what it is. So they decide to ask a simple random sample of 160 students if they have experienced extreme levels of stress during the past month. Subsequently, they find that 10% of the sample replied yes to the question. Assuming the true proportion is 15%, which they tell us up here, they say 15% of the population of the 1,750 students actually have experienced extreme levels of stress during the past month. So that is the true proportion. So let me just write that. The true proportion for our population is 0.15."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "Assuming the true proportion is 15%, which they tell us up here, they say 15% of the population of the 1,750 students actually have experienced extreme levels of stress during the past month. So that is the true proportion. So let me just write that. The true proportion for our population is 0.15. What is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month? So pause this video and see if you can answer it on your own, and there are four choices here. I'll scroll down a little bit and see if you can answer this on your own."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "The true proportion for our population is 0.15. What is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month? So pause this video and see if you can answer it on your own, and there are four choices here. I'll scroll down a little bit and see if you can answer this on your own. So the way that we're going to tackle this is we're gonna think about the sampling distribution of our sample proportions. And first, we're gonna say, well, is this sampling distribution approximately normal? Is it approximately normal?"}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "I'll scroll down a little bit and see if you can answer this on your own. So the way that we're going to tackle this is we're gonna think about the sampling distribution of our sample proportions. And first, we're gonna say, well, is this sampling distribution approximately normal? Is it approximately normal? And if it is, then we can use its mean and its standard deviation and create a normal distribution that has that same mean and standard deviation in order to approximate the probability that they're asking for. So first, this first part, how do we decide this? Well, the rule of thumb we have here, and it is a rule of thumb, is that if we take our sample size times our population proportion, and that is greater than or equal to 10, and our sample size times one minus our population proportion is greater than or equal to 10, then if both of these are true, then our sampling distribution of our sample proportions is going to be approximately normal."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "Is it approximately normal? And if it is, then we can use its mean and its standard deviation and create a normal distribution that has that same mean and standard deviation in order to approximate the probability that they're asking for. So first, this first part, how do we decide this? Well, the rule of thumb we have here, and it is a rule of thumb, is that if we take our sample size times our population proportion, and that is greater than or equal to 10, and our sample size times one minus our population proportion is greater than or equal to 10, then if both of these are true, then our sampling distribution of our sample proportions is going to be approximately normal. So in this case, the newspaper is asking 160 students. That's the sample size. So 160, the true population proportion is 0.15, and that needs to be greater than or equal to 10."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "Well, the rule of thumb we have here, and it is a rule of thumb, is that if we take our sample size times our population proportion, and that is greater than or equal to 10, and our sample size times one minus our population proportion is greater than or equal to 10, then if both of these are true, then our sampling distribution of our sample proportions is going to be approximately normal. So in this case, the newspaper is asking 160 students. That's the sample size. So 160, the true population proportion is 0.15, and that needs to be greater than or equal to 10. And so let's see, this is going to be 16 plus eight, which is 24, and 24 is indeed greater than or equal to 10, so that checks out. And then if I take our sample size times one minus P, well, one minus 15 hundredths is going to be 85 hundredths. And this is definitely going to be greater than or equal to 10."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "So 160, the true population proportion is 0.15, and that needs to be greater than or equal to 10. And so let's see, this is going to be 16 plus eight, which is 24, and 24 is indeed greater than or equal to 10, so that checks out. And then if I take our sample size times one minus P, well, one minus 15 hundredths is going to be 85 hundredths. And this is definitely going to be greater than or equal to 10. Let's see, this is going to be 24 less than 160, so this is going to be 136, which is way larger than 10, so that checks out. And so the sampling distribution of our sample proportions is approximately going to be normal. And so what is the mean and standard deviation of our sampling distribution?"}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And this is definitely going to be greater than or equal to 10. Let's see, this is going to be 24 less than 160, so this is going to be 136, which is way larger than 10, so that checks out. And so the sampling distribution of our sample proportions is approximately going to be normal. And so what is the mean and standard deviation of our sampling distribution? So the mean of our sampling distribution is just going to be our population proportion. We've seen that in other videos, which is equal to 0.15. And our standard deviation of our sampling distribution, of our sample proportions, is going to be equal to the square root of P times one minus P over N, which is equal to the square root of 0.15 times 0.85, all of that over our sample size, 160."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And so what is the mean and standard deviation of our sampling distribution? So the mean of our sampling distribution is just going to be our population proportion. We've seen that in other videos, which is equal to 0.15. And our standard deviation of our sampling distribution, of our sample proportions, is going to be equal to the square root of P times one minus P over N, which is equal to the square root of 0.15 times 0.85, all of that over our sample size, 160. So now let's get our calculator out. So I'm gonna take the square root of 0.15 times 0.85 divided by 160, and let me close those parentheses. And so what is this going to give me?"}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And our standard deviation of our sampling distribution, of our sample proportions, is going to be equal to the square root of P times one minus P over N, which is equal to the square root of 0.15 times 0.85, all of that over our sample size, 160. So now let's get our calculator out. So I'm gonna take the square root of 0.15 times 0.85 divided by 160, and let me close those parentheses. And so what is this going to give me? So it's going to give me approximately 0.028, and I'll go to the thousands place here. So this is approximately 0.028. This is going to be approximately a normal distribution."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And so what is this going to give me? So it's going to give me approximately 0.028, and I'll go to the thousands place here. So this is approximately 0.028. This is going to be approximately a normal distribution. So you could draw your classic bell curve for a normal distribution, so something like this. And our normal distribution is going to have a mean. It's going to have a mean right over here of, so this is the mean of our sampling distribution."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "This is going to be approximately a normal distribution. So you could draw your classic bell curve for a normal distribution, so something like this. And our normal distribution is going to have a mean. It's going to have a mean right over here of, so this is the mean of our sampling distribution. So this is going to be equal to the same thing as our population proportion, 0.15. And we also know that our standard deviation here is going to be approximately equal to 0.028. And what we wanna know is what is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month?"}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "It's going to have a mean right over here of, so this is the mean of our sampling distribution. So this is going to be equal to the same thing as our population proportion, 0.15. And we also know that our standard deviation here is going to be approximately equal to 0.028. And what we wanna know is what is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month? So we could say that 10% would be right over here. I'll say 0.10. And so the probability that in a sample of 160, you get a proportion for that sample, a sample proportion that is larger than 10% would be this area right over here."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And what we wanna know is what is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month? So we could say that 10% would be right over here. I'll say 0.10. And so the probability that in a sample of 160, you get a proportion for that sample, a sample proportion that is larger than 10% would be this area right over here. So this right over here would be the probability that your sample proportion is greater than, they say is more than 10%, is more than 0.1. I could write one zero just like that. And then to calculate it, I can get out our calculator again."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And so the probability that in a sample of 160, you get a proportion for that sample, a sample proportion that is larger than 10% would be this area right over here. So this right over here would be the probability that your sample proportion is greater than, they say is more than 10%, is more than 0.1. I could write one zero just like that. And then to calculate it, I can get out our calculator again. So here I'm gonna go to my distribution menu right over there. And then I'm gonna do a normal cumulative distribution function. So let me click Enter there."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And then to calculate it, I can get out our calculator again. So here I'm gonna go to my distribution menu right over there. And then I'm gonna do a normal cumulative distribution function. So let me click Enter there. And so what is my lower bound? Well, my lower bound is 10%, 0.1. What is my upper bound?"}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "So let me click Enter there. And so what is my lower bound? Well, my lower bound is 10%, 0.1. What is my upper bound? Well, we'll just make this one because that is the highest proportion you could have for a sampling distribution of sample proportions. Now what is our mean? Well, we already know that's 0.15."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "What is my upper bound? Well, we'll just make this one because that is the highest proportion you could have for a sampling distribution of sample proportions. Now what is our mean? Well, we already know that's 0.15. What is the standard deviation of our sampling distribution? Well, it's approximately 0.028. And then I can click Enter."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "Well, we already know that's 0.15. What is the standard deviation of our sampling distribution? Well, it's approximately 0.028. And then I can click Enter. And if you're taking an AP exam, you actually should write this. You should say, you should tell the graders what you're actually typing in in your normal CDF function. But if we click Enter right over here, and then Enter, there we have it."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "And then I can click Enter. And if you're taking an AP exam, you actually should write this. You should say, you should tell the graders what you're actually typing in in your normal CDF function. But if we click Enter right over here, and then Enter, there we have it. It's approximately 96%. So this is approximately 0.96. And then out of our choices, it would be this one right over here."}, {"video_title": "Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3", "Sentence": "But if we click Enter right over here, and then Enter, there we have it. It's approximately 96%. So this is approximately 0.96. And then out of our choices, it would be this one right over here. If you're taking this on the AP exam, you would say that called, called normal, normal CDF, where you have your lower bound, lower bound, and you would put in your 0.10. You would say that you use an upper bound, upper bound of one. You would say that you gave a mean of 0.15, and then you gave a standard deviation of 0.028, just so people know that you knew what you were doing."}, {"video_title": "Constructing hypotheses for two proportions AP Statistics Khan Academy.mp3", "Sentence": "At the end of each month, he obtains data from a random sample of adults on whether or not they currently approve of the prime minister's performance, using a separate sample each month. Derek wants to test if the proportion of adults who approved was significantly lower in December than it was in November. Which of the following is an appropriate set of hypotheses for Derek's significance test? Pause this video and see if you can figure it out on your own. All right, so let's think about ways to write a null hypothesis first. So remember, your null hypothesis is assuming that there's no news here, there's no difference. So one way to say it is that the true proportion in December is equal to your true proportion in November."}, {"video_title": "Constructing hypotheses for two proportions AP Statistics Khan Academy.mp3", "Sentence": "Pause this video and see if you can figure it out on your own. All right, so let's think about ways to write a null hypothesis first. So remember, your null hypothesis is assuming that there's no news here, there's no difference. So one way to say it is that the true proportion in December is equal to your true proportion in November. Another way to write that exact same thing is to say that the difference between the true proportion in December, and let's see, they say Derek wants to test if the proportion of adults who approved was significantly lower in December than it was in November. And so you could write it as, because he's wanting to see if November is higher or not, I'll put November first. So another way to say this exact same thing is that the true proportion in November minus the true proportion in December is equal to zero."}, {"video_title": "Constructing hypotheses for two proportions AP Statistics Khan Academy.mp3", "Sentence": "So one way to say it is that the true proportion in December is equal to your true proportion in November. Another way to write that exact same thing is to say that the difference between the true proportion in December, and let's see, they say Derek wants to test if the proportion of adults who approved was significantly lower in December than it was in November. And so you could write it as, because he's wanting to see if November is higher or not, I'll put November first. So another way to say this exact same thing is that the true proportion in November minus the true proportion in December is equal to zero. So each of these would be legitimate null hypotheses. And so let's see, this one looks good, this one looks good, this one looks good. This one is not a legitimate null hypothesis for what we're trying to do, so we could rule out D. And then the other one is this, Derek wants to test if the proportion of adults who approved was significantly lower in December than it was in November."}, {"video_title": "Constructing hypotheses for two proportions AP Statistics Khan Academy.mp3", "Sentence": "So another way to say this exact same thing is that the true proportion in November minus the true proportion in December is equal to zero. So each of these would be legitimate null hypotheses. And so let's see, this one looks good, this one looks good, this one looks good. This one is not a legitimate null hypothesis for what we're trying to do, so we could rule out D. And then the other one is this, Derek wants to test if the proportion of adults who approved was significantly lower in December than it was in November. So the news here would be, if this actually is the case, if we have evidence that the proportion of adults who approved was significantly lower in December than it was in November. So the alternative hypothesis could look something like this, that the proportion in December was less than the proportion in November. Or it could be that the proportion in November is greater than the proportion in December."}, {"video_title": "Constructing hypotheses for two proportions AP Statistics Khan Academy.mp3", "Sentence": "This one is not a legitimate null hypothesis for what we're trying to do, so we could rule out D. And then the other one is this, Derek wants to test if the proportion of adults who approved was significantly lower in December than it was in November. So the news here would be, if this actually is the case, if we have evidence that the proportion of adults who approved was significantly lower in December than it was in November. So the alternative hypothesis could look something like this, that the proportion in December was less than the proportion in November. Or it could be that the proportion in November is greater than the proportion in December. And if we look at these choices, the proportion in December is less than the proportion in November, that's what I wrote right over here, so that looks good as well. Here they swapped it, here they're saying our alternative hypothesis is that the true proportion in December is more than the true proportion in November, which is the opposite of what's saying here, so we rule that one out. Here they're just saying that we actually have a difference in proportions."}, {"video_title": "Constructing hypotheses for two proportions AP Statistics Khan Academy.mp3", "Sentence": "Or it could be that the proportion in November is greater than the proportion in December. And if we look at these choices, the proportion in December is less than the proportion in November, that's what I wrote right over here, so that looks good as well. Here they swapped it, here they're saying our alternative hypothesis is that the true proportion in December is more than the true proportion in November, which is the opposite of what's saying here, so we rule that one out. Here they're just saying that we actually have a difference in proportions. And many times you will see something like this, but here Derek wants to test if the proportion of adults who approved was significantly lower in December than in November. He's not interested in the other way around. If it said Derek wants to test if the proportion of adults who approved was significantly different in December than November, then you would pick choice C instead of choice A."}, {"video_title": "Constructing hypotheses for two proportions AP Statistics Khan Academy.mp3", "Sentence": "Here they're just saying that we actually have a difference in proportions. And many times you will see something like this, but here Derek wants to test if the proportion of adults who approved was significantly lower in December than in November. He's not interested in the other way around. If it said Derek wants to test if the proportion of adults who approved was significantly different in December than November, then you would pick choice C instead of choice A. But given the way it was phrased, I would pick choice A. Let's do another example. Here it says that Kylie has a dime and a nickel, and she wonders if they have the same likelihood of showing heads when they are flipped."}, {"video_title": "Constructing hypotheses for two proportions AP Statistics Khan Academy.mp3", "Sentence": "If it said Derek wants to test if the proportion of adults who approved was significantly different in December than November, then you would pick choice C instead of choice A. But given the way it was phrased, I would pick choice A. Let's do another example. Here it says that Kylie has a dime and a nickel, and she wonders if they have the same likelihood of showing heads when they are flipped. She flips each coin 100 times to test if there is a significant difference in the proportion of flips that they each land showing heads, which of the following is an appropriate set of hypotheses for Kylie's significance test? So once again, pause the video. Try to do it on your own."}, {"video_title": "Constructing hypotheses for two proportions AP Statistics Khan Academy.mp3", "Sentence": "Here it says that Kylie has a dime and a nickel, and she wonders if they have the same likelihood of showing heads when they are flipped. She flips each coin 100 times to test if there is a significant difference in the proportion of flips that they each land showing heads, which of the following is an appropriate set of hypotheses for Kylie's significance test? So once again, pause the video. Try to do it on your own. All right, well, your null hypothesis would be that there is no difference, so that the proportion of getting heads with your dime is the same as the proportion of heads with your nickel. And then your alternative hypothesis, so it says here she wants to test if there is a significant difference. She's not trying to say if the proportion of dimes coming up head is significantly lower or significantly larger."}, {"video_title": "Constructing hypotheses for two proportions AP Statistics Khan Academy.mp3", "Sentence": "Try to do it on your own. All right, well, your null hypothesis would be that there is no difference, so that the proportion of getting heads with your dime is the same as the proportion of heads with your nickel. And then your alternative hypothesis, so it says here she wants to test if there is a significant difference. She's not trying to say if the proportion of dimes coming up head is significantly lower or significantly larger. She just cares about the difference, if there's a significant difference in the proportion of flips. So her alternative hypothesis is that there is a difference, that these two proportions are not equal to each other. And so if we look at the choices, so this null hypothesis looks good."}, {"video_title": "Constructing hypotheses for two proportions AP Statistics Khan Academy.mp3", "Sentence": "She's not trying to say if the proportion of dimes coming up head is significantly lower or significantly larger. She just cares about the difference, if there's a significant difference in the proportion of flips. So her alternative hypothesis is that there is a difference, that these two proportions are not equal to each other. And so if we look at the choices, so this null hypothesis looks good. This null hypothesis does not look good. Remember, your null hypothesis, you're trying to assume that, man, there's no news here. So all of these null hypotheses, these A, B, and D's null hypotheses look good."}, {"video_title": "Constructing hypotheses for two proportions AP Statistics Khan Academy.mp3", "Sentence": "And so if we look at the choices, so this null hypothesis looks good. This null hypothesis does not look good. Remember, your null hypothesis, you're trying to assume that, man, there's no news here. So all of these null hypotheses, these A, B, and D's null hypotheses look good. And then the alternative hypothesis, this is exactly what we wrote before, is for choice D. Choice A's alternative hypothesis would work if here it said she flips each coin 100 times to test if the proportion of heads with the dime is significantly lower than the proportion of heads with the nickel or something like that. And then if it was the reverse, then choice B would look good. But she just wants to see if there's a difference, not if one is lower than the other."}, {"video_title": "Calculating t statistic for slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "He noticed a positive linear relationship between the times on each task. Here is computer output on the sample data. So we have some statistics calculated on the reaction time, on the memory time, and then he had his computer do a regression for the data that he collected, and then we're told assume that all conditions for inference have been met, calculate the test statistic that should be used for testing a null hypothesis that the population slope is actually zero. So pause this video and have a go at it. All right, so let's just make sure we understand what is going on. So let's first think about the population. So I'll do that right over here."}, {"video_title": "Calculating t statistic for slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "So pause this video and have a go at it. All right, so let's just make sure we understand what is going on. So let's first think about the population. So I'll do that right over here. So in the population, there might be some true linear relationship. So in theory, on our x-axis, we would have our reaction time, and on our y-axis, you have your memory time. If you were able to plot every single possible data point, it might even be an infinite or near infinite, so it would be very hard to do it, but if there was just some truth in the universe that says, yes, there actually is a positive linear relationship, and it looks like this, and you could describe that regression line as y hat, it's a regression line, is equal to some true population parameter, which would be this y-intercept, so we could call that alpha, plus some true population parameter that would be the slope of this regression line, we could call that beta, times x."}, {"video_title": "Calculating t statistic for slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "So I'll do that right over here. So in the population, there might be some true linear relationship. So in theory, on our x-axis, we would have our reaction time, and on our y-axis, you have your memory time. If you were able to plot every single possible data point, it might even be an infinite or near infinite, so it would be very hard to do it, but if there was just some truth in the universe that says, yes, there actually is a positive linear relationship, and it looks like this, and you could describe that regression line as y hat, it's a regression line, is equal to some true population parameter, which would be this y-intercept, so we could call that alpha, plus some true population parameter that would be the slope of this regression line, we could call that beta, times x. Now, we don't know what this truth of the universe is, of the linear relationship between reaction time and memory time, but we can try to estimate it, and that's what Jian is trying to do. So he's taking a sample of 24, so samples, samples, 24 data, data points, and that's much easier to then, you could even visualize it on a scatter plot like this, so you'd have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24. You input those data points into a computer, and it does a regression line, and it's trying to minimize the squared distance to all of these points, and so let's say it gets a regression line that looks something like this, where this regression line can be described as some estimate of the true y-intercept, so this would actually be a statistic right over here that's estimating this parameter, plus some estimate of the true slope of the regression line, so this is just a statistic."}, {"video_title": "Calculating t statistic for slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "If you were able to plot every single possible data point, it might even be an infinite or near infinite, so it would be very hard to do it, but if there was just some truth in the universe that says, yes, there actually is a positive linear relationship, and it looks like this, and you could describe that regression line as y hat, it's a regression line, is equal to some true population parameter, which would be this y-intercept, so we could call that alpha, plus some true population parameter that would be the slope of this regression line, we could call that beta, times x. Now, we don't know what this truth of the universe is, of the linear relationship between reaction time and memory time, but we can try to estimate it, and that's what Jian is trying to do. So he's taking a sample of 24, so samples, samples, 24 data, data points, and that's much easier to then, you could even visualize it on a scatter plot like this, so you'd have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24. You input those data points into a computer, and it does a regression line, and it's trying to minimize the squared distance to all of these points, and so let's say it gets a regression line that looks something like this, where this regression line can be described as some estimate of the true y-intercept, so this would actually be a statistic right over here that's estimating this parameter, plus some estimate of the true slope of the regression line, so this is just a statistic. This b is just a statistic that is trying to estimate the true parameter beta. Now, when we went and inputted these data points into a computer, we got values for a and b right over here. A is equal to this, the constant coefficient, and then the reaction coefficient, this is just telling us, hey, for every incremental change in the reaction, how much would we expect the memory time to change, or for every change in x, how much would we expect for a change in y, so this is actually our estimate of the slope of the regression line."}, {"video_title": "Calculating t statistic for slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "You input those data points into a computer, and it does a regression line, and it's trying to minimize the squared distance to all of these points, and so let's say it gets a regression line that looks something like this, where this regression line can be described as some estimate of the true y-intercept, so this would actually be a statistic right over here that's estimating this parameter, plus some estimate of the true slope of the regression line, so this is just a statistic. This b is just a statistic that is trying to estimate the true parameter beta. Now, when we went and inputted these data points into a computer, we got values for a and b right over here. A is equal to this, the constant coefficient, and then the reaction coefficient, this is just telling us, hey, for every incremental change in the reaction, how much would we expect the memory time to change, or for every change in x, how much would we expect for a change in y, so this is actually our estimate of the slope of the regression line. Now, you could imagine, every time you take a different sample, you might get a different estimate of these things, and when we're doing inferential statistics, we set up hypotheses, you set up a null and an alternative hypothesis, and the null hypothesis is always the no news here, and no news, when you're dealing with regressions, is that even though you might suspect there's a positive linear relationship, even though you might see it in the data you got, it's, for your null hypothesis, you wanna assume that there is no positive linear relationship, so our null hypothesis here would be that the true slope of the true regression line, this, the parameter right over here, is equal to zero, so beta is equal to zero, so our null hypothesis is actually, might be that our true regression line might look something like this, that what y is is somewhat independent of what x is, and that if you suspect that there is a positive linear relationship, you could say something like, well, my alternative hypothesis is that my beta is greater than zero, or if you suspect that there's just some linear relationship, you don't know if it's positive or negative, then you might say that the beta is not equal to zero, but here it says he noticed, or he suspects, a positive linear relationship, so this would be his alternative hypothesis, but what you then want to do to test your null hypothesis, which we've done multiple, multiple times, is find a test statistic that is associated with the statistic for b that you actually got. Now, ideally, you would take your b, you would take your b, and from that, subtract the slope assumed in the null hypothesis, so the slope of the regression line you get, minus the slope that's assumed from the null hypothesis, and then divide by the standard deviation of the sampling distribution of the slope of the regression line, and if you did this, you would get a, it would be appropriate to use a z statistic over here. Now, the problem is is that we don't know exactly what the standard deviation of the sampling distribution is, but we can estimate it."}, {"video_title": "Calculating t statistic for slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "A is equal to this, the constant coefficient, and then the reaction coefficient, this is just telling us, hey, for every incremental change in the reaction, how much would we expect the memory time to change, or for every change in x, how much would we expect for a change in y, so this is actually our estimate of the slope of the regression line. Now, you could imagine, every time you take a different sample, you might get a different estimate of these things, and when we're doing inferential statistics, we set up hypotheses, you set up a null and an alternative hypothesis, and the null hypothesis is always the no news here, and no news, when you're dealing with regressions, is that even though you might suspect there's a positive linear relationship, even though you might see it in the data you got, it's, for your null hypothesis, you wanna assume that there is no positive linear relationship, so our null hypothesis here would be that the true slope of the true regression line, this, the parameter right over here, is equal to zero, so beta is equal to zero, so our null hypothesis is actually, might be that our true regression line might look something like this, that what y is is somewhat independent of what x is, and that if you suspect that there is a positive linear relationship, you could say something like, well, my alternative hypothesis is that my beta is greater than zero, or if you suspect that there's just some linear relationship, you don't know if it's positive or negative, then you might say that the beta is not equal to zero, but here it says he noticed, or he suspects, a positive linear relationship, so this would be his alternative hypothesis, but what you then want to do to test your null hypothesis, which we've done multiple, multiple times, is find a test statistic that is associated with the statistic for b that you actually got. Now, ideally, you would take your b, you would take your b, and from that, subtract the slope assumed in the null hypothesis, so the slope of the regression line you get, minus the slope that's assumed from the null hypothesis, and then divide by the standard deviation of the sampling distribution of the slope of the regression line, and if you did this, you would get a, it would be appropriate to use a z statistic over here. Now, the problem is is that we don't know exactly what the standard deviation of the sampling distribution is, but we can estimate it. We can calculate the slope that we got for our sample regression line, minus the slope we're assuming in our null hypothesis, which is going to be equal to zero, so we know what we're assuming, and we can calculate the standard error of the sampling distribution. In fact, our computer has already done it for us, and this is an estimate of this, and we know what number that is, so we know what all of these numbers are, but if you're using an estimate of the standard deviation of the sampling distribution, and we've seen this before when we've done inferential statistics using for means, it is appropriate to use a t statistic, but with that said, pause the video. What is this going to be equal to?"}, {"video_title": "Calculating t statistic for slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "Now, the problem is is that we don't know exactly what the standard deviation of the sampling distribution is, but we can estimate it. We can calculate the slope that we got for our sample regression line, minus the slope we're assuming in our null hypothesis, which is going to be equal to zero, so we know what we're assuming, and we can calculate the standard error of the sampling distribution. In fact, our computer has already done it for us, and this is an estimate of this, and we know what number that is, so we know what all of these numbers are, but if you're using an estimate of the standard deviation of the sampling distribution, and we've seen this before when we've done inferential statistics using for means, it is appropriate to use a t statistic, but with that said, pause the video. What is this going to be equal to? Well, this is going to be equal to the slope for our sample regression line. We know it's 14.686, minus our assumed true population parameter, the slope of the true regression line. Well, we're assuming that is zero, so minus zero, and then we divide that by the standard error, which is going to be, we could view this as a standard error for B, and so this is divided by 13.329, so it's just gonna be 14.686 divided by 13.329, and if we assume, if we're doing a one-sided test here, what we would then do is take this t statistic and think about the degrees of freedom, and then say, and then calculate a p value."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "What I want to do in this video is review much of what we've already talked about and then hopefully build some of the intuition on why we divide by n minus 1 if we want to have an unbiased estimate of the population variance when we're calculating the sample variance. So let's think about a population. So let's say this is the population right over here and it is of size capital N. And we also have a sample of that population. So a sample of that population and at its size we have lowercase n data points. So let's think about all of the parameters and statistics that we know about so far. So the first is the idea of the mean. So if we're trying to calculate the mean for the population, is that going to be a parameter or a statistic?"}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "So a sample of that population and at its size we have lowercase n data points. So let's think about all of the parameters and statistics that we know about so far. So the first is the idea of the mean. So if we're trying to calculate the mean for the population, is that going to be a parameter or a statistic? Well, when we're trying to calculate it on the population, we are calculating a parameter. So let me write this down. So this is going to be, so for the population, we are calculating a parameter."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "So if we're trying to calculate the mean for the population, is that going to be a parameter or a statistic? Well, when we're trying to calculate it on the population, we are calculating a parameter. So let me write this down. So this is going to be, so for the population, we are calculating a parameter. And when we attempt to calculate something for a sample, we would call that a statistic. So how do we think about the mean for a population? Well, first of all, we denote it with the Greek letter mu."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "So this is going to be, so for the population, we are calculating a parameter. And when we attempt to calculate something for a sample, we would call that a statistic. So how do we think about the mean for a population? Well, first of all, we denote it with the Greek letter mu. And we essentially take every data point in our population. So we take the sum of every data point. So we start at the first data point and we go all the way to the capital Nth data point."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "Well, first of all, we denote it with the Greek letter mu. And we essentially take every data point in our population. So we take the sum of every data point. So we start at the first data point and we go all the way to the capital Nth data point. So every data point we add up. So this is the ith data point. So X sub 1 plus X sub 2 all the way to X sub capital N. And then we divide by the total number of data points we have."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "So we start at the first data point and we go all the way to the capital Nth data point. So every data point we add up. So this is the ith data point. So X sub 1 plus X sub 2 all the way to X sub capital N. And then we divide by the total number of data points we have. Well, how do we calculate the sample mean? Well, the sample mean, we do a very similar thing with the sample. And we denote it with an X with a bar over it."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "So X sub 1 plus X sub 2 all the way to X sub capital N. And then we divide by the total number of data points we have. Well, how do we calculate the sample mean? Well, the sample mean, we do a very similar thing with the sample. And we denote it with an X with a bar over it. And that's going to be taking every data point in the sample, so going up to lowercase n, adding them up. So these are the sum of all the data points in our sample. And then dividing by the number of data points that we actually had."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "And we denote it with an X with a bar over it. And that's going to be taking every data point in the sample, so going up to lowercase n, adding them up. So these are the sum of all the data points in our sample. And then dividing by the number of data points that we actually had. Now, the other thing that we're trying to calculate for the population, which was a parameter, and then we'll also try to calculate it for the sample and estimate it for the population, was the variance, which was a measure of how dispersed or how much the data points vary from the mean. So let's write variance right over here. And how do we denote and calculate variance for a population?"}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "And then dividing by the number of data points that we actually had. Now, the other thing that we're trying to calculate for the population, which was a parameter, and then we'll also try to calculate it for the sample and estimate it for the population, was the variance, which was a measure of how dispersed or how much the data points vary from the mean. So let's write variance right over here. And how do we denote and calculate variance for a population? Well, for a population, we'd say that the variance, we use the Greek letter sigma squared, is equal to, and you could view it as the mean of the squared distances from the population mean. But what we do is we take, for each data point, so I equal 1 all the way to N, we take that data point, subtract from it the population mean. So if you want to calculate this, you'd want to figure this out."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "And how do we denote and calculate variance for a population? Well, for a population, we'd say that the variance, we use the Greek letter sigma squared, is equal to, and you could view it as the mean of the squared distances from the population mean. But what we do is we take, for each data point, so I equal 1 all the way to N, we take that data point, subtract from it the population mean. So if you want to calculate this, you'd want to figure this out. Or that's one way to do it. We'll see there's other ways to do it, where you can kind of calculate them at the same time. But you would, the easiest or the most intuitive, calculate this first."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "So if you want to calculate this, you'd want to figure this out. Or that's one way to do it. We'll see there's other ways to do it, where you can kind of calculate them at the same time. But you would, the easiest or the most intuitive, calculate this first. And for each of the data points, take the data point and subtract it from that. Subtract the mean from that. Square it."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "But you would, the easiest or the most intuitive, calculate this first. And for each of the data points, take the data point and subtract it from that. Subtract the mean from that. Square it. And then divide by the total number of data points you have. Now we get to the interesting part, sample variance. There are several ways where, when people talk about sample variance, there are several, I guess, tools in their toolkits, or there are several ways to calculate it."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "Square it. And then divide by the total number of data points you have. Now we get to the interesting part, sample variance. There are several ways where, when people talk about sample variance, there are several, I guess, tools in their toolkits, or there are several ways to calculate it. One way is the biased sample variance, the non-unbiased estimator of the population variance. And that's denoted, usually denoted, by S with a subscript N. And what is the biased estimator? How would we calculate it?"}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "There are several ways where, when people talk about sample variance, there are several, I guess, tools in their toolkits, or there are several ways to calculate it. One way is the biased sample variance, the non-unbiased estimator of the population variance. And that's denoted, usually denoted, by S with a subscript N. And what is the biased estimator? How would we calculate it? Well, we would calculate it very similar to how we calculated the variance right over here, but we would do it for our sample, not our population. So for every data point in our sample, so we have N of them, we take that data point and from it we subtract our sample mean, we subtract our sample mean, square it, and then divide by the number of data points that we have. But we already talked about in the last video, how would we find what is our best unbiased estimate of the population variance?"}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "How would we calculate it? Well, we would calculate it very similar to how we calculated the variance right over here, but we would do it for our sample, not our population. So for every data point in our sample, so we have N of them, we take that data point and from it we subtract our sample mean, we subtract our sample mean, square it, and then divide by the number of data points that we have. But we already talked about in the last video, how would we find what is our best unbiased estimate of the population variance? This is usually what we're trying to get at. We're trying to find an unbiased estimate of the population variance. Well, in the last video we talked about that if we want to have an unbiased estimate, and here in this video I want to give you a sense of the intuition why, we would take the sum, so we're going to go through every data point in our sample, we're going to take that data point, subtract from it the sample mean, square that, but instead of dividing by N, we divide by N minus 1."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "But we already talked about in the last video, how would we find what is our best unbiased estimate of the population variance? This is usually what we're trying to get at. We're trying to find an unbiased estimate of the population variance. Well, in the last video we talked about that if we want to have an unbiased estimate, and here in this video I want to give you a sense of the intuition why, we would take the sum, so we're going to go through every data point in our sample, we're going to take that data point, subtract from it the sample mean, square that, but instead of dividing by N, we divide by N minus 1. We're dividing by a smaller number. And when you divide by a smaller number, you're going to get a larger value. So this is going to be larger, this is going to be larger, this is going to be smaller."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "Well, in the last video we talked about that if we want to have an unbiased estimate, and here in this video I want to give you a sense of the intuition why, we would take the sum, so we're going to go through every data point in our sample, we're going to take that data point, subtract from it the sample mean, square that, but instead of dividing by N, we divide by N minus 1. We're dividing by a smaller number. And when you divide by a smaller number, you're going to get a larger value. So this is going to be larger, this is going to be larger, this is going to be smaller. And this one we refer to the unbiased estimate. And this one we refer to the biased estimate. If people just write this, they're talking about the sample variance, it's a good idea to clarify which one they're talking about, but if you had to guess and people give you no further information, they're probably talking about the unbiased estimate of the variance."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "So this is going to be larger, this is going to be larger, this is going to be smaller. And this one we refer to the unbiased estimate. And this one we refer to the biased estimate. If people just write this, they're talking about the sample variance, it's a good idea to clarify which one they're talking about, but if you had to guess and people give you no further information, they're probably talking about the unbiased estimate of the variance. So you'd probably divide by N minus 1. But let's think about why this estimate would be biased, and why we might want to have an estimate like this that is larger, and then maybe in the future we could have a computer program or something that really makes us feel better that dividing by N minus 1 gives us a better estimate of the true population variance. So let's imagine all of the data in a population, and I'm just going to plot them on a number line."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "If people just write this, they're talking about the sample variance, it's a good idea to clarify which one they're talking about, but if you had to guess and people give you no further information, they're probably talking about the unbiased estimate of the variance. So you'd probably divide by N minus 1. But let's think about why this estimate would be biased, and why we might want to have an estimate like this that is larger, and then maybe in the future we could have a computer program or something that really makes us feel better that dividing by N minus 1 gives us a better estimate of the true population variance. So let's imagine all of the data in a population, and I'm just going to plot them on a number line. All the data. So this is my number line, this is my number line, and let me plot all of the data points in my population. So this is some data, this is some data, here's some data, and here is some data here, and I can just do as many points as I want."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "So let's imagine all of the data in a population, and I'm just going to plot them on a number line. All the data. So this is my number line, this is my number line, and let me plot all of the data points in my population. So this is some data, this is some data, here's some data, and here is some data here, and I can just do as many points as I want. So these are just points on the number line. Now let's say I take a sample of this. So this is my entire population."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "So this is some data, this is some data, here's some data, and here is some data here, and I can just do as many points as I want. So these are just points on the number line. Now let's say I take a sample of this. So this is my entire population. So let's see how many I have. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. So in this case, what would be my big N?"}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "So this is my entire population. So let's see how many I have. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. So in this case, what would be my big N? My big N would be 14. Now let's say I take a sample, a lowercase N of, let's say my sample size is 3. I could take, well, before I even think about that, let's think about roughly where the mean of this population would sit."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "So in this case, what would be my big N? My big N would be 14. Now let's say I take a sample, a lowercase N of, let's say my sample size is 3. I could take, well, before I even think about that, let's think about roughly where the mean of this population would sit. So the way I drew it, and I'm not going to calculate it exactly, it looks like the mean might sit someplace roughly right over here. So the mean, the true population mean, the parameter is going to sit right over here. Now let's think about what happens when we sample."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "I could take, well, before I even think about that, let's think about roughly where the mean of this population would sit. So the way I drew it, and I'm not going to calculate it exactly, it looks like the mean might sit someplace roughly right over here. So the mean, the true population mean, the parameter is going to sit right over here. Now let's think about what happens when we sample. And I'm going to do just a very small sample size just to give us the intuition, but this is true of any sample size. So let's say we have sample size of 3. So there is some possibility when we take our sample size of 3 that we happen to sample it in a way that our sample mean is pretty close to our population mean."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "Now let's think about what happens when we sample. And I'm going to do just a very small sample size just to give us the intuition, but this is true of any sample size. So let's say we have sample size of 3. So there is some possibility when we take our sample size of 3 that we happen to sample it in a way that our sample mean is pretty close to our population mean. So for example, if we sample to that point, that point, and that point, I could imagine our sample mean might actually sit pretty close to our population mean. But there's a distinct possibility that maybe when I take a sample, I sample that, that, and that. And the key idea here is when you take a sample, your sample mean is always going to sit within your sample."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "So there is some possibility when we take our sample size of 3 that we happen to sample it in a way that our sample mean is pretty close to our population mean. So for example, if we sample to that point, that point, and that point, I could imagine our sample mean might actually sit pretty close to our population mean. But there's a distinct possibility that maybe when I take a sample, I sample that, that, and that. And the key idea here is when you take a sample, your sample mean is always going to sit within your sample. And so there is a possibility that when you take your sample, your mean could even be outside of the sample. And so in this situation, and this is just to give you an intuition, so here your sample mean is going to be sitting someplace in there. And so if you were to just calculate the distance from each of these points to the sample mean, so this distance, that distance, and you square it, and you were to divide by the number of data points you have, this is going to be a much lower estimate than the true variance from the actual population mean, where these things are much, much, much further."}, {"video_title": "Review and intuition why we divide by n-1 for the unbiased sample Khan Academy.mp3", "Sentence": "And the key idea here is when you take a sample, your sample mean is always going to sit within your sample. And so there is a possibility that when you take your sample, your mean could even be outside of the sample. And so in this situation, and this is just to give you an intuition, so here your sample mean is going to be sitting someplace in there. And so if you were to just calculate the distance from each of these points to the sample mean, so this distance, that distance, and you square it, and you were to divide by the number of data points you have, this is going to be a much lower estimate than the true variance from the actual population mean, where these things are much, much, much further. Now you're always not going to have the true population mean outside of your sample, but it's possible that you do. So in general, when you just take your points, find the square to distance to your sample mean, which is always going to sit inside of your data, even though the true population mean could be outside of it, or it could be at one end of your data, however you might want to think about it, you are likely to be underestimating the true population variance. So this right over here is an underestimate."}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "So I have two different random variables here, and what I want to do is think about what type of random variables they are. So this first random variable, x, it's equal to the number of sixes after 12 rolls of a fair die. Well, this looks pretty much like a binomial random variable. In fact, I'm pretty confident it is a binomial random variable, and we could just go down the checklist. The outcome of each trial can be a success or failure. So trial, outcome, success, or failure. It's either gonna go either way."}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "In fact, I'm pretty confident it is a binomial random variable, and we could just go down the checklist. The outcome of each trial can be a success or failure. So trial, outcome, success, or failure. It's either gonna go either way. The result of each trial is independent from the other ones. Whether I get a six on the third trial is independent on whether I got a six on the first or the second trial. So result, let me write this trial, I'll just do a shorthand trial, results, results, independent."}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "It's either gonna go either way. The result of each trial is independent from the other ones. Whether I get a six on the third trial is independent on whether I got a six on the first or the second trial. So result, let me write this trial, I'll just do a shorthand trial, results, results, independent. Independent, that's an important condition. Let's see, there are a fixed number of trials, fixed number of trials. In this case, we're gonna have 12 trials."}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "So result, let me write this trial, I'll just do a shorthand trial, results, results, independent. Independent, that's an important condition. Let's see, there are a fixed number of trials, fixed number of trials. In this case, we're gonna have 12 trials. And then the last one is we have the same probability on each trial. Same probability of success. Probability on each trial."}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "In this case, we're gonna have 12 trials. And then the last one is we have the same probability on each trial. Same probability of success. Probability on each trial. So yes, indeed, this met all the conditions for being a binomial, binomial random, random variable. And this was all just a little bit of review about things that we have talked about in other videos. But what about this thing in the salmon color, the random variable Y?"}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "Probability on each trial. So yes, indeed, this met all the conditions for being a binomial, binomial random, random variable. And this was all just a little bit of review about things that we have talked about in other videos. But what about this thing in the salmon color, the random variable Y? So this says the number of rolls until we get a six on a fair die. So this one strikes us as a little bit different, but let's see where it is actually different. So does it meet that the trial outcomes, that there's a clear success or failure for each trial?"}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "But what about this thing in the salmon color, the random variable Y? So this says the number of rolls until we get a six on a fair die. So this one strikes us as a little bit different, but let's see where it is actually different. So does it meet that the trial outcomes, that there's a clear success or failure for each trial? Well, yeah, we're just gonna keep rolling. So each time we roll, it's a trial. And success is when we get a six, failure is when we don't get a six."}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "So does it meet that the trial outcomes, that there's a clear success or failure for each trial? Well, yeah, we're just gonna keep rolling. So each time we roll, it's a trial. And success is when we get a six, failure is when we don't get a six. So the outcome of each trial can be classified as either a success or failure. So it meets, and let me put the checks right over here, it meets this first constraint. Are the results of each trial independent?"}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "And success is when we get a six, failure is when we don't get a six. So the outcome of each trial can be classified as either a success or failure. So it meets, and let me put the checks right over here, it meets this first constraint. Are the results of each trial independent? Well, whether I get a six on the first roll or the second roll or the third roll or the fourth roll or the third roll, the probabilities shouldn't be dependent on whether I did or didn't get a six on a previous roll. So we have the independence. And we also have the same probability of success on each trial."}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "Are the results of each trial independent? Well, whether I get a six on the first roll or the second roll or the third roll or the fourth roll or the third roll, the probabilities shouldn't be dependent on whether I did or didn't get a six on a previous roll. So we have the independence. And we also have the same probability of success on each trial. In every case, it's a 1 6th probability that I get a six. So this stays constant. And I skipped this third condition for a reason."}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "And we also have the same probability of success on each trial. In every case, it's a 1 6th probability that I get a six. So this stays constant. And I skipped this third condition for a reason. Because we clearly don't have a fixed number of trials. Over here, we could roll 50 times until we get a six. The probability that we'd have to roll 50 times is very low, but we might have to roll 500 times in order to get a six."}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "And I skipped this third condition for a reason. Because we clearly don't have a fixed number of trials. Over here, we could roll 50 times until we get a six. The probability that we'd have to roll 50 times is very low, but we might have to roll 500 times in order to get a six. In fact, think about what the minimum value of y is and what the maximum value of y is. So the minimum value that this random variable can take, I'll just call it min y, is equal to what? Well, it's gonna take at least one roll, so that's the minimum value."}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "The probability that we'd have to roll 50 times is very low, but we might have to roll 500 times in order to get a six. In fact, think about what the minimum value of y is and what the maximum value of y is. So the minimum value that this random variable can take, I'll just call it min y, is equal to what? Well, it's gonna take at least one roll, so that's the minimum value. But what is the maximum value for y? And I'll let you think about that. Well, I've assumed you've thought about it if you paused the video."}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, it's gonna take at least one roll, so that's the minimum value. But what is the maximum value for y? And I'll let you think about that. Well, I've assumed you've thought about it if you paused the video. Well, there is no max value. You can't say, oh, it's a billion, because there's some probability that it might take a billion and one rolls. That is a very, very, very, very, very, very small probability, but there's some probability it could take a Google rolls, a Googleplex rolls, so you can imagine where this is going."}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, I've assumed you've thought about it if you paused the video. Well, there is no max value. You can't say, oh, it's a billion, because there's some probability that it might take a billion and one rolls. That is a very, very, very, very, very, very small probability, but there's some probability it could take a Google rolls, a Googleplex rolls, so you can imagine where this is going. So this type of random variable, where it meets a lot of the constraints of a binomial random variable, each trial has a clear success or failure outcome, the probability of success on each trial is constant, the trial results are independent of each other, but we don't have a fixed number of trials. In fact, it's a situation where we're saying, how many trials do we need to get, do we need to have until we get success? Maybe that's a general way of framing this type of random variable."}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "That is a very, very, very, very, very, very small probability, but there's some probability it could take a Google rolls, a Googleplex rolls, so you can imagine where this is going. So this type of random variable, where it meets a lot of the constraints of a binomial random variable, each trial has a clear success or failure outcome, the probability of success on each trial is constant, the trial results are independent of each other, but we don't have a fixed number of trials. In fact, it's a situation where we're saying, how many trials do we need to get, do we need to have until we get success? Maybe that's a general way of framing this type of random variable. How many trials until success, while the binomial random variable was, how many trials, or how many successes, I should say, how many successes in finite number of trials? So if you see this general form and it meets these conditions, you can feel good it's a binomial random variable, but if you're meeting this condition, clear success or failure outcome, independent trials, constant probability, but we're not talking about the successes in a finite number of trials, we're talking about how many trials until success, then this type of random variable is called a geometric, geometric random variable. And we will see why in future videos it is called geometric because the math that involves the probabilities of various outcomes looks a lot like geometric growth or geometric sequences and series that we look at in other types of mathematics."}, {"video_title": "Geometric random variables introduction Random variables AP Statistics Khan Academy.mp3", "Sentence": "Maybe that's a general way of framing this type of random variable. How many trials until success, while the binomial random variable was, how many trials, or how many successes, I should say, how many successes in finite number of trials? So if you see this general form and it meets these conditions, you can feel good it's a binomial random variable, but if you're meeting this condition, clear success or failure outcome, independent trials, constant probability, but we're not talking about the successes in a finite number of trials, we're talking about how many trials until success, then this type of random variable is called a geometric, geometric random variable. And we will see why in future videos it is called geometric because the math that involves the probabilities of various outcomes looks a lot like geometric growth or geometric sequences and series that we look at in other types of mathematics. And in case I forgot to mention, the reason why we call binomial random variables is because when you think about the probabilities of different outcomes, you have these things called binomial coefficients based on combinatorics, and those come out of things like Pascal's triangle and when you take a binomial to ever-increasing powers. So that's where those words come from. But in the next few videos, the important thing is to recognize the difference between the two, and then we're gonna start thinking about how do we deal with geometric random variables."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "And what I'm going to ask you is, which of these intervals, interval A, B, or C, which one contains the median of the scores, and which one, or give an estimate of which one contains the mean of the scores? Pause this video and see if you can figure that out. So let's just start with the median. Remember, the median you could view as the middle number, or if you have an even number of data points, it would be the average of the middle two. Here we have an odd number of data points, so it would be the middle number. So what would be the middle number if you were to order them from least to greatest? Well, it would be the one that has 15 on either side, so it would be the 16th data point."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "Remember, the median you could view as the middle number, or if you have an even number of data points, it would be the average of the middle two. Here we have an odd number of data points, so it would be the middle number. So what would be the middle number if you were to order them from least to greatest? Well, it would be the one that has 15 on either side, so it would be the 16th data point. 16th data point. And so we could just think about which interval here contains the 16th data point. You could view it for the 16th from the highest, or the 16th from the lowest."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "Well, it would be the one that has 15 on either side, so it would be the 16th data point. 16th data point. And so we could just think about which interval here contains the 16th data point. You could view it for the 16th from the highest, or the 16th from the lowest. It is the middle one. All right, so let's start from the highest. So this interval C contains the 13 highest data points, and then interval B goes from the 14th highest all the way to the 18th highest."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "You could view it for the 16th from the highest, or the 16th from the lowest. It is the middle one. All right, so let's start from the highest. So this interval C contains the 13 highest data points, and then interval B goes from the 14th highest all the way to the 18th highest. So this B contains the median. It contains the 16th highest data point, or if you started from the left, it would also be the 16th lowest data point. So that's where the median is, the median."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "So this interval C contains the 13 highest data points, and then interval B goes from the 14th highest all the way to the 18th highest. So this B contains the median. It contains the 16th highest data point, or if you started from the left, it would also be the 16th lowest data point. So that's where the median is, the median. Now what about an estimate for the mean? Well, you have calculated the mean in the past, but when you're looking at a distribution like this, when you're looking at a histogram, one way to think about the mean is it would be the balancing point. If you imagine that this histogram was made out of some material of, let's say, uniform density, where would you put a fulcrum in order to balance it?"}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "So that's where the median is, the median. Now what about an estimate for the mean? Well, you have calculated the mean in the past, but when you're looking at a distribution like this, when you're looking at a histogram, one way to think about the mean is it would be the balancing point. If you imagine that this histogram was made out of some material of, let's say, uniform density, where would you put a fulcrum in order to balance it? If you put the fulcrum right over here, it feels like you would have, it feels like you would tip over to the left because this is a left-skewed distribution. You have this long tail to the left. If you really wanted to balance it out, it seems like you would have to move your fulcrum in the direction of that left skew, in the direction of the tail."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "If you imagine that this histogram was made out of some material of, let's say, uniform density, where would you put a fulcrum in order to balance it? If you put the fulcrum right over here, it feels like you would have, it feels like you would tip over to the left because this is a left-skewed distribution. You have this long tail to the left. If you really wanted to balance it out, it seems like you would have to move your fulcrum in the direction of that left skew, in the direction of the tail. And so I would estimate to balance it out, it would actually be closer to that, which would be interval A. Interval A would contain the mean. The intention of this type of exercise isn't for you to try to calculate every data point. In fact, they don't give you all the information here and add them all up and then divide by 31."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "If you really wanted to balance it out, it seems like you would have to move your fulcrum in the direction of that left skew, in the direction of the tail. And so I would estimate to balance it out, it would actually be closer to that, which would be interval A. Interval A would contain the mean. The intention of this type of exercise isn't for you to try to calculate every data point. In fact, they don't give you all the information here and add them all up and then divide by 31. It's really to estimate and to also get the intuition that when you have a left-skewed distribution like this, you will often see a situation where your mean is to the left of the median. If you have a right-skewed distribution, it would be the other way around. And as we will see, when you see a symmetric distribution, the mean and the median will be awfully close to each other or when you have a roughly symmetric distribution."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "In fact, they don't give you all the information here and add them all up and then divide by 31. It's really to estimate and to also get the intuition that when you have a left-skewed distribution like this, you will often see a situation where your mean is to the left of the median. If you have a right-skewed distribution, it would be the other way around. And as we will see, when you see a symmetric distribution, the mean and the median will be awfully close to each other or when you have a roughly symmetric distribution. If you have a perfectly symmetric distribution, they might be exactly in the same place. So let's do another example. So here it says we have the ages of 14 coworkers and what I want you to do is say roughly where is the mean and roughly where is the median?"}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "And as we will see, when you see a symmetric distribution, the mean and the median will be awfully close to each other or when you have a roughly symmetric distribution. If you have a perfectly symmetric distribution, they might be exactly in the same place. So let's do another example. So here it says we have the ages of 14 coworkers and what I want you to do is say roughly where is the mean and roughly where is the median? Is it roughly at A, is it roughly at B, or is it roughly at C? Pause this video and try to figure it out. So let's first start off with the median."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "So here it says we have the ages of 14 coworkers and what I want you to do is say roughly where is the mean and roughly where is the median? Is it roughly at A, is it roughly at B, or is it roughly at C? Pause this video and try to figure it out. So let's first start off with the median. We have 14 data points. So this would be the average of the middle two data points. It would be the average of the seventh and eighth data point."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "So let's first start off with the median. We have 14 data points. So this would be the average of the middle two data points. It would be the average of the seventh and eighth data point. Well, you could say one, two, three, four, five, six, seven, and then the eighth one is here. So the seventh data point is a 30. The eighth one is in the 31 bucket."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "It would be the average of the seventh and eighth data point. Well, you could say one, two, three, four, five, six, seven, and then the eighth one is here. So the seventh data point is a 30. The eighth one is in the 31 bucket. So the average of the two would get you to B. Another way that you could think about it is you can just eyeball it and see you have just as many data points below B as you do have above B and so that also gives you a good indication that B would be where the median is. So that is where the median is."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "The eighth one is in the 31 bucket. So the average of the two would get you to B. Another way that you could think about it is you can just eyeball it and see you have just as many data points below B as you do have above B and so that also gives you a good indication that B would be where the median is. So that is where the median is. Now what about the mean? Well, this is a perfectly symmetric distribution. If I wanted to balance it, I would put the fulcrum right in the middle."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let's say that you have a cherry pie store, and you've noticed that there is variability in the number of cherries on each pie that you sell. Some pies might have over 100 cherries, while other pies might have fewer than 50 cherries. So what you're curious about is what is the distribution? How many of the different types of pies do you have? How many pies do you have that have a lot of cherries? How many pies do you have that have very few cherries? How many pies are in between?"}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "How many of the different types of pies do you have? How many pies do you have that have a lot of cherries? How many pies do you have that have very few cherries? How many pies are in between? And so to do that, you set up a histogram. What you do is you take each pie in your store. Let's see if I can draw a pie of some kind."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "How many pies are in between? And so to do that, you set up a histogram. What you do is you take each pie in your store. Let's see if I can draw a pie of some kind. It's a cherry pie. I don't know if this is an adequate drawing of a pie. But you take each of the pies in your store, and you count the number of cherries on it."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let's see if I can draw a pie of some kind. It's a cherry pie. I don't know if this is an adequate drawing of a pie. But you take each of the pies in your store, and you count the number of cherries on it. So this pie right over here is one, two, three, four, five, six, seven, eight, nine, ten. Let's see, you keep counting, and let's say it has 32 cherries. And you do that for every pie."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "But you take each of the pies in your store, and you count the number of cherries on it. So this pie right over here is one, two, three, four, five, six, seven, eight, nine, ten. Let's see, you keep counting, and let's say it has 32 cherries. And you do that for every pie. And then you created buckets, because you don't want to create just a graph of how many have exactly 32. You just want to get a general sense of things. So you create buckets of 30."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And you do that for every pie. And then you created buckets, because you don't want to create just a graph of how many have exactly 32. You just want to get a general sense of things. So you create buckets of 30. You say, how many pies have between zero and 29 cherries? How many pies have between 30 and 59, including 30 and 59? How many pies have at least 60 and at most 89 cherries?"}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So you create buckets of 30. You say, how many pies have between zero and 29 cherries? How many pies have between 30 and 59, including 30 and 59? How many pies have at least 60 and at most 89 cherries? How many pies have at least 90 and at most 119? And then how many pies have at least 120 and at most 149? And you know that you don't have any pies that have more than 149 cherries."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "How many pies have at least 60 and at most 89 cherries? How many pies have at least 90 and at most 119? And then how many pies have at least 120 and at most 149? And you know that you don't have any pies that have more than 149 cherries. So this should account for everything. And then you count them. So for example, you say, okay, five pies have 30 to 59 cherries."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And you know that you don't have any pies that have more than 149 cherries. So this should account for everything. And then you count them. So for example, you say, okay, five pies have 30 to 59 cherries. And so we create a histogram, or you create a histogram, and you make this magenta bar go up to five. So that's how you would construct this histogram. That's what this pies at different cherry levels histogram is telling us."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So for example, you say, okay, five pies have 30 to 59 cherries. And so we create a histogram, or you create a histogram, and you make this magenta bar go up to five. So that's how you would construct this histogram. That's what this pies at different cherry levels histogram is telling us. So now that we know how to construct it, let's see if we can interpret it based on the information given in the histogram. So the first question is, based on just this information, can you figure out the total number of pies in your store, assuming that they're all accounted for by this histogram? And I encourage you to pause the video and try to figure it out on your own."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "That's what this pies at different cherry levels histogram is telling us. So now that we know how to construct it, let's see if we can interpret it based on the information given in the histogram. So the first question is, based on just this information, can you figure out the total number of pies in your store, assuming that they're all accounted for by this histogram? And I encourage you to pause the video and try to figure it out on your own. Well, what's the total number of pies? Well, let's see. There's five pies that have more than, or 30 or more, have at least 30 cherries, but no more than 59."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And I encourage you to pause the video and try to figure it out on your own. Well, what's the total number of pies? Well, let's see. There's five pies that have more than, or 30 or more, have at least 30 cherries, but no more than 59. You have eight pies in this blue bucket. You have four pies in this green bucket. And then you have, what is this, five, no, this is three pies that have at least 120, but no more than 149 cherries."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "There's five pies that have more than, or 30 or more, have at least 30 cherries, but no more than 59. You have eight pies in this blue bucket. You have four pies in this green bucket. And then you have, what is this, five, no, this is three pies that have at least 120, but no more than 149 cherries. And this accounts for all of the pies. So the total number of pies you have at this store are five plus eight plus four plus three, which is what? Five plus eight is 13, plus four is 17, plus three is 20."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then you have, what is this, five, no, this is three pies that have at least 120, but no more than 149 cherries. And this accounts for all of the pies. So the total number of pies you have at this store are five plus eight plus four plus three, which is what? Five plus eight is 13, plus four is 17, plus three is 20. So there are 20 pies in this store. But then you can ask more nuanced questions. What if you wanted to know the number of pies with more than 60 cherries?"}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Five plus eight is 13, plus four is 17, plus three is 20. So there are 20 pies in this store. But then you can ask more nuanced questions. What if you wanted to know the number of pies with more than 60 cherries? The number of pies with more than 60. So number of pies with, I'll say, let's say 60 or more. 60 or more, 60 or more cherries."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "What if you wanted to know the number of pies with more than 60 cherries? The number of pies with more than 60. So number of pies with, I'll say, let's say 60 or more. 60 or more, 60 or more cherries. So let's think about it. Well, this magenta bar doesn't apply because these all have less than 60, but all of these other bars are counting pies that have 60 or more cherries. This is 60 to 89, this is 90 to 119, this is 120 to 149."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "60 or more, 60 or more cherries. So let's think about it. Well, this magenta bar doesn't apply because these all have less than 60, but all of these other bars are counting pies that have 60 or more cherries. This is 60 to 89, this is 90 to 119, this is 120 to 149. So it's going to be these eight cherries that are, sorry, these eight pies that are in this bucket plus these four pies, plus these three pies. So it is going to be essentially everything but this first bucket. Everything but all the pies except for these five pies have 60 or more cherries."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "This is 60 to 89, this is 90 to 119, this is 120 to 149. So it's going to be these eight cherries that are, sorry, these eight pies that are in this bucket plus these four pies, plus these three pies. So it is going to be essentially everything but this first bucket. Everything but all the pies except for these five pies have 60 or more cherries. So it should be five less than 20, and so let's see, eight plus four is 12, plus three is 15, which is five less than 20. So using this histogram, we can answer a really interesting question. We can say, well, wait, how many more pies do we have that have 60 to 89 cherries than 120 to 149 cherries?"}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Everything but all the pies except for these five pies have 60 or more cherries. So it should be five less than 20, and so let's see, eight plus four is 12, plus three is 15, which is five less than 20. So using this histogram, we can answer a really interesting question. We can say, well, wait, how many more pies do we have that have 60 to 89 cherries than 120 to 149 cherries? Well, we say, well, we have eight pies that have 60 to 89 cherries, three that have 120 to 149. So we have five more pies in the 60 to 89 category than we do in the 120 to 149 category. So a lot of questions that we can start to answer, and hopefully this gives you a sense of how you can interpret histograms."}, {"video_title": "Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3", "Sentence": "This one has a population of 383, and then it calculates the parameters for that population directly from it. The mean is 10.9, the variance is 25.5. And then it uses that population and samples from it, and it does samples of size two, three, four, five, all the way up to 10, and it keeps sampling from it, calculates the statistics for those samples, so the sample mean and the sample variance, in particular the biased sample variance, and it starts telling us some things about us that give us some intuition. And you can actually click on each of these and zoom in to really be able to study these graphs in detail. So I've already taken a screenshot of this and put it on my little doodle pad, so it can really delve into some of the math and the intuition of what this is actually showing us. So here I took a screenshot, and you see for this case right over here, the population was 529, population mean was 10.6. And down here in this chart, he plots the population mean right here at 10.6, right over there, and you see that the population variance is at 36.8, and right here he plots that right over here, 36.8."}, {"video_title": "Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3", "Sentence": "And you can actually click on each of these and zoom in to really be able to study these graphs in detail. So I've already taken a screenshot of this and put it on my little doodle pad, so it can really delve into some of the math and the intuition of what this is actually showing us. So here I took a screenshot, and you see for this case right over here, the population was 529, population mean was 10.6. And down here in this chart, he plots the population mean right here at 10.6, right over there, and you see that the population variance is at 36.8, and right here he plots that right over here, 36.8. So this first chart on the bottom left tells us a couple of interesting things. And just to be clear, this is the biased sample variance that he's calculating. This is the biased sample variance."}, {"video_title": "Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3", "Sentence": "And down here in this chart, he plots the population mean right here at 10.6, right over there, and you see that the population variance is at 36.8, and right here he plots that right over here, 36.8. So this first chart on the bottom left tells us a couple of interesting things. And just to be clear, this is the biased sample variance that he's calculating. This is the biased sample variance. So he's calculating it. That is being calculated for each of our data points, so starting with our first data point in each of our samples, going to our nth data point in the sample. You're taking that data point, subtracting out the sample mean, squaring it, and then dividing the whole thing not by n minus one, but by lowercase n. And this tells us several interesting things."}, {"video_title": "Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3", "Sentence": "This is the biased sample variance. So he's calculating it. That is being calculated for each of our data points, so starting with our first data point in each of our samples, going to our nth data point in the sample. You're taking that data point, subtracting out the sample mean, squaring it, and then dividing the whole thing not by n minus one, but by lowercase n. And this tells us several interesting things. The first thing it shows us is that the cases where we are significantly underestimating the sample variance, when we're getting sample variances close to zero, these are also the cases, these are also the cases, or they're disproportionately the cases, where the means for those samples are way far off from the true sample mean. Or you could view that the other way around. The cases where the mean is way far off from the sample mean, it seems like you're much more likely to underestimate the sample variance in those situations."}, {"video_title": "Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3", "Sentence": "You're taking that data point, subtracting out the sample mean, squaring it, and then dividing the whole thing not by n minus one, but by lowercase n. And this tells us several interesting things. The first thing it shows us is that the cases where we are significantly underestimating the sample variance, when we're getting sample variances close to zero, these are also the cases, these are also the cases, or they're disproportionately the cases, where the means for those samples are way far off from the true sample mean. Or you could view that the other way around. The cases where the mean is way far off from the sample mean, it seems like you're much more likely to underestimate the sample variance in those situations. The other thing that might pop out at you is the realization that the pinker dots are the ones for smaller sample size, while the bluer dots are the ones of a larger sample size. And you see here these two little, the two little, I guess the tails, so to speak, of this hump, that at these ends, you disproportionately, it's more of a reddish color. That most of the bluish or the purplish dots are focused right in the middle right over here, that they are giving us better estimates."}, {"video_title": "Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3", "Sentence": "The cases where the mean is way far off from the sample mean, it seems like you're much more likely to underestimate the sample variance in those situations. The other thing that might pop out at you is the realization that the pinker dots are the ones for smaller sample size, while the bluer dots are the ones of a larger sample size. And you see here these two little, the two little, I guess the tails, so to speak, of this hump, that at these ends, you disproportionately, it's more of a reddish color. That most of the bluish or the purplish dots are focused right in the middle right over here, that they are giving us better estimates. There are some red ones here, and that's why it gives us that purplish color. But out here on these tails, it's almost purely some of these red. Every now and then, by happenstance, you get a little blue one."}, {"video_title": "Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3", "Sentence": "That most of the bluish or the purplish dots are focused right in the middle right over here, that they are giving us better estimates. There are some red ones here, and that's why it gives us that purplish color. But out here on these tails, it's almost purely some of these red. Every now and then, by happenstance, you get a little blue one. But this is disproportionately far more red, which really makes sense. When you have a smaller sample size, you're more likely to get a sample mean that is a bad estimate of the population mean, that's far from the population mean, and you're more likely to significantly underestimate the sample variance. Now this next chart really gets to the meat, really gets to the meat of the issue, because what it's telling us is that for each of these sample sizes, so this right over here, for sample size two, if we keep taking sample size two and we keep calculating the by sample variances and dividing that by the population variance and finding the mean over all of those, you see that over many, many, many trials, many, many samples of size two, that that by sample variance over population variance, it's approaching half of the true population variance."}, {"video_title": "Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3", "Sentence": "Every now and then, by happenstance, you get a little blue one. But this is disproportionately far more red, which really makes sense. When you have a smaller sample size, you're more likely to get a sample mean that is a bad estimate of the population mean, that's far from the population mean, and you're more likely to significantly underestimate the sample variance. Now this next chart really gets to the meat, really gets to the meat of the issue, because what it's telling us is that for each of these sample sizes, so this right over here, for sample size two, if we keep taking sample size two and we keep calculating the by sample variances and dividing that by the population variance and finding the mean over all of those, you see that over many, many, many trials, many, many samples of size two, that that by sample variance over population variance, it's approaching half of the true population variance. When sample size is three, it's approaching 2 3rds, 66.6% of the true population variance. When sample size is four, it's approaching 3 4ths of the true population variance. And so we can come up with a general theme that's happening."}, {"video_title": "Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3", "Sentence": "Now this next chart really gets to the meat, really gets to the meat of the issue, because what it's telling us is that for each of these sample sizes, so this right over here, for sample size two, if we keep taking sample size two and we keep calculating the by sample variances and dividing that by the population variance and finding the mean over all of those, you see that over many, many, many trials, many, many samples of size two, that that by sample variance over population variance, it's approaching half of the true population variance. When sample size is three, it's approaching 2 3rds, 66.6% of the true population variance. When sample size is four, it's approaching 3 4ths of the true population variance. And so we can come up with a general theme that's happening. When we use the biased estimate, when we use the biased estimate, we're not approaching the population variance, we're approaching n minus one, let me write this down, we're approaching n minus one over n times the population variance. When n was two, this approached 1 1.5, 1 1.5. When n is three, this is 2 3rds."}, {"video_title": "Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3", "Sentence": "And so we can come up with a general theme that's happening. When we use the biased estimate, when we use the biased estimate, we're not approaching the population variance, we're approaching n minus one, let me write this down, we're approaching n minus one over n times the population variance. When n was two, this approached 1 1.5, 1 1.5. When n is three, this is 2 3rds. When n is four, this is 3 4ths. So this is giving us a biased estimate. So how would we unbiased this?"}, {"video_title": "Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3", "Sentence": "When n is three, this is 2 3rds. When n is four, this is 3 4ths. So this is giving us a biased estimate. So how would we unbiased this? Well, if we really wanna get our best estimate of the true population variance, not n minus one over n times the population variance, we would wanna multiply, let me do this in a color I haven't used yet, we would wanna multiply times n over n minus one. We would wanna multiply n over n minus one to get an unbiased estimate. Here, these cancel out, and you are left just with your population variance."}, {"video_title": "Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3", "Sentence": "So how would we unbiased this? Well, if we really wanna get our best estimate of the true population variance, not n minus one over n times the population variance, we would wanna multiply, let me do this in a color I haven't used yet, we would wanna multiply times n over n minus one. We would wanna multiply n over n minus one to get an unbiased estimate. Here, these cancel out, and you are left just with your population variance. That's what we want to estimate. And over here, you are left with our unbiased estimate, our unbiased estimate of population variance, our unbiased sample variance, which is equal to, and this is what we see, what we saw in the last several videos, what you see in statistics books, and sometimes it's confusing why. Hopefully, Peter's simulation gives you a good idea of why, or at least convinces you that it is the case."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "And just as a bit of a review, we have the formula here. And it looks a bit intimidating, but in that video we saw all it is is an average of the product of the z-scores for each of those pairs. And as we said, if r is equal to one, you have a perfect positive correlation. If r is equal to negative one, you have a perfect negative correlation. And if r is equal to zero, you don't have a correlation. But for this particular bivariate data set, we got an r of 0.946, which means we have a fairly strong positive correlation. What we're going to do on this video is build on this notion and actually come up with the equation for the least squares line that tries to fit these points."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "If r is equal to negative one, you have a perfect negative correlation. And if r is equal to zero, you don't have a correlation. But for this particular bivariate data set, we got an r of 0.946, which means we have a fairly strong positive correlation. What we're going to do on this video is build on this notion and actually come up with the equation for the least squares line that tries to fit these points. So before I do that, let's just visualize some of the statistics that we have here for these data points. We clearly have the four data points plotted, but let's plot the statistics for x. So the sample mean and the sample standard deviation for x are here in red."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "What we're going to do on this video is build on this notion and actually come up with the equation for the least squares line that tries to fit these points. So before I do that, let's just visualize some of the statistics that we have here for these data points. We clearly have the four data points plotted, but let's plot the statistics for x. So the sample mean and the sample standard deviation for x are here in red. And actually, let me box these off in red so that you know that's what is going on here. So the sample mean for x, and it's easy to calculate, is one plus two plus two plus three divided by four is eight divided by four, which is two. So we have x equals two right over here."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "So the sample mean and the sample standard deviation for x are here in red. And actually, let me box these off in red so that you know that's what is going on here. So the sample mean for x, and it's easy to calculate, is one plus two plus two plus three divided by four is eight divided by four, which is two. So we have x equals two right over here. And then this is one sample standard deviation above the mean. This is one sample standard deviation below the mean. And then we could do the same thing for the y variables."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "So we have x equals two right over here. And then this is one sample standard deviation above the mean. This is one sample standard deviation below the mean. And then we could do the same thing for the y variables. So the mean is three. And this is one sample standard deviation for y above the mean. And this is one sample standard deviation for y below the mean."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "And then we could do the same thing for the y variables. So the mean is three. And this is one sample standard deviation for y above the mean. And this is one sample standard deviation for y below the mean. And visualizing these means, especially their intersection, and also their standard deviations, will help us build an intuition for the equation of the least squares line. So generally speaking, the equation for any line is going to be y is equal to mx plus b, where this is the slope and this is the y-intercept. For the regression line, we'll put a little hat over it."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "And this is one sample standard deviation for y below the mean. And visualizing these means, especially their intersection, and also their standard deviations, will help us build an intuition for the equation of the least squares line. So generally speaking, the equation for any line is going to be y is equal to mx plus b, where this is the slope and this is the y-intercept. For the regression line, we'll put a little hat over it. So this, you would literally say y hat. This tells you that this is a regression line that we're trying to fit to these points. First, what is going to be the slope?"}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "For the regression line, we'll put a little hat over it. So this, you would literally say y hat. This tells you that this is a regression line that we're trying to fit to these points. First, what is going to be the slope? Well, the slope is going to be r times the ratio between the sample standard deviation in the y direction over the sample standard deviation in the x direction. This might not seem intuitive at first, but we'll talk about it in a few seconds, and hopefully it'll make a lot more sense. But the next thing we need to know is, all right, if we can calculate our slope, how do we calculate our y-intercept?"}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "First, what is going to be the slope? Well, the slope is going to be r times the ratio between the sample standard deviation in the y direction over the sample standard deviation in the x direction. This might not seem intuitive at first, but we'll talk about it in a few seconds, and hopefully it'll make a lot more sense. But the next thing we need to know is, all right, if we can calculate our slope, how do we calculate our y-intercept? Well, like you first learned in Algebra I, you can calculate the y-intercept if you already know the slope by saying, well, what point is definitely going to be on my line? And for a least squares regression line, you're definitely going to have the point sample mean of x, comma, sample mean of y. So you're definitely going to go through that point."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "But the next thing we need to know is, all right, if we can calculate our slope, how do we calculate our y-intercept? Well, like you first learned in Algebra I, you can calculate the y-intercept if you already know the slope by saying, well, what point is definitely going to be on my line? And for a least squares regression line, you're definitely going to have the point sample mean of x, comma, sample mean of y. So you're definitely going to go through that point. So before I even calculate for this particular example where in previous videos we calculated the r to be 0.946, or roughly equal to that, let's just think about what's going on. So our least squares line is definitely going to go through that point. Now, if r were one, if we had a perfect positive correlation, then our slope would be the standard deviation of y over the standard deviation of x."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "So you're definitely going to go through that point. So before I even calculate for this particular example where in previous videos we calculated the r to be 0.946, or roughly equal to that, let's just think about what's going on. So our least squares line is definitely going to go through that point. Now, if r were one, if we had a perfect positive correlation, then our slope would be the standard deviation of y over the standard deviation of x. So if you were to start at this point, and if you were to run your standard deviation of x and rise your standard deviation of y, well, with a perfect positive correlation, your line would look like this. And that makes a lot of sense because you're looking at your spread of y over your spread of x. If r were equal to one, this would be your slope, standard deviation of y over standard deviation of x."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "Now, if r were one, if we had a perfect positive correlation, then our slope would be the standard deviation of y over the standard deviation of x. So if you were to start at this point, and if you were to run your standard deviation of x and rise your standard deviation of y, well, with a perfect positive correlation, your line would look like this. And that makes a lot of sense because you're looking at your spread of y over your spread of x. If r were equal to one, this would be your slope, standard deviation of y over standard deviation of x. That has parallels to when you first learn about slope. Change in y over change in x. Here you're seeing the, you could say the average spread in y over the average spread in x."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "If r were equal to one, this would be your slope, standard deviation of y over standard deviation of x. That has parallels to when you first learn about slope. Change in y over change in x. Here you're seeing the, you could say the average spread in y over the average spread in x. And this would be the case when r is one. So let me write that down. This would be the case if r is equal to one."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "Here you're seeing the, you could say the average spread in y over the average spread in x. And this would be the case when r is one. So let me write that down. This would be the case if r is equal to one. What if r were equal to negative one? It would look like this. That would be our line if we had a perfect negative correlation."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "This would be the case if r is equal to one. What if r were equal to negative one? It would look like this. That would be our line if we had a perfect negative correlation. Now what if r were zero? Then your slope would be zero, and then your line would just be this line, y is equal to the mean of y. So you would just go through that right over there."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "That would be our line if we had a perfect negative correlation. Now what if r were zero? Then your slope would be zero, and then your line would just be this line, y is equal to the mean of y. So you would just go through that right over there. But now let's think about this scenario. In this scenario, our r is 0.946. So we have a fairly strong correlation."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "So you would just go through that right over there. But now let's think about this scenario. In this scenario, our r is 0.946. So we have a fairly strong correlation. This is pretty close to one. And so if you were to take 0.946 and multiply it by this ratio, if you were to move forward in x by the standard deviation in x, for this case, how much would you move up in y? Well, you would move up r times the standard deviation of y."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "So we have a fairly strong correlation. This is pretty close to one. And so if you were to take 0.946 and multiply it by this ratio, if you were to move forward in x by the standard deviation in x, for this case, how much would you move up in y? Well, you would move up r times the standard deviation of y. And as we said, if r was one, you would get all the way up to this perfect correlation line but here it's 0.946. So you would get up about 95% of the way to that. And so our line, without even looking at the equation, is going to look something like this, which we can see is a pretty good fit for those points."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "Well, you would move up r times the standard deviation of y. And as we said, if r was one, you would get all the way up to this perfect correlation line but here it's 0.946. So you would get up about 95% of the way to that. And so our line, without even looking at the equation, is going to look something like this, which we can see is a pretty good fit for those points. I'm not proving it here in this video. But now that we have an intuition for these things, hopefully you appreciate this isn't just coming out of nowhere and it's some strange formula. It actually makes intuitive sense."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "And so our line, without even looking at the equation, is going to look something like this, which we can see is a pretty good fit for those points. I'm not proving it here in this video. But now that we have an intuition for these things, hopefully you appreciate this isn't just coming out of nowhere and it's some strange formula. It actually makes intuitive sense. Let's calculate it for this particular set of data. M is going to be equal to r, 0.946, times the sample standard deviation of y, 2.160, over the sample standard deviation of x, 0.816. We can get our calculator out to calculate that."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "It actually makes intuitive sense. Let's calculate it for this particular set of data. M is going to be equal to r, 0.946, times the sample standard deviation of y, 2.160, over the sample standard deviation of x, 0.816. We can get our calculator out to calculate that. So we have 0.946 times 2.160 divided by 0.816. It gets us to 2.50. Let's just round to the nearest hundredth for simplicity here."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "We can get our calculator out to calculate that. So we have 0.946 times 2.160 divided by 0.816. It gets us to 2.50. Let's just round to the nearest hundredth for simplicity here. So this is approximately equal to 2.50. And so how do we figure out the y-intercept? Well, remember, we go through this point."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "Let's just round to the nearest hundredth for simplicity here. So this is approximately equal to 2.50. And so how do we figure out the y-intercept? Well, remember, we go through this point. So we're going to have 2.50 times our x-mean. So our x-mean is two times two. Remember, this right over here is our x-mean."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "Well, remember, we go through this point. So we're going to have 2.50 times our x-mean. So our x-mean is two times two. Remember, this right over here is our x-mean. Plus b is going to be equal to our y-mean. Our y-mean, we see right over here, is three. And so what do we get?"}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "Remember, this right over here is our x-mean. Plus b is going to be equal to our y-mean. Our y-mean, we see right over here, is three. And so what do we get? We get three is equal to five plus b. Five plus b. And so what is b?"}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "And so what do we get? We get three is equal to five plus b. Five plus b. And so what is b? Well, if you subtract five from both sides, you get b is equal to negative two. And so there you have it. The equation for our regression line."}, {"video_title": "Calculating the equation of a regression line AP Statistics Khan Academy.mp3", "Sentence": "And so what is b? Well, if you subtract five from both sides, you get b is equal to negative two. And so there you have it. The equation for our regression line. We deserve a little bit of a drum roll here. We would say y-hat, the hat tells us that this is the equation for a regression line, is equal to 2.50 times x minus two. Minus two."}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "She chooses a confidence level of 94%. A random sample of 200 computers show that 12 computers have the defect. What critical value, z star, should Elena use to construct this confidence interval? So before I even ask you to pause this video, let me just give you a little reminder of what a critical value is. Remember, the whole point behind confidence intervals are we have some true population parameter. In this case, it is the proportion of computers that have a defect. So there's some true population proportion."}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "So before I even ask you to pause this video, let me just give you a little reminder of what a critical value is. Remember, the whole point behind confidence intervals are we have some true population parameter. In this case, it is the proportion of computers that have a defect. So there's some true population proportion. We don't know what that is, but we try to estimate it. We take a sample. In this case, it's a sample, a random sample of 200 computers."}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "So there's some true population proportion. We don't know what that is, but we try to estimate it. We take a sample. In this case, it's a sample, a random sample of 200 computers. We take a random sample, and then we estimate this by calculating the sample proportion. But then we also wanna construct a confidence interval. And remember, a confidence interval at a 94% confidence level means that if we were to keep doing this, and if we were to keep creating intervals around these statistics, so maybe that's the confidence interval around that one."}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "In this case, it's a sample, a random sample of 200 computers. We take a random sample, and then we estimate this by calculating the sample proportion. But then we also wanna construct a confidence interval. And remember, a confidence interval at a 94% confidence level means that if we were to keep doing this, and if we were to keep creating intervals around these statistics, so maybe that's the confidence interval around that one. Maybe if we were to do it again, that's the confidence interval around that one, that 94%, that roughly, as I keep doing this over and over again, that roughly 94% of these intervals are going to overlap with our true population parameter. And the way that we do this is we take the statistic. Let me just write this in general form, even if we're not talking about a proportion."}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "And remember, a confidence interval at a 94% confidence level means that if we were to keep doing this, and if we were to keep creating intervals around these statistics, so maybe that's the confidence interval around that one. Maybe if we were to do it again, that's the confidence interval around that one, that 94%, that roughly, as I keep doing this over and over again, that roughly 94% of these intervals are going to overlap with our true population parameter. And the way that we do this is we take the statistic. Let me just write this in general form, even if we're not talking about a proportion. It could be if we're trying to estimate the population mean. So we take our statistic, statistic, and then we go plus or minus around that statistic, plus or minus around that statistic. And then we say, okay, how many standard deviations for the sampling distribution do we wanna go above or beyond?"}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "Let me just write this in general form, even if we're not talking about a proportion. It could be if we're trying to estimate the population mean. So we take our statistic, statistic, and then we go plus or minus around that statistic, plus or minus around that statistic. And then we say, okay, how many standard deviations for the sampling distribution do we wanna go above or beyond? So the number of standard deviations we wanna go, that is our critical value. And then we multiply that times the standard deviation of the statistic, of the statistic. Now in this particular situation, our statistic is p hat from this one sample that Elena made."}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "And then we say, okay, how many standard deviations for the sampling distribution do we wanna go above or beyond? So the number of standard deviations we wanna go, that is our critical value. And then we multiply that times the standard deviation of the statistic, of the statistic. Now in this particular situation, our statistic is p hat from this one sample that Elena made. So it's that one sample proportion that she was able to calculate, plus or minus z star. And we're gonna think about which z star, because that's essentially the question, the critical value. So plus or minus some critical value times, and what we do, because in order to actually calculate the true standard deviation of the sampling distribution, of the sample proportions, well, then you actually have to know the population parameter."}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "Now in this particular situation, our statistic is p hat from this one sample that Elena made. So it's that one sample proportion that she was able to calculate, plus or minus z star. And we're gonna think about which z star, because that's essentially the question, the critical value. So plus or minus some critical value times, and what we do, because in order to actually calculate the true standard deviation of the sampling distribution, of the sample proportions, well, then you actually have to know the population parameter. But we don't know that, so we multiply that times the standard error of the statistic. And we've done this in previous videos. But the key question here is, what is our z star?"}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "So plus or minus some critical value times, and what we do, because in order to actually calculate the true standard deviation of the sampling distribution, of the sample proportions, well, then you actually have to know the population parameter. But we don't know that, so we multiply that times the standard error of the statistic. And we've done this in previous videos. But the key question here is, what is our z star? And what we really needed to think about is, assuming that the sampling distribution is roughly normal, and this is the mean of it, which would actually be our true population parameter, which we do not know. But how many standard deviations above and below the mean in order to capture 94% of the probability? 94% of the area."}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "But the key question here is, what is our z star? And what we really needed to think about is, assuming that the sampling distribution is roughly normal, and this is the mean of it, which would actually be our true population parameter, which we do not know. But how many standard deviations above and below the mean in order to capture 94% of the probability? 94% of the area. So this distance right over here, where this is 94%, this number of standard deviations, that is z star right over here. Now, all we have to really do is look it up on a z table, but even there we have to be careful. And you should always be careful which type of z table you're using, or if you're using a calculator function, what your calculator function does."}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "94% of the area. So this distance right over here, where this is 94%, this number of standard deviations, that is z star right over here. Now, all we have to really do is look it up on a z table, but even there we have to be careful. And you should always be careful which type of z table you're using, or if you're using a calculator function, what your calculator function does. Because a lot of z tables will actually do something like this. For a given z, they'll say, what is the total area going all the way from negative infinity up to including z standard deviations above the mean? So when you look up a lot of z tables, they will give you this area."}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "And you should always be careful which type of z table you're using, or if you're using a calculator function, what your calculator function does. Because a lot of z tables will actually do something like this. For a given z, they'll say, what is the total area going all the way from negative infinity up to including z standard deviations above the mean? So when you look up a lot of z tables, they will give you this area. So one way to think about this, we wanna find the critical value, we wanna find the z that leaves not 6% unshaded in, but leaves 3% unshaded in. Where did I get that from? Well, 100% minus 94% is 6%."}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "So when you look up a lot of z tables, they will give you this area. So one way to think about this, we wanna find the critical value, we wanna find the z that leaves not 6% unshaded in, but leaves 3% unshaded in. Where did I get that from? Well, 100% minus 94% is 6%. But remember, this is going to be symmetric on the left and the right. So you're gonna want 3% not shaded in over here, and 3% not shaded in over here. So when I look at a traditional z table that is viewing it from this point of view, this cumulative area, what I really wanna do is find the z that is leaving 3% open over here, which would mean the z that is filling in 97% over here, not 94%."}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "Well, 100% minus 94% is 6%. But remember, this is going to be symmetric on the left and the right. So you're gonna want 3% not shaded in over here, and 3% not shaded in over here. So when I look at a traditional z table that is viewing it from this point of view, this cumulative area, what I really wanna do is find the z that is leaving 3% open over here, which would mean the z that is filling in 97% over here, not 94%. But if I find this z, but if I were to stop it right over there as well, then I would have 3% available there, and then the true area that we're filling in would be 94%. So with that out of the way, let's look that up. What z gives us, fills us, fills in 97% of the area?"}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "So when I look at a traditional z table that is viewing it from this point of view, this cumulative area, what I really wanna do is find the z that is leaving 3% open over here, which would mean the z that is filling in 97% over here, not 94%. But if I find this z, but if I were to stop it right over there as well, then I would have 3% available there, and then the true area that we're filling in would be 94%. So with that out of the way, let's look that up. What z gives us, fills us, fills in 97% of the area? So I got a z table. This is actually the one that you would see if you were, say, taking AP Statistics. And we would just look up, where do we get to 97%?"}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "What z gives us, fills us, fills in 97% of the area? So I got a z table. This is actually the one that you would see if you were, say, taking AP Statistics. And we would just look up, where do we get to 97%? And so it is 97%, looks like it is right about here. That looks like the closest number. This is 6 10,000ths above it."}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "And we would just look up, where do we get to 97%? And so it is 97%, looks like it is right about here. That looks like the closest number. This is 6 10,000ths above it. This is only 1 10,000th below it. And so this is, let's see, you would look at the row first. If we look at the row, it is 1.8, 1.88 is our z."}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "This is 6 10,000ths above it. This is only 1 10,000th below it. And so this is, let's see, you would look at the row first. If we look at the row, it is 1.8, 1.88 is our z. So going back to this right over here, if our z is equal to 1.88, so this is equal to 1.88, then all of this area, up to and including 1.88 standard deviations above the mean, that would be 97%. But if you were to go 1.88 standard deviations above the mean and 1.88 standard deviations below the mean, that would leave 3% open on either side. And so this would be 94%."}, {"video_title": "Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3", "Sentence": "If we look at the row, it is 1.8, 1.88 is our z. So going back to this right over here, if our z is equal to 1.88, so this is equal to 1.88, then all of this area, up to and including 1.88 standard deviations above the mean, that would be 97%. But if you were to go 1.88 standard deviations above the mean and 1.88 standard deviations below the mean, that would leave 3% open on either side. And so this would be 94%. So this would be 94%. But to answer their question, what critical value z star? Well, this is going to be 1.88."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "The sociologist wants to sample people and create a two-sample z interval. In other videos, we introduce what that idea is to estimate the difference between the proportion of men who have received a speeding ticket and the proportion of women who have received a speeding ticket. Which of the following are conditions for this type of interval? Choose all answers that apply. So like always, pause this video and see if you can answer it on your own. All righty, let's review our conditions for inference. So you have your random condition, and these are the same ones that we have talked about when we were dealing with one sample, but now we just have to make sure that it applies to both samples, that both samples, we feel good, are randomly selected."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "Choose all answers that apply. So like always, pause this video and see if you can answer it on your own. All righty, let's review our conditions for inference. So you have your random condition, and these are the same ones that we have talked about when we were dealing with one sample, but now we just have to make sure that it applies to both samples, that both samples, we feel good, are randomly selected. The second one is the normal condition. And this is to feel good that the sampling distribution of the sample proportion for each of the samples is roughly normal. And so what you have to do is you take the sample size of the first sample times the sample proportion of the first sample, and that needs to be greater than or equal to 10."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "So you have your random condition, and these are the same ones that we have talked about when we were dealing with one sample, but now we just have to make sure that it applies to both samples, that both samples, we feel good, are randomly selected. The second one is the normal condition. And this is to feel good that the sampling distribution of the sample proportion for each of the samples is roughly normal. And so what you have to do is you take the sample size of the first sample times the sample proportion of the first sample, and that needs to be greater than or equal to 10. You take the sample size of the first sample times one minus the sample proportion of the first sample. That should also be greater than or equal to 10. Another way to think about it is your best sense of the expected number of successes and failures should be greater than or equal to 10."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "And so what you have to do is you take the sample size of the first sample times the sample proportion of the first sample, and that needs to be greater than or equal to 10. You take the sample size of the first sample times one minus the sample proportion of the first sample. That should also be greater than or equal to 10. Another way to think about it is your best sense of the expected number of successes and failures should be greater than or equal to 10. And then you do this with the second sample. So the sample size of the second sample, these don't have to be the same, times the sample proportion of the second sample should be greater than or equal to 10 as well. And the sample size of the second sample times one minus the sample proportion of the second sample, that needs to be greater than or equal to 10."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "Another way to think about it is your best sense of the expected number of successes and failures should be greater than or equal to 10. And then you do this with the second sample. So the sample size of the second sample, these don't have to be the same, times the sample proportion of the second sample should be greater than or equal to 10 as well. And the sample size of the second sample times one minus the sample proportion of the second sample, that needs to be greater than or equal to 10. This has to be and. All of this needs to be true. And the final one is the independence condition."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "And the sample size of the second sample times one minus the sample proportion of the second sample, that needs to be greater than or equal to 10. This has to be and. All of this needs to be true. And the final one is the independence condition. And we meet that if individual observations in these samples are done with replacement, or even if they're not done with replacement, but if the samples are no more than 10% of the population, then we meet the independence condition. And once again, you've seen this before. We're now just doing it with two samples."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "And the final one is the independence condition. And we meet that if individual observations in these samples are done with replacement, or even if they're not done with replacement, but if the samples are no more than 10% of the population, then we meet the independence condition. And once again, you've seen this before. We're now just doing it with two samples. So let's see. Which of the following are conditions for this type of interval? So the samples both include at least 10 people who have received a speeding ticket and at least 10 people who haven't."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "We're now just doing it with two samples. So let's see. Which of the following are conditions for this type of interval? So the samples both include at least 10 people who have received a speeding ticket and at least 10 people who haven't. Yeah, that's right. This is, you could view this as the expected number of people who have received a speeding ticket, and this is the expected number of people who haven't received a speeding ticket, or our estimate of the expected number, because we're using the sample proportion instead of the true proportion. So these need to be greater than or equal to 10 in both samples."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "So the samples both include at least 10 people who have received a speeding ticket and at least 10 people who haven't. Yeah, that's right. This is, you could view this as the expected number of people who have received a speeding ticket, and this is the expected number of people who haven't received a speeding ticket, or our estimate of the expected number, because we're using the sample proportion instead of the true proportion. So these need to be greater than or equal to 10 in both samples. So this is absolutely true. The people in each sample can be considered independent. Yeah, we have that independence condition."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "So these need to be greater than or equal to 10 in both samples. So this is absolutely true. The people in each sample can be considered independent. Yeah, we have that independence condition. Either they're sampled with replacement, or we are sampling no more than 10% of the population. So this is important. And then last but not least, they take separate random samples of men and women."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "Yeah, we have that independence condition. Either they're sampled with replacement, or we are sampling no more than 10% of the population. So this is important. And then last but not least, they take separate random samples of men and women. Yeah, that's the random condition right over here. So they have all three of them right over here. We have our normal condition, our independent condition, and our random condition."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "And then last but not least, they take separate random samples of men and women. Yeah, that's the random condition right over here. So they have all three of them right over here. We have our normal condition, our independent condition, and our random condition. Let's do another example. A biologist is studying a certain disease affecting oak trees in a forest. They are curious if there's a difference in the proportion of trees that are infected in the north and south sections of the forest."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "We have our normal condition, our independent condition, and our random condition. Let's do another example. A biologist is studying a certain disease affecting oak trees in a forest. They are curious if there's a difference in the proportion of trees that are infected in the north and south sections of the forest. They want to take a sample of trees from each section and do a two-sample z-test to test their hypotheses. Which of the following are conditions for this type of test? So pause the video again and see if you can answer this."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "They are curious if there's a difference in the proportion of trees that are infected in the north and south sections of the forest. They want to take a sample of trees from each section and do a two-sample z-test to test their hypotheses. Which of the following are conditions for this type of test? So pause the video again and see if you can answer this. Okay, so we've already reviewed our conditions for inference. So let's see which of these are the actual conditions for inference. So both samples include at least 30 trees."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "So pause the video again and see if you can answer this. Okay, so we've already reviewed our conditions for inference. So let's see which of these are the actual conditions for inference. So both samples include at least 30 trees. So this might have been tempting because this 30 number shows up when we're thinking about conditions for inference when we're dealing with means. But this does not come up when we're dealing with proportions. Both samples do not need to include at least 30 trees."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "So both samples include at least 30 trees. So this might have been tempting because this 30 number shows up when we're thinking about conditions for inference when we're dealing with means. But this does not come up when we're dealing with proportions. Both samples do not need to include at least 30 trees. So this would not be one of our choices. They sample an equal number of trees from each region of the forest. This is a very common misconception that when you're doing a two-sample z-test or when you're doing a two-sample z-interval or confidence interval, that the both samples have to have the same sample size."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "Both samples do not need to include at least 30 trees. So this would not be one of our choices. They sample an equal number of trees from each region of the forest. This is a very common misconception that when you're doing a two-sample z-test or when you're doing a two-sample z-interval or confidence interval, that the both samples have to have the same sample size. But that is actually not the case. So we can rule that one out. They observe at least 10 trees with the disease and at least 10 trees without the disease in each sample."}, {"video_title": "Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3", "Sentence": "This is a very common misconception that when you're doing a two-sample z-test or when you're doing a two-sample z-interval or confidence interval, that the both samples have to have the same sample size. But that is actually not the case. So we can rule that one out. They observe at least 10 trees with the disease and at least 10 trees without the disease in each sample. Yes, this is the normal condition that we just looked at. So this would be our only choice. And we're done."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "His results are displayed in the table below. Alright, this is interesting. These columns, on time, delayed, and the total. So for example, when it was sunny, there's a total of 170 sunny days that year, 167 of which the train was on time, three of which the train was delayed. And we can look at that by the different types of weather conditions. And then they say, for these days, are the events delayed and snowy independent? So to think about this, and remember, we're only going to be able to figure out experimental probabilities, and you should always view experimental probabilities as somewhat suspect."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "So for example, when it was sunny, there's a total of 170 sunny days that year, 167 of which the train was on time, three of which the train was delayed. And we can look at that by the different types of weather conditions. And then they say, for these days, are the events delayed and snowy independent? So to think about this, and remember, we're only going to be able to figure out experimental probabilities, and you should always view experimental probabilities as somewhat suspect. The more experiments you're able to take, the more likely it is to approximate the true theoretical probability, but there's always some chance that they might be different or even quite different. Let's use this data to try to calculate the experimental probability. So the key question here is, what is the probability that the train is delayed?"}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "So to think about this, and remember, we're only going to be able to figure out experimental probabilities, and you should always view experimental probabilities as somewhat suspect. The more experiments you're able to take, the more likely it is to approximate the true theoretical probability, but there's always some chance that they might be different or even quite different. Let's use this data to try to calculate the experimental probability. So the key question here is, what is the probability that the train is delayed? And then we want to think about, what is the probability that the train is delayed given that it is snowy? If we knew the theoretical probabilities, and if they were exactly the same, if the probability of being delayed was exactly the same as the probability of being delayed given snowy, then being delayed or being snowy would be independent. But if we knew the theoretical probabilities and the probability of being delayed given snowy were different than the probability of being delayed, then we would not say that these are independent variables."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "So the key question here is, what is the probability that the train is delayed? And then we want to think about, what is the probability that the train is delayed given that it is snowy? If we knew the theoretical probabilities, and if they were exactly the same, if the probability of being delayed was exactly the same as the probability of being delayed given snowy, then being delayed or being snowy would be independent. But if we knew the theoretical probabilities and the probability of being delayed given snowy were different than the probability of being delayed, then we would not say that these are independent variables. Now, we don't know the theoretical probabilities. We're just going to calculate the experimental probabilities. And we do have a good number of experiments here."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "But if we knew the theoretical probabilities and the probability of being delayed given snowy were different than the probability of being delayed, then we would not say that these are independent variables. Now, we don't know the theoretical probabilities. We're just going to calculate the experimental probabilities. And we do have a good number of experiments here. So if these are quite different, I would feel confident saying that they are dependent. If they are pretty close with the experimental probability, I would say that it would be hard to make the statement that they are dependent and that you would probably lean towards independence. But let's calculate this."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "And we do have a good number of experiments here. So if these are quite different, I would feel confident saying that they are dependent. If they are pretty close with the experimental probability, I would say that it would be hard to make the statement that they are dependent and that you would probably lean towards independence. But let's calculate this. What is the probability that the train is just delayed? Pause this video and try to figure that out. Well, let's see."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "But let's calculate this. What is the probability that the train is just delayed? Pause this video and try to figure that out. Well, let's see. If we just think in general, we have a total of 365 trials or 365 experiments. And of them, the train was delayed 35 times. Now, what's the probability that the train is delayed given that it is snowy?"}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "Well, let's see. If we just think in general, we have a total of 365 trials or 365 experiments. And of them, the train was delayed 35 times. Now, what's the probability that the train is delayed given that it is snowy? Pause the video and try to figure that out. Well, let's see. We have a total of 20 snowy days."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "Now, what's the probability that the train is delayed given that it is snowy? Pause the video and try to figure that out. Well, let's see. We have a total of 20 snowy days. And we are delayed 12 of those 20 snowy days. And so this is going to be a probability. 12 20ths is the same thing as, if we multiply both the numerator and the denominator by five, this is a 60% probability, or I could say a 0.6 probability of being delayed when it is snowy."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "We have a total of 20 snowy days. And we are delayed 12 of those 20 snowy days. And so this is going to be a probability. 12 20ths is the same thing as, if we multiply both the numerator and the denominator by five, this is a 60% probability, or I could say a 0.6 probability of being delayed when it is snowy. This is, of course, an experimental probability, which is much higher than this. This is less than 10% right over here. This right over here is less than 0.1."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "12 20ths is the same thing as, if we multiply both the numerator and the denominator by five, this is a 60% probability, or I could say a 0.6 probability of being delayed when it is snowy. This is, of course, an experimental probability, which is much higher than this. This is less than 10% right over here. This right over here is less than 0.1. I could get a calculator to calculate it exactly. It'll be 9 point something percent or 0.9 something. But clearly, this, you are much more likely, at least from the experimental data, it seems like you have a much higher proportion of your snowy days are delayed than just general days in general, than just general days."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "This right over here is less than 0.1. I could get a calculator to calculate it exactly. It'll be 9 point something percent or 0.9 something. But clearly, this, you are much more likely, at least from the experimental data, it seems like you have a much higher proportion of your snowy days are delayed than just general days in general, than just general days. And so based on this data, because the experimental probability of being delayed given snowy is so much higher than the experimental probability of just being delayed, I would make the statement that these are not independent. So for these days, are the events delayed and snowy independent? No."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "Maybe it looks something like this. I'll do discrete distribution. Really, to model anything, at some point you have to make it discrete. It could be a very granular discrete distribution. But let's say something crazy that looks like this. This is clearly not a normal distribution. But we saw in the first video, if you take, let's say, sample sizes of four."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "It could be a very granular discrete distribution. But let's say something crazy that looks like this. This is clearly not a normal distribution. But we saw in the first video, if you take, let's say, sample sizes of four. So if you took four numbers from this distribution, four random numbers where, let's say, this is the probability of a 1, 2, 3, 4, 5, 6, 7, 8, 9. If you took four numbers at a time and averaged them, let me do that here. If you took four numbers at a time, let's say we use this distribution to generate four random numbers."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "But we saw in the first video, if you take, let's say, sample sizes of four. So if you took four numbers from this distribution, four random numbers where, let's say, this is the probability of a 1, 2, 3, 4, 5, 6, 7, 8, 9. If you took four numbers at a time and averaged them, let me do that here. If you took four numbers at a time, let's say we use this distribution to generate four random numbers. We're very likely to get a 9. We're definitely not going to get any 7 or 8's. We're definitely not going to get a 4."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "If you took four numbers at a time, let's say we use this distribution to generate four random numbers. We're very likely to get a 9. We're definitely not going to get any 7 or 8's. We're definitely not going to get a 4. We might get a 1 or 2. 3 is also very likely. 5 is very likely."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "We're definitely not going to get a 4. We might get a 1 or 2. 3 is also very likely. 5 is very likely. So we use this function to essentially generate random numbers for us. And we take samples of four, and then we average them up. So let's say our first average is like, I don't know, it's a 9, it's a 5, it's another 9, and then it's a 1."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "5 is very likely. So we use this function to essentially generate random numbers for us. And we take samples of four, and then we average them up. So let's say our first average is like, I don't know, it's a 9, it's a 5, it's another 9, and then it's a 1. So what is that? That's 14 plus 10, 24 divided by 4. The average for this first trial, for this first sample of four, is going to be 6."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "So let's say our first average is like, I don't know, it's a 9, it's a 5, it's another 9, and then it's a 1. So what is that? That's 14 plus 10, 24 divided by 4. The average for this first trial, for this first sample of four, is going to be 6. They add up to 24 divided by 4. So we would plot it right here. Our average was 6 that time."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "The average for this first trial, for this first sample of four, is going to be 6. They add up to 24 divided by 4. So we would plot it right here. Our average was 6 that time. Just like that, and we'll just keep doing it. And we've seen in the past that if you just keep doing this, this is going to start looking something like a normal distribution. So maybe we do it again."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "Our average was 6 that time. Just like that, and we'll just keep doing it. And we've seen in the past that if you just keep doing this, this is going to start looking something like a normal distribution. So maybe we do it again. The average is 6 again. Maybe we do it again. The average is 5."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "So maybe we do it again. The average is 6 again. Maybe we do it again. The average is 5. We do it again. The average is 7. We do it again."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "The average is 5. We do it again. The average is 7. We do it again. The average is 6. And then if you just do this a ton, a ton of times, your distribution might look something that looks very much like a normal distribution. So if these boxes are really small, so we just do a bunch of these trials, at some point it might look a lot like a normal distribution."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "We do it again. The average is 6. And then if you just do this a ton, a ton of times, your distribution might look something that looks very much like a normal distribution. So if these boxes are really small, so we just do a bunch of these trials, at some point it might look a lot like a normal distribution. Obviously, there's some average values. It won't be a perfect normal distribution, because you can't ever get anything less than 0, or anything less than 1, really, as an average. You can't get 0 as an average."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "So if these boxes are really small, so we just do a bunch of these trials, at some point it might look a lot like a normal distribution. Obviously, there's some average values. It won't be a perfect normal distribution, because you can't ever get anything less than 0, or anything less than 1, really, as an average. You can't get 0 as an average. And you can't get anything more than 9. So it's not going to have infinitely long tails. But at least for the middle part of it, a normal distribution might be a good approximation."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "You can't get 0 as an average. And you can't get anything more than 9. So it's not going to have infinitely long tails. But at least for the middle part of it, a normal distribution might be a good approximation. In this video, what I want to think about is what happens as we change n. So in this case, n was 4. n is our sample size. Every time we do a trial, we took 4. And we took their average, and we plotted it."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "But at least for the middle part of it, a normal distribution might be a good approximation. In this video, what I want to think about is what happens as we change n. So in this case, n was 4. n is our sample size. Every time we do a trial, we took 4. And we took their average, and we plotted it. We could have had n equal 10. We could have taken 10 samples from this population, you could say, or from this random variable, averaged them, and then plotted them here. And in the last video, we ran the simulation."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "And we took their average, and we plotted it. We could have had n equal 10. We could have taken 10 samples from this population, you could say, or from this random variable, averaged them, and then plotted them here. And in the last video, we ran the simulation. I'm going to go back to that simulation in a second. We saw a couple of things. And I'll show it to you in a little bit more depth this time."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "And in the last video, we ran the simulation. I'm going to go back to that simulation in a second. We saw a couple of things. And I'll show it to you in a little bit more depth this time. When n is pretty small, it doesn't approach a normal distribution that well. So when n is small, I mean, let's take the extreme case. What happens when n is equal to 1?"}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "And I'll show it to you in a little bit more depth this time. When n is pretty small, it doesn't approach a normal distribution that well. So when n is small, I mean, let's take the extreme case. What happens when n is equal to 1? That literally just means I take one instance of this random variable and average it. Well, it's just going to be that thing. So if I just take a bunch of trials from this thing and plot it over time, what's it going to look like?"}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "What happens when n is equal to 1? That literally just means I take one instance of this random variable and average it. Well, it's just going to be that thing. So if I just take a bunch of trials from this thing and plot it over time, what's it going to look like? Well, it's definitely not going to look like a normal distribution. It's going to look as though you're going to have a couple of 1s, you're going to have a couple of 2s, you're going to have more 3s like that. You're going to have no 4s."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "So if I just take a bunch of trials from this thing and plot it over time, what's it going to look like? Well, it's definitely not going to look like a normal distribution. It's going to look as though you're going to have a couple of 1s, you're going to have a couple of 2s, you're going to have more 3s like that. You're going to have no 4s. You're going to have a bunch of 5s. You're going to have some 6s that look like that. And you're going to have a bunch of 9s."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "You're going to have no 4s. You're going to have a bunch of 5s. You're going to have some 6s that look like that. And you're going to have a bunch of 9s. So there your sampling distribution of the sample mean for n of 1 is going to look, I don't care how many trials you do, it's not going to look like a normal distribution. So the central limit theorem, although I you do a bunch of trials that look like a normal distribution, it definitely doesn't work for n equals 1. As n gets larger, though, it starts to make sense."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "And you're going to have a bunch of 9s. So there your sampling distribution of the sample mean for n of 1 is going to look, I don't care how many trials you do, it's not going to look like a normal distribution. So the central limit theorem, although I you do a bunch of trials that look like a normal distribution, it definitely doesn't work for n equals 1. As n gets larger, though, it starts to make sense. Let's say if we got n equals 2, and I'm all just doing this in my head, I don't know what the actual distributions would look like, but then it still would be difficult for it to become an exact normal distribution, but then you can get more instances, you could get more, you know, you might get things from all of the above, but you can only get 2 in each of your baskets that you're averaging, you're only going to get 2 numbers, right? So you're never going to, let's see, I mean, for example, you're never going to get a 7.5 in your sampling distribution of the sample mean for n is equal to 2, because it's impossible to get a 7, and it's impossible to get an 8. So you're never going to get 7.5 as, so maybe when you plot it, maybe it looks like this, but there'll be a gap at 7.5, because that's impossible, and, you know, maybe it looks something like that."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "As n gets larger, though, it starts to make sense. Let's say if we got n equals 2, and I'm all just doing this in my head, I don't know what the actual distributions would look like, but then it still would be difficult for it to become an exact normal distribution, but then you can get more instances, you could get more, you know, you might get things from all of the above, but you can only get 2 in each of your baskets that you're averaging, you're only going to get 2 numbers, right? So you're never going to, let's see, I mean, for example, you're never going to get a 7.5 in your sampling distribution of the sample mean for n is equal to 2, because it's impossible to get a 7, and it's impossible to get an 8. So you're never going to get 7.5 as, so maybe when you plot it, maybe it looks like this, but there'll be a gap at 7.5, because that's impossible, and, you know, maybe it looks something like that. So it still won't be a normal distribution when n is equal to 2. So there's a couple of interesting things here. So one thing, and I didn't mention this the first time, just because I really wanted to get the gut sense of what the central limit theorem is."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "So you're never going to get 7.5 as, so maybe when you plot it, maybe it looks like this, but there'll be a gap at 7.5, because that's impossible, and, you know, maybe it looks something like that. So it still won't be a normal distribution when n is equal to 2. So there's a couple of interesting things here. So one thing, and I didn't mention this the first time, just because I really wanted to get the gut sense of what the central limit theorem is. The central limit theorem says as n approaches, really as it approaches infinity, then is when you get the real normal distribution. Normal distribution. But in kind of everyday practice, you don't have to get that much beyond n equals 2."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "So one thing, and I didn't mention this the first time, just because I really wanted to get the gut sense of what the central limit theorem is. The central limit theorem says as n approaches, really as it approaches infinity, then is when you get the real normal distribution. Normal distribution. But in kind of everyday practice, you don't have to get that much beyond n equals 2. If you get to n equals 10 or n equals 15, you're getting very close to a normal distribution. So this converges to a normal distribution very quickly. Distribution."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "But in kind of everyday practice, you don't have to get that much beyond n equals 2. If you get to n equals 10 or n equals 15, you're getting very close to a normal distribution. So this converges to a normal distribution very quickly. Distribution. Now the other thing is you obviously want many, many trials. So this is your sample size. That is your sample size."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "Distribution. Now the other thing is you obviously want many, many trials. So this is your sample size. That is your sample size. That's the size of each of your baskets. In the very first video I did on this, I took a sample size of 4. In the simulation I did in the last video, we did sample sizes of 4 and 10 and whatever else."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "That is your sample size. That's the size of each of your baskets. In the very first video I did on this, I took a sample size of 4. In the simulation I did in the last video, we did sample sizes of 4 and 10 and whatever else. This is a sample size of 1. So that's our sample size. So as that approaches infinity, your actual sampling distribution of the sample mean will approach a normal distribution."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "In the simulation I did in the last video, we did sample sizes of 4 and 10 and whatever else. This is a sample size of 1. So that's our sample size. So as that approaches infinity, your actual sampling distribution of the sample mean will approach a normal distribution. Now in order to actually see that normal distribution and actually prove it to yourself, you would have to do this many, many times. Remember, the normal distribution happens... This is essentially..."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "So as that approaches infinity, your actual sampling distribution of the sample mean will approach a normal distribution. Now in order to actually see that normal distribution and actually prove it to yourself, you would have to do this many, many times. Remember, the normal distribution happens... This is essentially... This is kind of the population, or this is the random variable. That tells you all of the possibilities. In real life, we seldom know all of the possibilities."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "This is essentially... This is kind of the population, or this is the random variable. That tells you all of the possibilities. In real life, we seldom know all of the possibilities. In fact, in real life, we seldom know the pure probability generating function, only if we're kind of writing it or if we're writing a computer program. Normally we're doing samples, and we're trying to estimate things. Normally there's some random variable, and then maybe we'll do a bunch of... We take a bunch of samples, we take their means, and we plot them, and then we're going to get some type of normal distribution."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "In real life, we seldom know all of the possibilities. In fact, in real life, we seldom know the pure probability generating function, only if we're kind of writing it or if we're writing a computer program. Normally we're doing samples, and we're trying to estimate things. Normally there's some random variable, and then maybe we'll do a bunch of... We take a bunch of samples, we take their means, and we plot them, and then we're going to get some type of normal distribution. Let's say we take samples of 100 and we average them. We're going to get some normal distribution. In theory, as we take those averages hundreds or thousands of times, our data set is going to more closely approximate that pure sampling distribution of the sample mean."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "Normally there's some random variable, and then maybe we'll do a bunch of... We take a bunch of samples, we take their means, and we plot them, and then we're going to get some type of normal distribution. Let's say we take samples of 100 and we average them. We're going to get some normal distribution. In theory, as we take those averages hundreds or thousands of times, our data set is going to more closely approximate that pure sampling distribution of the sample mean. This thing is a real distribution. It's a real distribution with a real mean. Its mean, it has a pure mean."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "In theory, as we take those averages hundreds or thousands of times, our data set is going to more closely approximate that pure sampling distribution of the sample mean. This thing is a real distribution. It's a real distribution with a real mean. Its mean, it has a pure mean. Its mean, so the mean of the sampling... The mean of the sampling distribution of the sample mean, we'll write it like that. Notice I didn't write it as just the x."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "Its mean, it has a pure mean. Its mean, so the mean of the sampling... The mean of the sampling distribution of the sample mean, we'll write it like that. Notice I didn't write it as just the x. What this is, is this is actually saying that this is a real population mean. This is a real random variable mean. If you looked at every possibility of all of the samples that you can take from your original distribution, from some other random original distribution, and you took all of the possibilities of, let's say, sample size."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "Notice I didn't write it as just the x. What this is, is this is actually saying that this is a real population mean. This is a real random variable mean. If you looked at every possibility of all of the samples that you can take from your original distribution, from some other random original distribution, and you took all of the possibilities of, let's say, sample size. Let's say we're dealing with a world where our sample size is 10. If you took all of the combinations of 10 samples from some original distribution, and you averaged them out, this would describe that function. Of course, in reality, if you don't know the original distribution, you can't take an infinite sample from it, so you won't know every combination."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "If you looked at every possibility of all of the samples that you can take from your original distribution, from some other random original distribution, and you took all of the possibilities of, let's say, sample size. Let's say we're dealing with a world where our sample size is 10. If you took all of the combinations of 10 samples from some original distribution, and you averaged them out, this would describe that function. Of course, in reality, if you don't know the original distribution, you can't take an infinite sample from it, so you won't know every combination. But if you did it with 1,000, if you did the trial 1,000 times, so 1,000 times you took 10 samples from some distribution, and took 1,000 averages, and then plotted them, you're going to get pretty close. Now, the next thing I want to touch on is what happens as n approaches infinity, it becomes more of a normal distribution. But as I said already, n equals 10 is pretty good, and n equals 20 is even better."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "Of course, in reality, if you don't know the original distribution, you can't take an infinite sample from it, so you won't know every combination. But if you did it with 1,000, if you did the trial 1,000 times, so 1,000 times you took 10 samples from some distribution, and took 1,000 averages, and then plotted them, you're going to get pretty close. Now, the next thing I want to touch on is what happens as n approaches infinity, it becomes more of a normal distribution. But as I said already, n equals 10 is pretty good, and n equals 20 is even better. But we saw something in the last video that at least I find pretty interesting. Let's say we start with this crazy distribution up here. It really doesn't matter what distribution we're starting with."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "But as I said already, n equals 10 is pretty good, and n equals 20 is even better. But we saw something in the last video that at least I find pretty interesting. Let's say we start with this crazy distribution up here. It really doesn't matter what distribution we're starting with. We saw in the simulation that when n is equal to 5, our graph after we take samples of 5, average them, and we do it 10,000 times, our graph looks something like this. It's kind of wide like that. And then when we did n is equal to 10, our graph looked a little bit, it was actually a little bit squeezed in like that a little bit more."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "It really doesn't matter what distribution we're starting with. We saw in the simulation that when n is equal to 5, our graph after we take samples of 5, average them, and we do it 10,000 times, our graph looks something like this. It's kind of wide like that. And then when we did n is equal to 10, our graph looked a little bit, it was actually a little bit squeezed in like that a little bit more. So not only was it more normal, that's what the central limit theorem tells us because we're taking larger sample sizes, but it had a smaller standard deviation or a smaller variance. The mean is going to be the same either case, but when our sample size was larger, our standard deviation became smaller. In fact, our standard deviation became smaller than our original population distribution or our original probability density function."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "And then when we did n is equal to 10, our graph looked a little bit, it was actually a little bit squeezed in like that a little bit more. So not only was it more normal, that's what the central limit theorem tells us because we're taking larger sample sizes, but it had a smaller standard deviation or a smaller variance. The mean is going to be the same either case, but when our sample size was larger, our standard deviation became smaller. In fact, our standard deviation became smaller than our original population distribution or our original probability density function. Let me show you that with a simulation. So let me clear everything. And this simulation is as good as any."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "In fact, our standard deviation became smaller than our original population distribution or our original probability density function. Let me show you that with a simulation. So let me clear everything. And this simulation is as good as any. So the first thing I want to show, or this distribution is as good as any. The first thing I want to show you is that n of 2 is really not that good. So let's compare an n of 2 to, let's say an n of 16."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "And this simulation is as good as any. So the first thing I want to show, or this distribution is as good as any. The first thing I want to show you is that n of 2 is really not that good. So let's compare an n of 2 to, let's say an n of 16. So when you compare an n of 2 to an n of 16, let's do it once. So you get one, two trials, you average them, and then it's going to do it 16. And then it's going to plot it down here and average there."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "So let's compare an n of 2 to, let's say an n of 16. So when you compare an n of 2 to an n of 16, let's do it once. So you get one, two trials, you average them, and then it's going to do it 16. And then it's going to plot it down here and average there. Let's do that 10,000 times. So notice, when you took an n of 2, even though we did it 10,000 times, this is not approaching a normal distribution. You can actually see it in the skew in kurtosis numbers."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "And then it's going to plot it down here and average there. Let's do that 10,000 times. So notice, when you took an n of 2, even though we did it 10,000 times, this is not approaching a normal distribution. You can actually see it in the skew in kurtosis numbers. It has a rightward positive skew, which means it has a longer tail to the right than to the left. And then it has a negative kurtosis, which means that it's a little bit, it has shorter tails and smaller peaks than a standard normal distribution. Now, when n is equal to 16, you do the same, so every time we took 16 samples from this distribution function up here and averaged them, and each of these dots represent an average, we did it 10,001 times."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "You can actually see it in the skew in kurtosis numbers. It has a rightward positive skew, which means it has a longer tail to the right than to the left. And then it has a negative kurtosis, which means that it's a little bit, it has shorter tails and smaller peaks than a standard normal distribution. Now, when n is equal to 16, you do the same, so every time we took 16 samples from this distribution function up here and averaged them, and each of these dots represent an average, we did it 10,001 times. Here, notice, the mean is the same in both places, but here all of a sudden, our kurtosis is much smaller and our skew is much smaller, so we are more normal in this situation. But even a more interesting thing is our standard deviation is smaller, right? This is more squeezed in than that is, and it's definitely more squeezed in than our original distribution."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "Now, when n is equal to 16, you do the same, so every time we took 16 samples from this distribution function up here and averaged them, and each of these dots represent an average, we did it 10,001 times. Here, notice, the mean is the same in both places, but here all of a sudden, our kurtosis is much smaller and our skew is much smaller, so we are more normal in this situation. But even a more interesting thing is our standard deviation is smaller, right? This is more squeezed in than that is, and it's definitely more squeezed in than our original distribution. Now, let me do it with two, let me clear everything again. I like this distribution because it's a very non-normal distribution. It looks like a bimodal distribution of some kind."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "This is more squeezed in than that is, and it's definitely more squeezed in than our original distribution. Now, let me do it with two, let me clear everything again. I like this distribution because it's a very non-normal distribution. It looks like a bimodal distribution of some kind. And let's take a scenario where I take an n of, let's take two good n's. Let's take an n of 16, that's a nice healthy n, and let's take an n of 25, and let's compare them a little bit. So if we, let's just do one trial animated, just to, it's always nice to see it."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "It looks like a bimodal distribution of some kind. And let's take a scenario where I take an n of, let's take two good n's. Let's take an n of 16, that's a nice healthy n, and let's take an n of 25, and let's compare them a little bit. So if we, let's just do one trial animated, just to, it's always nice to see it. So first it's going to do 16 of these trials and average them, and there we go. Then it's going to do 25 of these trials and then average them, and then there we go. Now let's do that, what I just did, animated."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "So if we, let's just do one trial animated, just to, it's always nice to see it. So first it's going to do 16 of these trials and average them, and there we go. Then it's going to do 25 of these trials and then average them, and then there we go. Now let's do that, what I just did, animated. Let's do it 10,000 times. Miracles of computers. Now notice something, this is 10,000 times, these are both pretty good approximations of normal distributions."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "Now let's do that, what I just did, animated. Let's do it 10,000 times. Miracles of computers. Now notice something, this is 10,000 times, these are both pretty good approximations of normal distributions. The n is equal to 25 is more normal, it has less skew, slightly less skew than n is equal to 16. It has slightly less kurtosis, which means it's closer to being a normal distribution than n is equal to 16. But even more interesting, it's more squeezed in, it has a lower standard deviation."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "Now notice something, this is 10,000 times, these are both pretty good approximations of normal distributions. The n is equal to 25 is more normal, it has less skew, slightly less skew than n is equal to 16. It has slightly less kurtosis, which means it's closer to being a normal distribution than n is equal to 16. But even more interesting, it's more squeezed in, it has a lower standard deviation. The standard deviation here is 2.1 and the standard deviation here is 2.64. So that's another, and I kind of touched on that in the last video, and it kind of makes sense. For every sample you do for your average, the more you put in that sample, the less standard deviation."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "But even more interesting, it's more squeezed in, it has a lower standard deviation. The standard deviation here is 2.1 and the standard deviation here is 2.64. So that's another, and I kind of touched on that in the last video, and it kind of makes sense. For every sample you do for your average, the more you put in that sample, the less standard deviation. Think of the extreme case. If instead of taking 16 samples from our distribution every time, or instead of taking 25, if I were to take a million samples from this distribution every time, if I were to take a million samples from this distribution every time, that sample mean is always going to be pretty darn close to my mean. If I take a million samples of everything, if I try to essentially try to estimate a mean by taking a million samples, I'm going to get a pretty good estimate of that mean."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "For every sample you do for your average, the more you put in that sample, the less standard deviation. Think of the extreme case. If instead of taking 16 samples from our distribution every time, or instead of taking 25, if I were to take a million samples from this distribution every time, if I were to take a million samples from this distribution every time, that sample mean is always going to be pretty darn close to my mean. If I take a million samples of everything, if I try to essentially try to estimate a mean by taking a million samples, I'm going to get a pretty good estimate of that mean. The probability that a bunch of the million numbers are all out here is very low. So if n is a million, of course all of my sample means when I average them are all going to be really tightly focused around the mean itself. Hopefully that kind of makes sense to you as well."}, {"video_title": "Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3", "Sentence": "If I take a million samples of everything, if I try to essentially try to estimate a mean by taking a million samples, I'm going to get a pretty good estimate of that mean. The probability that a bunch of the million numbers are all out here is very low. So if n is a million, of course all of my sample means when I average them are all going to be really tightly focused around the mean itself. Hopefully that kind of makes sense to you as well. Just think about it, or even use this tool and experiment with it just so you can trust that that is really the case. It actually turns out that there's a very clean formula that relates the standard deviation of the original probability distribution function to the standard deviation of the sampling distribution of the sample mean. As you can imagine, it is a function of your sample size, of how many samples you take out in every basket before you average them."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "And the reason why it's so neat is we can start with any distribution that has a well-defined mean and variance. Actually, I wrote the standard deviation here in the last video, that should be the mean. And let's say it has some variance. I could write it like that, or I could write the standard deviation there. But as long as it has a well-defined mean and standard deviation, I don't care what the distribution looks like. What I can do is take samples in the last video of, say, size 4. So that means I take literally four instances of this random variable."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "I could write it like that, or I could write the standard deviation there. But as long as it has a well-defined mean and standard deviation, I don't care what the distribution looks like. What I can do is take samples in the last video of, say, size 4. So that means I take literally four instances of this random variable. This is one example. I take their mean, and I consider this the sample mean for my first trial. Or you could almost say for my first sample."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "So that means I take literally four instances of this random variable. This is one example. I take their mean, and I consider this the sample mean for my first trial. Or you could almost say for my first sample. I know it's very confusing, because you can consider that a sample, the set to be a sample, or you could consider each member of the set as a sample. So that can be a little bit confusing there. But I have this first sample mean, and then I keep doing that over and over."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "Or you could almost say for my first sample. I know it's very confusing, because you can consider that a sample, the set to be a sample, or you could consider each member of the set as a sample. So that can be a little bit confusing there. But I have this first sample mean, and then I keep doing that over and over. In my second sample, my sample size is 4. I got four instances of this random variable. I average them."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "But I have this first sample mean, and then I keep doing that over and over. In my second sample, my sample size is 4. I got four instances of this random variable. I average them. I have another sample mean. And the cool thing about the central limit theorem is as I keep plotting the frequency distribution of my sample means, it starts to approach something that approximates the normal distribution. And it's going to do a better job of approximating that normal distribution as n gets larger."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "I average them. I have another sample mean. And the cool thing about the central limit theorem is as I keep plotting the frequency distribution of my sample means, it starts to approach something that approximates the normal distribution. And it's going to do a better job of approximating that normal distribution as n gets larger. And just so we have a little terminology on our belt, this frequency distribution right here that I've plotted out, or here, or up here, that I started plotting out, that is called, and it's kind of confusing because we use the word sample so much, that is called the sampling distribution of the sample mean. And let's dissect this a little bit, just so that this long description of this distribution starts to make a little bit of sense. When we say it's the sampling distribution, that's telling us that it's being derived from, it's the distribution of some statistic, which in this case happens to be the sample mean, and we're deriving it from samples of an original distribution."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "And it's going to do a better job of approximating that normal distribution as n gets larger. And just so we have a little terminology on our belt, this frequency distribution right here that I've plotted out, or here, or up here, that I started plotting out, that is called, and it's kind of confusing because we use the word sample so much, that is called the sampling distribution of the sample mean. And let's dissect this a little bit, just so that this long description of this distribution starts to make a little bit of sense. When we say it's the sampling distribution, that's telling us that it's being derived from, it's the distribution of some statistic, which in this case happens to be the sample mean, and we're deriving it from samples of an original distribution. So each of these, so this is my first sample. My sample size is 4. I'm using the statistic, the mean."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "When we say it's the sampling distribution, that's telling us that it's being derived from, it's the distribution of some statistic, which in this case happens to be the sample mean, and we're deriving it from samples of an original distribution. So each of these, so this is my first sample. My sample size is 4. I'm using the statistic, the mean. I actually could have done it with other things. I could have done the mode, or the range, or other statistics, but the sampling distribution of the sample mean is the most common one. It's probably, in my mind, the best place to start learning about the central limit theorem, and even, frankly, sampling distribution."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "I'm using the statistic, the mean. I actually could have done it with other things. I could have done the mode, or the range, or other statistics, but the sampling distribution of the sample mean is the most common one. It's probably, in my mind, the best place to start learning about the central limit theorem, and even, frankly, sampling distribution. So that's what it's called. And just as a little bit of background, and I'll prove this to you experimentally, not mathematically, but I think the experimental is, on some levels, more satisfying with statistics, that this will have the same mean as your original distribution right here. So it has the same mean, but we'll see in the next video that this is actually going to start approximating a normal distribution, even though my original distribution that this is kind of generated from is completely non-normal."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "It's probably, in my mind, the best place to start learning about the central limit theorem, and even, frankly, sampling distribution. So that's what it's called. And just as a little bit of background, and I'll prove this to you experimentally, not mathematically, but I think the experimental is, on some levels, more satisfying with statistics, that this will have the same mean as your original distribution right here. So it has the same mean, but we'll see in the next video that this is actually going to start approximating a normal distribution, even though my original distribution that this is kind of generated from is completely non-normal. So let's do that with this app right here. And just to give proper credit where credit is due, I think it was developed at Rice University. This is from onlinestatbook.com."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "So it has the same mean, but we'll see in the next video that this is actually going to start approximating a normal distribution, even though my original distribution that this is kind of generated from is completely non-normal. So let's do that with this app right here. And just to give proper credit where credit is due, I think it was developed at Rice University. This is from onlinestatbook.com. This is their app, which I think is a really neat app, because it really helps you to visualize what a sampling distribution of the sample mean is. So I can literally create my own custom distribution here. So let me make something kind of crazy."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "This is from onlinestatbook.com. This is their app, which I think is a really neat app, because it really helps you to visualize what a sampling distribution of the sample mean is. So I can literally create my own custom distribution here. So let me make something kind of crazy. So you could do this, in theory, with a discrete or a continuous probability density function. But what they have here, we could take on one of 32 values and I'm just going to set the different probabilities of getting any of those 32 values. So clearly, this right here is not a normal distribution."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "So let me make something kind of crazy. So you could do this, in theory, with a discrete or a continuous probability density function. But what they have here, we could take on one of 32 values and I'm just going to set the different probabilities of getting any of those 32 values. So clearly, this right here is not a normal distribution. It looks a little bit bimodal, but it doesn't have long tails. But what I want to do is first just use the simulation to understand, or to better understand, what the sampling distribution is all about. So what I'm going to do is I'm going to take, well, let's start with 5 at a time."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "So clearly, this right here is not a normal distribution. It looks a little bit bimodal, but it doesn't have long tails. But what I want to do is first just use the simulation to understand, or to better understand, what the sampling distribution is all about. So what I'm going to do is I'm going to take, well, let's start with 5 at a time. So my sample size is going to be 5. And so when I click Animated, what it's going to do is it's going to take 5 samples from this probability distribution function, it's going to take 5 samples, and you're going to see them when I click Animated, it's going to average them and plot the average down here. And then I'm going to click it again, it's going to do it again."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "So what I'm going to do is I'm going to take, well, let's start with 5 at a time. So my sample size is going to be 5. And so when I click Animated, what it's going to do is it's going to take 5 samples from this probability distribution function, it's going to take 5 samples, and you're going to see them when I click Animated, it's going to average them and plot the average down here. And then I'm going to click it again, it's going to do it again. So there you go. It got 5 samples from there, it averaged them, and it hit there. What did I just do?"}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "And then I'm going to click it again, it's going to do it again. So there you go. It got 5 samples from there, it averaged them, and it hit there. What did I just do? I clicked, oh, I wanted to clear that. Let me make this bottom one none. So let me do that over again."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "What did I just do? I clicked, oh, I wanted to clear that. Let me make this bottom one none. So let me do that over again. So I'm going to take 5 at a time. So I took 5 samples from up here, and then it took its mean and plotted the mean there. Let me do it again."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "So let me do that over again. So I'm going to take 5 at a time. So I took 5 samples from up here, and then it took its mean and plotted the mean there. Let me do it again. 5 samples from this probability distribution function, plotted it right there. I could keep doing it, it'll take some time, but as you can see, I plotted it right there. Now, I could do this 1,000 times, it's going to take forever."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "Let me do it again. 5 samples from this probability distribution function, plotted it right there. I could keep doing it, it'll take some time, but as you can see, I plotted it right there. Now, I could do this 1,000 times, it's going to take forever. Let's say I just wanted to do it 1,000 times. So this program, just to be clear, it's actually generating the random numbers. This isn't like a rigged program."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "Now, I could do this 1,000 times, it's going to take forever. Let's say I just wanted to do it 1,000 times. So this program, just to be clear, it's actually generating the random numbers. This isn't like a rigged program. It's actually going to generate the random numbers according to this probability distribution function. It's going to take 5 at a time, find their means, and plot the mean. So if I click 10,000, it's going to do that 10,000 times."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "This isn't like a rigged program. It's actually going to generate the random numbers according to this probability distribution function. It's going to take 5 at a time, find their means, and plot the mean. So if I click 10,000, it's going to do that 10,000 times. So it's going to take 5 numbers from here 10,000 times and find their means 10,000 times, and then plot the 10,000 means here, so let's do that. So there you go. And notice, it's already looking a lot like a normal distribution."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "So if I click 10,000, it's going to do that 10,000 times. So it's going to take 5 numbers from here 10,000 times and find their means 10,000 times, and then plot the 10,000 means here, so let's do that. So there you go. And notice, it's already looking a lot like a normal distribution. And like I said, the original mean of my crazy distribution here was 14.45, and the mean of after doing 10,000 samples, or 10,000 trials, my mean here is 14.42. So I'm already getting pretty close to the mean there. My standard deviation, you might notice, is less than that, we'll talk about that in a future video."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "And notice, it's already looking a lot like a normal distribution. And like I said, the original mean of my crazy distribution here was 14.45, and the mean of after doing 10,000 samples, or 10,000 trials, my mean here is 14.42. So I'm already getting pretty close to the mean there. My standard deviation, you might notice, is less than that, we'll talk about that in a future video. And the skew and kurtosis, these are things that help us measure how normal a distribution is. And I've talked a little bit about it in the past, and let me actually just diverge a little bit, just so it's interesting. And they're fairly straightforward concepts."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "My standard deviation, you might notice, is less than that, we'll talk about that in a future video. And the skew and kurtosis, these are things that help us measure how normal a distribution is. And I've talked a little bit about it in the past, and let me actually just diverge a little bit, just so it's interesting. And they're fairly straightforward concepts. Skew literally tells, so if this is, let me do it in a different color. If this is a perfect normal distribution, and clearly my drawing is very far from perfect, if that's a perfect distribution, this would have a skew of 0. If you have a positive skew, that means you have a larger right tail than you would have otherwise expect."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "And they're fairly straightforward concepts. Skew literally tells, so if this is, let me do it in a different color. If this is a perfect normal distribution, and clearly my drawing is very far from perfect, if that's a perfect distribution, this would have a skew of 0. If you have a positive skew, that means you have a larger right tail than you would have otherwise expect. So something with a positive skew might look like this. It would have a large tail to the right. So this would be a positive skew, which makes it a little less than ideal for normal distribution."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "If you have a positive skew, that means you have a larger right tail than you would have otherwise expect. So something with a positive skew might look like this. It would have a large tail to the right. So this would be a positive skew, which makes it a little less than ideal for normal distribution. And a negative skew would look like this. It has a long tail to the left. So a negative skew might look like that."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "So this would be a positive skew, which makes it a little less than ideal for normal distribution. And a negative skew would look like this. It has a long tail to the left. So a negative skew might look like that. So that is a negative skew. If you have trouble remembering it, just remember which direction the tail is going. This tail is going towards the negative direction."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "So a negative skew might look like that. So that is a negative skew. If you have trouble remembering it, just remember which direction the tail is going. This tail is going towards the negative direction. This tail is going to the positive direction. So if something has no skew, that means that it's nice and symmetrical around its mean. Now kurtosis, which sounds like a very fancy word, is similarly not that fancy of an idea."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "This tail is going towards the negative direction. This tail is going to the positive direction. So if something has no skew, that means that it's nice and symmetrical around its mean. Now kurtosis, which sounds like a very fancy word, is similarly not that fancy of an idea. Kurtosis. So once again, if I were to draw a perfect normal distribution, remember, there's no one normal distribution. You can have different means and different standard deviations."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "Now kurtosis, which sounds like a very fancy word, is similarly not that fancy of an idea. Kurtosis. So once again, if I were to draw a perfect normal distribution, remember, there's no one normal distribution. You can have different means and different standard deviations. Let's say that's a perfect normal distribution. If I have positive kurtosis, what's going to happen is I'm going to have fatter tails. Let me draw it a little nicer than that."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "You can have different means and different standard deviations. Let's say that's a perfect normal distribution. If I have positive kurtosis, what's going to happen is I'm going to have fatter tails. Let me draw it a little nicer than that. I'm going to have fatter tails, but I'm going to have a more pointy peak. I didn't even have to draw it that pointy. Let me draw it like this."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "Let me draw it a little nicer than that. I'm going to have fatter tails, but I'm going to have a more pointy peak. I didn't even have to draw it that pointy. Let me draw it like this. I'm going to have fatter tails, and I'm going to have a more pointy peak than a normal distribution. So this right here is positive kurtosis. So something that has positive kurtosis, depending on how positive it is, it tells you it's a little bit more pointy than a real normal distribution."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "Let me draw it like this. I'm going to have fatter tails, and I'm going to have a more pointy peak than a normal distribution. So this right here is positive kurtosis. So something that has positive kurtosis, depending on how positive it is, it tells you it's a little bit more pointy than a real normal distribution. Positive kurtosis. And negative kurtosis has smaller tails, but it's smoother near the middle. So it's like this."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "So something that has positive kurtosis, depending on how positive it is, it tells you it's a little bit more pointy than a real normal distribution. Positive kurtosis. And negative kurtosis has smaller tails, but it's smoother near the middle. So it's like this. So something like this would have negative kurtosis. And maybe in future videos we'll explore that in more detail. But in the context of this simulation, it's just telling us how normal this distribution is."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "So it's like this. So something like this would have negative kurtosis. And maybe in future videos we'll explore that in more detail. But in the context of this simulation, it's just telling us how normal this distribution is. So when our sample size was n equals 5 and we did 10,000 trials, we got pretty close to normal distribution. Let's do another 10,000 trials just to see what happens. It looks even more like a normal distribution."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "But in the context of this simulation, it's just telling us how normal this distribution is. So when our sample size was n equals 5 and we did 10,000 trials, we got pretty close to normal distribution. Let's do another 10,000 trials just to see what happens. It looks even more like a normal distribution. Our mean is now the exact same number, but we still have a little bit of skew and a little bit of kurtosis. Now let's see what happens if we were to do the same thing with a larger sample size. And we can actually do them simultaneously."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "It looks even more like a normal distribution. Our mean is now the exact same number, but we still have a little bit of skew and a little bit of kurtosis. Now let's see what happens if we were to do the same thing with a larger sample size. And we can actually do them simultaneously. So here's n equals 5. Let's do here n equals 25. Let me clear them."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "And we can actually do them simultaneously. So here's n equals 5. Let's do here n equals 25. Let me clear them. I'm going to do the sampling distribution of the sample mean, and I'm going to run 10,000 trials. So I'll do one animated trial, just so you remember what's going on. So I'm literally taking first five samples from up here, find their mean."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "Let me clear them. I'm going to do the sampling distribution of the sample mean, and I'm going to run 10,000 trials. So I'll do one animated trial, just so you remember what's going on. So I'm literally taking first five samples from up here, find their mean. Now I'm taking 25 samples from up here, find its mean, and then plotting it down here. So here the sample size is 25, here it's 5. I'll do it one more time."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "So I'm literally taking first five samples from up here, find their mean. Now I'm taking 25 samples from up here, find its mean, and then plotting it down here. So here the sample size is 25, here it's 5. I'll do it one more time. I take 5, get the mean, plot it. Take 25, get the mean, and then plot it down there. This is a larger sample size."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "I'll do it one more time. I take 5, get the mean, plot it. Take 25, get the mean, and then plot it down there. This is a larger sample size. Now that thing that I just did, I'm going to do 10,000 times, and that's interesting. Remember, our first distribution was just this really crazy, very non-normal distribution. But once we did it, so here what's interesting."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "This is a larger sample size. Now that thing that I just did, I'm going to do 10,000 times, and that's interesting. Remember, our first distribution was just this really crazy, very non-normal distribution. But once we did it, so here what's interesting. I mean, they both look a little normal, but if you look at the skew and the kurtosis when our sample size is larger, it's more normal. This has a lower skew than when our sample size was only 5, and it has a less negative kurtosis than when our sample size was 5. So this is a more normal distribution."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "But once we did it, so here what's interesting. I mean, they both look a little normal, but if you look at the skew and the kurtosis when our sample size is larger, it's more normal. This has a lower skew than when our sample size was only 5, and it has a less negative kurtosis than when our sample size was 5. So this is a more normal distribution. And one thing that we're going to explore further in a future video is not only is it more normal in its shape, but it's also a tighter fit around the mean. And you can even think about why that kind of makes sense. When your sample size is larger, your odds of getting really far away from the mean is lower, because it's very low likelihood if you're taking 25 samples or 100 samples that you're just going to get a bunch of stuff way out here, or a bunch of stuff way out here."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "So this is a more normal distribution. And one thing that we're going to explore further in a future video is not only is it more normal in its shape, but it's also a tighter fit around the mean. And you can even think about why that kind of makes sense. When your sample size is larger, your odds of getting really far away from the mean is lower, because it's very low likelihood if you're taking 25 samples or 100 samples that you're just going to get a bunch of stuff way out here, or a bunch of stuff way out here. You're very likely to get a reasonable spread of things. So it makes sense that your mean, your sample mean, is less likely to be far away from the mean. We're going to talk a little bit more about that in the future."}, {"video_title": "Sampling distribution of the sample mean Probability and Statistics Khan Academy.mp3", "Sentence": "When your sample size is larger, your odds of getting really far away from the mean is lower, because it's very low likelihood if you're taking 25 samples or 100 samples that you're just going to get a bunch of stuff way out here, or a bunch of stuff way out here. You're very likely to get a reasonable spread of things. So it makes sense that your mean, your sample mean, is less likely to be far away from the mean. We're going to talk a little bit more about that in the future. But hopefully this kind of satisfies you that, at least experimentally, I haven't proven it to you with mathematical rigor, which hopefully we'll do in the future, but hopefully this satisfies you at least experimentally that the central limit theorem really does apply to any distribution. I mean, this is a crazy distribution. I encourage you to use this applet at onlinestatbook.com and experiment with other crazy distributions to believe it for yourself."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Before applying to medical school, students need to take an exam called the MCAT. Here are some summary statistics for each exam. So the LSAT, the mean score is 151 with a standard deviation of 10. And the MCAT, the mean score is 25.1 with a standard deviation of 6.4. Juwan took both exams. He scored 172 on the LSAT and 37 on the MCAT. Which exam did he do relatively better on?"}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And the MCAT, the mean score is 25.1 with a standard deviation of 6.4. Juwan took both exams. He scored 172 on the LSAT and 37 on the MCAT. Which exam did he do relatively better on? So pause this video and see if you can figure it out. So the way I would think about it is, you can't just look at the absolute score because they are on different scales and they have different distributions. But we can use this information, if we assume it's a normal distribution or relatively close to a normal distribution with a mean centered at this mean, we can think about, well, how many standard deviations from the mean did he score in each of these situations?"}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Which exam did he do relatively better on? So pause this video and see if you can figure it out. So the way I would think about it is, you can't just look at the absolute score because they are on different scales and they have different distributions. But we can use this information, if we assume it's a normal distribution or relatively close to a normal distribution with a mean centered at this mean, we can think about, well, how many standard deviations from the mean did he score in each of these situations? So in both cases, he scored above the mean, but how many standard deviations above the mean? So let's see if we can figure that out. So on the LSAT, let's see, let me write this down."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "But we can use this information, if we assume it's a normal distribution or relatively close to a normal distribution with a mean centered at this mean, we can think about, well, how many standard deviations from the mean did he score in each of these situations? So in both cases, he scored above the mean, but how many standard deviations above the mean? So let's see if we can figure that out. So on the LSAT, let's see, let me write this down. On the LSAT, he scored 172. So how many standard deviations is that going to be? Well, let's take 172, his score, minus the mean."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So on the LSAT, let's see, let me write this down. On the LSAT, he scored 172. So how many standard deviations is that going to be? Well, let's take 172, his score, minus the mean. So this is the absolute number that he scored above the mean. And now let's divide that by the standard deviation. So on the LSAT, this is what?"}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Well, let's take 172, his score, minus the mean. So this is the absolute number that he scored above the mean. And now let's divide that by the standard deviation. So on the LSAT, this is what? This is going to be 21 divided by 10. So this is 2.1 standard deviations. Deviations above the mean, above the mean."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So on the LSAT, this is what? This is going to be 21 divided by 10. So this is 2.1 standard deviations. Deviations above the mean, above the mean. You could view this as a z-score. It's a z-score of 2.1. We are 2.1 above the mean in this situation."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Deviations above the mean, above the mean. You could view this as a z-score. It's a z-score of 2.1. We are 2.1 above the mean in this situation. Now let's think about how he did on the MCAT. On the MCAT, he scored a 37. He scored a 37."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "We are 2.1 above the mean in this situation. Now let's think about how he did on the MCAT. On the MCAT, he scored a 37. He scored a 37. The mean is a 25.1. And there is a standard deviation of 6.4. So let's see, 37.1 minus 25 would be 12, but now it's gonna be 11.9."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "He scored a 37. The mean is a 25.1. And there is a standard deviation of 6.4. So let's see, 37.1 minus 25 would be 12, but now it's gonna be 11.9. 11.9 divided by 6.4. So without even looking at this, this is going to be approximately, well, this is gonna be a little bit less than two. This is going to be less than two."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So let's see, 37.1 minus 25 would be 12, but now it's gonna be 11.9. 11.9 divided by 6.4. So without even looking at this, this is going to be approximately, well, this is gonna be a little bit less than two. This is going to be less than two. So based on this information, and we could figure out the exact number here. In fact, let me get my calculator out. So you get the calculator."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "This is going to be less than two. So based on this information, and we could figure out the exact number here. In fact, let me get my calculator out. So you get the calculator. So if we do 11.9 divided by 6.4, that's gonna get us to 1 point, I'll just say approximately 1.86. So approximately 1.86. So relatively speaking, he did slightly better on the LSAT."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So you get the calculator. So if we do 11.9 divided by 6.4, that's gonna get us to 1 point, I'll just say approximately 1.86. So approximately 1.86. So relatively speaking, he did slightly better on the LSAT. He did more standard deviations, although this is close. I would say they're comparable. He did roughly two standard deviations if we were around to the nearest standard deviation, but if you wanted to get precise, he did a little bit better, relatively speaking, on the LSAT."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So relatively speaking, he did slightly better on the LSAT. He did more standard deviations, although this is close. I would say they're comparable. He did roughly two standard deviations if we were around to the nearest standard deviation, but if you wanted to get precise, he did a little bit better, relatively speaking, on the LSAT. He did 2.1 standard deviations here, while over here he did 1.86 or 1.9 standard deviations. But in everyday language, you would probably say, well, this is comparable. If this was three standard deviations and this is one standard deviation, then you'd be like, oh, he definitely did better on the LSAT."}, {"video_title": "Interpreting computer regression data AP Statistics Khan Academy.mp3", "Sentence": "So here it tells us Cheryl Dixon is interested to see if students who consume more caffeine tend to study more as well. She randomly selects 20 students at her school and records their caffeine intake in milligrams and the number of hours spent studying. A scatterplot of the data showed a linear relationship. This is a computer output from a least squares regression analysis on the data. So we have these things called the predictors, coefficient, and then we have these other things, standard error of coefficient, T and P, and then all of these things down here. How do we make sense of this in order to come up with an equation for our linear regression? So let's just get straight on our variables."}, {"video_title": "Interpreting computer regression data AP Statistics Khan Academy.mp3", "Sentence": "This is a computer output from a least squares regression analysis on the data. So we have these things called the predictors, coefficient, and then we have these other things, standard error of coefficient, T and P, and then all of these things down here. How do we make sense of this in order to come up with an equation for our linear regression? So let's just get straight on our variables. Let's just say that we say that Y is the thing that we're trying to predict. So this is the hours spent studying, hours studying, and then let's say X is what we think explains the hours studying or is one of the things that explains the hours studying, and this is the amount of caffeine ingested. So this is caffeine consumed in milligrams."}, {"video_title": "Interpreting computer regression data AP Statistics Khan Academy.mp3", "Sentence": "So let's just get straight on our variables. Let's just say that we say that Y is the thing that we're trying to predict. So this is the hours spent studying, hours studying, and then let's say X is what we think explains the hours studying or is one of the things that explains the hours studying, and this is the amount of caffeine ingested. So this is caffeine consumed in milligrams. And so our regression line would have the form Y hat. This tells us that this is a linear regression. It's trying to estimate the actual Y values for given Xs."}, {"video_title": "Interpreting computer regression data AP Statistics Khan Academy.mp3", "Sentence": "So this is caffeine consumed in milligrams. And so our regression line would have the form Y hat. This tells us that this is a linear regression. It's trying to estimate the actual Y values for given Xs. It's going to be equal to MX plus B. Now how do we figure out what M and B are based on this computer output? So when you look at this table here, this first column says predictor, and it says constant, and it has caffeine."}, {"video_title": "Interpreting computer regression data AP Statistics Khan Academy.mp3", "Sentence": "It's trying to estimate the actual Y values for given Xs. It's going to be equal to MX plus B. Now how do we figure out what M and B are based on this computer output? So when you look at this table here, this first column says predictor, and it says constant, and it has caffeine. And so all this is saying is when you're trying to predict the number of hours studying, when you're trying to predict Y, there's essentially two inputs there. There is the constant value, and there is your variable, in this case caffeine, that you're using to predict the amount that you study. And so this tells you the coefficients on each."}, {"video_title": "Interpreting computer regression data AP Statistics Khan Academy.mp3", "Sentence": "So when you look at this table here, this first column says predictor, and it says constant, and it has caffeine. And so all this is saying is when you're trying to predict the number of hours studying, when you're trying to predict Y, there's essentially two inputs there. There is the constant value, and there is your variable, in this case caffeine, that you're using to predict the amount that you study. And so this tells you the coefficients on each. The coefficient on a constant is the constant. You could view this as the coefficient on the X to the zeroth term. And so the coefficient on the constant, that is the constant, 2.544."}, {"video_title": "Interpreting computer regression data AP Statistics Khan Academy.mp3", "Sentence": "And so this tells you the coefficients on each. The coefficient on a constant is the constant. You could view this as the coefficient on the X to the zeroth term. And so the coefficient on the constant, that is the constant, 2.544. And then the coefficient on the caffeine, well, we just said that X is the caffeine consumed, so this is that coefficient, 0.164. So just like that, we actually have the equation for the regression line. That is why these computer things are useful."}, {"video_title": "Interpreting computer regression data AP Statistics Khan Academy.mp3", "Sentence": "And so the coefficient on the constant, that is the constant, 2.544. And then the coefficient on the caffeine, well, we just said that X is the caffeine consumed, so this is that coefficient, 0.164. So just like that, we actually have the equation for the regression line. That is why these computer things are useful. So we can just write it out. Y hat is equal to 0.164X plus 2.544, 2.544. So that's the regression line."}, {"video_title": "Interpreting computer regression data AP Statistics Khan Academy.mp3", "Sentence": "That is why these computer things are useful. So we can just write it out. Y hat is equal to 0.164X plus 2.544, 2.544. So that's the regression line. What is this other information they give us? Well, I won't give you a very satisfying answer because all of this is actually useful for inferential statistics. To think about things like, well, what is the probability that this is chance that we got something to fit this well?"}, {"video_title": "Interpreting computer regression data AP Statistics Khan Academy.mp3", "Sentence": "So that's the regression line. What is this other information they give us? Well, I won't give you a very satisfying answer because all of this is actually useful for inferential statistics. To think about things like, well, what is the probability that this is chance that we got something to fit this well? So this right over here is the R squared. And if you wanted to figure out the R from this, you would just take the square root here. We could say that R is going to be equal to the square root of 0.60032, depending on how much precision you have."}, {"video_title": "Interpreting computer regression data AP Statistics Khan Academy.mp3", "Sentence": "To think about things like, well, what is the probability that this is chance that we got something to fit this well? So this right over here is the R squared. And if you wanted to figure out the R from this, you would just take the square root here. We could say that R is going to be equal to the square root of 0.60032, depending on how much precision you have. But you might say, well, how do we know if R is a positive square root or the negative square root of that? R can take on values between negative one and positive one. And the answer is, you would look at the slope here."}, {"video_title": "Interpreting computer regression data AP Statistics Khan Academy.mp3", "Sentence": "We could say that R is going to be equal to the square root of 0.60032, depending on how much precision you have. But you might say, well, how do we know if R is a positive square root or the negative square root of that? R can take on values between negative one and positive one. And the answer is, you would look at the slope here. We have a positive slope, which tells us that R is going to be positive. If we had a negative slope, then we would take the negative square root. Now, this right here is the adjusted R squared."}, {"video_title": "Interpreting computer regression data AP Statistics Khan Academy.mp3", "Sentence": "And the answer is, you would look at the slope here. We have a positive slope, which tells us that R is going to be positive. If we had a negative slope, then we would take the negative square root. Now, this right here is the adjusted R squared. And we really don't have to worry about it too much when we're thinking about just bivariate data. We're talking about caffeine and hour studying in this case. If we started to have more variables that tried to explain the hour studying, then we would care about adjusted R squared, but we're not gonna do that just yet."}, {"video_title": "Interpreting computer regression data AP Statistics Khan Academy.mp3", "Sentence": "Now, this right here is the adjusted R squared. And we really don't have to worry about it too much when we're thinking about just bivariate data. We're talking about caffeine and hour studying in this case. If we started to have more variables that tried to explain the hour studying, then we would care about adjusted R squared, but we're not gonna do that just yet. Last but not least, this S variable, this is the standard deviation of the residuals, which we study in other videos. And why is that useful? Well, that's a measure of how well the regression line fits the data."}, {"video_title": "Interpreting computer regression data AP Statistics Khan Academy.mp3", "Sentence": "If we started to have more variables that tried to explain the hour studying, then we would care about adjusted R squared, but we're not gonna do that just yet. Last but not least, this S variable, this is the standard deviation of the residuals, which we study in other videos. And why is that useful? Well, that's a measure of how well the regression line fits the data. It's a measure of, we could say, the typical error. So big takeaway, computers are useful. They'll give you a lot of data."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So let's say you were to go to a restaurant, and just out of curiosity, you want to see what the makeup of the ages at the restaurant are. So you go around the restaurant, and you write down everyone's age. And so these are the ages of everyone in the restaurant at that moment. And so you're interested in somehow presenting this, somehow visualizing the distribution of the ages, because you want to just say, well, are there more young people, are there more teenagers, are there more middle-aged people, are there more seniors here? And so when you just look at these numbers, it really doesn't give you a good sense of it. It's just a bunch of numbers. And so how could you do that?"}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And so you're interested in somehow presenting this, somehow visualizing the distribution of the ages, because you want to just say, well, are there more young people, are there more teenagers, are there more middle-aged people, are there more seniors here? And so when you just look at these numbers, it really doesn't give you a good sense of it. It's just a bunch of numbers. And so how could you do that? Well, one way to think about it is is to put these ages into different buckets, and then to think about how many people are there in each of those buckets, or sometimes someone might say, how many in each of those bins? So let's do that. So let's do buckets or categories."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And so how could you do that? Well, one way to think about it is is to put these ages into different buckets, and then to think about how many people are there in each of those buckets, or sometimes someone might say, how many in each of those bins? So let's do that. So let's do buckets or categories. So I like, sometimes it's called a bin. So the bucket, I like to think of it more as a bucket. The bucket, and then the number in the bucket."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So let's do buckets or categories. So I like, sometimes it's called a bin. So the bucket, I like to think of it more as a bucket. The bucket, and then the number in the bucket. The number in the bucket. Number, I'll just write the number, whoops. It's the, whoops."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "The bucket, and then the number in the bucket. The number in the bucket. Number, I'll just write the number, whoops. It's the, whoops. It's the number. It's the number in the bucket. All right, so let's just make buckets, let's make them 10-year ranges."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It's the, whoops. It's the number. It's the number in the bucket. All right, so let's just make buckets, let's make them 10-year ranges. So let's say the first one is ages zero to nine. So how many people, and we just define all of the buckets here. So the next one is ages 10 to 19, then 20 to 29, then 30 to 39, and 40 to 49, 50 to 59."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "All right, so let's just make buckets, let's make them 10-year ranges. So let's say the first one is ages zero to nine. So how many people, and we just define all of the buckets here. So the next one is ages 10 to 19, then 20 to 29, then 30 to 39, and 40 to 49, 50 to 59. Make sure you can read that properly. Then you have 60 to 69. I think that covers everyone."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So the next one is ages 10 to 19, then 20 to 29, then 30 to 39, and 40 to 49, 50 to 59. Make sure you can read that properly. Then you have 60 to 69. I think that covers everyone. I don't see anyone 70 years old or older here. So then how many people fall into the zero to nine-year-old bucket? Well, it's gonna be one, two, three, four, five, six people fall into that bucket."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "I think that covers everyone. I don't see anyone 70 years old or older here. So then how many people fall into the zero to nine-year-old bucket? Well, it's gonna be one, two, three, four, five, six people fall into that bucket. How many people fall into the, how many people fall into the 10 to 19-year-old bucket? Well, let's see. One, two, three."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, it's gonna be one, two, three, four, five, six people fall into that bucket. How many people fall into the, how many people fall into the 10 to 19-year-old bucket? Well, let's see. One, two, three. Three people. And I think you see where this is going. What about 20 to 29?"}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "One, two, three. Three people. And I think you see where this is going. What about 20 to 29? So it's one, two, three, four, five people. Five people fall into that bucket. All right, what about 30 to 39?"}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "What about 20 to 29? So it's one, two, three, four, five people. Five people fall into that bucket. All right, what about 30 to 39? We have one, and that's it. Only one person in that 30 to 39 bin or bucket or category. All right, what about 40 to 49?"}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "All right, what about 30 to 39? We have one, and that's it. Only one person in that 30 to 39 bin or bucket or category. All right, what about 40 to 49? We have one, two people. Two people are in that bucket. And then 50 to 59."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "All right, what about 40 to 49? We have one, two people. Two people are in that bucket. And then 50 to 59. So you have one, two people. Two people. And then finally, finally, ages 60 to 69, let me do it in a different color, 60 to 69, there is one person right over there."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then 50 to 59. So you have one, two people. Two people. And then finally, finally, ages 60 to 69, let me do it in a different color, 60 to 69, there is one person right over there. So this is one way of thinking about how the ages are distributed, but let's actually make a visualization of this. And the visualization that we're gonna create, this is called a histogram. Histogram."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then finally, finally, ages 60 to 69, let me do it in a different color, 60 to 69, there is one person right over there. So this is one way of thinking about how the ages are distributed, but let's actually make a visualization of this. And the visualization that we're gonna create, this is called a histogram. Histogram. Histogram. We're taking data that can take on a whole bunch of different values. We're putting them into categories, and we're gonna plot how many folks are in each category."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Histogram. Histogram. We're taking data that can take on a whole bunch of different values. We're putting them into categories, and we're gonna plot how many folks are in each category. How big are each of those categories? And actually, I wrote histogram. I wrote histograph, I should have written histogram."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We're putting them into categories, and we're gonna plot how many folks are in each category. How big are each of those categories? And actually, I wrote histogram. I wrote histograph, I should have written histogram. So a histogram. So let's do this. All right, so on this axis, let's see, the largest category has six."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "I wrote histograph, I should have written histogram. So a histogram. So let's do this. All right, so on this axis, let's see, the largest category has six. So this is the number. Number of folks, and it's gonna go one, two, three, four, five, six. One, two, three, four, five, six."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "All right, so on this axis, let's see, the largest category has six. So this is the number. Number of folks, and it's gonna go one, two, three, four, five, six. One, two, three, four, five, six. This is the number, and on this axis, I'm gonna make the buckets. The buckets, and let me scroll up a little bit now that I have my data here, I don't have to look at my data set again. So I have one bucket."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "One, two, three, four, five, six. This is the number, and on this axis, I'm gonna make the buckets. The buckets, and let me scroll up a little bit now that I have my data here, I don't have to look at my data set again. So I have one bucket. This is going to be the zero to nine bucket, right over here. Zero to nine. Then I'm going to have the three, actually, let me just plot them, since I have my pen that color."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So I have one bucket. This is going to be the zero to nine bucket, right over here. Zero to nine. Then I'm going to have the three, actually, let me just plot them, since I have my pen that color. So zero to nine, there are six people. Zero to nine, there are six people. So I'll just plot it like that."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Then I'm going to have the three, actually, let me just plot them, since I have my pen that color. So zero to nine, there are six people. Zero to nine, there are six people. So I'll just plot it like that. And then we have the 10 to 19. There are three people. So 10 to 19, there are three people."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So I'll just plot it like that. And then we have the 10 to 19. There are three people. So 10 to 19, there are three people. So I'll do a bar like this. Then 20 to 29, I have five people. 20 to 29, which is gonna be this one, which is getting, I'm writing too big."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So 10 to 19, there are three people. So I'll do a bar like this. Then 20 to 29, I have five people. 20 to 29, which is gonna be this one, which is getting, I'm writing too big. So 20 to 29, this is gonna be this bar. There's five people. Five people there."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "20 to 29, which is gonna be this one, which is getting, I'm writing too big. So 20 to 29, this is gonna be this bar. There's five people. Five people there. So it'll look like this. I should have made the bars wide enough so I could write below them, but I've already, that train has already left. All right, then 30 to 39."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Five people there. So it'll look like this. I should have made the bars wide enough so I could write below them, but I've already, that train has already left. All right, then 30 to 39. I'll try to write smaller. 30 to 39, that's gonna be this bar right over here. We have one person."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "All right, then 30 to 39. I'll try to write smaller. 30 to 39, that's gonna be this bar right over here. We have one person. One person. And then we have 40 to 49. We have two people."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We have one person. One person. And then we have 40 to 49. We have two people. 40 to 49, two people. So it looks like this. 40 to 49, two people."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We have two people. 40 to 49, two people. So it looks like this. 40 to 49, two people. Almost there, 50 to 59. We have two people. 50 to 59, we also have two people."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "40 to 49, two people. Almost there, 50 to 59. We have two people. 50 to 59, we also have two people. So that's that right over there. That's this category. And then finally, 60 to 69, we have one person."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "50 to 59, we also have two people. So that's that right over there. That's this category. And then finally, 60 to 69, we have one person. 60 to 69, we have one person. We have one person. And what I have just constructed, I took our data, I put it into buckets that are kind of representative of the categories I care about, zero to nine is kind of young kids, 10 to 19, I guess you could call them adolescents or roughly teenagers, although obviously if you're 10 you're not quite a teenager yet."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then finally, 60 to 69, we have one person. 60 to 69, we have one person. We have one person. And what I have just constructed, I took our data, I put it into buckets that are kind of representative of the categories I care about, zero to nine is kind of young kids, 10 to 19, I guess you could call them adolescents or roughly teenagers, although obviously if you're 10 you're not quite a teenager yet. And then all the different age groups. And then when I counted the number in each bucket and I plotted it, now I can visually get a sense of how are the ages distributed in this restaurant. This must be some type of a restaurant that gives away toys or something because there's a lot of younger people."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And what I have just constructed, I took our data, I put it into buckets that are kind of representative of the categories I care about, zero to nine is kind of young kids, 10 to 19, I guess you could call them adolescents or roughly teenagers, although obviously if you're 10 you're not quite a teenager yet. And then all the different age groups. And then when I counted the number in each bucket and I plotted it, now I can visually get a sense of how are the ages distributed in this restaurant. This must be some type of a restaurant that gives away toys or something because there's a lot of younger people. Maybe it's very family friendly. So every adult that comes in, maybe there's a lot of young adults with kids or maybe grandparents up here, and they just bring a lot of kids to this restaurant. So it gives you a view of what's going on here."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "This must be some type of a restaurant that gives away toys or something because there's a lot of younger people. Maybe it's very family friendly. So every adult that comes in, maybe there's a lot of young adults with kids or maybe grandparents up here, and they just bring a lot of kids to this restaurant. So it gives you a view of what's going on here. Just a lot of kids here, a lot fewer senior citizens. So once again, this is just a way of visualizing things. We took a lot of data that can take multiple data points."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So it gives you a view of what's going on here. Just a lot of kids here, a lot fewer senior citizens. So once again, this is just a way of visualizing things. We took a lot of data that can take multiple data points. Instead of plotting each data point like we might do in a dot plot, instead of saying how many one year olds are there, well there's only one one year old, how many three year olds are there? There's only one three year old. That wouldn't give us much information."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We took a lot of data that can take multiple data points. Instead of plotting each data point like we might do in a dot plot, instead of saying how many one year olds are there, well there's only one one year old, how many three year olds are there? There's only one three year old. That wouldn't give us much information. We would just have like these single dots if we were doing a dot plot. But as a histogram, we're able to put them into buckets. And we're just like, hey, generally between the ages zero and nine, we have six people."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "That wouldn't give us much information. We would just have like these single dots if we were doing a dot plot. But as a histogram, we're able to put them into buckets. And we're just like, hey, generally between the ages zero and nine, we have six people. And so you see that plotted out just like that. And obviously this doesn't apply just to ages of people in a restaurant. It applies to all sorts of data that you might want to collect and observe."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Accident frequency, and I'm just making this up. And I could just show these data points, maybe for some kind of statistical survey, that when the age is this, whatever number this is, maybe this is 20 years old, this is the accident frequency, and it could be a number of accidents per hundred. And that when the age is 21 years old, this is the frequency. And so these data scientists or statisticians went and plotted all of these in this scatter plot. This is often known as bivariate data, which is a very fancy way of saying, hey, you're plotting things that take two variables into consideration, and you're trying to see whether there's a pattern with how they relate. And what we're going to do in this video is think about, well, can we try to fit a line? Does it look like there's a linear or nonlinear relationship between the variables on the different axes?"}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And so these data scientists or statisticians went and plotted all of these in this scatter plot. This is often known as bivariate data, which is a very fancy way of saying, hey, you're plotting things that take two variables into consideration, and you're trying to see whether there's a pattern with how they relate. And what we're going to do in this video is think about, well, can we try to fit a line? Does it look like there's a linear or nonlinear relationship between the variables on the different axes? How strong is that variable? Is it a positive, is it a negative relationship? And then we'll think about this idea of outliers."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Does it look like there's a linear or nonlinear relationship between the variables on the different axes? How strong is that variable? Is it a positive, is it a negative relationship? And then we'll think about this idea of outliers. So let's just first think about whether there's a linear or nonlinear relationship. And I'll get my little ruler tool out here. So this data right over here, it looks like I could put a line through it that gets pretty close through the data."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And then we'll think about this idea of outliers. So let's just first think about whether there's a linear or nonlinear relationship. And I'll get my little ruler tool out here. So this data right over here, it looks like I could put a line through it that gets pretty close through the data. You're not gonna, it's very unlikely you're gonna be able to go through all of the data points, but you can try to get a line, and I'm just doing this. There's more numerical, more precise ways of doing this, but I'm just eyeballing it right over here. And it looks like I could plot a line that looks something like that that goes roughly through the data."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So this data right over here, it looks like I could put a line through it that gets pretty close through the data. You're not gonna, it's very unlikely you're gonna be able to go through all of the data points, but you can try to get a line, and I'm just doing this. There's more numerical, more precise ways of doing this, but I'm just eyeballing it right over here. And it looks like I could plot a line that looks something like that that goes roughly through the data. So this looks pretty linear. So I would call this a linear relationship. And since as we increase one variable, it looks like the other variable decreases."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And it looks like I could plot a line that looks something like that that goes roughly through the data. So this looks pretty linear. So I would call this a linear relationship. And since as we increase one variable, it looks like the other variable decreases. This is a downward-sloping line. I would say this is a negative, this is a negative linear relationship. But this one looks pretty strong."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And since as we increase one variable, it looks like the other variable decreases. This is a downward-sloping line. I would say this is a negative, this is a negative linear relationship. But this one looks pretty strong. So because the dots aren't that far from my line, this one gets a little bit further, but it's not, you know, there's not some dots way out there. So most of them are pretty close to the line. So I'll call this a negative, reasonably strong linear relationship."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "But this one looks pretty strong. So because the dots aren't that far from my line, this one gets a little bit further, but it's not, you know, there's not some dots way out there. So most of them are pretty close to the line. So I'll call this a negative, reasonably strong linear relationship. Negative, strong. I call reasonably, I ought to say strong, but reasonably strong linear, linear relationship between these two variables. Now let's look at this one, and pause this video and think about what this one would be for you."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So I'll call this a negative, reasonably strong linear relationship. Negative, strong. I call reasonably, I ought to say strong, but reasonably strong linear, linear relationship between these two variables. Now let's look at this one, and pause this video and think about what this one would be for you. Well, let's see. I'll get my ruler tool out again. It looks like I can try to put a line."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Now let's look at this one, and pause this video and think about what this one would be for you. Well, let's see. I'll get my ruler tool out again. It looks like I can try to put a line. It looks like generally speaking, as one variable increases, the other variable increases as well. So something like this goes through the data and approximates the direction. And this looks positive."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "It looks like I can try to put a line. It looks like generally speaking, as one variable increases, the other variable increases as well. So something like this goes through the data and approximates the direction. And this looks positive. As one variable increases, the other variable increases, roughly. So this is a positive relationship. But this is weak."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And this looks positive. As one variable increases, the other variable increases, roughly. So this is a positive relationship. But this is weak. A lot of the data is off, well off of the line. So positive, weak. But I'd say this is still linear."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "But this is weak. A lot of the data is off, well off of the line. So positive, weak. But I'd say this is still linear. It seems that as we increase one, the other one increases at roughly the same rate, although these data points are all over the place. So I would still call this linear. Now there's also this notion of outliers."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "But I'd say this is still linear. It seems that as we increase one, the other one increases at roughly the same rate, although these data points are all over the place. So I would still call this linear. Now there's also this notion of outliers. If I said, hey, this line is trying to describe the data, well, we have some data that is fairly off the line. So for example, even though we're saying it's a positive, weak, linear relationship, this one over here is reasonably high on the vertical variable, but it's low on the horizontal variable. And so this one right over here is an outlier."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Now there's also this notion of outliers. If I said, hey, this line is trying to describe the data, well, we have some data that is fairly off the line. So for example, even though we're saying it's a positive, weak, linear relationship, this one over here is reasonably high on the vertical variable, but it's low on the horizontal variable. And so this one right over here is an outlier. It's quite far away from the line. You could view that as an outlier. And this is a little bit subjective."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And so this one right over here is an outlier. It's quite far away from the line. You could view that as an outlier. And this is a little bit subjective. Outliers, well, what looks pretty far from the rest of the data? This could also be an outlier. Let me label these."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And this is a little bit subjective. Outliers, well, what looks pretty far from the rest of the data? This could also be an outlier. Let me label these. Outlier. Now pause the video and see if you can think about this one. Is this positive or negative?"}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Let me label these. Outlier. Now pause the video and see if you can think about this one. Is this positive or negative? Is it linear, non-linear? Is it strong or weak? I'll get my ruler tool out here."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Is this positive or negative? Is it linear, non-linear? Is it strong or weak? I'll get my ruler tool out here. So this goes here. It seems like I can fit a line pretty well to this. So I could fit, maybe I'll do the line in purple."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "I'll get my ruler tool out here. So this goes here. It seems like I can fit a line pretty well to this. So I could fit, maybe I'll do the line in purple. I could fit a line that looks like that. And so this one looks like it's positive. As one variable increases, the other one does for these data points."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So I could fit, maybe I'll do the line in purple. I could fit a line that looks like that. And so this one looks like it's positive. As one variable increases, the other one does for these data points. So it's a positive. I'd say this is pretty strong. The dots are pretty close to the line."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "As one variable increases, the other one does for these data points. So it's a positive. I'd say this is pretty strong. The dots are pretty close to the line. It really does look like a little bit of a fat line if you just look at the dots. So positive, strong, linear, linear relationship. And none of these data points are really strong outliers."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "The dots are pretty close to the line. It really does look like a little bit of a fat line if you just look at the dots. So positive, strong, linear, linear relationship. And none of these data points are really strong outliers. This one's a little bit further out. But they're all pretty close to the line and seem to describe that trend roughly. All right, now let's look at this data right over here."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And none of these data points are really strong outliers. This one's a little bit further out. But they're all pretty close to the line and seem to describe that trend roughly. All right, now let's look at this data right over here. So let me get my line tool out again. So it looks like I can fit a line. So it looks like it's a positive relationship."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "All right, now let's look at this data right over here. So let me get my line tool out again. So it looks like I can fit a line. So it looks like it's a positive relationship. The line would be upward sloping. It would look something like this. And once again, I'm eyeballing it."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So it looks like it's a positive relationship. The line would be upward sloping. It would look something like this. And once again, I'm eyeballing it. You can use computers and other methods to actually find a more precise line that minimizes the collective distance to all of the points. But it looks like there is a positive. But I would say this one is a weak linear relationship because you have a lot of points that are far off the line."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And once again, I'm eyeballing it. You can use computers and other methods to actually find a more precise line that minimizes the collective distance to all of the points. But it looks like there is a positive. But I would say this one is a weak linear relationship because you have a lot of points that are far off the line. So not so strong. So I would call this a positive, weak, linear relationship. And there's a lot of outliers here."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "But I would say this one is a weak linear relationship because you have a lot of points that are far off the line. So not so strong. So I would call this a positive, weak, linear relationship. And there's a lot of outliers here. You know, this one over here is pretty far out. Now let's look at this one. Pause this video and think about is it positive, negative?"}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And there's a lot of outliers here. You know, this one over here is pretty far out. Now let's look at this one. Pause this video and think about is it positive, negative? Is it strong or weak? Is this linear, nonlinear? Well, the first thing we wanna do is just think about it in linear, nonlinear."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Pause this video and think about is it positive, negative? Is it strong or weak? Is this linear, nonlinear? Well, the first thing we wanna do is just think about it in linear, nonlinear. I could try to put a line on it. But if I try to put a line on it, it's actually quite difficult. If I try to do a line like this, notice everything is kind of bending away from the line."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Well, the first thing we wanna do is just think about it in linear, nonlinear. I could try to put a line on it. But if I try to put a line on it, it's actually quite difficult. If I try to do a line like this, notice everything is kind of bending away from the line. It looks like generally as one variable increases, the other variable decreases. But they're not doing it in a linear fashion. It looks like there's some other type of curve at play."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "If I try to do a line like this, notice everything is kind of bending away from the line. It looks like generally as one variable increases, the other variable decreases. But they're not doing it in a linear fashion. It looks like there's some other type of curve at play. So I could try to do a fancier curve that looks something like this. And this seems to fit the data a lot better. So this one I would describe as nonlinear."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "It looks like there's some other type of curve at play. So I could try to do a fancier curve that looks something like this. And this seems to fit the data a lot better. So this one I would describe as nonlinear. And it is a negative relationship. As one variable increases, the other variable decreases. So this is a negative, I would say reasonably strong nonlinear relationship."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So this one I would describe as nonlinear. And it is a negative relationship. As one variable increases, the other variable decreases. So this is a negative, I would say reasonably strong nonlinear relationship. Pretty strong. Pretty strong. Once again, this is subjective."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So this is a negative, I would say reasonably strong nonlinear relationship. Pretty strong. Pretty strong. Once again, this is subjective. So I'll say negative, reasonably strong, nonlinear relationship. And maybe you could call this one an outlier, but it's not that far. And I might even be able to fit a curve that gets a little bit closer to that."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Once again, this is subjective. So I'll say negative, reasonably strong, nonlinear relationship. And maybe you could call this one an outlier, but it's not that far. And I might even be able to fit a curve that gets a little bit closer to that. Once again, I'm eyeballing this. Now let's do this last one. So this one looks like a negative linear relationship to me."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And I might even be able to fit a curve that gets a little bit closer to that. Once again, I'm eyeballing this. Now let's do this last one. So this one looks like a negative linear relationship to me. A fairly strong negative linear relationship, although there's some outliers. So let me draw this line. So that seems to fit the data pretty good."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So this one looks like a negative linear relationship to me. A fairly strong negative linear relationship, although there's some outliers. So let me draw this line. So that seems to fit the data pretty good. So this is a negative, reasonably strong, reasonably strong linear relationship. But these are very clear outliers. These are well away from the data or from the cluster of where most of the points are."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So that seems to fit the data pretty good. So this is a negative, reasonably strong, reasonably strong linear relationship. But these are very clear outliers. These are well away from the data or from the cluster of where most of the points are. So with some significant, with at least these two significant outliers here. So hopefully this makes you a little bit familiar with some of this terminology. And it's important to keep in mind, this is a little bit subjective."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "These are well away from the data or from the cluster of where most of the points are. So with some significant, with at least these two significant outliers here. So hopefully this makes you a little bit familiar with some of this terminology. And it's important to keep in mind, this is a little bit subjective. There'll be some cases that are more obvious than others. So for, and oftentimes you wanna make a comparison. That this is a stronger linear, positive linear relationship than this one is, right over here, because you can see most of the data is closer to the line."}, {"video_title": "Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "Miriam was testing her null hypothesis that the population mean of some data set is equal to 18 versus her alternative hypothesis is that the mean is less than 18 with a sample of seven observations. Her test statistic, I can never say that right, was t is equal to negative 1.9. Assume that the conditions for inference were met. What is the approximate p-value for Miriam's test? So pause this video and see if you can figure this out on your own. All right, well I always just like to remind ourselves what's going on here before I just go ahead and calculate the p-value. So there's some data set, some population here, and the null hypothesis is that the true mean is 18."}, {"video_title": "Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "What is the approximate p-value for Miriam's test? So pause this video and see if you can figure this out on your own. All right, well I always just like to remind ourselves what's going on here before I just go ahead and calculate the p-value. So there's some data set, some population here, and the null hypothesis is that the true mean is 18. The alternative is that it's less than 18. So to test that null hypothesis, Miriam takes a sample. Sample size is equal to seven."}, {"video_title": "Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "So there's some data set, some population here, and the null hypothesis is that the true mean is 18. The alternative is that it's less than 18. So to test that null hypothesis, Miriam takes a sample. Sample size is equal to seven. From that, she would calculate her sample mean and her sample standard deviation. And from that, she would calculate this t-statistic. The way she would do that, or if they didn't tell us ahead of time what that was, they would say, okay, well we would say the t-statistic is equal to her sample mean minus the assumed mean from the null hypothesis, that's what we have over here, divided by, and this is a mouthful, our approximation of the standard error of the mean."}, {"video_title": "Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "Sample size is equal to seven. From that, she would calculate her sample mean and her sample standard deviation. And from that, she would calculate this t-statistic. The way she would do that, or if they didn't tell us ahead of time what that was, they would say, okay, well we would say the t-statistic is equal to her sample mean minus the assumed mean from the null hypothesis, that's what we have over here, divided by, and this is a mouthful, our approximation of the standard error of the mean. And the way we get that approximation, we take our sample standard deviation and divide it by the square root of our sample size. Well, they've calculated this ahead of time for us. This is equal to negative 1.9."}, {"video_title": "Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "The way she would do that, or if they didn't tell us ahead of time what that was, they would say, okay, well we would say the t-statistic is equal to her sample mean minus the assumed mean from the null hypothesis, that's what we have over here, divided by, and this is a mouthful, our approximation of the standard error of the mean. And the way we get that approximation, we take our sample standard deviation and divide it by the square root of our sample size. Well, they've calculated this ahead of time for us. This is equal to negative 1.9. And so if we think about a t-distribution, I'll try to hand-draw a rough t-distribution really fast, and if this is the mean of the t-distribution, what we are curious about, because our alternative hypothesis is that the mean is less than 18, so what we care about is, well, what is the probability of getting a t-value that is more than 1.9 below the mean? So this right over here, negative 1.9. So it's this area right there."}, {"video_title": "Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "This is equal to negative 1.9. And so if we think about a t-distribution, I'll try to hand-draw a rough t-distribution really fast, and if this is the mean of the t-distribution, what we are curious about, because our alternative hypothesis is that the mean is less than 18, so what we care about is, well, what is the probability of getting a t-value that is more than 1.9 below the mean? So this right over here, negative 1.9. So it's this area right there. And so I'm gonna do this with a TI-84, at least an emulator of a TI-84, and all we have to do is we would go to second distribution and then I would use the t-cumulative distribution function. So let's go there, that's the number six right there, click Enter. And so my lower bound, yeah, I essentially want it to be negative infinity, and so we can just call that negative infinity, it's an approximation of negative infinity, very, very low number."}, {"video_title": "Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "So it's this area right there. And so I'm gonna do this with a TI-84, at least an emulator of a TI-84, and all we have to do is we would go to second distribution and then I would use the t-cumulative distribution function. So let's go there, that's the number six right there, click Enter. And so my lower bound, yeah, I essentially want it to be negative infinity, and so we can just call that negative infinity, it's an approximation of negative infinity, very, very low number. Our upper bound would be negative 1.9, negative 1.9. And then our degrees of freedom, that's our sample size minus one. Our sample size is seven, so our degrees of freedom would be six, and so there we have it."}, {"video_title": "Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3", "Sentence": "And so my lower bound, yeah, I essentially want it to be negative infinity, and so we can just call that negative infinity, it's an approximation of negative infinity, very, very low number. Our upper bound would be negative 1.9, negative 1.9. And then our degrees of freedom, that's our sample size minus one. Our sample size is seven, so our degrees of freedom would be six, and so there we have it. And then so this would be, our p-value would be approximately 0.053. So our p-value would be approximately 0.053. And then what Miriam would do is, would compare this p-value to her preset significance level, to alpha."}, {"video_title": "Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "What we're going to do in this video is talk about type one errors and type two, type two errors. And this is in the context of significance testing. So just as a little bit of review, in order to do a significance test, we first come up with a null and an alternative hypothesis. And we'll do this on some population in question. These will say some hypotheses about a true parameter for this population. And the null hypothesis tends to be kind of what was always assumed or the status quo, while the alternative hypothesis, hey, there's news here, there's something alternative here. And to test it, and we're really testing the null hypothesis, we're gonna decide whether we want to reject or fail to reject the null hypothesis, we take a sample."}, {"video_title": "Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "And we'll do this on some population in question. These will say some hypotheses about a true parameter for this population. And the null hypothesis tends to be kind of what was always assumed or the status quo, while the alternative hypothesis, hey, there's news here, there's something alternative here. And to test it, and we're really testing the null hypothesis, we're gonna decide whether we want to reject or fail to reject the null hypothesis, we take a sample. We take a sample from this population. Using that sample, we calculate a statistic, we calculate a statistic that's trying to estimate the parameter in question. And then using that statistic, we try to come up with the probability of getting that statistic, the probability of getting that statistic that we just calculated from that sample of a certain size, given, if we were to assume that our null hypothesis, if our null hypothesis is true."}, {"video_title": "Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "And to test it, and we're really testing the null hypothesis, we're gonna decide whether we want to reject or fail to reject the null hypothesis, we take a sample. We take a sample from this population. Using that sample, we calculate a statistic, we calculate a statistic that's trying to estimate the parameter in question. And then using that statistic, we try to come up with the probability of getting that statistic, the probability of getting that statistic that we just calculated from that sample of a certain size, given, if we were to assume that our null hypothesis, if our null hypothesis is true. And if this probability, which is often known as a p-value, is below some threshold that we set ahead of time, which is known as the significance level, then we reject the null hypothesis. Let me write this down. So this right over here, this is our p-value."}, {"video_title": "Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "And then using that statistic, we try to come up with the probability of getting that statistic, the probability of getting that statistic that we just calculated from that sample of a certain size, given, if we were to assume that our null hypothesis, if our null hypothesis is true. And if this probability, which is often known as a p-value, is below some threshold that we set ahead of time, which is known as the significance level, then we reject the null hypothesis. Let me write this down. So this right over here, this is our p-value. This should be all we review, we introduce it in other videos. We have seen in other videos if our p-value is less than our significance level, then we reject, reject our null hypothesis. And if our p-value is greater than or equal to our significance level, alpha, then we fail to reject, fail to reject our null hypothesis."}, {"video_title": "Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "So this right over here, this is our p-value. This should be all we review, we introduce it in other videos. We have seen in other videos if our p-value is less than our significance level, then we reject, reject our null hypothesis. And if our p-value is greater than or equal to our significance level, alpha, then we fail to reject, fail to reject our null hypothesis. And when we reject our null hypothesis, some people say that might suggest the alternative hypothesis. And the reason why this makes sense is if the probability of getting this statistic from a sample of a certain size, if we assume that the null hypothesis is true, is reasonably low, if it's below a threshold, maybe this threshold is 5%, if the probability of that happening was less than 5%, then hey, maybe it's reasonable to reject it. But we might be wrong in either of these scenarios, and that's where these errors come into play."}, {"video_title": "Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "And if our p-value is greater than or equal to our significance level, alpha, then we fail to reject, fail to reject our null hypothesis. And when we reject our null hypothesis, some people say that might suggest the alternative hypothesis. And the reason why this makes sense is if the probability of getting this statistic from a sample of a certain size, if we assume that the null hypothesis is true, is reasonably low, if it's below a threshold, maybe this threshold is 5%, if the probability of that happening was less than 5%, then hey, maybe it's reasonable to reject it. But we might be wrong in either of these scenarios, and that's where these errors come into play. Let's make a grid to make this clear. So there's the reality. Let me put reality up here."}, {"video_title": "Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "But we might be wrong in either of these scenarios, and that's where these errors come into play. Let's make a grid to make this clear. So there's the reality. Let me put reality up here. So the reality is there's two possible scenarios in reality. One is is that the null hypothesis is true, and the other is that the null hypothesis is false. And then based on our significance test, there's two things that we might do."}, {"video_title": "Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "Let me put reality up here. So the reality is there's two possible scenarios in reality. One is is that the null hypothesis is true, and the other is that the null hypothesis is false. And then based on our significance test, there's two things that we might do. We might reject the null hypothesis, or we might fail to reject the null hypothesis. And so let's put a little grid here to think about the different combinations, the different scenarios here. So in a scenario where the null hypothesis is true, but we reject it, that feels like an error."}, {"video_title": "Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "And then based on our significance test, there's two things that we might do. We might reject the null hypothesis, or we might fail to reject the null hypothesis. And so let's put a little grid here to think about the different combinations, the different scenarios here. So in a scenario where the null hypothesis is true, but we reject it, that feels like an error. We shouldn't reject something that is true, and that indeed is a type one error. Type one error. You shouldn't reject the null hypothesis if it was true."}, {"video_title": "Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "So in a scenario where the null hypothesis is true, but we reject it, that feels like an error. We shouldn't reject something that is true, and that indeed is a type one error. Type one error. You shouldn't reject the null hypothesis if it was true. You should reject the null hypothesis if it was true. And you could even figure out what is the probability of getting a type one error? Well, that's gonna be your significance level."}, {"video_title": "Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "You shouldn't reject the null hypothesis if it was true. You should reject the null hypothesis if it was true. And you could even figure out what is the probability of getting a type one error? Well, that's gonna be your significance level. Because if your null hypothesis is true, let's say that your significance level is 5%. Well, 5% of the time, even if your null hypothesis is true, you're going to get a statistic that's going to make you reject the null hypothesis. So one way to think about the probability of a type one error is your significance level."}, {"video_title": "Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "Well, that's gonna be your significance level. Because if your null hypothesis is true, let's say that your significance level is 5%. Well, 5% of the time, even if your null hypothesis is true, you're going to get a statistic that's going to make you reject the null hypothesis. So one way to think about the probability of a type one error is your significance level. Now, if your null hypothesis is true and you fail to reject it, well, that's good. This, we could write this as, this is a correct, correct conclusion. The good thing just happened to happen this time."}, {"video_title": "Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3", "Sentence": "So one way to think about the probability of a type one error is your significance level. Now, if your null hypothesis is true and you fail to reject it, well, that's good. This, we could write this as, this is a correct, correct conclusion. The good thing just happened to happen this time. Now, if your null hypothesis is false and you reject it, that's also good. That is the correct, correct conclusion. But if your null hypothesis is false and you fail to reject it, well, then that is a type two error."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Flavia wanted to estimate the mean age of the faculty members at her large university. She took an SRS, or simple random sample, of 20 of the approximately 700 faculty members, and each faculty member in the sample provided Flavia with their age. The data were skewed to the right with a sample mean of 38.75. She's considering using her data to make a confidence interval to estimate the mean age of faculty members at her university. Which conditions for constructing a t-interval have been met? So pause this video and see if you can answer this on your own. Okay, now let's try to answer this together."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "She's considering using her data to make a confidence interval to estimate the mean age of faculty members at her university. Which conditions for constructing a t-interval have been met? So pause this video and see if you can answer this on your own. Okay, now let's try to answer this together. So there's 700 faculty members over here. She's trying to estimate the population mean, the mean age. She can't talk to all 700, so she takes a sample, a simple random sample of 20."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Okay, now let's try to answer this together. So there's 700 faculty members over here. She's trying to estimate the population mean, the mean age. She can't talk to all 700, so she takes a sample, a simple random sample of 20. So the n is equal to 20 here. From this 20, she calculates a sample mean of 38.75. Now ideally, she wants to construct a t-interval, a confidence interval using the t-statistic."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "She can't talk to all 700, so she takes a sample, a simple random sample of 20. So the n is equal to 20 here. From this 20, she calculates a sample mean of 38.75. Now ideally, she wants to construct a t-interval, a confidence interval using the t-statistic. And so that interval would look something like this. It would be the sample mean, plus or minus the critical value, times the sample standard deviation divided by the square root of n. And we use a t-statistic like this, and a t-table, and a t-distribution when we are trying to create confidence intervals for means where we don't have access to the standard deviation of the sampling distribution, but we can compute the sample standard deviation. Now in order for this to hold true, there's three conditions, just like what we saw when we thought about z-intervals."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Now ideally, she wants to construct a t-interval, a confidence interval using the t-statistic. And so that interval would look something like this. It would be the sample mean, plus or minus the critical value, times the sample standard deviation divided by the square root of n. And we use a t-statistic like this, and a t-table, and a t-distribution when we are trying to create confidence intervals for means where we don't have access to the standard deviation of the sampling distribution, but we can compute the sample standard deviation. Now in order for this to hold true, there's three conditions, just like what we saw when we thought about z-intervals. The first is, is that our sample is random. Well, they tell us that here, that she took a simple random sample of 20. And so we know that we are meeting that constraint, and that's actually choice A."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Now in order for this to hold true, there's three conditions, just like what we saw when we thought about z-intervals. The first is, is that our sample is random. Well, they tell us that here, that she took a simple random sample of 20. And so we know that we are meeting that constraint, and that's actually choice A. The data is a random sample from the population of interest. So we can circle that in. So the next condition is the normal condition."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And so we know that we are meeting that constraint, and that's actually choice A. The data is a random sample from the population of interest. So we can circle that in. So the next condition is the normal condition. Now the normal condition when we're doing a t-interval is a little bit more involved, because we do need to assume that the sampling distribution of the sample means is roughly normal. Now there's a couple of ways that we can get there. Either our sample size is greater than or equal to 30."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So the next condition is the normal condition. Now the normal condition when we're doing a t-interval is a little bit more involved, because we do need to assume that the sampling distribution of the sample means is roughly normal. Now there's a couple of ways that we can get there. Either our sample size is greater than or equal to 30. The central limit theorem tells us that then our sampling distribution, regardless of what the distribution is in the population, that the sampling distribution actually would then be approximately normal. She didn't meet that constraint right over here. Here, her sample size is only 20."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Either our sample size is greater than or equal to 30. The central limit theorem tells us that then our sampling distribution, regardless of what the distribution is in the population, that the sampling distribution actually would then be approximately normal. She didn't meet that constraint right over here. Here, her sample size is only 20. So, so far, this isn't looking good. Now that's not the only way to meet the normal condition. Another way to meet the normal condition, if we have a smaller sample size, smaller than 30, is one, if the original distribution of ages is normal."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Here, her sample size is only 20. So, so far, this isn't looking good. Now that's not the only way to meet the normal condition. Another way to meet the normal condition, if we have a smaller sample size, smaller than 30, is one, if the original distribution of ages is normal. So original, distribution, normal. Or even if it's roughly symmetric around the mean. So approximately symmetric."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Another way to meet the normal condition, if we have a smaller sample size, smaller than 30, is one, if the original distribution of ages is normal. So original, distribution, normal. Or even if it's roughly symmetric around the mean. So approximately symmetric. But if you look at this, they tell us that it has a right skew. They say the data were skewed to the right with the sample mean of 38.75. So that tells us that the data set that we're getting in our sample is not symmetric."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So approximately symmetric. But if you look at this, they tell us that it has a right skew. They say the data were skewed to the right with the sample mean of 38.75. So that tells us that the data set that we're getting in our sample is not symmetric. And the original distribution is unlikely to be normal. Think about it. It's not going to be, you're likely to have people who are, you could have faculty members who are 30 years older than this, 68 and 3 quarters, but you're very unlikely to have faculty members who are 30 years younger than this."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So that tells us that the data set that we're getting in our sample is not symmetric. And the original distribution is unlikely to be normal. Think about it. It's not going to be, you're likely to have people who are, you could have faculty members who are 30 years older than this, 68 and 3 quarters, but you're very unlikely to have faculty members who are 30 years younger than this. And that's actually what's causing that skew to the right. So this one does not meet the normal condition. We can't feel good that our sampling distribution of the sample means is going to be normal."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "It's not going to be, you're likely to have people who are, you could have faculty members who are 30 years older than this, 68 and 3 quarters, but you're very unlikely to have faculty members who are 30 years younger than this. And that's actually what's causing that skew to the right. So this one does not meet the normal condition. We can't feel good that our sampling distribution of the sample means is going to be normal. So I'm not gonna fill that one in. Choice C, individual observations can be considered independent. So there's two ways to meet this constraint."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "We can't feel good that our sampling distribution of the sample means is going to be normal. So I'm not gonna fill that one in. Choice C, individual observations can be considered independent. So there's two ways to meet this constraint. One is, is if we sample with replacement. Every faculty member we look at after asking them their age, we say, hey, go back into the pool and we might pick them again until we get our sample of 20. It does not look like she did that."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So there's two ways to meet this constraint. One is, is if we sample with replacement. Every faculty member we look at after asking them their age, we say, hey, go back into the pool and we might pick them again until we get our sample of 20. It does not look like she did that. It doesn't look like she sampled with replacement. And so even if you're sampling without replacement, the 10% rule says that, look, as long as this is less than 10% or less than or equal to 10% of the population, then we're good. And the 10% of this population is 70."}, {"video_title": "Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "It does not look like she did that. It doesn't look like she sampled with replacement. And so even if you're sampling without replacement, the 10% rule says that, look, as long as this is less than 10% or less than or equal to 10% of the population, then we're good. And the 10% of this population is 70. 70 is 10% of 700. And so this is definitely less than or equal to 10%. And so it can be considered independent."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "We're told that a board game has players roll two three-sided dice. These exist, and actually I looked it up, they do exist and they're actually fascinating. And subtract the numbers showing on the faces. The game only looks at non-negative differences. For example, if a player rolls a one and a three, the difference is two. Let D represent the difference in a given roll. Construct the theoretical probability distribution of D. So pause this video and see if you can have a go at that before we work through it together."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "The game only looks at non-negative differences. For example, if a player rolls a one and a three, the difference is two. Let D represent the difference in a given roll. Construct the theoretical probability distribution of D. So pause this video and see if you can have a go at that before we work through it together. All right, now let's work through it together. So let's just think about all of the scenarios for the two die. So let me draw a little table here."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "Construct the theoretical probability distribution of D. So pause this video and see if you can have a go at that before we work through it together. All right, now let's work through it together. So let's just think about all of the scenarios for the two die. So let me draw a little table here. So let me do it like that. And let me do it like this. And then let me put a little divider right over here."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "So let me draw a little table here. So let me do it like that. And let me do it like this. And then let me put a little divider right over here. And for this top, this is going to be die one, and then this is going to be die two. Die one can take on one, two, or three. And die two could be one, two, or three."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "And then let me put a little divider right over here. And for this top, this is going to be die one, and then this is going to be die two. Die one can take on one, two, or three. And die two could be one, two, or three. And so let me finish making this a bit of a table here. And what we wanna do is look at the difference, but the non-negative difference. So we'll always subtract the lower die from the higher die."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "And die two could be one, two, or three. And so let me finish making this a bit of a table here. And what we wanna do is look at the difference, but the non-negative difference. So we'll always subtract the lower die from the higher die. So what's the difference here? Well, this is going to be zero if I roll a one and a one. Now, what if I roll a two and a one?"}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "So we'll always subtract the lower die from the higher die. So what's the difference here? Well, this is going to be zero if I roll a one and a one. Now, what if I roll a two and a one? Well, here the difference is going to be two minus one, which is one. Here the difference is three minus one, which is two. Now, what about right over here?"}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "Now, what if I roll a two and a one? Well, here the difference is going to be two minus one, which is one. Here the difference is three minus one, which is two. Now, what about right over here? Well, here the higher die is two, the lower one is one right over here. So two minus one is one, two minus two is zero. And now this is going to be the higher roll."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "Now, what about right over here? Well, here the higher die is two, the lower one is one right over here. So two minus one is one, two minus two is zero. And now this is going to be the higher roll. Die one is gonna have the higher roll in this scenario. Three minus two is one. And then right over here, three minus one is two."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "And now this is going to be the higher roll. Die one is gonna have the higher roll in this scenario. Three minus two is one. And then right over here, three minus one is two. Now, if die one rolls a two, die two rolls a three. Die three is higher, three minus two is one. And then three minus three is zero."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "And then right over here, three minus one is two. Now, if die one rolls a two, die two rolls a three. Die three is higher, three minus two is one. And then three minus three is zero. So we've come up with all of the scenarios, and we can see that we're either gonna end up with a zero or one or a two when we look at the positive difference. So there's a scenario of getting a zero, a one, or a two. Those are the different differences that we could actually get."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "And then three minus three is zero. So we've come up with all of the scenarios, and we can see that we're either gonna end up with a zero or one or a two when we look at the positive difference. So there's a scenario of getting a zero, a one, or a two. Those are the different differences that we could actually get. And so let's think about the probability of each of them. What's the probability that the difference is zero? Well, we can see that one, two, three of the nine equally likely outcomes result in a difference of zero."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "Those are the different differences that we could actually get. And so let's think about the probability of each of them. What's the probability that the difference is zero? Well, we can see that one, two, three of the nine equally likely outcomes result in a difference of zero. So it's gonna be three out of nine or 1 3rd. What about a difference of, let me use blue, one? Well, we could see there are one, two, three, four of the nine scenarios have that."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "Well, we can see that one, two, three of the nine equally likely outcomes result in a difference of zero. So it's gonna be three out of nine or 1 3rd. What about a difference of, let me use blue, one? Well, we could see there are one, two, three, four of the nine scenarios have that. So there is a 4 9th probability. And then last but not least, a difference of two. Well, there's two out of the nine scenarios that have that."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "Well, we could see there are one, two, three, four of the nine scenarios have that. So there is a 4 9th probability. And then last but not least, a difference of two. Well, there's two out of the nine scenarios that have that. So there is a 2 9th probability right over there. And we're done. We've constructed the theoretical probability distribution of D."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "So let's say we define some random variable, let's call it x, and let's call it the number of rolls until we get a one. One. So what's the probability that x is equal to one? Pause this video and think about it. All right, the probability that x is equal to one means that it only takes us one roll to get a one. Well, that's going to be a 1 1\u2076 probability. Well, what's the probability that x is equal to two?"}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "Pause this video and think about it. All right, the probability that x is equal to one means that it only takes us one roll to get a one. Well, that's going to be a 1 1\u2076 probability. Well, what's the probability that x is equal to two? Well, that means that on the first roll, we get something other than a one, so that is going to be 5 6. And then on the second roll, we get a one, so that has a 1 6th probability. And we could keep going."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "Well, what's the probability that x is equal to two? Well, that means that on the first roll, we get something other than a one, so that is going to be 5 6. And then on the second roll, we get a one, so that has a 1 6th probability. And we could keep going. What's the probability that x is equal to three? Pause the video and think about that. Well, that means we miss on the first two, so we have a 5 6th chance of getting something other than a one on the first two rolls."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "And we could keep going. What's the probability that x is equal to three? Pause the video and think about that. Well, that means we miss on the first two, so we have a 5 6th chance of getting something other than a one on the first two rolls. So we could say that's 5 6th times 5 6th, so we could write 5 6th squared. And then on the third roll, we have the 1 6th chance of getting the one, so times 1 6th. And I think you see a pattern here, and you might recognize what type of random variable this is."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "Well, that means we miss on the first two, so we have a 5 6th chance of getting something other than a one on the first two rolls. So we could say that's 5 6th times 5 6th, so we could write 5 6th squared. And then on the third roll, we have the 1 6th chance of getting the one, so times 1 6th. And I think you see a pattern here, and you might recognize what type of random variable this is. This is a geometric variable. Now, how do we know that? Well, each trial or each roll is either a success or a failure."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "And I think you see a pattern here, and you might recognize what type of random variable this is. This is a geometric variable. Now, how do we know that? Well, each trial or each roll is either a success or a failure. Every time we roll, we either get a one or we don't. We have the same probability of success of rolling a one each trial. These are independent trials."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "Well, each trial or each roll is either a success or a failure. Every time we roll, we either get a one or we don't. We have the same probability of success of rolling a one each trial. These are independent trials. And that there's no set number of trials. It could take us an arbitrary number of trials to get the first success. So that's what tells us that we're dealing with the geometric random variable."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "These are independent trials. And that there's no set number of trials. It could take us an arbitrary number of trials to get the first success. So that's what tells us that we're dealing with the geometric random variable. Now, one question is, is what is going to be the mean of this geometric random variable? Well, we prove it in another video where we talk about the expected value of a geometric random variable. We're really talking about the mean of a geometric random variable."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "So that's what tells us that we're dealing with the geometric random variable. Now, one question is, is what is going to be the mean of this geometric random variable? Well, we prove it in another video where we talk about the expected value of a geometric random variable. We're really talking about the mean of a geometric random variable. And it is a little bit intuitive. If you were to just guess, what is the mean of a geometric random variable where the chance of success on each roll is one sixth, you might say, well, maybe on average, it takes you about six tries. And you would be correct."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "We're really talking about the mean of a geometric random variable. And it is a little bit intuitive. If you were to just guess, what is the mean of a geometric random variable where the chance of success on each roll is one sixth, you might say, well, maybe on average, it takes you about six tries. And you would be correct. The mean of a geometric random variable is one over the probability of success on each trial. So in this situation, the mean is going to be one over, the probability of success in each trial is one over six. So it's equal to six."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "And you would be correct. The mean of a geometric random variable is one over the probability of success on each trial. So in this situation, the mean is going to be one over, the probability of success in each trial is one over six. So it's equal to six. So one way to think about it is, on average, you would have six trials until you get a one. Now, another question is, what's a measure of the spread of a geometric random variable? And we don't prove this in another video."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "So it's equal to six. So one way to think about it is, on average, you would have six trials until you get a one. Now, another question is, what's a measure of the spread of a geometric random variable? And we don't prove this in another video. Maybe I'll do it eventually. That the standard deviation of a geometric random variable is the mean times the square root of one minus P, or you could just write this as a square root of one minus P over P. Now, in this situation, what would this be? Well, the standard deviation of this random variable, this geometric random variable, it's going to be the square root of one minus one sixth, all of that over one sixth."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "And we don't prove this in another video. Maybe I'll do it eventually. That the standard deviation of a geometric random variable is the mean times the square root of one minus P, or you could just write this as a square root of one minus P over P. Now, in this situation, what would this be? Well, the standard deviation of this random variable, this geometric random variable, it's going to be the square root of one minus one sixth, all of that over one sixth. So this is going to be equal to the square root of five sixths over one sixth, which is equal to six times the square root of five sixths. And this is going to be approximately equal to, five divided by six is equal to that. We'll take the square root of that and then multiply that times six, gets us to about 5.5."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "Well, the standard deviation of this random variable, this geometric random variable, it's going to be the square root of one minus one sixth, all of that over one sixth. So this is going to be equal to the square root of five sixths over one sixth, which is equal to six times the square root of five sixths. And this is going to be approximately equal to, five divided by six is equal to that. We'll take the square root of that and then multiply that times six, gets us to about 5.5. So approximately equal to 5.5. And what's interesting about a geometric random variable, obviously the lowest value here in this case is one, two, three, it can go higher and higher, but it can go arbitrary. You could get really unlucky and it might take you a thousand rolls in order to get that one."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "We'll take the square root of that and then multiply that times six, gets us to about 5.5. So approximately equal to 5.5. And what's interesting about a geometric random variable, obviously the lowest value here in this case is one, two, three, it can go higher and higher, but it can go arbitrary. You could get really unlucky and it might take you a thousand rolls in order to get that one. It could take you a million rolls, very low probability, but it could take you a million rolls in order to get that one. And so another thing to realize about a geometric random variable's distribution, it tends to look something like this, where the mean might be over here. And so you have a very long tail to the right of your mean, and this is classic right skew."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "You could get really unlucky and it might take you a thousand rolls in order to get that one. It could take you a million rolls, very low probability, but it could take you a million rolls in order to get that one. And so another thing to realize about a geometric random variable's distribution, it tends to look something like this, where the mean might be over here. And so you have a very long tail to the right of your mean, and this is classic right skew. And so all geometric random variables distributions are right skewed. They have a long tail of values, an infinitely long tail of values they can take to the right. Now, one last question."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "And so you have a very long tail to the right of your mean, and this is classic right skew. And so all geometric random variables distributions are right skewed. They have a long tail of values, an infinitely long tail of values they can take to the right. Now, one last question. Instead of dealing with a six-sided die, what would be the situation if we were dealing with a 12-sided die? What would then be the mean of our random variable? And what would be the standard deviation of our random variable?"}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "Now, one last question. Instead of dealing with a six-sided die, what would be the situation if we were dealing with a 12-sided die? What would then be the mean of our random variable? And what would be the standard deviation of our random variable? Pause this video and think about that. Well, the mean would be one over 112, because you have a probability of 112 every time of getting a one. We're assuming we're playing the same game now with a 12-sided die."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "And what would be the standard deviation of our random variable? Pause this video and think about that. Well, the mean would be one over 112, because you have a probability of 112 every time of getting a one. We're assuming we're playing the same game now with a 12-sided die. So one over 112 would be 12. So on average, it would take 12 rolls to get that first one. And then our standard deviation is going to be, essentially, this times the square root of one minus 112."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "We're assuming we're playing the same game now with a 12-sided die. So one over 112 would be 12. So on average, it would take 12 rolls to get that first one. And then our standard deviation is going to be, essentially, this times the square root of one minus 112. Let me write it this way. It's one minus 112 over one over 12, which is the same thing as 12 times the square root of 11 twelfths. 11 divided by 12 is equal to, take the square root and then multiply that times 12, and you get about 11.5, 11.5."}, {"video_title": "Geometric distribution mean and standard deviation AP Statistics Khan Academy.mp3", "Sentence": "And then our standard deviation is going to be, essentially, this times the square root of one minus 112. Let me write it this way. It's one minus 112 over one over 12, which is the same thing as 12 times the square root of 11 twelfths. 11 divided by 12 is equal to, take the square root and then multiply that times 12, and you get about 11.5, 11.5. And so you can see with a 12-sided die, it has the same pattern, where you have your mean of your random variable, and then you have a standard deviation that goes a reasonable bit on either side of the mean. It's almost equal to the mean, actually, in both situations. It's a little bit lower than the mean."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Let's say that we run a school, and in that school there is a population of students right over here. That is our population. And we want to get a sense of how these students feel about the quality of math instruction at this school. So we construct a survey, and we just need to decide who are we going to get to actually answer this survey. One option is to just go to every member of the population, but let's just say it's a really large school. Let's say we're a college, and there's 10,000 people in the college. We say, well, we can't just talk to everyone."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So we construct a survey, and we just need to decide who are we going to get to actually answer this survey. One option is to just go to every member of the population, but let's just say it's a really large school. Let's say we're a college, and there's 10,000 people in the college. We say, well, we can't just talk to everyone. So instead we say, let's sample this population to get an indication of how the entire school feels. So we are going to sample it. We're going to sample that population."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "We say, well, we can't just talk to everyone. So instead we say, let's sample this population to get an indication of how the entire school feels. So we are going to sample it. We're going to sample that population. Now, in order to avoid having bias in our response, in order for it to have the best chance of it being indicative of the entire population, we want our sample to be random. So our sample could either be random, random, or not random, not random. And it might seem at first pretty straightforward to do a random sample, but when you actually get down to it, it's not always as straightforward as you would think."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "We're going to sample that population. Now, in order to avoid having bias in our response, in order for it to have the best chance of it being indicative of the entire population, we want our sample to be random. So our sample could either be random, random, or not random, not random. And it might seem at first pretty straightforward to do a random sample, but when you actually get down to it, it's not always as straightforward as you would think. So one type of random sample is just a simple random sample. So simple, simple, random, random sample. And this is saying, all right, let me maybe assign a number to every person in the school."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And it might seem at first pretty straightforward to do a random sample, but when you actually get down to it, it's not always as straightforward as you would think. So one type of random sample is just a simple random sample. So simple, simple, random, random sample. And this is saying, all right, let me maybe assign a number to every person in the school. Maybe they already have a student ID number. And I'm just going to get a computer, a random number generator, to generate the 100 people, the 100 students. So let's say there's a sample of 100 students that I'm going to apply the survey to."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And this is saying, all right, let me maybe assign a number to every person in the school. Maybe they already have a student ID number. And I'm just going to get a computer, a random number generator, to generate the 100 people, the 100 students. So let's say there's a sample of 100 students that I'm going to apply the survey to. So that would be a simple random sample. We are just going into this whole population and randomly, let me just draw this. So this is the population."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So let's say there's a sample of 100 students that I'm going to apply the survey to. So that would be a simple random sample. We are just going into this whole population and randomly, let me just draw this. So this is the population. We are just randomly picking people out. And we know it's random because a random number generator, or we have a string of numbers or something like that, that is allowing us to pick these students. Now that's pretty good."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So this is the population. We are just randomly picking people out. And we know it's random because a random number generator, or we have a string of numbers or something like that, that is allowing us to pick these students. Now that's pretty good. It's unlikely that you're going to have bias from this sample. But there is some probability that just by chance, your random number generator just happened to select maybe a disproportionate number of boys over girls, or a disproportionate number of freshmen, or a disproportionate number of engineering majors versus English majors. And that's a possibility."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Now that's pretty good. It's unlikely that you're going to have bias from this sample. But there is some probability that just by chance, your random number generator just happened to select maybe a disproportionate number of boys over girls, or a disproportionate number of freshmen, or a disproportionate number of engineering majors versus English majors. And that's a possibility. So even though you're taking a simple random sample and it's truly random, once again, it's some probability that's not indicative of the entire population. And so to mitigate that, there are other techniques at our disposal. One technique is a stratified sample."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And that's a possibility. So even though you're taking a simple random sample and it's truly random, once again, it's some probability that's not indicative of the entire population. And so to mitigate that, there are other techniques at our disposal. One technique is a stratified sample. Stratified. And so this is the idea of taking our entire population and essentially stratifying it. So let's say we take that same population, we take that same population, I'll draw it as a square here just for convenience, and we're gonna stratify it by, let's say we're concerned that we get an appropriate sample of freshmen, sophomores, juniors, and seniors."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "One technique is a stratified sample. Stratified. And so this is the idea of taking our entire population and essentially stratifying it. So let's say we take that same population, we take that same population, I'll draw it as a square here just for convenience, and we're gonna stratify it by, let's say we're concerned that we get an appropriate sample of freshmen, sophomores, juniors, and seniors. So we'll stratify it by freshmen, sophomores, juniors, and seniors. And then we sample 25 from each of these groups. So these are the stratifications."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So let's say we take that same population, we take that same population, I'll draw it as a square here just for convenience, and we're gonna stratify it by, let's say we're concerned that we get an appropriate sample of freshmen, sophomores, juniors, and seniors. So we'll stratify it by freshmen, sophomores, juniors, and seniors. And then we sample 25 from each of these groups. So these are the stratifications. This is freshmen, sophomore, juniors, and seniors. And instead of just sampling 100 out of the entire pool, we sample 25 from each of these. So just like that."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So these are the stratifications. This is freshmen, sophomore, juniors, and seniors. And instead of just sampling 100 out of the entire pool, we sample 25 from each of these. So just like that. And so that makes sure that you are getting indicative responses from at least all of the different age groups or levels within your university. Now there might be another issue where you say, well, I'm actually more concerned that we have accurate representation of males and females in the school. And there is some probability, if I do 100 random people, it's very likely that it's close to 50-50, but there's some chance just due to randomness that it's disproportionately male or disproportionately female."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So just like that. And so that makes sure that you are getting indicative responses from at least all of the different age groups or levels within your university. Now there might be another issue where you say, well, I'm actually more concerned that we have accurate representation of males and females in the school. And there is some probability, if I do 100 random people, it's very likely that it's close to 50-50, but there's some chance just due to randomness that it's disproportionately male or disproportionately female. And that's even possible in the stratified case. And so what you might say is, well, you know what I'm gonna do? I'm going to, there's a technique called a clustered sample."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And there is some probability, if I do 100 random people, it's very likely that it's close to 50-50, but there's some chance just due to randomness that it's disproportionately male or disproportionately female. And that's even possible in the stratified case. And so what you might say is, well, you know what I'm gonna do? I'm going to, there's a technique called a clustered sample. Let me write this right over here. Clustered, a clustered sample. And what we do is we sample groups."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "I'm going to, there's a technique called a clustered sample. Let me write this right over here. Clustered, a clustered sample. And what we do is we sample groups. Each of those groups, we feel confident, has a good balance of male, females. So for example, we might, instead of sampling individuals from the entire population, we might say, look, you know, on Tuesdays and Thursdays, and this, well, even there, as you can tell, this is not a trivial thing to do. We'll, let's just say that we can split our, let's say we can split our population into groups."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And what we do is we sample groups. Each of those groups, we feel confident, has a good balance of male, females. So for example, we might, instead of sampling individuals from the entire population, we might say, look, you know, on Tuesdays and Thursdays, and this, well, even there, as you can tell, this is not a trivial thing to do. We'll, let's just say that we can split our, let's say we can split our population into groups. Maybe these are classrooms. And each of these classrooms have an even distribution of males and females, or pretty close to even distributions. And so what we do is we sample the actual classrooms."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "We'll, let's just say that we can split our, let's say we can split our population into groups. Maybe these are classrooms. And each of these classrooms have an even distribution of males and females, or pretty close to even distributions. And so what we do is we sample the actual classrooms. So that's why it's called cluster, or cluster technique, or clustered random sample, because we're going to randomly sample our classrooms, each of which have a close, or maybe an exact balance of males and females. So we know that we're gonna get good representation, but we are still sampling. We are sampling from the clusters, but then we're gonna survey every single person in each of these clusters, every single person in one of these classrooms."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And so what we do is we sample the actual classrooms. So that's why it's called cluster, or cluster technique, or clustered random sample, because we're going to randomly sample our classrooms, each of which have a close, or maybe an exact balance of males and females. So we know that we're gonna get good representation, but we are still sampling. We are sampling from the clusters, but then we're gonna survey every single person in each of these clusters, every single person in one of these classrooms. So once again, these are all forms of random surveys, or random samples. You have the simple random sample, you can stratify, or you can cluster, and then randomly pick the clusters, and then survey everyone in that cluster. Now, if these are all random samples, what are the non-random things like?"}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "We are sampling from the clusters, but then we're gonna survey every single person in each of these clusters, every single person in one of these classrooms. So once again, these are all forms of random surveys, or random samples. You have the simple random sample, you can stratify, or you can cluster, and then randomly pick the clusters, and then survey everyone in that cluster. Now, if these are all random samples, what are the non-random things like? Well, one case of non-random, you could have a voluntary, voluntary survey, or voluntary sample. This might just be, you tell every student at the school, hey, here's a web address, if you're interested, come and fill out this survey. And that's likely to introduce bias, because you might have, maybe, the students who really like the math instruction at their school, more likely to fill it out."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Now, if these are all random samples, what are the non-random things like? Well, one case of non-random, you could have a voluntary, voluntary survey, or voluntary sample. This might just be, you tell every student at the school, hey, here's a web address, if you're interested, come and fill out this survey. And that's likely to introduce bias, because you might have, maybe, the students who really like the math instruction at their school, more likely to fill it out. Maybe the students who really don't like it, more likely to fill it out. Maybe it's just the kids who have more time, more likely to fill it out. So this has a good chance of introducing bias."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And that's likely to introduce bias, because you might have, maybe, the students who really like the math instruction at their school, more likely to fill it out. Maybe the students who really don't like it, more likely to fill it out. Maybe it's just the kids who have more time, more likely to fill it out. So this has a good chance of introducing bias. The students who fill out the survey might be just more skewed one way or the other, because they volunteered for it. Another non-random sample would be called a, a, you're introducing bias because of convenience, is a term that's often used. And this might say, well, let's just sample the hundred first students who show up in school."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So this has a good chance of introducing bias. The students who fill out the survey might be just more skewed one way or the other, because they volunteered for it. Another non-random sample would be called a, a, you're introducing bias because of convenience, is a term that's often used. And this might say, well, let's just sample the hundred first students who show up in school. And that's just convenient for me, because I didn't have to do these random numbers, or do the stratification, or doing any of this clustering. But you can understand how this also would introduce bias. Because the first hundred students who show up at school, maybe those are the most diligent students, maybe they all take an early math class that has a very good instructor, or they're all happy about it."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And this might say, well, let's just sample the hundred first students who show up in school. And that's just convenient for me, because I didn't have to do these random numbers, or do the stratification, or doing any of this clustering. But you can understand how this also would introduce bias. Because the first hundred students who show up at school, maybe those are the most diligent students, maybe they all take an early math class that has a very good instructor, or they're all happy about it. Or it might go the other way. The instructor there isn't the best one, and so it might introduce bias the other way. So if you let people volunteer, or you just say, oh, let me go to the first N students, or you say, hey, let me just talk to all the students who happen to be in front of me right now."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Because the first hundred students who show up at school, maybe those are the most diligent students, maybe they all take an early math class that has a very good instructor, or they're all happy about it. Or it might go the other way. The instructor there isn't the best one, and so it might introduce bias the other way. So if you let people volunteer, or you just say, oh, let me go to the first N students, or you say, hey, let me just talk to all the students who happen to be in front of me right now. They might be in front of you out of convenience, but they might not be a true random sample. Now, there's other reasons why you might introduce bias, and it might not be because of the sampling. You might introduce bias because of the wording of your survey."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So if you let people volunteer, or you just say, oh, let me go to the first N students, or you say, hey, let me just talk to all the students who happen to be in front of me right now. They might be in front of you out of convenience, but they might not be a true random sample. Now, there's other reasons why you might introduce bias, and it might not be because of the sampling. You might introduce bias because of the wording of your survey. You could imagine a survey that says, do you consider yourself lucky to get a math education that very few other people in the world have access to? Well, that might bias you to say, well, yeah, I guess I feel lucky. Well, if the wording was, do you like the fact that a disproportionate, more students at your school tend to fail algebra than our surrounding schools?"}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "You might introduce bias because of the wording of your survey. You could imagine a survey that says, do you consider yourself lucky to get a math education that very few other people in the world have access to? Well, that might bias you to say, well, yeah, I guess I feel lucky. Well, if the wording was, do you like the fact that a disproportionate, more students at your school tend to fail algebra than our surrounding schools? Well, that might bias you negatively. So the wording really, really, really matters in surveys, and there's a lot that would go into this. And the other one is just people's, it's called response bias."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Well, if the wording was, do you like the fact that a disproportionate, more students at your school tend to fail algebra than our surrounding schools? Well, that might bias you negatively. So the wording really, really, really matters in surveys, and there's a lot that would go into this. And the other one is just people's, it's called response bias. And once again, this isn't about response bias. And this is just people not wanting to tell the truth or maybe not wanting to respond at all. Maybe they're afraid that somehow their response is gonna show up in front of their math teacher or the administrators, or if they're too negative, it might be taken out on them in some way."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And the other one is just people's, it's called response bias. And once again, this isn't about response bias. And this is just people not wanting to tell the truth or maybe not wanting to respond at all. Maybe they're afraid that somehow their response is gonna show up in front of their math teacher or the administrators, or if they're too negative, it might be taken out on them in some way. And because of that, they might not be truthful. And so they might be overly positive or not fill it out at all. So anyway, this is a very high-level overview of how you could think about sampling."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Maybe they're afraid that somehow their response is gonna show up in front of their math teacher or the administrators, or if they're too negative, it might be taken out on them in some way. And because of that, they might not be truthful. And so they might be overly positive or not fill it out at all. So anyway, this is a very high-level overview of how you could think about sampling. You wanna go random because it lowers the probability of their introducing some bias into it. And then these are some techniques. And also think about whether you're falling into some of these pitfalls that have a good chance of introducing bias."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And what I have here are five different statements. And I want you to look at these statements. Pause the video, look at these statements, and think about which of these, based on the information in the box and whiskers plot, which of these are for sure true, which of these are for sure false, and which of these we don't have enough information. It could go either way. All right, so let's work through these. So the first statement is that all of the students are less than 17 years old. Well, we see right over here that the maximum age, that's the right end of this right whisker, is 16."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It could go either way. All right, so let's work through these. So the first statement is that all of the students are less than 17 years old. Well, we see right over here that the maximum age, that's the right end of this right whisker, is 16. So it is the case that all of the students are less than 17 years old. So this is definitely going to be true. The next statement, at least 75% of the students are 10 years old or older."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, we see right over here that the maximum age, that's the right end of this right whisker, is 16. So it is the case that all of the students are less than 17 years old. So this is definitely going to be true. The next statement, at least 75% of the students are 10 years old or older. So when you look at this, this feels right because 10 is the value, that is at the beginning of the second quartile. This is the second quartile right over there. And actually, let me do this in a different color."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "The next statement, at least 75% of the students are 10 years old or older. So when you look at this, this feels right because 10 is the value, that is at the beginning of the second quartile. This is the second quartile right over there. And actually, let me do this in a different color. So this is the second quartile. So 25% of the value of the numbers are in the second, or roughly, sometimes it's not exactly, so approximately, I'll say roughly, 25% are going to be in the second quartile. Approximately 25% are going to be in the third quartile."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And actually, let me do this in a different color. So this is the second quartile. So 25% of the value of the numbers are in the second, or roughly, sometimes it's not exactly, so approximately, I'll say roughly, 25% are going to be in the second quartile. Approximately 25% are going to be in the third quartile. And approximately 25% are going to be in the fourth quartile. So it seems reasonable for saying 10 years old or older that this is going to be true. In fact, you could even have a couple of values in the first quartile that are 10."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Approximately 25% are going to be in the third quartile. And approximately 25% are going to be in the fourth quartile. So it seems reasonable for saying 10 years old or older that this is going to be true. In fact, you could even have a couple of values in the first quartile that are 10. But to make that a little bit more tangible, let's look at some, so I'm feeling good that this is true, but let's look at a few more examples to make this a little bit more concrete. So they don't know, we don't know, based on the information here, exactly how many students are at the party. We'll have to construct some scenarios."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "In fact, you could even have a couple of values in the first quartile that are 10. But to make that a little bit more tangible, let's look at some, so I'm feeling good that this is true, but let's look at a few more examples to make this a little bit more concrete. So they don't know, we don't know, based on the information here, exactly how many students are at the party. We'll have to construct some scenarios. So we could do a scenario. Let's see if we can do, we could do a scenario where, well, let's see, let's see if I can construct something where, let's see, the median is 13. We know that for sure."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We'll have to construct some scenarios. So we could do a scenario. Let's see if we can do, we could do a scenario where, well, let's see, let's see if I can construct something where, let's see, the median is 13. We know that for sure. The median is 13. So if I have an odd number, I would have 13 in the middle, just like that. And maybe I have three on each side."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We know that for sure. The median is 13. So if I have an odd number, I would have 13 in the middle, just like that. And maybe I have three on each side. And I'm just making that number up. I'm just trying to see what I can learn about the different types of data sets that could be described by this box and whiskers plot. So 10 is going to be the middle of the bottom half."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And maybe I have three on each side. And I'm just making that number up. I'm just trying to see what I can learn about the different types of data sets that could be described by this box and whiskers plot. So 10 is going to be the middle of the bottom half. So that's 10 right over there. And 15 is going to be the middle of the top half. That's what this box and whiskers plot is telling us."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So 10 is going to be the middle of the bottom half. So that's 10 right over there. And 15 is going to be the middle of the top half. That's what this box and whiskers plot is telling us. And they of course tell us what the minimum, the minimum is seven, and they tell us that the maximum is 16. So we know that's seven, and then that is 16. And then this right over here could be anything."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "That's what this box and whiskers plot is telling us. And they of course tell us what the minimum, the minimum is seven, and they tell us that the maximum is 16. So we know that's seven, and then that is 16. And then this right over here could be anything. It could be 10, it could be 11, it could be 12, it could be 13. It wouldn't change what these medians are. It wouldn't change this box and whiskers plot."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then this right over here could be anything. It could be 10, it could be 11, it could be 12, it could be 13. It wouldn't change what these medians are. It wouldn't change this box and whiskers plot. Similarly, this could be 13, it could be 14, it could be 15. And so any of those values wouldn't change it. And so 75% are 10 or older."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It wouldn't change this box and whiskers plot. Similarly, this could be 13, it could be 14, it could be 15. And so any of those values wouldn't change it. And so 75% are 10 or older. Well, this value, in this case, six out of seven are 10 years old or older. And we could try it out with other scenarios where let's try to minimize the number of 10s given this data set. Well, we could do something like, let's say that we have eight."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And so 75% are 10 or older. Well, this value, in this case, six out of seven are 10 years old or older. And we could try it out with other scenarios where let's try to minimize the number of 10s given this data set. Well, we could do something like, let's say that we have eight. So let's see, one, two, three, four, five, six, seven, eight. And so here, we know that the minimum, we know that the minimum is seven, we know that the maximum is 16. We know that the mean of these middle two values, we have an even number now."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, we could do something like, let's say that we have eight. So let's see, one, two, three, four, five, six, seven, eight. And so here, we know that the minimum, we know that the minimum is seven, we know that the maximum is 16. We know that the mean of these middle two values, we have an even number now. So the median is going to be the mean of these two values. So it's going to be the mean of this and this is going to be 13. And we know that the mean of, we know that the mean of this and this is going to be 10, and that the mean of this and this is going to be 15."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We know that the mean of these middle two values, we have an even number now. So the median is going to be the mean of these two values. So it's going to be the mean of this and this is going to be 13. And we know that the mean of, we know that the mean of this and this is going to be 10, and that the mean of this and this is going to be 15. So what could we construct? Well, actually, we don't even have to construct to answer this question. We know that this is going to have to be 10 or larger, and then all of these other things are going to be 10 or larger."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And we know that the mean of, we know that the mean of this and this is going to be 10, and that the mean of this and this is going to be 15. So what could we construct? Well, actually, we don't even have to construct to answer this question. We know that this is going to have to be 10 or larger, and then all of these other things are going to be 10 or larger. So this is exactly 75%, exactly 75%, if we assume that this is less than 10, are going to be 10 years old or older. So feeling very good, very good, about this one right over here. And actually, just to make this concrete, I'll put in some values here."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We know that this is going to have to be 10 or larger, and then all of these other things are going to be 10 or larger. So this is exactly 75%, exactly 75%, if we assume that this is less than 10, are going to be 10 years old or older. So feeling very good, very good, about this one right over here. And actually, just to make this concrete, I'll put in some values here. You know, this could be a nine and an 11. This could be a 12 and a 14. This could be a 14 and a 16."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And actually, just to make this concrete, I'll put in some values here. You know, this could be a nine and an 11. This could be a 12 and a 14. This could be a 14 and a 16. Or it could be a 15 and a 15. You could think about it in any of those ways. But feeling very good that this is definitely going to be true based on the information given in this plot."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "This could be a 14 and a 16. Or it could be a 15 and a 15. You could think about it in any of those ways. But feeling very good that this is definitely going to be true based on the information given in this plot. Now they say there's only one seven-year-old at the party. One seven-year-old at the party. Well, this first possibility that we looked at, that was the case."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "But feeling very good that this is definitely going to be true based on the information given in this plot. Now they say there's only one seven-year-old at the party. One seven-year-old at the party. Well, this first possibility that we looked at, that was the case. There was only one seven-year-old at the party, and there was one 16-year-old at the party. And actually, that was the next statement. There's only one 16-year-old at the party."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, this first possibility that we looked at, that was the case. There was only one seven-year-old at the party, and there was one 16-year-old at the party. And actually, that was the next statement. There's only one 16-year-old at the party. So both of these seem like we can definitely construct data that's consistent with this box plot, box and whiskers plot, where this is true. But could we construct one where it's not true? Well, sure."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "There's only one 16-year-old at the party. So both of these seem like we can definitely construct data that's consistent with this box plot, box and whiskers plot, where this is true. But could we construct one where it's not true? Well, sure. Let's imagine, let's see, we have our median at 13. Median at 13. And then we have, let's see, one, two, three, four, five."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, sure. Let's imagine, let's see, we have our median at 13. Median at 13. And then we have, let's see, one, two, three, four, five. This is going to be the 10, the median of this bottom half. This is going to be 15. This is going to be 7."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then we have, let's see, one, two, three, four, five. This is going to be the 10, the median of this bottom half. This is going to be 15. This is going to be 7. This is going to be 16. Well, this could also be 7. It doesn't have to be."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "This is going to be 7. This is going to be 16. Well, this could also be 7. It doesn't have to be. It could be 7, 8, 9, or 10. This could also be 16. It doesn't have to be."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It doesn't have to be. It could be 7, 8, 9, or 10. This could also be 16. It doesn't have to be. It could be 15 as well. But just like that, I've constructed a data set. And these could be 10, 11, 12, 13."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It doesn't have to be. It could be 15 as well. But just like that, I've constructed a data set. And these could be 10, 11, 12, 13. This could be 10, 11, 12, 13. This could be 13, 14, 15. This one also could be 13, 14, 15."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And these could be 10, 11, 12, 13. This could be 10, 11, 12, 13. This could be 13, 14, 15. This one also could be 13, 14, 15. But the simple thing is, or the basic idea here, I can have a data set where I have multiple 7's and multiple 16's. Or I could have a data set where I only have one 7, or only one 16. So both of these statements, we just plain don't know."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "This one also could be 13, 14, 15. But the simple thing is, or the basic idea here, I can have a data set where I have multiple 7's and multiple 16's. Or I could have a data set where I only have one 7, or only one 16. So both of these statements, we just plain don't know. Now the next statement, exactly half the students are older than 13. Well, if you look at this possibility up here, we saw that 3 out of the 7 are older than 13. So that's not exactly half."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So both of these statements, we just plain don't know. Now the next statement, exactly half the students are older than 13. Well, if you look at this possibility up here, we saw that 3 out of the 7 are older than 13. So that's not exactly half. 3 7's is not 1 half. But in this one over here, we did see that exactly half are older than 13. In fact, if you're saying exactly half, well, in this one, we're saying that exactly half are older than 13."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So that's not exactly half. 3 7's is not 1 half. But in this one over here, we did see that exactly half are older than 13. In fact, if you're saying exactly half, well, in this one, we're saying that exactly half are older than 13. We have an even number right over here. So it is exactly half. So it's possible that it's true."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "In fact, if you're saying exactly half, well, in this one, we're saying that exactly half are older than 13. We have an even number right over here. So it is exactly half. So it's possible that it's true. It's possible that it's not true based on the information given. We once again do not know. Anyway, hopefully you found this interesting."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So it's possible that it's true. It's possible that it's not true based on the information given. We once again do not know. Anyway, hopefully you found this interesting. The whole point of me doing this is, when you look at statistics, sometimes it's easy to kind of say, OK, I think it roughly means that. And that's sometimes OK. But it's very important to think about what types of actual statements you can make and what you can't make."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3", "Sentence": "We're told a company produces processing chips for cell phones. At one of its large factories, 2% of the chips produced are defective in some way. A quality check involves randomly selecting and testing 500 chips. What are the mean and standard deviation of the number of defective processing chips in these samples? So like always, try to pause this video and have a go at it on your own, and then we will work through it together. All right, so let me define a random variable that we're gonna find the mean and standard deviation of. And I'm gonna make that random variable the number of defective processing chips in a 500-chip sample."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3", "Sentence": "What are the mean and standard deviation of the number of defective processing chips in these samples? So like always, try to pause this video and have a go at it on your own, and then we will work through it together. All right, so let me define a random variable that we're gonna find the mean and standard deviation of. And I'm gonna make that random variable the number of defective processing chips in a 500-chip sample. So let's let X be equal to the number of defective chips in 500-chip sample. So the first thing to recognize is that this will be a binomial variable. This is binomial."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3", "Sentence": "And I'm gonna make that random variable the number of defective processing chips in a 500-chip sample. So let's let X be equal to the number of defective chips in 500-chip sample. So the first thing to recognize is that this will be a binomial variable. This is binomial. How do we know it's binomial? Well, it's made up of 500, it's a finite number of trials right over here. The probability of getting a defective chip, you could view this as a probability of success."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3", "Sentence": "This is binomial. How do we know it's binomial? Well, it's made up of 500, it's a finite number of trials right over here. The probability of getting a defective chip, you could view this as a probability of success. It's a little bit counterintuitive that a defective chip would be a success, but we're summing up the defective chips. So we would view the probability of a defect, or I should say a defective chip, it is constant across these 500 trials, and we will assume that they are independent of each other, 0.02. You might be saying, hey, well, are we replacing the chips before or after?"}, {"video_title": "Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3", "Sentence": "The probability of getting a defective chip, you could view this as a probability of success. It's a little bit counterintuitive that a defective chip would be a success, but we're summing up the defective chips. So we would view the probability of a defect, or I should say a defective chip, it is constant across these 500 trials, and we will assume that they are independent of each other, 0.02. You might be saying, hey, well, are we replacing the chips before or after? But we're assuming it's from a functionally infinite population, or if you wanna make it feel better, you could say, well, maybe you are replacing the chips. They're not really telling us that right over here. So we'll assume that each of these trials are independent of each other, and that the probability of getting a defective chip stays constant here."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3", "Sentence": "You might be saying, hey, well, are we replacing the chips before or after? But we're assuming it's from a functionally infinite population, or if you wanna make it feel better, you could say, well, maybe you are replacing the chips. They're not really telling us that right over here. So we'll assume that each of these trials are independent of each other, and that the probability of getting a defective chip stays constant here. And so this is a binomial random variable, or binomial variable, and we know the formulas for the mean and standard deviation of a binomial variable. So the mean, the mean of x, which is the same thing as the expected value of x, is going to be equal to the number of trials, n, times the probability of success on each trial, times p. So what is this going to be? Well, this is going to be equal to, we have 500 trials, and then the probability of success on each of these trials is 0.02."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3", "Sentence": "So we'll assume that each of these trials are independent of each other, and that the probability of getting a defective chip stays constant here. And so this is a binomial random variable, or binomial variable, and we know the formulas for the mean and standard deviation of a binomial variable. So the mean, the mean of x, which is the same thing as the expected value of x, is going to be equal to the number of trials, n, times the probability of success on each trial, times p. So what is this going to be? Well, this is going to be equal to, we have 500 trials, and then the probability of success on each of these trials is 0.02. So it's 500 times 0.02. And what is this going to be? 500 times 2 hundredths is going to be, it's going to be equal to 10."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3", "Sentence": "Well, this is going to be equal to, we have 500 trials, and then the probability of success on each of these trials is 0.02. So it's 500 times 0.02. And what is this going to be? 500 times 2 hundredths is going to be, it's going to be equal to 10. So that is your expected value of the number of defective processing chips, or the mean. Now what about the standard deviation? So the standard deviation of our random variable, x, well that's just going to be equal to the square root of the variance of our random variable, x."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3", "Sentence": "500 times 2 hundredths is going to be, it's going to be equal to 10. So that is your expected value of the number of defective processing chips, or the mean. Now what about the standard deviation? So the standard deviation of our random variable, x, well that's just going to be equal to the square root of the variance of our random variable, x. So I could just write it, I'm just writing it all the different ways that you might see it, because sometimes the notation is the most confusing part in statistics. And so this is going to be the square root of what? Well, the variance of a binomial variable is going to be equal to the number of trials, times the probability of success in each trial, times one minus the probability of success in each trial."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3", "Sentence": "So the standard deviation of our random variable, x, well that's just going to be equal to the square root of the variance of our random variable, x. So I could just write it, I'm just writing it all the different ways that you might see it, because sometimes the notation is the most confusing part in statistics. And so this is going to be the square root of what? Well, the variance of a binomial variable is going to be equal to the number of trials, times the probability of success in each trial, times one minus the probability of success in each trial. And so in this context, this is going to be equal to, you're gonna have the 500, 500, times 0.02, 0.02, times one minus 0.02 is.98. So times 0.98, and all of this is under the radical sign. I didn't make that radical sign long enough."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3", "Sentence": "Well, the variance of a binomial variable is going to be equal to the number of trials, times the probability of success in each trial, times one minus the probability of success in each trial. And so in this context, this is going to be equal to, you're gonna have the 500, 500, times 0.02, 0.02, times one minus 0.02 is.98. So times 0.98, and all of this is under the radical sign. I didn't make that radical sign long enough. And so what is this going to be? Well, let's see, 500 times 0.02, we already said that this is going to be 10, 10 times 0.98, this is going to be equal to the square root of 9.8. So it's going to be, I don't know, three point something."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3", "Sentence": "I didn't make that radical sign long enough. And so what is this going to be? Well, let's see, 500 times 0.02, we already said that this is going to be 10, 10 times 0.98, this is going to be equal to the square root of 9.8. So it's going to be, I don't know, three point something. If we want, we can get a calculator out to feel a little bit better about this value. So I'm gonna take 9.8, and then take the square root of it, and I get three point, if I round to the nearest hundredth, 3.13. So this is approximately 3.13 for the standard deviation."}, {"video_title": "Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3", "Sentence": "So it's going to be, I don't know, three point something. If we want, we can get a calculator out to feel a little bit better about this value. So I'm gonna take 9.8, and then take the square root of it, and I get three point, if I round to the nearest hundredth, 3.13. So this is approximately 3.13 for the standard deviation. If I wanted the variance, it would be 9.8. But they ask for the standard deviation, so that's why we got that. All right, hopefully you enjoyed that."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "What is the smallest sample size required to obtain the desired margin of error? So let's just remind ourselves what the confidence interval will look like and what part of it is the margin of error and then we can think about what is her sample size that she would need. So she wants to estimate the true population proportion that favor a tax increase. She doesn't know what this is so she is going to take a sample size of size n and in fact this question is all about what n does she need in order to have the desired margin of error. Well whatever sample she takes there she's going to calculate a sample proportion and then the confidence interval that she's going to construct is going to be that sample proportion plus or minus critical value and this critical value is based on the confidence level. We'll talk about that in a second. What z-star, what critical value would correspond to a 95% confidence level times and then you would have times the standard error of her statistic and so in this case it would be the square root."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "She doesn't know what this is so she is going to take a sample size of size n and in fact this question is all about what n does she need in order to have the desired margin of error. Well whatever sample she takes there she's going to calculate a sample proportion and then the confidence interval that she's going to construct is going to be that sample proportion plus or minus critical value and this critical value is based on the confidence level. We'll talk about that in a second. What z-star, what critical value would correspond to a 95% confidence level times and then you would have times the standard error of her statistic and so in this case it would be the square root. It would be the standard error of her sample proportion which is the sample proportion times one minus the sample proportion, all of that over her sample size. Now she wants the margin of error to be no more than 2%. So the margin of error is this part right over here."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "What z-star, what critical value would correspond to a 95% confidence level times and then you would have times the standard error of her statistic and so in this case it would be the square root. It would be the standard error of her sample proportion which is the sample proportion times one minus the sample proportion, all of that over her sample size. Now she wants the margin of error to be no more than 2%. So the margin of error is this part right over here. So this part right over there she wants to be no more than 2%. Has to be less than or equal to 2%. That green color is kind of too shocking."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "So the margin of error is this part right over here. So this part right over there she wants to be no more than 2%. Has to be less than or equal to 2%. That green color is kind of too shocking. It's unpleasant. All right, less than or equal to 2% right over here. So how do we figure that out?"}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "That green color is kind of too shocking. It's unpleasant. All right, less than or equal to 2% right over here. So how do we figure that out? Well the first thing, let's just make sure we incorporate the 95% confidence level. So we could look at a z-table. Remember, 95% confidence level, that means if we have a normal distribution here, if we have a normal distribution here, 95% confidence level means the number of standard deviations we need to go above and beyond this in order to capture 95% of the area right over here."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "So how do we figure that out? Well the first thing, let's just make sure we incorporate the 95% confidence level. So we could look at a z-table. Remember, 95% confidence level, that means if we have a normal distribution here, if we have a normal distribution here, 95% confidence level means the number of standard deviations we need to go above and beyond this in order to capture 95% of the area right over here. So this would be 2.5% that is unshaded at the top right over there, and then this would be 2.5% right over here. And we could look up in a z-table, and if you were to look up in a z-table, you would not look up 95%. You would look up the percentage that would leave 2.5% unshaded at the top."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "Remember, 95% confidence level, that means if we have a normal distribution here, if we have a normal distribution here, 95% confidence level means the number of standard deviations we need to go above and beyond this in order to capture 95% of the area right over here. So this would be 2.5% that is unshaded at the top right over there, and then this would be 2.5% right over here. And we could look up in a z-table, and if you were to look up in a z-table, you would not look up 95%. You would look up the percentage that would leave 2.5% unshaded at the top. So you would actually look up 97.5%. But it's good to know in general that at a 95% confidence level, you're looking at a critical value of 1.96. And that's just something good to know."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "You would look up the percentage that would leave 2.5% unshaded at the top. So you would actually look up 97.5%. But it's good to know in general that at a 95% confidence level, you're looking at a critical value of 1.96. And that's just something good to know. We could of course look it up on a z-table. So this is 1.96. And so this is going to be 1.96 right over here."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "And that's just something good to know. We could of course look it up on a z-table. So this is 1.96. And so this is going to be 1.96 right over here. But what about p hat? We don't know what p hat is until we actually take the sample, but this whole question is, how large of a sample should we take? Well, remember, we want this stuff right over here that I'm now circling or squaring in this less, less bright color, this blue color."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "And so this is going to be 1.96 right over here. But what about p hat? We don't know what p hat is until we actually take the sample, but this whole question is, how large of a sample should we take? Well, remember, we want this stuff right over here that I'm now circling or squaring in this less, less bright color, this blue color. We want this thing to be less than or equal to 2%. This is our margin of error. And so what we could do is we could pick a sample proportion."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "Well, remember, we want this stuff right over here that I'm now circling or squaring in this less, less bright color, this blue color. We want this thing to be less than or equal to 2%. This is our margin of error. And so what we could do is we could pick a sample proportion. We don't know if that's what it's going to be, that maximizes this right over here. Because if we maximize this, we know that we're essentially figuring out the largest thing that this could end up being, and then we'll be safe. So the p hat, the maximum p hat."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "And so what we could do is we could pick a sample proportion. We don't know if that's what it's going to be, that maximizes this right over here. Because if we maximize this, we know that we're essentially figuring out the largest thing that this could end up being, and then we'll be safe. So the p hat, the maximum p hat. And so if you wanna maximize p hat times one minus p hat, you could do some trial and error here. This is a fairly simple quadratic. It's actually going to be p hat is 0.5."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "So the p hat, the maximum p hat. And so if you wanna maximize p hat times one minus p hat, you could do some trial and error here. This is a fairly simple quadratic. It's actually going to be p hat is 0.5. And I wanna emphasize, we don't know. She didn't even perform the sample yet. She didn't even take the random sample and calculate the sample proportion."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "It's actually going to be p hat is 0.5. And I wanna emphasize, we don't know. She didn't even perform the sample yet. She didn't even take the random sample and calculate the sample proportion. But we wanna figure out what n to take. And so to be safe, she says, okay, well, what sample proportion would maximize my margin of error? And so let me just assume that, and then let me calculate n. So let me set up an inequality here."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "She didn't even take the random sample and calculate the sample proportion. But we wanna figure out what n to take. And so to be safe, she says, okay, well, what sample proportion would maximize my margin of error? And so let me just assume that, and then let me calculate n. So let me set up an inequality here. We want 1.96, that's our critical value, times the square root of, we're just going to assume 0.5 for our sample proportion, although, of course, we don't know what it is yet until we actually take the sample. So that's our sample proportion. That's one minus our sample proportion."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "And so let me just assume that, and then let me calculate n. So let me set up an inequality here. We want 1.96, that's our critical value, times the square root of, we're just going to assume 0.5 for our sample proportion, although, of course, we don't know what it is yet until we actually take the sample. So that's our sample proportion. That's one minus our sample proportion. All of that over n needs to be less than or equal to 2%. We don't want our margin of error to be any larger than 2%. And let me just write this as a decimal, 0.02."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "That's one minus our sample proportion. All of that over n needs to be less than or equal to 2%. We don't want our margin of error to be any larger than 2%. And let me just write this as a decimal, 0.02. And now we just have to do a little bit of algebra to calculate this. So let's see how we could do this. So this could be rewritten as, we could divide both sides by 1.96, 1.96, one over 1.96."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "And let me just write this as a decimal, 0.02. And now we just have to do a little bit of algebra to calculate this. So let's see how we could do this. So this could be rewritten as, we could divide both sides by 1.96, 1.96, one over 1.96. And so this would be equal to, on the left-hand side, we'd have the square root of all of this. But that's the same thing as the square root of 0.5 times 0.5, so that'd just be 0.5 over the square root of n. Needs to be less than or equal to, actually, let me write it this way. This is the same thing as two over 100."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "So this could be rewritten as, we could divide both sides by 1.96, 1.96, one over 1.96. And so this would be equal to, on the left-hand side, we'd have the square root of all of this. But that's the same thing as the square root of 0.5 times 0.5, so that'd just be 0.5 over the square root of n. Needs to be less than or equal to, actually, let me write it this way. This is the same thing as two over 100. So two over 100 times one over 1.96 needs to be less than or equal to two over 196. Let me scroll down a little bit. This is fancier algebra than we typically do in statistics, or at least in introductory statistics class."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "This is the same thing as two over 100. So two over 100 times one over 1.96 needs to be less than or equal to two over 196. Let me scroll down a little bit. This is fancier algebra than we typically do in statistics, or at least in introductory statistics class. All right, so let's see. We could take the reciprocal of both sides. We could say the square root of n over 0.5 and 1.96 over two."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "This is fancier algebra than we typically do in statistics, or at least in introductory statistics class. All right, so let's see. We could take the reciprocal of both sides. We could say the square root of n over 0.5 and 1.96 over two. And let's see, what's 196 divided by two? That is going to be 98. So this would be 98."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "We could say the square root of n over 0.5 and 1.96 over two. And let's see, what's 196 divided by two? That is going to be 98. So this would be 98. And so if we take the reciprocal of both sides, then you're gonna swap the inequality, so it's gonna be greater than or equal to. Let's see, I can multiply both sides of this by 0.5. So 0.5, that's what my, I said 0.5, but my fingers wrote down 0.4."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "So this would be 98. And so if we take the reciprocal of both sides, then you're gonna swap the inequality, so it's gonna be greater than or equal to. Let's see, I can multiply both sides of this by 0.5. So 0.5, that's what my, I said 0.5, but my fingers wrote down 0.4. Let's see, 0.5. And so there we get the square root of n needs to be greater than or equal to 49, or n needs to be greater than or equal to 49 squared. And what's 49 squared?"}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "So 0.5, that's what my, I said 0.5, but my fingers wrote down 0.4. Let's see, 0.5. And so there we get the square root of n needs to be greater than or equal to 49, or n needs to be greater than or equal to 49 squared. And what's 49 squared? Well, you know 50 squared is 2,500, so you know it's going to be close to that, so you can already make a pretty good estimate that it's going to be d. But if you wanna multiply it out, we can. 49 times 49, nine times nine is 81. Nine times four is 36, plus eight is 44."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "And what's 49 squared? Well, you know 50 squared is 2,500, so you know it's going to be close to that, so you can already make a pretty good estimate that it's going to be d. But if you wanna multiply it out, we can. 49 times 49, nine times nine is 81. Nine times four is 36, plus eight is 44. Four times nine, 36. Four times four is 16 plus three, we have 19. And then you add all of that together, and you indeed do get, so that's 10."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "Nine times four is 36, plus eight is 44. Four times nine, 36. Four times four is 16 plus three, we have 19. And then you add all of that together, and you indeed do get, so that's 10. And so this is a 14. You do indeed get 2,401. So that's the minimum sample size that Della should take if she genuinely wanted her margin of error to be no more than 2%."}, {"video_title": "Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3", "Sentence": "And then you add all of that together, and you indeed do get, so that's 10. And so this is a 14. You do indeed get 2,401. So that's the minimum sample size that Della should take if she genuinely wanted her margin of error to be no more than 2%. Now, it might turn out that her margin of error, when she actually takes the sample of size 2,401, if her sample proportion is less than 0.5, or greater than 0.5, well, then she's going to be in a situation where her margin of error might be less than this, but she just wanted it to be no more than that. Another important thing to appreciate is, it just, the math all worked out very nicely just now, where I got our n to be actually a whole number, but if I got 2,401.5, then you would have to round up to the nearest whole number because you can't have a, your sample size is always going to be a whole number value. So I will leave you there."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "If the significance level was lowered to 1 hundredth, which of the following would be true? So pause this video and see if you can answer it on your own. Okay, now let's do this together. And let's see, they're talking about how the probability of a type two error or the power, and or the power would change. So before I even look at the choices, let's think about this. We have talked about in previous videos that if we increase our level of significance, that will increase our power. And power is the probability of not making a type two error."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "And let's see, they're talking about how the probability of a type two error or the power, and or the power would change. So before I even look at the choices, let's think about this. We have talked about in previous videos that if we increase our level of significance, that will increase our power. And power is the probability of not making a type two error. So that would decrease the probability of making a type two error. But in this question, we're going the other way. We're decreasing the level of significance, which would lower the probability of making a type one error, but this would decrease the power."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "And power is the probability of not making a type two error. So that would decrease the probability of making a type two error. But in this question, we're going the other way. We're decreasing the level of significance, which would lower the probability of making a type one error, but this would decrease the power. It actually would increase, it actually would increase the probability of making a type two, a type two error. And so which of these choices are consistent with that? Well, choice A says that both the type two error and the power would decrease."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "We're decreasing the level of significance, which would lower the probability of making a type one error, but this would decrease the power. It actually would increase, it actually would increase the probability of making a type two, a type two error. And so which of these choices are consistent with that? Well, choice A says that both the type two error and the power would decrease. Well, those don't, these two things don't move together. If one increases, the other decreases. So we rule that one out."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "Well, choice A says that both the type two error and the power would decrease. Well, those don't, these two things don't move together. If one increases, the other decreases. So we rule that one out. Choice B also has these two things moving together, which can't be true. If one increases, the other decreases. Choice C, the probability of a type two error would increase."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So we rule that one out. Choice B also has these two things moving together, which can't be true. If one increases, the other decreases. Choice C, the probability of a type two error would increase. That's consistent with what we have here. And the power of the test would decrease. Yep, that's consistent with what we have here."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "Choice C, the probability of a type two error would increase. That's consistent with what we have here. And the power of the test would decrease. Yep, that's consistent with what we have here. So that looks good. And choice D is the opposite of that. The probability of a type two error would decrease."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "Yep, that's consistent with what we have here. So that looks good. And choice D is the opposite of that. The probability of a type two error would decrease. So this is, they're talking about this scenario over here, and that would have happened if they increased our significance level, not decreased it. So we could rule that one out as well. Let's do another example."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "The probability of a type two error would decrease. So this is, they're talking about this scenario over here, and that would have happened if they increased our significance level, not decreased it. So we could rule that one out as well. Let's do another example. Asha owns a car wash and is trying to decide whether or not to purchase a vending machine so that customers can buy coffee while they wait. She'll get the machine if she's convinced that more than 30% of her customers would buy coffee. She plans on taking a random sample of N customers and asking them whether or not they would buy coffee from the machine."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "Let's do another example. Asha owns a car wash and is trying to decide whether or not to purchase a vending machine so that customers can buy coffee while they wait. She'll get the machine if she's convinced that more than 30% of her customers would buy coffee. She plans on taking a random sample of N customers and asking them whether or not they would buy coffee from the machine. And she'll then do a significance test using alpha equals 0.05 to see if the sample proportion who say yes is significantly greater than 30%. Which situation below would result in the highest power for her test? So again, pause this video and try to answer it."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "She plans on taking a random sample of N customers and asking them whether or not they would buy coffee from the machine. And she'll then do a significance test using alpha equals 0.05 to see if the sample proportion who say yes is significantly greater than 30%. Which situation below would result in the highest power for her test? So again, pause this video and try to answer it. Well, before I even look at the choices, we could think about what her hypotheses would be. Her null hypothesis is, you could kind of view it as a status quo, no news here. And that would be that the true population proportion of people who want to buy coffee is 30%."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So again, pause this video and try to answer it. Well, before I even look at the choices, we could think about what her hypotheses would be. Her null hypothesis is, you could kind of view it as a status quo, no news here. And that would be that the true population proportion of people who want to buy coffee is 30%. And that her alternative hypothesis is that no, the true population proportion, the true population parameter there, is greater than, is greater than 30%. And so if we're talking about what would result in the highest power for her test. So a high power, a high power means the lowest probability of making a type two error."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "And that would be that the true population proportion of people who want to buy coffee is 30%. And that her alternative hypothesis is that no, the true population proportion, the true population parameter there, is greater than, is greater than 30%. And so if we're talking about what would result in the highest power for her test. So a high power, a high power means the lowest probability of making a type two error. And in other videos, we've talked about it. It looks like she's dealing with the sample size and what is the true proportion of customers that would buy coffee. And the sample size is under her control."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So a high power, a high power means the lowest probability of making a type two error. And in other videos, we've talked about it. It looks like she's dealing with the sample size and what is the true proportion of customers that would buy coffee. And the sample size is under her control. The true proportion isn't. Don't wanna make it seem like somehow you can change the true proportion in order to get a higher power. You can change the sample size."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "And the sample size is under her control. The true proportion isn't. Don't wanna make it seem like somehow you can change the true proportion in order to get a higher power. You can change the sample size. But the general principle is, the higher the sample size, the higher the power. So you want a highest possible sample size. And you're going to have a higher power if the true proportion is further from your hypothesis, your null hypothesis proportion."}, {"video_title": "Examples thinking about power in significance tests AP Statistics Khan Academy.mp3", "Sentence": "You can change the sample size. But the general principle is, the higher the sample size, the higher the power. So you want a highest possible sample size. And you're going to have a higher power if the true proportion is further from your hypothesis, your null hypothesis proportion. And so we want the highest possible N, and that looks like an N of 200, which is there and there. And we want a true proportion of customers that would actually buy coffee as far away as possible from our null hypothesis, which once again would not be under Asha's control. But you can clearly see that 50% is further from 30 than 32 is, so this one, choice D, is the one that looks good."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "The filling machines can be set to the labeled amount. Variability in the filling process causes the actual contents of milk containers to be normally distributed. A random sample of 12 containers of milk was drawn from the milk processing line in a plant, and the amount of milk in each container was recorded. The sample mean and standard deviation of this sample of 12 containers of milk were 127.2 ounces and 2.1 ounces, respectively. Is there sufficient evidence to conclude that the packaging plant is not in compliance with the regulations? Provide statistical justification for your answer. So pause this video and see if you can have a go at it."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "The sample mean and standard deviation of this sample of 12 containers of milk were 127.2 ounces and 2.1 ounces, respectively. Is there sufficient evidence to conclude that the packaging plant is not in compliance with the regulations? Provide statistical justification for your answer. So pause this video and see if you can have a go at it. All right, now let's do this together. So first, let's say what we're talking about. So let me define mu, and this is going to be the mean amount amount of milk in population, population of containers, containers at the plant that we care about."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "So pause this video and see if you can have a go at it. All right, now let's do this together. So first, let's say what we're talking about. So let me define mu, and this is going to be the mean amount amount of milk in population, population of containers, containers at the plant that we care about. And so then we can set up our hypotheses. Our null hypothesis over here is that we are in compliance. We could say that the mean for our population of containers is actually 128."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "So let me define mu, and this is going to be the mean amount amount of milk in population, population of containers, containers at the plant that we care about. And so then we can set up our hypotheses. Our null hypothesis over here is that we are in compliance. We could say that the mean for our population of containers is actually 128. That's our minimum we need to be in compliance. And that our alternative hypothesis is that we are not in compliance. So that's that our mean, the true population mean, is less than 128 fluid ounces."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "We could say that the mean for our population of containers is actually 128. That's our minimum we need to be in compliance. And that our alternative hypothesis is that we are not in compliance. So that's that our mean, the true population mean, is less than 128 fluid ounces. And so this is a situation where we are not in compliance, not in compliance, compliance in the alternative hypothesis. Now, if you're going to do a significance test, you need to set a significance level. So let's do that over here."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "So that's that our mean, the true population mean, is less than 128 fluid ounces. And so this is a situation where we are not in compliance, not in compliance, compliance in the alternative hypothesis. Now, if you're going to do a significance test, you need to set a significance level. So let's do that over here. Significance level. And if you haven't noticed, I'm doing, I'm trying to do in this video what would be expected of you on a test. And this is an actual question from an AP exam."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "So let's do that over here. Significance level. And if you haven't noticed, I'm doing, I'm trying to do in this video what would be expected of you on a test. And this is an actual question from an AP exam. So our significance level here, I'll just pick it to be 0.05 because, well, that's a fairly typical one. And since they didn't give it one to us, it's important to set one ahead of time. And now we wanna check our conditions for inference."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "And this is an actual question from an AP exam. So our significance level here, I'll just pick it to be 0.05 because, well, that's a fairly typical one. And since they didn't give it one to us, it's important to set one ahead of time. And now we wanna check our conditions for inference. So let me do that over here. Conditions, conditions for inference. And this is to feel good that the sample that we're using to make our inference, to do our significance test, that it's a reasonable one to make inferences from."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "And now we wanna check our conditions for inference. So let me do that over here. Conditions, conditions for inference. And this is to feel good that the sample that we're using to make our inference, to do our significance test, that it's a reasonable one to make inferences from. And so the first one is the random condition. And do we meet that? Well, they tell us here it's a random sample of 12 containers of milk."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "And this is to feel good that the sample that we're using to make our inference, to do our significance test, that it's a reasonable one to make inferences from. And so the first one is the random condition. And do we meet that? Well, they tell us here it's a random sample of 12 containers of milk. If I was doing this on the AP exam, I would write it out here. So I would say in the passage or in question, in the question, they say, they say a random, a random sample of 12. And then they go on to say more things."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "Well, they tell us here it's a random sample of 12 containers of milk. If I was doing this on the AP exam, I would write it out here. So I would say in the passage or in question, in the question, they say, they say a random, a random sample of 12. And then they go on to say more things. And so I would say that meets condition, meets condition. Now the next one we wanna care about is our normal condition and this is to feel good that our sampling distribution is roughly normal. Now there's a couple of ways that we could do that."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "And then they go on to say more things. And so I would say that meets condition, meets condition. Now the next one we wanna care about is our normal condition and this is to feel good that our sampling distribution is roughly normal. Now there's a couple of ways that we could do that. One is if our sample size is greater than 30 or greater than or equal to 30, then we say, okay, our sampling distribution is going to be roughly normal. But in this situation, our sample size n, so sample size, sample size is less than 30, but, but there's another way to meet the normal condition and that's if the underlying parent data is normally distributed. And they actually say it right over here."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "Now there's a couple of ways that we could do that. One is if our sample size is greater than 30 or greater than or equal to 30, then we say, okay, our sampling distribution is going to be roughly normal. But in this situation, our sample size n, so sample size, sample size is less than 30, but, but there's another way to meet the normal condition and that's if the underlying parent data is normally distributed. And they actually say it right over here. Variability in the filling process causes the actual contents of milk to be normally distributed. So we could say in passage, in passage says, and let's see, I could quote part of this. So actual contents, actual contents, and then dot, dot, dot, normally distributed, normally distributed."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "And they actually say it right over here. Variability in the filling process causes the actual contents of milk to be normally distributed. So we could say in passage, in passage says, and let's see, I could quote part of this. So actual contents, actual contents, and then dot, dot, dot, normally distributed, normally distributed. So that meets condition, meets condition. And then the last condition we wanna think about is the independence condition, independence. And this is to feel good that the observations, the individual observations in our sample can be considered to be roughly independent."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "So actual contents, actual contents, and then dot, dot, dot, normally distributed, normally distributed. So that meets condition, meets condition. And then the last condition we wanna think about is the independence condition, independence. And this is to feel good that the observations, the individual observations in our sample can be considered to be roughly independent. Now one way is if they were sampling with replacement, which they're not doing here. It looks like they took all 12 containers at once. But another way is if this is less than 10% of the overall population, then you could say, okay, they're gonna, you can view them as roughly independent."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "And this is to feel good that the observations, the individual observations in our sample can be considered to be roughly independent. Now one way is if they were sampling with replacement, which they're not doing here. It looks like they took all 12 containers at once. But another way is if this is less than 10% of the overall population, then you could say, okay, they're gonna, you can view them as roughly independent. And so you can say didn't, didn't sample with replacement, with replacement, but, but assume, assume that 12 is less than 10% of the population. And in that case, you would meet condition, meet this condition as well. So it looks like we are, we've met these three conditions that we need to make for inference, or we can assume we've done it."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "But another way is if this is less than 10% of the overall population, then you could say, okay, they're gonna, you can view them as roughly independent. And so you can say didn't, didn't sample with replacement, with replacement, but, but assume, assume that 12 is less than 10% of the population. And in that case, you would meet condition, meet this condition as well. So it looks like we are, we've met these three conditions that we need to make for inference, or we can assume we've done it. They haven't given us any information to the contrary. And so now what we can do is calculate a t-statistic, and then from that, calculate our p-value, compare our p-value to our significance level, and see what kind of conclusions we can make. And so our t-statistic, right over here, and once again, if at any point you're inspired, and if you haven't done so already, try to do it on your own."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "So it looks like we are, we've met these three conditions that we need to make for inference, or we can assume we've done it. They haven't given us any information to the contrary. And so now what we can do is calculate a t-statistic, and then from that, calculate our p-value, compare our p-value to our significance level, and see what kind of conclusions we can make. And so our t-statistic, right over here, and once again, if at any point you're inspired, and if you haven't done so already, try to do it on your own. Our t-statistic is going to be our sample mean minus the assumed mean from the null hypothesis. And let me, since I'm introducing this notation, this little sub zero, I'll say that's the assumed mean from my null hypothesis. So I'll do that, and then I'll divide."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "And so our t-statistic, right over here, and once again, if at any point you're inspired, and if you haven't done so already, try to do it on your own. Our t-statistic is going to be our sample mean minus the assumed mean from the null hypothesis. And let me, since I'm introducing this notation, this little sub zero, I'll say that's the assumed mean from my null hypothesis. So I'll do that, and then I'll divide. Ideally, if I was doing a z-statistic, I would divide by the standard deviation of the sampling distribution of the sample mean, which is often known as the standard error of the mean. But the whole reason why I'm doing a t-statistic is, well, I don't know exactly what that is, but I could estimate the standard deviation of the sampling distribution of the sample mean using the sample standard deviation divided by the square root of n. And once again, it's always good, if you're doing this on a test, to explain what n is or what some of these things are. If you're using standard notation, people might assume what they are, but if you have time on these tests, you can always explain more of what these actual variables are."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "So I'll do that, and then I'll divide. Ideally, if I was doing a z-statistic, I would divide by the standard deviation of the sampling distribution of the sample mean, which is often known as the standard error of the mean. But the whole reason why I'm doing a t-statistic is, well, I don't know exactly what that is, but I could estimate the standard deviation of the sampling distribution of the sample mean using the sample standard deviation divided by the square root of n. And once again, it's always good, if you're doing this on a test, to explain what n is or what some of these things are. If you're using standard notation, people might assume what they are, but if you have time on these tests, you can always explain more of what these actual variables are. But in this case, this is going to be 127.2, that is our sample mean, minus our assumed mean from our null hypothesis, minus 128, all of that over, our sample standard deviation is 2.1, divided by the square root of 12. And so this is going to be approximately equal to, got a calculator out here, and so we have, see in the numerator we have 127.2 minus 128, and then we're gonna divide that by, I'll do another parentheses, 2.1 divided by the square root of 12, and then let me close my parentheses. Did I type that in correctly?"}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "If you're using standard notation, people might assume what they are, but if you have time on these tests, you can always explain more of what these actual variables are. But in this case, this is going to be 127.2, that is our sample mean, minus our assumed mean from our null hypothesis, minus 128, all of that over, our sample standard deviation is 2.1, divided by the square root of 12. And so this is going to be approximately equal to, got a calculator out here, and so we have, see in the numerator we have 127.2 minus 128, and then we're gonna divide that by, I'll do another parentheses, 2.1 divided by the square root of 12, and then let me close my parentheses. Did I type that in correctly? Yeah, that looks right. Click Enter. And so this is negative, I'll say it's approximately negative 1.32."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "Did I type that in correctly? Yeah, that looks right. Click Enter. And so this is negative, I'll say it's approximately negative 1.32. So negative 1.32. And now we can figure out our p-value, our p-value, which is the same thing as the probability of getting a t-statistic this low or lower. So we could say t is less than or equal to negative 1.32 is equal to, so I'll get my calculator back out."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "And so this is negative, I'll say it's approximately negative 1.32. So negative 1.32. And now we can figure out our p-value, our p-value, which is the same thing as the probability of getting a t-statistic this low or lower. So we could say t is less than or equal to negative 1.32 is equal to, so I'll get my calculator back out. And so here, what I would use is I would use the cumulative distribution function for t-statistic, so that's that right over there. And so I do care about the left tail, so I care about the area under the curve from negative infinity up to and including negative 1.32. So let's do negative 1.32."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "So we could say t is less than or equal to negative 1.32 is equal to, so I'll get my calculator back out. And so here, what I would use is I would use the cumulative distribution function for t-statistic, so that's that right over there. And so I do care about the left tail, so I care about the area under the curve from negative infinity up to and including negative 1.32. So let's do negative 1.32. And then my degrees of freedom, well, it's gonna be my sample size minus one. My sample size was 12, so that minus one is 11. And then I do paste."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "So let's do negative 1.32. And then my degrees of freedom, well, it's gonna be my sample size minus one. My sample size was 12, so that minus one is 11. And then I do paste. And so I have this tcdf from negative E99 to negative 1.32 comma 11. And actually, you'd wanna write this down on your exam if you were doing it, just so they know where you got that from. And so this is, this is equal to 0.107."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "And then I do paste. And so I have this tcdf from negative E99 to negative 1.32 comma 11. And actually, you'd wanna write this down on your exam if you were doing it, just so they know where you got that from. And so this is, this is equal to 0.107. So let me write it. This is approximately 0.107. And it's important to say how you calculated this."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "And so this is, this is equal to 0.107. So let me write it. This is approximately 0.107. And it's important to say how you calculated this. So used, used tcdf, and we went from negative one times 10 to the 99th power, and we went up to negative 1.32, and then we had 11 degrees of freedom to get this result right over here. And it also might be good practice to draw your t-distribution right over here. So that's our t-distribution."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "And it's important to say how you calculated this. So used, used tcdf, and we went from negative one times 10 to the 99th power, and we went up to negative 1.32, and then we had 11 degrees of freedom to get this result right over here. And it also might be good practice to draw your t-distribution right over here. So that's our t-distribution. That's the mean of our t-distribution. So we say that this is the area that we care about. So that is that right over there, just to make sure people know what we're talking about."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "So that's our t-distribution. That's the mean of our t-distribution. So we say that this is the area that we care about. So that is that right over there, just to make sure people know what we're talking about. And so here, now we're ready to make a conclusion. We can compare this to our significance level. And so we can say since, since 0.107 is greater than, our significance level is greater than 0.05, which is alpha, we fail, we fail to reject, reject the null hypothesis."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "So that is that right over there, just to make sure people know what we're talking about. And so here, now we're ready to make a conclusion. We can compare this to our significance level. And so we can say since, since 0.107 is greater than, our significance level is greater than 0.05, which is alpha, we fail, we fail to reject, reject the null hypothesis. And so let's just make sure we read their question right. Is there sufficient evidence to conclude that the packaging plant is not in compliance with the regulations? And so another way of saying this is there is not, there is not sufficient, sufficient, I'm gonna have to scroll down a little bit."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy (2).mp3", "Sentence": "And so we can say since, since 0.107 is greater than, our significance level is greater than 0.05, which is alpha, we fail, we fail to reject, reject the null hypothesis. And so let's just make sure we read their question right. Is there sufficient evidence to conclude that the packaging plant is not in compliance with the regulations? And so another way of saying this is there is not, there is not sufficient, sufficient, I'm gonna have to scroll down a little bit. I'm trying to squeeze it on the page, but I'm gonna have to go down. There is not sufficient evidence, sufficient evidence to conclude, to conclude that the plant is not in compliance with regulations. And then we are done."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "A group of four friends likes to bowl together, and each friend keeps track of his all-time highest score in a single game. Their high scores are all between 180 and 220, except for Adam, whose high score is 250. Adam then bowls a great game and has a new high score of 290. How will increasing Adam's high score affect the mean and median? Now, like always, pause this video and see if you can figure this out yourself. All right, so let's just think about what they're saying. We have four friends, and they're each going to have, they each keep track of their all-time high score."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "How will increasing Adam's high score affect the mean and median? Now, like always, pause this video and see if you can figure this out yourself. All right, so let's just think about what they're saying. We have four friends, and they're each going to have, they each keep track of their all-time high score. So we're gonna have four data points, an all-time high score for each of the friends. So let's see, this is the lowest score of the friends. This is the second lowest, second to highest, and this is the highest scoring of the friends."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "We have four friends, and they're each going to have, they each keep track of their all-time high score. So we're gonna have four data points, an all-time high score for each of the friends. So let's see, this is the lowest score of the friends. This is the second lowest, second to highest, and this is the highest scoring of the friends. So let's see. Their high scores are all between 100 and 220, except for Adam, whose high score is 250. So before Adam bowls this super awesome game, the scores look something like this."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "This is the second lowest, second to highest, and this is the highest scoring of the friends. So let's see. Their high scores are all between 100 and 220, except for Adam, whose high score is 250. So before Adam bowls this super awesome game, the scores look something like this. The lowest score is 180. Adam scores 250. And if you take Adam out of the picture, the highest score is a 220."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "So before Adam bowls this super awesome game, the scores look something like this. The lowest score is 180. Adam scores 250. And if you take Adam out of the picture, the highest score is a 220. And we actually don't know what this score right over there is. Now, after Adam bowls a great new game and has a new high score of 290, what does the data set look like? Well, this guy's high score hasn't changed."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "And if you take Adam out of the picture, the highest score is a 220. And we actually don't know what this score right over there is. Now, after Adam bowls a great new game and has a new high score of 290, what does the data set look like? Well, this guy's high score hasn't changed. This guy's high score hasn't changed. This guy's high score hasn't changed. But now Adam has a new high score."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "Well, this guy's high score hasn't changed. This guy's high score hasn't changed. This guy's high score hasn't changed. But now Adam has a new high score. Instead of 250, it is now 290. So my question is, well, the first question is, does this change the median? Well, remember, the median is the middle number."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "But now Adam has a new high score. Instead of 250, it is now 290. So my question is, well, the first question is, does this change the median? Well, remember, the median is the middle number. And if we're looking at four numbers here, the median is going to be the average of the middle two numbers. So we're going to take the average of whatever this question mark is in 220. That's going to be the median."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "Well, remember, the median is the middle number. And if we're looking at four numbers here, the median is going to be the average of the middle two numbers. So we're going to take the average of whatever this question mark is in 220. That's going to be the median. Now, over here, after Adam has scored a new high score, how would we calculate the median? Well, we still have four numbers, and the middle two are still the same two middle numbers, whatever this friend's high score was, and it hasn't changed. And so we're going to have the same median."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "That's going to be the median. Now, over here, after Adam has scored a new high score, how would we calculate the median? Well, we still have four numbers, and the middle two are still the same two middle numbers, whatever this friend's high score was, and it hasn't changed. And so we're going to have the same median. It's going to be 220 plus question mark divided by two. It's going to be halfway between question mark and 220. So our median won't change."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "And so we're going to have the same median. It's going to be 220 plus question mark divided by two. It's going to be halfway between question mark and 220. So our median won't change. So median no change. So let's think about the mean now. Well, the mean, you take the sum of all these numbers and then you divide by four."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "So our median won't change. So median no change. So let's think about the mean now. Well, the mean, you take the sum of all these numbers and then you divide by four. And then you take the sum of all these numbers and divide by four. So which sum is going to be higher? Well, the first three numbers are the same, but in the second list, you have a higher number, 290."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "Well, the mean, you take the sum of all these numbers and then you divide by four. And then you take the sum of all these numbers and divide by four. So which sum is going to be higher? Well, the first three numbers are the same, but in the second list, you have a higher number, 290. 290 is higher than 250. So if you take these four and divide by four, you're going to have a larger value than if you take these four and divide by four because their sum is going to be larger. And so the mean is going to go up."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "Well, the first three numbers are the same, but in the second list, you have a higher number, 290. 290 is higher than 250. So if you take these four and divide by four, you're going to have a larger value than if you take these four and divide by four because their sum is going to be larger. And so the mean is going to go up. The mean will increase. So median no change and mean increase. All right, so this says both increase."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "And so the mean is going to go up. The mean will increase. So median no change and mean increase. All right, so this says both increase. No, that's not right. The median will increase. No, median doesn't change."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "All right, so this says both increase. No, that's not right. The median will increase. No, median doesn't change. The mean will increase, yep, and the median will stay the same. Yep, that's exactly what we're talking about. And if you want to make it a little bit more tangible, you could replace question mark with some number."}, {"video_title": "Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3", "Sentence": "No, median doesn't change. The mean will increase, yep, and the median will stay the same. Yep, that's exactly what we're talking about. And if you want to make it a little bit more tangible, you could replace question mark with some number. You could replace it, maybe this question mark is 200. And if you try it out with 200 just to make things tangible, you're going to see that that is indeed going to be the case. The median would be halfway between these two numbers, and I just arbitrarily picked 200."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "In this video I want to do some examples looking at distributions, in particular different features in distributions like clusters, gaps, and peaks. So over here I want to do some examples. Which of the following are accurate descriptions of the distribution below? Select all that apply. So the first statement is the distribution has an outlier. So an outlier is a data point that's way off of where the other data points are. It's way larger or way smaller than where all of the other data points seem to be clustered."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "Select all that apply. So the first statement is the distribution has an outlier. So an outlier is a data point that's way off of where the other data points are. It's way larger or way smaller than where all of the other data points seem to be clustered. And if we look over here, we have a lot of data points between zero and six. And just think about what they're measuring. This is shelf time for each apple at Gorge's Grocer."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "It's way larger or way smaller than where all of the other data points seem to be clustered. And if we look over here, we have a lot of data points between zero and six. And just think about what they're measuring. This is shelf time for each apple at Gorge's Grocer. So for example, we see there's one, two, three, four, five, six, seven apples that have a shelf life of zero days. So they're about to go bad. You see you have one, two, three, four, five, six, seven, eight apples that are gonna be good for another day."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "This is shelf time for each apple at Gorge's Grocer. So for example, we see there's one, two, three, four, five, six, seven apples that have a shelf life of zero days. So they're about to go bad. You see you have one, two, three, four, five, six, seven, eight apples that are gonna be good for another day. You have two apples that are gonna be good for another six days. And you have one apple that's gonna be good for 10 days. And this is unusual."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "You see you have one, two, three, four, five, six, seven, eight apples that are gonna be good for another day. You have two apples that are gonna be good for another six days. And you have one apple that's gonna be good for 10 days. And this is unusual. This is an outlier here. It has a way larger shelf life than all of the other data. So I would say this definitely does have an outlier."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "And this is unusual. This is an outlier here. It has a way larger shelf life than all of the other data. So I would say this definitely does have an outlier. We just have this one data point sitting all the way to the right. Way larger, way more shelf life than everything else. So it definitely has an outlier."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "So I would say this definitely does have an outlier. We just have this one data point sitting all the way to the right. Way larger, way more shelf life than everything else. So it definitely has an outlier. And this one would be the outlier. The distribution has a cluster from four to six days. And we indeed do see a cluster from four to six days."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "So it definitely has an outlier. And this one would be the outlier. The distribution has a cluster from four to six days. And we indeed do see a cluster from four to six days. A cluster, you can imagine it's a grouping of data that's sitting there, or you have a grouping of apples that have a shelf life between four and six days. And you definitely do see that cluster there. And since I already selected two things, I'm definitely not gonna select none of the above."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "And we indeed do see a cluster from four to six days. A cluster, you can imagine it's a grouping of data that's sitting there, or you have a grouping of apples that have a shelf life between four and six days. And you definitely do see that cluster there. And since I already selected two things, I'm definitely not gonna select none of the above. So let me check my answer. Let me do a few more of these. Which of the following are accurate descriptions of the distribution below?"}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "And since I already selected two things, I'm definitely not gonna select none of the above. So let me check my answer. Let me do a few more of these. Which of the following are accurate descriptions of the distribution below? And once again, we're going to select all that apply. So the distribution has an outlier. So let's see, this distribution, I do have a data point here that's at the high end, and I have another data point here that's at the low end."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "Which of the following are accurate descriptions of the distribution below? And once again, we're going to select all that apply. So the distribution has an outlier. So let's see, this distribution, I do have a data point here that's at the high end, and I have another data point here that's at the low end. But I don't have any data points that are sitting far above or far below the bulk of the data. If I had a data point that was out here, then yeah, I would say that was an outlier to the right, or a positive outlier. If I had a data point way to the left off the screen over here, maybe that would be an outlier."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "So let's see, this distribution, I do have a data point here that's at the high end, and I have another data point here that's at the low end. But I don't have any data points that are sitting far above or far below the bulk of the data. If I had a data point that was out here, then yeah, I would say that was an outlier to the right, or a positive outlier. If I had a data point way to the left off the screen over here, maybe that would be an outlier. But I don't really see any obvious outliers. All of the data, it's pretty clustered together. So I would not say that the distribution has an outlier."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "If I had a data point way to the left off the screen over here, maybe that would be an outlier. But I don't really see any obvious outliers. All of the data, it's pretty clustered together. So I would not say that the distribution has an outlier. The distribution has a peak at 22 degrees. Yeah, it does indeed look like we have, and let's just look at what we're actually measuring, high temperature each day in Edgton, Iowa in July. So it does indeed look like we have the most number of days that had a high temperature at 22, most number of days in July had a high temperature at 22 degrees Celsius."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "So I would not say that the distribution has an outlier. The distribution has a peak at 22 degrees. Yeah, it does indeed look like we have, and let's just look at what we're actually measuring, high temperature each day in Edgton, Iowa in July. So it does indeed look like we have the most number of days that had a high temperature at 22, most number of days in July had a high temperature at 22 degrees Celsius. So that is a peak, and you can see it. If you imagine this is kind of a mountain, this is a peak right here. This is a high point."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "So it does indeed look like we have the most number of days that had a high temperature at 22, most number of days in July had a high temperature at 22 degrees Celsius. So that is a peak, and you can see it. If you imagine this is kind of a mountain, this is a peak right here. This is a high point. You have, at least locally, you have the most number of days at 22 degrees Celsius. So I would say it definitely has a peak there. Since I selected something, I'm not gonna select none of the above."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "This is a high point. You have, at least locally, you have the most number of days at 22 degrees Celsius. So I would say it definitely has a peak there. Since I selected something, I'm not gonna select none of the above. Let's do a couple more of these. Which of the following are accurate descriptions of the distribution below? So the first one, the distribution has an outlier."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "Since I selected something, I'm not gonna select none of the above. Let's do a couple more of these. Which of the following are accurate descriptions of the distribution below? So the first one, the distribution has an outlier. So let's see, this number of guests by day at Seth's Sandwich Shop. So let's see, the lowest, they have, so they have no days, no days where he had between zero and 19 guests. No days where he had between 20 and 39 guests."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "So the first one, the distribution has an outlier. So let's see, this number of guests by day at Seth's Sandwich Shop. So let's see, the lowest, they have, so they have no days, no days where he had between zero and 19 guests. No days where he had between 20 and 39 guests. Looks like there's about nine days where he had between 40 and 59 guests. Looks like 20 days where he had between 60 and 79 guests. All the way, it looks like this is maybe eight days that he had between 180 and 199 guests."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "No days where he had between 20 and 39 guests. Looks like there's about nine days where he had between 40 and 59 guests. Looks like 20 days where he had between 60 and 79 guests. All the way, it looks like this is maybe eight days that he had between 180 and 199 guests. But the question of outliers, there doesn't seem to be any day where he had an unusual number of guests. There's not a day that's way out here where he had like 500 guests. So I would say this distribution does not have an outlier."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "All the way, it looks like this is maybe eight days that he had between 180 and 199 guests. But the question of outliers, there doesn't seem to be any day where he had an unusual number of guests. There's not a day that's way out here where he had like 500 guests. So I would say this distribution does not have an outlier. The distribution has a cluster from zero to 39 guests. So zero to 39 guests is right over here, zero to 39 guests. And there's no days where he had between zero and 39 guests, either zero to 19 or 20 to 39."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "So I would say this distribution does not have an outlier. The distribution has a cluster from zero to 39 guests. So zero to 39 guests is right over here, zero to 39 guests. And there's no days where he had between zero and 39 guests, either zero to 19 or 20 to 39. So there's definitely not a cluster there. I would say that the cluster would be between, were days that had between 40 and 199 guests. Definitely not zero and 39."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "And there's no days where he had between zero and 39 guests, either zero to 19 or 20 to 39. So there's definitely not a cluster there. I would say that the cluster would be between, were days that had between 40 and 199 guests. Definitely not zero and 39. There was no days that were between zero and 39 guests. So I would say none of the above, very confidently. Let's do one more of these."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "Definitely not zero and 39. There was no days that were between zero and 39 guests. So I would say none of the above, very confidently. Let's do one more of these. Which of the following are accurate descriptions of the distribution below? All right. The distribution has a peak from 12 to 13 points."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "Let's do one more of these. Which of the following are accurate descriptions of the distribution below? All right. The distribution has a peak from 12 to 13 points. Let me see what this is measuring or what this data is about. Test scores by student in Ms. Friend's class."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "The distribution has a peak from 12 to 13 points. Let me see what this is measuring or what this data is about. Test scores by student in Ms. Friend's class. So you had one student who got between a zero and a one on the 20 point scale. So got between, I guess you may be out of 20 questions, got between zero and one points. And then you see that those students got between two and three or four and five or six and seven."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "Friend's class. So you had one student who got between a zero and a one on the 20 point scale. So got between, I guess you may be out of 20 questions, got between zero and one points. And then you see that those students got between two and three or four and five or six and seven. Then we have another student who got between eight and nine. Looks like three students got between 10 and 11. And then we keep increasing."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "And then you see that those students got between two and three or four and five or six and seven. Then we have another student who got between eight and nine. Looks like three students got between 10 and 11. And then we keep increasing. This looks like it's about 12 students got either a 16 or a 17 or something in between, maybe if you could get decimal points on that test. And then it looks like 10 students got from 18 to 19. All right, so this says the distribution has a peak from 12 to 13 points."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "And then we keep increasing. This looks like it's about 12 students got either a 16 or a 17 or something in between, maybe if you could get decimal points on that test. And then it looks like 10 students got from 18 to 19. All right, so this says the distribution has a peak from 12 to 13 points. 12 to 13 points. There were five students, but this isn't a peak. If you just go to 14 to 15 points, you have more students."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "All right, so this says the distribution has a peak from 12 to 13 points. 12 to 13 points. There were five students, but this isn't a peak. If you just go to 14 to 15 points, you have more students. So this is definitely not a peak. If you were looking at this as a mountain of some kind, you definitely wouldn't describe this point as a peak. You would say this distribution has a peak, has the most number of students who got between 16 and 17 points."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "If you just go to 14 to 15 points, you have more students. So this is definitely not a peak. If you were looking at this as a mountain of some kind, you definitely wouldn't describe this point as a peak. You would say this distribution has a peak, has the most number of students who got between 16 and 17 points. So that's the peak right there, not 12 to 13 points. So I would not select that first choice. The distribution has an outlier."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "You would say this distribution has a peak, has the most number of students who got between 16 and 17 points. So that's the peak right there, not 12 to 13 points. So I would not select that first choice. The distribution has an outlier. Well, yeah, look at this. You have this outlier. Most of the students scored between eight and 19 points."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "The distribution has an outlier. Well, yeah, look at this. You have this outlier. Most of the students scored between eight and 19 points. And then you have this one student who got between zero and one. It's really an outlier. You even see this, when you look at it visually, it's not even connected to the rest of the distribution."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "Most of the students scored between eight and 19 points. And then you have this one student who got between zero and one. It's really an outlier. You even see this, when you look at it visually, it's not even connected to the rest of the distribution. It's way to the left. If something's way to the left or way to the right, that's an outlier. If it's unusually low or unusually high."}, {"video_title": "Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3", "Sentence": "You even see this, when you look at it visually, it's not even connected to the rest of the distribution. It's way to the left. If something's way to the left or way to the right, that's an outlier. If it's unusually low or unusually high. So I would say this distribution definitely does have an outlier. And I'm not gonna pick none of the above since I found a choice. And I think we're all done."}, {"video_title": "Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3", "Sentence": "And so the idea of statistical inference is new to you, or hypothesis testing, once again, watch those videos as well. But let's say we think there's a positive association between shoe size and height. And so what we might wanna do is, we could, here on the horizontal axis, that is shoe size, our sizes could go size one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, and it could keep going up from there. And then on this height, on this axis, our y-axis, this would be height. So one foot, two feet, three feet, four feet, five feet, six feet, seven feet. And then you could, to see if there's an association, you might take a sample, let's say you take a random sample of 20 people from the population, and in future videos, we'll talk about the conditions necessary for making appropriate inferences. Well, let's say those 20 people are these 20 data points."}, {"video_title": "Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3", "Sentence": "And then on this height, on this axis, our y-axis, this would be height. So one foot, two feet, three feet, four feet, five feet, six feet, seven feet. And then you could, to see if there's an association, you might take a sample, let's say you take a random sample of 20 people from the population, and in future videos, we'll talk about the conditions necessary for making appropriate inferences. Well, let's say those 20 people are these 20 data points. So there's a young child, then maybe there's a grown adult with bigger feet, and who's taller, and then three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and so you have these 20 data points, and then what you're likely to do is input them into a computer. You could do it by hand, but we have computers now to do that for us usually. And the computer could try to fit a regression line."}, {"video_title": "Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3", "Sentence": "Well, let's say those 20 people are these 20 data points. So there's a young child, then maybe there's a grown adult with bigger feet, and who's taller, and then three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and so you have these 20 data points, and then what you're likely to do is input them into a computer. You could do it by hand, but we have computers now to do that for us usually. And the computer could try to fit a regression line. And there's many techniques for doing it, but one typical technique is to try to overall minimize the square distance between these points and that line. And this regression line will have an equation as any line would have, and we tend to show that as saying y hat, this hat tells us that this is a regression line, is equal to the y-intercept, a, plus the slope times our x variable. So this right over here would be a."}, {"video_title": "Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3", "Sentence": "And the computer could try to fit a regression line. And there's many techniques for doing it, but one typical technique is to try to overall minimize the square distance between these points and that line. And this regression line will have an equation as any line would have, and we tend to show that as saying y hat, this hat tells us that this is a regression line, is equal to the y-intercept, a, plus the slope times our x variable. So this right over here would be a. Now to be clear, if you took another sample, you might get different results here. In fact, let's call this y sub one for our first sample, a sub one, b sub one, and this is a sub one. If you were to take another sample of 20 folks, so let's do that, maybe you get one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and then you tried to fit a line to that, that line might look something like this."}, {"video_title": "Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3", "Sentence": "So this right over here would be a. Now to be clear, if you took another sample, you might get different results here. In fact, let's call this y sub one for our first sample, a sub one, b sub one, and this is a sub one. If you were to take another sample of 20 folks, so let's do that, maybe you get one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and then you tried to fit a line to that, that line might look something like this. It might have a slightly different y-intercept and a slightly different slope, so we could call that for the second sample, y sub two or y hat sub two is equal to a sub two plus b sub two times x. And so every time you take a sample, you are likely to get different results for these values, which are essentially statistics. Remember, statistics are things that we can get from samples and we're trying to estimate true population parameters."}, {"video_title": "Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3", "Sentence": "If you were to take another sample of 20 folks, so let's do that, maybe you get one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and then you tried to fit a line to that, that line might look something like this. It might have a slightly different y-intercept and a slightly different slope, so we could call that for the second sample, y sub two or y hat sub two is equal to a sub two plus b sub two times x. And so every time you take a sample, you are likely to get different results for these values, which are essentially statistics. Remember, statistics are things that we can get from samples and we're trying to estimate true population parameters. Well, what would be the true population parameters we're trying to estimate? Well, imagine a world, imagine a world here that you're able to find out the true linear relationship, or maybe there is some true linear relationship between shoe size and height. You could get it if, theoretically, you could measure every human being on the planet and depending what you define as the population, it could be all living people or all people who'll ever live."}, {"video_title": "Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3", "Sentence": "Remember, statistics are things that we can get from samples and we're trying to estimate true population parameters. Well, what would be the true population parameters we're trying to estimate? Well, imagine a world, imagine a world here that you're able to find out the true linear relationship, or maybe there is some true linear relationship between shoe size and height. You could get it if, theoretically, you could measure every human being on the planet and depending what you define as the population, it could be all living people or all people who'll ever live. This isn't practical, but let's just say that you actually could. You would have billions of data points here for the true population, and then if you were to fit a regression line to that, you could view this as the true population regression line. And so that would be y hat is equal to, and to make it clear that here, the y-intercept and the slope, this would be the true population parameters."}, {"video_title": "Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3", "Sentence": "You could get it if, theoretically, you could measure every human being on the planet and depending what you define as the population, it could be all living people or all people who'll ever live. This isn't practical, but let's just say that you actually could. You would have billions of data points here for the true population, and then if you were to fit a regression line to that, you could view this as the true population regression line. And so that would be y hat is equal to, and to make it clear that here, the y-intercept and the slope, this would be the true population parameters. Instead of saying a, we say alpha, and instead of saying b, we say beta times x. But it's very hard to come up exactly with what alpha and beta are, and so that's why we estimate it with a's and b's based on a sample. Now, what's interesting with this in mind is we can start to make inferences based on our sample."}, {"video_title": "Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3", "Sentence": "And so that would be y hat is equal to, and to make it clear that here, the y-intercept and the slope, this would be the true population parameters. Instead of saying a, we say alpha, and instead of saying b, we say beta times x. But it's very hard to come up exactly with what alpha and beta are, and so that's why we estimate it with a's and b's based on a sample. Now, what's interesting with this in mind is we can start to make inferences based on our sample. So we know that, for example, b sub two is unlikely to be exactly beta, but how confident can we be that there is at least a positive linear relationship or a non-zero linear relationship? Or can we create a confidence interval around this statistic in order to have a good sense of where the true parameter might actually be? And the simple answer is yes."}, {"video_title": "Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3", "Sentence": "Now, what's interesting with this in mind is we can start to make inferences based on our sample. So we know that, for example, b sub two is unlikely to be exactly beta, but how confident can we be that there is at least a positive linear relationship or a non-zero linear relationship? Or can we create a confidence interval around this statistic in order to have a good sense of where the true parameter might actually be? And the simple answer is yes. And to do so, we'll use the same exact ideas that we did when we made inferences based on proportions or based on means. The way that you can make an inference, for example, for your true population slope of your regression line, you say, okay, I took a sample. I got this slope right over here, so I'll just call that b two, and then I could create a confidence interval around that."}, {"video_title": "Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3", "Sentence": "And the simple answer is yes. And to do so, we'll use the same exact ideas that we did when we made inferences based on proportions or based on means. The way that you can make an inference, for example, for your true population slope of your regression line, you say, okay, I took a sample. I got this slope right over here, so I'll just call that b two, and then I could create a confidence interval around that. And so that confidence interval is going to be based on some critical value times ideally the standard deviation of the sampling distribution of your sample statistic. In this case, it would be the sample regression line slope. But because we don't know exactly what this is, we can't figure out precisely what this is going to be from a sample, we are going to estimate it with what's known as the standard error of the statistic, and we'll go into more depth in this in future videos."}, {"video_title": "Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3", "Sentence": "I got this slope right over here, so I'll just call that b two, and then I could create a confidence interval around that. And so that confidence interval is going to be based on some critical value times ideally the standard deviation of the sampling distribution of your sample statistic. In this case, it would be the sample regression line slope. But because we don't know exactly what this is, we can't figure out precisely what this is going to be from a sample, we are going to estimate it with what's known as the standard error of the statistic, and we'll go into more depth in this in future videos. And since we're estimating here, we're going to use a critical t value here, which we have studied before. And so based on your confidence level you wanna have, let's say it's 95%, based on the degrees of freedom, which we'll see will come out of how many data points we have, we can figure this out. And from our sample, we can figure this out, and we can figure this out, and then we would have constructed a confidence interval."}, {"video_title": "Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3", "Sentence": "But because we don't know exactly what this is, we can't figure out precisely what this is going to be from a sample, we are going to estimate it with what's known as the standard error of the statistic, and we'll go into more depth in this in future videos. And since we're estimating here, we're going to use a critical t value here, which we have studied before. And so based on your confidence level you wanna have, let's say it's 95%, based on the degrees of freedom, which we'll see will come out of how many data points we have, we can figure this out. And from our sample, we can figure this out, and we can figure this out, and then we would have constructed a confidence interval. We'll also see that you could do hypothesis testing here. You could say, hey, let's set up a null hypothesis, and the null hypothesis is going to be that there's no non-zero linear relationship, or that the true population slope of the regression line, or slope of the population regression line, is equal to zero, and that the alternative hypothesis is that the true relationship could either be greater than zero, it's a positive linear relationship, or that it's just non-zero. And then what you could do is, assuming this, you could see what's the probability of getting a statistic that is at least this extreme, or more extreme."}, {"video_title": "Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3", "Sentence": "And from our sample, we can figure this out, and we can figure this out, and then we would have constructed a confidence interval. We'll also see that you could do hypothesis testing here. You could say, hey, let's set up a null hypothesis, and the null hypothesis is going to be that there's no non-zero linear relationship, or that the true population slope of the regression line, or slope of the population regression line, is equal to zero, and that the alternative hypothesis is that the true relationship could either be greater than zero, it's a positive linear relationship, or that it's just non-zero. And then what you could do is, assuming this, you could see what's the probability of getting a statistic that is at least this extreme, or more extreme. And if that's below some threshold, you might reject the null hypothesis, which would suggest the alternative. So this and this are things that we have done before, where you're creating a confidence interval around a statistic, or you're doing hypothesis testing, making assumptions about a true parameter. The only difference here is that the parameter that we're trying to estimate are going to be the parameters for a theoretical population regression line, and we're going to do that using sample statistics for a sample regression line."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "So let's say the probability of scoring, score, you know, free throw, is equal to, is going to be, let's say 70%. If we want to write it as a percent, or we could write it as 0.7 if we write it as a decimal. And let's say the probability of missing a free throw then, and this is just gonna come straight out of what we just wrote down. So the probability of missing, of missing a free throw is just going to be 100% minus this. You're either gonna make or miss, you're either gonna score or miss. I don't wanna use make and miss because they both start with M. So this is going to be a 30% probability, or if we write it as a decimal, 0.3. One minus 0.7."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "So the probability of missing, of missing a free throw is just going to be 100% minus this. You're either gonna make or miss, you're either gonna score or miss. I don't wanna use make and miss because they both start with M. So this is going to be a 30% probability, or if we write it as a decimal, 0.3. One minus 0.7. These are the only two possibilities, so they have to add up to 100%, or they have to add up to one. Now let's say that you are going to take six attempts. And what we're curious about, what we're curious about is the probability of exactly, exactly two scores, two scores in six attempts."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "One minus 0.7. These are the only two possibilities, so they have to add up to 100%, or they have to add up to one. Now let's say that you are going to take six attempts. And what we're curious about, what we're curious about is the probability of exactly, exactly two scores, two scores in six attempts. In six, in six attempts. So let's think about what that is. And I encourage you, if you get inspired at any point in this video, you should pause it and you should try to work through what we're asking right now."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "And what we're curious about, what we're curious about is the probability of exactly, exactly two scores, two scores in six attempts. In six, in six attempts. So let's think about what that is. And I encourage you, if you get inspired at any point in this video, you should pause it and you should try to work through what we're asking right now. So this is what we wanna figure out. The probability of exactly two scores in six attempts. So let's think about the way, let's think about the particular ways of getting two scores in six attempts, and think about the probability for any one of those particular ways, and then we can think about, well how many ways can we get two scores in six attempts?"}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "And I encourage you, if you get inspired at any point in this video, you should pause it and you should try to work through what we're asking right now. So this is what we wanna figure out. The probability of exactly two scores in six attempts. So let's think about the way, let's think about the particular ways of getting two scores in six attempts, and think about the probability for any one of those particular ways, and then we can think about, well how many ways can we get two scores in six attempts? So for example, you could get, you could make the first two free throws, so it could be score, score, and then you miss the next four. So score, score, and then it's miss, miss, miss, and miss. So what's the probability of this exact thing happening, this exact thing?"}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "So let's think about the way, let's think about the particular ways of getting two scores in six attempts, and think about the probability for any one of those particular ways, and then we can think about, well how many ways can we get two scores in six attempts? So for example, you could get, you could make the first two free throws, so it could be score, score, and then you miss the next four. So score, score, and then it's miss, miss, miss, and miss. So what's the probability of this exact thing happening, this exact thing? Well you have a 0.7 chance of making, of scoring on the first one. Then you have a 0.7 chance of scoring on the second one. And then you have a 0.3 chance of missing the next four."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "So what's the probability of this exact thing happening, this exact thing? Well you have a 0.7 chance of making, of scoring on the first one. Then you have a 0.7 chance of scoring on the second one. And then you have a 0.3 chance of missing the next four. So the probability of this exact circumstance is going to be what I'm writing down. And hopefully you don't get the multiplication symbols confused with the decimals. I'm trying to write them a little bit higher."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "And then you have a 0.3 chance of missing the next four. So the probability of this exact circumstance is going to be what I'm writing down. And hopefully you don't get the multiplication symbols confused with the decimals. I'm trying to write them a little bit higher. So times 0.3. And what is this going to be equal to? Well this is going to be equal to, this is going to be 0.7 squared times 0.3, times 0.3 to the one, two, three, fourth, to the fourth power."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "I'm trying to write them a little bit higher. So times 0.3. And what is this going to be equal to? Well this is going to be equal to, this is going to be 0.7 squared times 0.3, times 0.3 to the one, two, three, fourth, to the fourth power. Now is this the only way to get two scores in six attempts? No, there's many ways of getting two scores in six attempts. For example, maybe you miss the first one, the first attempt, and then you make the second attempt, you score."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "Well this is going to be equal to, this is going to be 0.7 squared times 0.3, times 0.3 to the one, two, three, fourth, to the fourth power. Now is this the only way to get two scores in six attempts? No, there's many ways of getting two scores in six attempts. For example, maybe you miss the first one, the first attempt, and then you make the second attempt, you score. Then you miss the third attempt. And then you, let's just say you make the fourth attempt. And then you miss the next two."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "For example, maybe you miss the first one, the first attempt, and then you make the second attempt, you score. Then you miss the third attempt. And then you, let's just say you make the fourth attempt. And then you miss the next two. So then you miss and you miss. This is another way to get two scores in six attempts. And what's the probability of this happening?"}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "And then you miss the next two. So then you miss and you miss. This is another way to get two scores in six attempts. And what's the probability of this happening? Well as we'll see, it's going to be exactly this, it's just we're multiplying in different order. This is going to be 0.3 times 0.7. You have a 30% chance of missing the first one, a 70% chance of making the second one."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "And what's the probability of this happening? Well as we'll see, it's going to be exactly this, it's just we're multiplying in different order. This is going to be 0.3 times 0.7. You have a 30% chance of missing the first one, a 70% chance of making the second one. And then times 0.3, 30% chance of missing the third, times a 70% chance of making the fourth, times a 30% chance for each of the next two misses. If you wanted this exact circumstance, this exact circumstance, this is once again going to be 0.7, if you just rearrange the order in which you're multiplying, this is going to be 0.7 squared times 0.3 to the fourth power. So for any one of these particular ways to get exactly two scores in six attempts, the probability is going to be this."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "You have a 30% chance of missing the first one, a 70% chance of making the second one. And then times 0.3, 30% chance of missing the third, times a 70% chance of making the fourth, times a 30% chance for each of the next two misses. If you wanted this exact circumstance, this exact circumstance, this is once again going to be 0.7, if you just rearrange the order in which you're multiplying, this is going to be 0.7 squared times 0.3 to the fourth power. So for any one of these particular ways to get exactly two scores in six attempts, the probability is going to be this. So the probability of getting exactly two scores in six attempts, well it's going to be any one of these probabilities times the number of ways you can get six scores, times the number of ways you can get two scores in six attempts. Well, if you have, out of six attempts, you're choosing two of them to have scores, how many ways are there? Well, as you can imagine, this is a combinatorics problem."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "So for any one of these particular ways to get exactly two scores in six attempts, the probability is going to be this. So the probability of getting exactly two scores in six attempts, well it's going to be any one of these probabilities times the number of ways you can get six scores, times the number of ways you can get two scores in six attempts. Well, if you have, out of six attempts, you're choosing two of them to have scores, how many ways are there? Well, as you can imagine, this is a combinatorics problem. So you could write this as, you could write this, and let me see how I could, you're gonna take six attempts, so you could write this as six choose, or we're trying to, if you're picking from six things, your six attempts, and you're picking two of them, or two of them are going to be, need to be made if you want to meet these circumstances. This is gonna tell us the number of different ways you can make two scores in six attempts. And of course, we can write this in kind of the binomial coefficient notation."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "Well, as you can imagine, this is a combinatorics problem. So you could write this as, you could write this, and let me see how I could, you're gonna take six attempts, so you could write this as six choose, or we're trying to, if you're picking from six things, your six attempts, and you're picking two of them, or two of them are going to be, need to be made if you want to meet these circumstances. This is gonna tell us the number of different ways you can make two scores in six attempts. And of course, we can write this in kind of the binomial coefficient notation. We could write this as six choose two, six choose two, and we could just apply the formula for combinations. And if this looks completely unfamiliar, I encourage you to look up combinations on Khan Academy, and we go into some detail on the reasoning behind this formula. It actually makes a lot of sense."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "And of course, we can write this in kind of the binomial coefficient notation. We could write this as six choose two, six choose two, and we could just apply the formula for combinations. And if this looks completely unfamiliar, I encourage you to look up combinations on Khan Academy, and we go into some detail on the reasoning behind this formula. It actually makes a lot of sense. This is going to be equal to six factorial over two factorial, over two factorial times six minus two factorial. So six minus two, six minus two factorial. I'll do the factorial in green again."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "It actually makes a lot of sense. This is going to be equal to six factorial over two factorial, over two factorial times six minus two factorial. So six minus two, six minus two factorial. I'll do the factorial in green again. Six minus two factorial, and what's this going to be equal to? This is going to be equal to six times five times four times three times two, and I'll just throw in the one there, although it doesn't change the value, over two times one, and six minus two is four, so that's going to be four factorial. So this right over here is four factorial, so times four times three times two times one."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "I'll do the factorial in green again. Six minus two factorial, and what's this going to be equal to? This is going to be equal to six times five times four times three times two, and I'll just throw in the one there, although it doesn't change the value, over two times one, and six minus two is four, so that's going to be four factorial. So this right over here is four factorial, so times four times three times two times one. Well, that and that is going to cancel. And then, let's see, six divided by two is three. So this is 15."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "So this right over here is four factorial, so times four times three times two times one. Well, that and that is going to cancel. And then, let's see, six divided by two is three. So this is 15. There's 15 different ways that you could get, that you could pick two things out of six, I guess is one way to say it, or there's 15, did I say 16? There's 15, the sixes and the fives, there's 15 different ways that you could pick two things out of six, or another way of thinking about it is there's 15 different ways to make two out of six free throws. Now, the probability for each of those is this right over here."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "So this is 15. There's 15 different ways that you could get, that you could pick two things out of six, I guess is one way to say it, or there's 15, did I say 16? There's 15, the sixes and the fives, there's 15 different ways that you could pick two things out of six, or another way of thinking about it is there's 15 different ways to make two out of six free throws. Now, the probability for each of those is this right over here. So the probability of exactly two scores in six attempts, well, this is where we deserve a little bit of a drumroll, this is going to be six choose two times 0.7, 0.7 squared. That's, this is two, two, you're gonna make two, you're gonna make two, and then it's 0.3, 0.3 to the, 0.3 to the fourth power. Notice, these will necessarily add up to six."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "Now, the probability for each of those is this right over here. So the probability of exactly two scores in six attempts, well, this is where we deserve a little bit of a drumroll, this is going to be six choose two times 0.7, 0.7 squared. That's, this is two, two, you're gonna make two, you're gonna make two, and then it's 0.3, 0.3 to the, 0.3 to the fourth power. Notice, these will necessarily add up to six. So if this right over here was a three, then this right over here would be a three, and then this would be six minus three, or three right over here. And now, what is this value? Well, it's going to be equal to, it's going to be equal to, we have our 15, three times five, so we have this business right over here, it's going to be 15 times, times, let's see, in yellow, 0.7 times 0.7 is going to be times 0.49."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "Notice, these will necessarily add up to six. So if this right over here was a three, then this right over here would be a three, and then this would be six minus three, or three right over here. And now, what is this value? Well, it's going to be equal to, it's going to be equal to, we have our 15, three times five, so we have this business right over here, it's going to be 15 times, times, let's see, in yellow, 0.7 times 0.7 is going to be times 0.49. And let's see, three to the fourth power would be 81. Three to the fourth power would be 81, but I'm multiplying four decimals, each of them have one space to the right of the decimal point, so I'm gonna have, this is gonna, I'm gonna have four spaces to the right of the decimal, so 0.0081. So, there you go, whatever this number is, and actually, I might as well get a calculator out and calculate it."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "Well, it's going to be equal to, it's going to be equal to, we have our 15, three times five, so we have this business right over here, it's going to be 15 times, times, let's see, in yellow, 0.7 times 0.7 is going to be times 0.49. And let's see, three to the fourth power would be 81. Three to the fourth power would be 81, but I'm multiplying four decimals, each of them have one space to the right of the decimal point, so I'm gonna have, this is gonna, I'm gonna have four spaces to the right of the decimal, so 0.0081. So, there you go, whatever this number is, and actually, I might as well get a calculator out and calculate it. So this is going to be, this is going to be, let me, so it's 15 times 0.49, times 0.0081, 0.0081, and we get 0.59535. So this is going to be equal to, let me write it down, and actually, maybe I'll, well, I wish I had a little bit more real estate right over here, but I'll write it in a very bold color. This is going to be, oh, actually, I'm kind of out of bold colors."}, {"video_title": "Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3", "Sentence": "So, there you go, whatever this number is, and actually, I might as well get a calculator out and calculate it. So this is going to be, this is going to be, let me, so it's 15 times 0.49, times 0.0081, 0.0081, and we get 0.59535. So this is going to be equal to, let me write it down, and actually, maybe I'll, well, I wish I had a little bit more real estate right over here, but I'll write it in a very bold color. This is going to be, oh, actually, I'm kind of out of bold colors. I'll write it in a slightly less bold color. This is going to be equal to 0.05935, if we wanted the exact number, or we could say this is approximately, if we round to the nearest percentage, this is approximately a 6% chance, 6% probability of getting exactly two scores in the six attempts. I didn't say two or more, I just said exactly two scores in the six attempts, and actually, it's a fairly low probability because I have a pretty high free throw percentage."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "We're defining it as the number of independent trials we need to get a success where the probability of success for each trial is lower case p. And we have seen this before when we introduced ourselves to geometric random variables. Now the goal of this video is to think about, well what is the expected value of a geometric random variable like this? And I'll tell you the answer in future videos when we will apply this formula. But in this video we're actually going to prove it to ourselves mathematically. But the expected value of a geometric random variable is gonna be one over the probability of success on any given trial. So now let's prove it to ourselves. So the expected value of any random variable is just going to be the probability weighted outcomes that you could have."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "But in this video we're actually going to prove it to ourselves mathematically. But the expected value of a geometric random variable is gonna be one over the probability of success on any given trial. So now let's prove it to ourselves. So the expected value of any random variable is just going to be the probability weighted outcomes that you could have. So you could say it is the probability, the probability that our random variable is equal to one times one plus the probability that our random variable is equal to two times two plus, and you get the general idea, it goes on and on and on. And a geometric random variable, it can only take on values one, two, three, four, so forth and so on. It will not take on the value zero because you cannot have a success if you have not had a trial yet."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "So the expected value of any random variable is just going to be the probability weighted outcomes that you could have. So you could say it is the probability, the probability that our random variable is equal to one times one plus the probability that our random variable is equal to two times two plus, and you get the general idea, it goes on and on and on. And a geometric random variable, it can only take on values one, two, three, four, so forth and so on. It will not take on the value zero because you cannot have a success if you have not had a trial yet. But what is this going to be equal to? Well this is going to be equal to, what's the probability that we have a success on our first trial? And actually let me just write it over here."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "It will not take on the value zero because you cannot have a success if you have not had a trial yet. But what is this going to be equal to? Well this is going to be equal to, what's the probability that we have a success on our first trial? And actually let me just write it over here. So this is going to be P. What is this going to be? What is the probability that we don't have a success on our first trial but we have one on our second trial? Well this is going to be one minus P, that's the first trial where we don't have a success, times a success on the second trial."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "And actually let me just write it over here. So this is going to be P. What is this going to be? What is the probability that we don't have a success on our first trial but we have one on our second trial? Well this is going to be one minus P, that's the first trial where we don't have a success, times a success on the second trial. And actually let me do a few more terms here. So let me erase this a little bit, do a few more terms. This is going to be the probability that X equals two, sorry, the probability that X equals three, times three."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "Well this is going to be one minus P, that's the first trial where we don't have a success, times a success on the second trial. And actually let me do a few more terms here. So let me erase this a little bit, do a few more terms. This is going to be the probability that X equals two, sorry, the probability that X equals three, times three. And we're going to keep going on and on and on. Well what's this going to be? Well the probability that X equals three is we're going to have to get two unsuccessful trials."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "This is going to be the probability that X equals two, sorry, the probability that X equals three, times three. And we're going to keep going on and on and on. Well what's this going to be? Well the probability that X equals three is we're going to have to get two unsuccessful trials. And so the probability of two unsuccessful trials is one minus P squared. And then one successful trial, just like that. So you get the general idea."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "Well the probability that X equals three is we're going to have to get two unsuccessful trials. And so the probability of two unsuccessful trials is one minus P squared. And then one successful trial, just like that. So you get the general idea. So if I wanted to rewrite this, I'm just going to rewrite it to make it a little bit simpler. So the expected, at least for the purposes of this proof, so the expected value of X is equal to, I'll write this as one P plus two P times one minus P plus three P times one minus P squared. And we're going to keep going on and on and on forever like that."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "So you get the general idea. So if I wanted to rewrite this, I'm just going to rewrite it to make it a little bit simpler. So the expected, at least for the purposes of this proof, so the expected value of X is equal to, I'll write this as one P plus two P times one minus P plus three P times one minus P squared. And we're going to keep going on and on and on forever like that. So how do we figure out this sum? Well now I'm going to do a little bit of mathematical trickery or gymnastics, but it's all valid. And if any of y'all have seen the proof of taking a infinite geometric series, then we're going to do a very similar technique."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "And we're going to keep going on and on and on forever like that. So how do we figure out this sum? Well now I'm going to do a little bit of mathematical trickery or gymnastics, but it's all valid. And if any of y'all have seen the proof of taking a infinite geometric series, then we're going to do a very similar technique. What I'm going to do here is I'm going to think about well what is one minus P times this expected value? So let's do that. So if I say one minus P times the expected value of X, what is that going to be equal to?"}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "And if any of y'all have seen the proof of taking a infinite geometric series, then we're going to do a very similar technique. What I'm going to do here is I'm going to think about well what is one minus P times this expected value? So let's do that. So if I say one minus P times the expected value of X, what is that going to be equal to? Well I would multiply every one of these terms by one minus P. So one P times one minus P would be one P times one minus P. You would get that right over there. What about two P times one minus P? What would that be equal to?"}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "So if I say one minus P times the expected value of X, what is that going to be equal to? Well I would multiply every one of these terms by one minus P. So one P times one minus P would be one P times one minus P. You would get that right over there. What about two P times one minus P? What would that be equal to? Well that would be two P times one minus P and now we're going to multiply it by one minus P again so you're going to get one minus P squared. And so I think you see where this is going and we're just going to keep adding and adding and adding from there. So now we're going to do something really fun and interesting at least from a mathematical point of view."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "What would that be equal to? Well that would be two P times one minus P and now we're going to multiply it by one minus P again so you're going to get one minus P squared. And so I think you see where this is going and we're just going to keep adding and adding and adding from there. So now we're going to do something really fun and interesting at least from a mathematical point of view. If this is equal to that, if the left hand side is equal to the right hand side, let's just subtract this value from both sides. So on the left hand side I would have the expected value of X, that's that, minus this. Minus one minus P times the expected value of X."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "So now we're going to do something really fun and interesting at least from a mathematical point of view. If this is equal to that, if the left hand side is equal to the right hand side, let's just subtract this value from both sides. So on the left hand side I would have the expected value of X, that's that, minus this. Minus one minus P times the expected value of X. So I'm just subtracting this from that side but let me subtract this from that side. Well I could subtract this expression from that but this is equivalent so I'm just going to subtract this from that. And so what do I get?"}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "Minus one minus P times the expected value of X. So I'm just subtracting this from that side but let me subtract this from that side. Well I could subtract this expression from that but this is equivalent so I'm just going to subtract this from that. And so what do I get? Well let's see, I'm going to have one minus P and then if I subtract one P times one minus P from two P times one minus P, well I'm just going to be left with plus one P times one minus P. And then if I subtract this from that, I'm going to be left with one P times one minus P squared. And we're just going to keep going on and on and on. And so let me simplify this a little bit."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "And so what do I get? Well let's see, I'm going to have one minus P and then if I subtract one P times one minus P from two P times one minus P, well I'm just going to be left with plus one P times one minus P. And then if I subtract this from that, I'm going to be left with one P times one minus P squared. And we're just going to keep going on and on and on. And so let me simplify this a little bit. If I distribute this negative, this could be plus and then this would be P minus one. And then if we distribute this expected value of X, we get on the left hand side, let me scroll up a little bit, I don't want to squinch it too much. So let's see, we have the expected value of X and then plus P times the expected value of X, P times the expected value of X minus the expected value of X."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "And so let me simplify this a little bit. If I distribute this negative, this could be plus and then this would be P minus one. And then if we distribute this expected value of X, we get on the left hand side, let me scroll up a little bit, I don't want to squinch it too much. So let's see, we have the expected value of X and then plus P times the expected value of X, P times the expected value of X minus the expected value of X. These cancel out. It's going to be equal to P plus P times one minus P plus P times one minus P squared. And it's going to keep going on and on and on."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "So let's see, we have the expected value of X and then plus P times the expected value of X, P times the expected value of X minus the expected value of X. These cancel out. It's going to be equal to P plus P times one minus P plus P times one minus P squared. And it's going to keep going on and on and on. Well on the left hand side, all I have is a P times the expected value of X. If I want to solve for the expected value of X, I just divide both sides by P. So I get, and this is kind of neat, through this mathematical gymnastics I now have, I'm just dividing everything by P, both sides. On the left hand side, I just have the expected value of X."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "And it's going to keep going on and on and on. Well on the left hand side, all I have is a P times the expected value of X. If I want to solve for the expected value of X, I just divide both sides by P. So I get, and this is kind of neat, through this mathematical gymnastics I now have, I'm just dividing everything by P, both sides. On the left hand side, I just have the expected value of X. If I divide all of these terms by P, this first term becomes one, the second term becomes one minus P. This third term, if I divide by P, becomes plus one minus P squared, so forth and so on. Now what's cool about this, this is a classic geometric series with a common ratio of one minus P. And if that term is completely unfamiliar to you, I encourage you, and this is why it's actually called a geometric, one of the reasons, arguments for why it's called a geometric random variable, but I encourage you to review what a geometric series is on Khan Academy if this looks completely unfamiliar. But in other places, we prove, using actually a very similar technique that we did up here, that this sum is going to be equal to one over one minus our common ratio."}, {"video_title": "Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3", "Sentence": "On the left hand side, I just have the expected value of X. If I divide all of these terms by P, this first term becomes one, the second term becomes one minus P. This third term, if I divide by P, becomes plus one minus P squared, so forth and so on. Now what's cool about this, this is a classic geometric series with a common ratio of one minus P. And if that term is completely unfamiliar to you, I encourage you, and this is why it's actually called a geometric, one of the reasons, arguments for why it's called a geometric random variable, but I encourage you to review what a geometric series is on Khan Academy if this looks completely unfamiliar. But in other places, we prove, using actually a very similar technique that we did up here, that this sum is going to be equal to one over one minus our common ratio. And our common ratio is one minus P. So what is this going to be equal to? And we are really in the home stretch right over here. This is going to be equal to one over one minus one plus P. One minus one plus P, which is indeed equal to one over P. So there you have it."}, {"video_title": "Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3", "Sentence": "He's curious about the difference of opinion between residents in the north and south parts of the city. He obtained separate random samples of voters from each region. Here are the results. So let's see, in the north, 54 out of the 120 said they want the school, 66 said they didn't. In the south, 77 said they wanted the school, 63 said they didn't. Duncan wants to use these results to construct a 90% confidence interval to estimate the difference in the proportion of residents in these regions who support the construction project P sub S minus P sub N. So these are the true parameters for the difference between these two populations. Assume that all of the conditions for inference have been met."}, {"video_title": "Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3", "Sentence": "So let's see, in the north, 54 out of the 120 said they want the school, 66 said they didn't. In the south, 77 said they wanted the school, 63 said they didn't. Duncan wants to use these results to construct a 90% confidence interval to estimate the difference in the proportion of residents in these regions who support the construction project P sub S minus P sub N. So these are the true parameters for the difference between these two populations. Assume that all of the conditions for inference have been met. All right, which of the following is a correct 90% confidence interval based on Duncan's samples? So pause this video and see if you can figure that out and you will need a calculator and depending on your calculator, you might need a Z table as well. In a previous video, we introduced the idea of a two-sample Z interval."}, {"video_title": "Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3", "Sentence": "Assume that all of the conditions for inference have been met. All right, which of the following is a correct 90% confidence interval based on Duncan's samples? So pause this video and see if you can figure that out and you will need a calculator and depending on your calculator, you might need a Z table as well. In a previous video, we introduced the idea of a two-sample Z interval. We talk about the conditions for inference. Lucky for us here, they say the conditions for inference have been met. So we can go straight to calculating the confidence interval."}, {"video_title": "Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3", "Sentence": "In a previous video, we introduced the idea of a two-sample Z interval. We talk about the conditions for inference. Lucky for us here, they say the conditions for inference have been met. So we can go straight to calculating the confidence interval. And that confidence interval is going to be the difference between the sample proportions, so P sub S hat, so the sample proportion in the south minus the sample proportion in the north. It's gonna be that difference plus or minus our critical value, Z star, times our estimate of the standard deviation of the sampling distribution of the difference between the sample proportions. And that is going to be, our estimate is going to be P hat sub S times one minus P hat sub S, all of that over the sample size in the south plus P hat sub N times one minus P hat sub N, all of that over the sample size in the north."}, {"video_title": "Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3", "Sentence": "So we can go straight to calculating the confidence interval. And that confidence interval is going to be the difference between the sample proportions, so P sub S hat, so the sample proportion in the south minus the sample proportion in the north. It's gonna be that difference plus or minus our critical value, Z star, times our estimate of the standard deviation of the sampling distribution of the difference between the sample proportions. And that is going to be, our estimate is going to be P hat sub S times one minus P hat sub S, all of that over the sample size in the south plus P hat sub N times one minus P hat sub N, all of that over the sample size in the north. Okay, so our sample proportion in the south, I'll later use a calculator to get a decimal value, but this is going to be, in the south, we have 77 out of 140 supported. So this is gonna be 77 out of 140. In the north, this is going to be 54 out of 120, 54 out of 120."}, {"video_title": "Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3", "Sentence": "And that is going to be, our estimate is going to be P hat sub S times one minus P hat sub S, all of that over the sample size in the south plus P hat sub N times one minus P hat sub N, all of that over the sample size in the north. Okay, so our sample proportion in the south, I'll later use a calculator to get a decimal value, but this is going to be, in the south, we have 77 out of 140 supported. So this is gonna be 77 out of 140. In the north, this is going to be 54 out of 120, 54 out of 120. What is my critical Z value? Well, here, I'm gonna have to either use a calculator or a Z table. Remember, we have a 90% confidence interval."}, {"video_title": "Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3", "Sentence": "In the north, this is going to be 54 out of 120, 54 out of 120. What is my critical Z value? Well, here, I'm gonna have to either use a calculator or a Z table. Remember, we have a 90% confidence interval. And so, let me see, I'll draw it right over here. If this is a normal distribution, and you wanna have a 90% confidence interval, that means you're containing 90% of the distribution, which means each of these tails, well, combined, they would have 10%, but each of them would have 5%, 5% of the distribution. And so, I'm gonna look at a Z table that figures out how many standard deviations below the mean do I need to be in order to get 5% right over here."}, {"video_title": "Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3", "Sentence": "Remember, we have a 90% confidence interval. And so, let me see, I'll draw it right over here. If this is a normal distribution, and you wanna have a 90% confidence interval, that means you're containing 90% of the distribution, which means each of these tails, well, combined, they would have 10%, but each of them would have 5%, 5% of the distribution. And so, I'm gonna look at a Z table that figures out how many standard deviations below the mean do I need to be in order to get 5% right over here. And then, that's going to tell me, well, if I'm that far below or above, that's gonna be my critical Z value. So let me get that Z table out. So I care about 5%, and I'm using this in a bit of a reverse direction, but let's see, 5%, so this is a little over 5%, getting closer to 5%, even closer to 5%, and now we've gotten right below 5%."}, {"video_title": "Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3", "Sentence": "And so, I'm gonna look at a Z table that figures out how many standard deviations below the mean do I need to be in order to get 5% right over here. And then, that's going to tell me, well, if I'm that far below or above, that's gonna be my critical Z value. So let me get that Z table out. So I care about 5%, and I'm using this in a bit of a reverse direction, but let's see, 5%, so this is a little over 5%, getting closer to 5%, even closer to 5%, and now we've gotten right below 5%. So we're gonna be in between this and this. I could just split the difference, and I could just say 1.6, let's just say 1.645 to go right in between. So this is going to be approximately equal to 1.645, and then, let's see, we know what P hat sub S is, we know what P hat sub N is."}, {"video_title": "Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3", "Sentence": "So I care about 5%, and I'm using this in a bit of a reverse direction, but let's see, 5%, so this is a little over 5%, getting closer to 5%, even closer to 5%, and now we've gotten right below 5%. So we're gonna be in between this and this. I could just split the difference, and I could just say 1.6, let's just say 1.645 to go right in between. So this is going to be approximately equal to 1.645, and then, let's see, we know what P hat sub S is, we know what P hat sub N is. In the south, our sample size is 140, and in the north, our sample size is 120. And so now, I just have to type all of this into the calculator, which is gonna get a little hairy, but we will do it together. For the sake of time, we'll accelerate this typing into the calculator, but I'm gonna start with calculating the upper bound, and then, we'll calculate the lower bound."}, {"video_title": "Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3", "Sentence": "So this is going to be approximately equal to 1.645, and then, let's see, we know what P hat sub S is, we know what P hat sub N is. In the south, our sample size is 140, and in the north, our sample size is 120. And so now, I just have to type all of this into the calculator, which is gonna get a little hairy, but we will do it together. For the sake of time, we'll accelerate this typing into the calculator, but I'm gonna start with calculating the upper bound, and then, we'll calculate the lower bound. And then, I think I've closed all my parentheses, and so I think we're ready to get the upper bound, is going to be equal to 0.218, or approximately 0.202. So we can immediately look at our choices and see where is that the upper bound. And so this one is looking pretty good, 0.202, but let's get the lower bound now."}, {"video_title": "Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3", "Sentence": "For the sake of time, we'll accelerate this typing into the calculator, but I'm gonna start with calculating the upper bound, and then, we'll calculate the lower bound. And then, I think I've closed all my parentheses, and so I think we're ready to get the upper bound, is going to be equal to 0.218, or approximately 0.202. So we can immediately look at our choices and see where is that the upper bound. And so this one is looking pretty good, 0.202, but let's get the lower bound now. So I got my calculator back, instead of retyping everything, I'm just gonna put a minus here. So I go to second, and just so you see what I'm doing, second, entry, I see the entry back, and then I can just change, I can just change the part where, right before the radical. So we are going to, all right, so this just needs to be a minus."}, {"video_title": "Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3", "Sentence": "And so this one is looking pretty good, 0.202, but let's get the lower bound now. So I got my calculator back, instead of retyping everything, I'm just gonna put a minus here. So I go to second, and just so you see what I'm doing, second, entry, I see the entry back, and then I can just change, I can just change the part where, right before the radical. So we are going to, all right, so this just needs to be a minus. Click Enter, and there you have it. Our lower bound is negative 0.002, and that is indeed this choice right over here. So there we go, we have picked our choice."}, {"video_title": "Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3", "Sentence": "Here is the summary of her results, or here is a summary of her results. And so they give the same data for both of these samples, and once again, they happen to have the same sample size. They don't need to. But over here, instead of giving us a p-value, we've gotten a confidence interval. Una wants to use these results to test her null hypothesis that the mean caloric content is the same versus her alternative hypothesis that they are different. Assume that all conditions for inference have been met. Based on the interval, what do we know about the corresponding p-value and conclusion at the alpha is equal to 0.01 level of significance?"}, {"video_title": "Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3", "Sentence": "But over here, instead of giving us a p-value, we've gotten a confidence interval. Una wants to use these results to test her null hypothesis that the mean caloric content is the same versus her alternative hypothesis that they are different. Assume that all conditions for inference have been met. Based on the interval, what do we know about the corresponding p-value and conclusion at the alpha is equal to 0.01 level of significance? So pause this video and see if you can figure that out. Remember what a 99% confidence interval is. That says that if we construct confidence intervals 100 times, that 99 of those times, we should overlap with the true parameter that we're trying to estimate."}, {"video_title": "Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3", "Sentence": "Based on the interval, what do we know about the corresponding p-value and conclusion at the alpha is equal to 0.01 level of significance? So pause this video and see if you can figure that out. Remember what a 99% confidence interval is. That says that if we construct confidence intervals 100 times, that 99 of those times, we should overlap with the true parameter that we're trying to estimate. In that case, the true parameter is the true difference of these means. Now, when we do a hypothesis test, we always start assuming that the null hypothesis is true. And so if we assume that the null hypothesis is true, well, another way of writing this null hypothesis, if the two means are equal, that's the same thing as the difference of the means equaling zero."}, {"video_title": "Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3", "Sentence": "That says that if we construct confidence intervals 100 times, that 99 of those times, we should overlap with the true parameter that we're trying to estimate. In that case, the true parameter is the true difference of these means. Now, when we do a hypothesis test, we always start assuming that the null hypothesis is true. And so if we assume that the null hypothesis is true, well, another way of writing this null hypothesis, if the two means are equal, that's the same thing as the difference of the means equaling zero. And since we're assuming this, and this is a 99% confidence interval, then 99 out of 100 times that we do this, we should see that this interval overlaps with what we're assuming is the true parameter right over here. Now, this interval does indeed overlap with zero. If you take four minus 6.44, you're going to get negative 2.44."}, {"video_title": "Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3", "Sentence": "And so if we assume that the null hypothesis is true, well, another way of writing this null hypothesis, if the two means are equal, that's the same thing as the difference of the means equaling zero. And since we're assuming this, and this is a 99% confidence interval, then 99 out of 100 times that we do this, we should see that this interval overlaps with what we're assuming is the true parameter right over here. Now, this interval does indeed overlap with zero. If you take four minus 6.44, you're going to get negative 2.44. So zero is definitely in the interval. And so another way to think about it, we're not in the 1% of the times where we don't overlap. If we were in the 1% of times where we don't overlap with the assumed difference, then we would reject the null hypothesis."}, {"video_title": "Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3", "Sentence": "If you take four minus 6.44, you're going to get negative 2.44. So zero is definitely in the interval. And so another way to think about it, we're not in the 1% of the times where we don't overlap. If we were in the 1% of times where we don't overlap with the assumed difference, then we would reject the null hypothesis. Or another way to think about it is our significance level, 0.01 right over here, it's one minus our confidence level right over here. If our 99% confidence interval overlaps, overlaps with mu from the Bosch pairs from the Bosch pairs minus the mean content of the Anju pairs equaling zero, then that means that the p-value is greater than 0.01. And so we could also say that our p-value is greater than our significance level because that is our significance level."}, {"video_title": "Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3", "Sentence": "If we were in the 1% of times where we don't overlap with the assumed difference, then we would reject the null hypothesis. Or another way to think about it is our significance level, 0.01 right over here, it's one minus our confidence level right over here. If our 99% confidence interval overlaps, overlaps with mu from the Bosch pairs from the Bosch pairs minus the mean content of the Anju pairs equaling zero, then that means that the p-value is greater than 0.01. And so we could also say that our p-value is greater than our significance level because that is our significance level. And because of that, we fail to reject our null hypothesis. If this did not overlap with our assumed difference in the means, if it did not overlap with zero, then we would be in that one in 100 scenario. And then that would tell us that, hey, our p-value is less than 0.01."}, {"video_title": "Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3", "Sentence": "And so we could also say that our p-value is greater than our significance level because that is our significance level. And because of that, we fail to reject our null hypothesis. If this did not overlap with our assumed difference in the means, if it did not overlap with zero, then we would be in that one in 100 scenario. And then that would tell us that, hey, our p-value is less than 0.01. Our p-value is less than one minus our confidence level. And in that case, we would reject the null hypothesis and it would suggest that there is a difference in caloric content. But because we fail to reject it, we can't conclude that there's a difference in caloric contents."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "If we only have a few experiments, it's very possible that our experimental probability could be different than our theoretical probability or even very different. But as we have many, many more experiments, thousands, millions, billions of experiments, the probability that the experimental and the theoretical probabilities are very different goes down dramatically. But let's get an intuitive sense for it. This right over here is a simulation created by Macmillan USA. I'll provide the link as an annotation. And what it does is it allows us to simulate many coin flips and figure out the proportion that are heads. So right over here, we can decide if we want our coin to be fair or not."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "This right over here is a simulation created by Macmillan USA. I'll provide the link as an annotation. And what it does is it allows us to simulate many coin flips and figure out the proportion that are heads. So right over here, we can decide if we want our coin to be fair or not. Right now it says that we have a 50% probability of getting heads. We can make it unfair by changing this, but I'll stick with the 50% probability. We want to show that on this graph here."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "So right over here, we can decide if we want our coin to be fair or not. Right now it says that we have a 50% probability of getting heads. We can make it unfair by changing this, but I'll stick with the 50% probability. We want to show that on this graph here. We can plot it. And what this says is at a time, how many tosses do we want to take? So let's say, let's just start with 10 tosses."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "We want to show that on this graph here. We can plot it. And what this says is at a time, how many tosses do we want to take? So let's say, let's just start with 10 tosses. So what this is going to do is take 10 simulated flips of coins with each one having a 50% chance of being heads. And then as we flip, we're gonna see our total proportion that are heads. So let's just talk through this together."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "So let's say, let's just start with 10 tosses. So what this is going to do is take 10 simulated flips of coins with each one having a 50% chance of being heads. And then as we flip, we're gonna see our total proportion that are heads. So let's just talk through this together. So starting to toss. And so what's going on here after 10 flips? So as you see, the first flip actually came out heads."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "So let's just talk through this together. So starting to toss. And so what's going on here after 10 flips? So as you see, the first flip actually came out heads. And if you wanted to say, what's your experimental probability after that one flip? You'd say, well, with only one experiment, I got one head. So it looks like 100% were heads."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "So as you see, the first flip actually came out heads. And if you wanted to say, what's your experimental probability after that one flip? You'd say, well, with only one experiment, I got one head. So it looks like 100% were heads. But in the second flip, it looks like it was a tails because now the proportion that was heads after two flips was 50%. But in the third flip, it looks like it was tails again because now only one out of three, or 33% of the flips have resulted in heads. Now by the fourth flip, we got a heads again, getting us back to 50th percentile."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "So it looks like 100% were heads. But in the second flip, it looks like it was a tails because now the proportion that was heads after two flips was 50%. But in the third flip, it looks like it was tails again because now only one out of three, or 33% of the flips have resulted in heads. Now by the fourth flip, we got a heads again, getting us back to 50th percentile. Now at the fifth flip, it looks like we got another heads. And so now we have three out of five, or 60% being heads. And so the general takeaway here is when you have one, two, three, four, five, or six experiments, it's completely plausible that your experimental proportion, your experimental probability, diverges from the real probability."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "Now by the fourth flip, we got a heads again, getting us back to 50th percentile. Now at the fifth flip, it looks like we got another heads. And so now we have three out of five, or 60% being heads. And so the general takeaway here is when you have one, two, three, four, five, or six experiments, it's completely plausible that your experimental proportion, your experimental probability, diverges from the real probability. And this even continues all the way until we get to our ninth or 10th tosses. But what happens if we do way more tosses? So now I'm gonna do another, well, let's just do another 200 tosses and see what happens."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "And so the general takeaway here is when you have one, two, three, four, five, or six experiments, it's completely plausible that your experimental proportion, your experimental probability, diverges from the real probability. And this even continues all the way until we get to our ninth or 10th tosses. But what happens if we do way more tosses? So now I'm gonna do another, well, let's just do another 200 tosses and see what happens. So I'm just gonna keep tossing here. And you can see, wow, look at this. There was a big run of getting a lot of heads right over here."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "So now I'm gonna do another, well, let's just do another 200 tosses and see what happens. So I'm just gonna keep tossing here. And you can see, wow, look at this. There was a big run of getting a lot of heads right over here. And then it looks like there's actually a run of getting a bunch of tails right over here, and then a little run of heads, tails, and then another run of heads. And notice, even after 215 tosses, our experimental probability is still reasonably different than our theoretical probability. So let's do another 200 and see if we can converge these over time."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "There was a big run of getting a lot of heads right over here. And then it looks like there's actually a run of getting a bunch of tails right over here, and then a little run of heads, tails, and then another run of heads. And notice, even after 215 tosses, our experimental probability is still reasonably different than our theoretical probability. So let's do another 200 and see if we can converge these over time. And what we're seeing in real time here should be the law of large numbers. As our number of tosses get larger and larger and larger, the probability that these two are very different goes down and down and down. Yes, you will get moments where you could even get 10 heads in a row or even 20 heads in a row, but over time, those will be balanced by the times where you're getting disproportionate number of tails."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "So let's do another 200 and see if we can converge these over time. And what we're seeing in real time here should be the law of large numbers. As our number of tosses get larger and larger and larger, the probability that these two are very different goes down and down and down. Yes, you will get moments where you could even get 10 heads in a row or even 20 heads in a row, but over time, those will be balanced by the times where you're getting disproportionate number of tails. So I'm just gonna keep going. We're now at almost 800 tosses, and you see now we are converging. Now this is, we're gonna cross 1,000 tosses soon."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "Yes, you will get moments where you could even get 10 heads in a row or even 20 heads in a row, but over time, those will be balanced by the times where you're getting disproportionate number of tails. So I'm just gonna keep going. We're now at almost 800 tosses, and you see now we are converging. Now this is, we're gonna cross 1,000 tosses soon. And you can see that our proportion here is now 51%. It's getting close now. We're at 50.6%, and I could just keep tossing."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "Now this is, we're gonna cross 1,000 tosses soon. And you can see that our proportion here is now 51%. It's getting close now. We're at 50.6%, and I could just keep tossing. This is 1,100. We're gonna approach 1,200 or 1,300 flips right over here. But as you can see, as we get many, many, many more flips, it was actually valuable to see even after 200 flips that there was a difference in the proportion between what we got from the experiment and what you would theoretically expect."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "We're at 50.6%, and I could just keep tossing. This is 1,100. We're gonna approach 1,200 or 1,300 flips right over here. But as you can see, as we get many, many, many more flips, it was actually valuable to see even after 200 flips that there was a difference in the proportion between what we got from the experiment and what you would theoretically expect. But as we get to many, many more flips, now we're at 1,210, we're getting pretty close to 50% of them turning out heads, but we could keep tossing it more and more and more. And what we'll see is as we get larger and larger and larger, it is likely that we're gonna get closer and closer and closer to 50%. It's not to say that it's impossible that we diverge again, but the likelihood of diverging gets lower and lower and lower the more tosses, the more experiments you make."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So just as a bit of review, a lot of what we do in confidence intervals is we are trying to estimate some population parameter. Let's say it's the proportion. Maybe it's the proportion that will vote for a candidate. We can't survey everyone, so we take a sample. And from that sample, maybe we calculate a sample proportion. And then using this sample proportion, we calculate a confidence interval on either side of that sample proportion. And what we know is that if we do this many, many, many times, every time we do it, we are very likely to have a different sample proportion."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "We can't survey everyone, so we take a sample. And from that sample, maybe we calculate a sample proportion. And then using this sample proportion, we calculate a confidence interval on either side of that sample proportion. And what we know is that if we do this many, many, many times, every time we do it, we are very likely to have a different sample proportion. So that'd be sample proportion one, sample proportion two. And every time we do it, we might get, maybe this is sample proportion two. Not only will we get a different, I guess you could say center of our interval, but the margin of error might change because we are using the sample proportion to calculate it."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And what we know is that if we do this many, many, many times, every time we do it, we are very likely to have a different sample proportion. So that'd be sample proportion one, sample proportion two. And every time we do it, we might get, maybe this is sample proportion two. Not only will we get a different, I guess you could say center of our interval, but the margin of error might change because we are using the sample proportion to calculate it. But the first assumption that has to be true and even to make any claims about this confidence interval with confidence is that your sample is random, so that you have a random sample. If you're trying to estimate the proportion of people that are gonna vote for a certain candidate, but you are only surveying people at a senior community, well, that would not be a truly random sample if we were only to survey people on a college campus. So like with all things with statistics, you really wanna make sure that you're dealing with a random sample and take great care to do that."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Not only will we get a different, I guess you could say center of our interval, but the margin of error might change because we are using the sample proportion to calculate it. But the first assumption that has to be true and even to make any claims about this confidence interval with confidence is that your sample is random, so that you have a random sample. If you're trying to estimate the proportion of people that are gonna vote for a certain candidate, but you are only surveying people at a senior community, well, that would not be a truly random sample if we were only to survey people on a college campus. So like with all things with statistics, you really wanna make sure that you're dealing with a random sample and take great care to do that. The second thing that we have to assume, and this is sometimes known as the normal condition, normal condition. Remember, the whole basis behind confidence intervals is we assume that the distribution of the sample proportions the sampling distribution of the sample proportions has roughly a normal shape like that. But in order to make that assumption that it's roughly normal, we have this normal condition."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So like with all things with statistics, you really wanna make sure that you're dealing with a random sample and take great care to do that. The second thing that we have to assume, and this is sometimes known as the normal condition, normal condition. Remember, the whole basis behind confidence intervals is we assume that the distribution of the sample proportions the sampling distribution of the sample proportions has roughly a normal shape like that. But in order to make that assumption that it's roughly normal, we have this normal condition. And the rule of thumb here is that you would expect per sample more than 10 successes, successes and, successes and failures each, each. So for example, if your sample size was only 10, let's say the true proportion was 50%, or 0.5, then you wouldn't meet that normal condition because you would expect five successes and five failures for each sample. Now, because usually when we're doing confidence intervals, we don't even know the true population parameter, what we would actually just do is look at our sample and just count how many successes and how many failures we have."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "But in order to make that assumption that it's roughly normal, we have this normal condition. And the rule of thumb here is that you would expect per sample more than 10 successes, successes and, successes and failures each, each. So for example, if your sample size was only 10, let's say the true proportion was 50%, or 0.5, then you wouldn't meet that normal condition because you would expect five successes and five failures for each sample. Now, because usually when we're doing confidence intervals, we don't even know the true population parameter, what we would actually just do is look at our sample and just count how many successes and how many failures we have. And if we have less than 10 on either one of those, then we are going to have a problem. So you wanna expect, you wanna have at least greater than or equal to 10 successes or failures on each. And you actually don't even have to say expect because you're going to get a sample and you could just count how many successes and failures you have."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Now, because usually when we're doing confidence intervals, we don't even know the true population parameter, what we would actually just do is look at our sample and just count how many successes and how many failures we have. And if we have less than 10 on either one of those, then we are going to have a problem. So you wanna expect, you wanna have at least greater than or equal to 10 successes or failures on each. And you actually don't even have to say expect because you're going to get a sample and you could just count how many successes and failures you have. If you don't see that, then the normal condition is not met and the statements you make about your confidence interval aren't necessarily going to be as valid. The last thing we wanna really make sure is known as the independence condition. Independence condition."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And you actually don't even have to say expect because you're going to get a sample and you could just count how many successes and failures you have. If you don't see that, then the normal condition is not met and the statements you make about your confidence interval aren't necessarily going to be as valid. The last thing we wanna really make sure is known as the independence condition. Independence condition. And this is the 10% rule. If we are sampling without replacement, and sometimes it's hard to do replacement. If you're serving people who are exiting a store, for example, you can't ask them to go back into the store or it might be very awkward to ask them to go back into the store."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Independence condition. And this is the 10% rule. If we are sampling without replacement, and sometimes it's hard to do replacement. If you're serving people who are exiting a store, for example, you can't ask them to go back into the store or it might be very awkward to ask them to go back into the store. And so the independence condition is that your sample size, so sample, let me just say n, n is less than 10% of the population size. And so let's say your population were 100,000 people. And if you surveyed 1,000 people, well, that was 1% of the population."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "If you're serving people who are exiting a store, for example, you can't ask them to go back into the store or it might be very awkward to ask them to go back into the store. And so the independence condition is that your sample size, so sample, let me just say n, n is less than 10% of the population size. And so let's say your population were 100,000 people. And if you surveyed 1,000 people, well, that was 1% of the population. So you'd feel pretty good that the independence condition is met. And once again, this is valuable when you are sampling without replacement. Now to appreciate how our confidence intervals don't do what we think they're going to do when any of these things are broken, and I'll focus on these latter two, the random sample condition."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And if you surveyed 1,000 people, well, that was 1% of the population. So you'd feel pretty good that the independence condition is met. And once again, this is valuable when you are sampling without replacement. Now to appreciate how our confidence intervals don't do what we think they're going to do when any of these things are broken, and I'll focus on these latter two, the random sample condition. That's super important, frankly, in all of statistics. So let's first look at a situation where our independence condition breaks down. So right over here, you can see that we are using our little gumball simulation."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Now to appreciate how our confidence intervals don't do what we think they're going to do when any of these things are broken, and I'll focus on these latter two, the random sample condition. That's super important, frankly, in all of statistics. So let's first look at a situation where our independence condition breaks down. So right over here, you can see that we are using our little gumball simulation. And in that gumball simulation, we have a true population proportion, but someone doing these samples might not know that. We're trying to construct confidence interval with a 95% confidence level. And what we've set up here is we aren't replacing."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So right over here, you can see that we are using our little gumball simulation. And in that gumball simulation, we have a true population proportion, but someone doing these samples might not know that. We're trying to construct confidence interval with a 95% confidence level. And what we've set up here is we aren't replacing. So every member of our sample, we're not looking at it and then putting it back in. We're just gonna take a sample of 200. And I've set up the population so that it's a far larger than 10% of the population."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And what we've set up here is we aren't replacing. So every member of our sample, we're not looking at it and then putting it back in. We're just gonna take a sample of 200. And I've set up the population so that it's a far larger than 10% of the population. And then when I drew a bunch of samples, so this is a situation where I did almost 1,500 samples here of size 200. What you can see here is the situations where our true population parameter was contained in the confidence interval that we calculated for that sample. And then you see in red the ones where it's not."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And I've set up the population so that it's a far larger than 10% of the population. And then when I drew a bunch of samples, so this is a situation where I did almost 1,500 samples here of size 200. What you can see here is the situations where our true population parameter was contained in the confidence interval that we calculated for that sample. And then you see in red the ones where it's not. And as you can see, we are only having a hit, so to speak. The overlap between the confidence interval that we're calculating in the true population parameter is happening about 93% of the time. And this is a pretty large number of samples."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And then you see in red the ones where it's not. And as you can see, we are only having a hit, so to speak. The overlap between the confidence interval that we're calculating in the true population parameter is happening about 93% of the time. And this is a pretty large number of samples. This is truly at a 95% confidence level. This should be happening 95% of the time. Similarly, we can look at a situation where our normal condition breaks down."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And this is a pretty large number of samples. This is truly at a 95% confidence level. This should be happening 95% of the time. Similarly, we can look at a situation where our normal condition breaks down. And our normal condition, we can see here that our sample size right here is 15. And actually, if I scroll down a little bit, you can see that the simulation even warns me. There are fewer than 10 expected successes."}, {"video_title": "Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "Similarly, we can look at a situation where our normal condition breaks down. And our normal condition, we can see here that our sample size right here is 15. And actually, if I scroll down a little bit, you can see that the simulation even warns me. There are fewer than 10 expected successes. And you can see that when I do, once again, I did a bunch of samples here. I did over 2,000 samples. Even though I'm trying to set up these confidence intervals that every time I compute it, that over time, that there's kind of a 95% hit rate, so to speak, here there's only a 94% hit rate."}, {"video_title": "Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "We're told that administrators at a school are considering a policy change. They survey a group of students, staff members, and parents about whether or not they agree with the new policy. The following mosaic plot summarizes their results. Which of the following statements can we justify from the mosaic plot? So pause this video and try to figure this out on your own. Pick which of these statements can be justified, and there could be more than one based on this mosaic plot. All right, now let's work through this together."}, {"video_title": "Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "Which of the following statements can we justify from the mosaic plot? So pause this video and try to figure this out on your own. Pick which of these statements can be justified, and there could be more than one based on this mosaic plot. All right, now let's work through this together. So before I even look at the choices, let me see if I can interpret this. So this mosaic plot, what it does above and beyond a segmented bar chart is it gives us the width that shows us how many, for example, students versus staff versus parents were sampled or surveyed. And it looks like more than half of the people surveyed were students, and then staff and parents seem similar."}, {"video_title": "Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "All right, now let's work through this together. So before I even look at the choices, let me see if I can interpret this. So this mosaic plot, what it does above and beyond a segmented bar chart is it gives us the width that shows us how many, for example, students versus staff versus parents were sampled or surveyed. And it looks like more than half of the people surveyed were students, and then staff and parents seem similar. In terms of who is agreeing with the policy, so that's that light blue color, it seems like students are not very likely to agree with the policy. It looks like staff is very likely to agree with the policy, that the bulk of staff is agreeing with it, and parents are kind of on both sides. So let's see which of these statements are consistent with what we just looked at."}, {"video_title": "Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "And it looks like more than half of the people surveyed were students, and then staff and parents seem similar. In terms of who is agreeing with the policy, so that's that light blue color, it seems like students are not very likely to agree with the policy. It looks like staff is very likely to agree with the policy, that the bulk of staff is agreeing with it, and parents are kind of on both sides. So let's see which of these statements are consistent with what we just looked at. Parents were the least likely to agree with the new policy. No, that's not true. The least likely to agree with the new policy, that's students right over here."}, {"video_title": "Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "So let's see which of these statements are consistent with what we just looked at. Parents were the least likely to agree with the new policy. No, that's not true. The least likely to agree with the new policy, that's students right over here. They were definitely the least likely. The lowest percentage of students are agreeing with the policy, so I don't like that choice. More than half of the total responses came from students."}, {"video_title": "Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "The least likely to agree with the new policy, that's students right over here. They were definitely the least likely. The lowest percentage of students are agreeing with the policy, so I don't like that choice. More than half of the total responses came from students. And that does look like the case, because if you view this entire width as the total responses, it looks like the student width right over here, that is more than half of it. It looks like it's about 50-something percent or even 60%. So I like this choice right over here."}, {"video_title": "Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "More than half of the total responses came from students. And that does look like the case, because if you view this entire width as the total responses, it looks like the student width right over here, that is more than half of it. It looks like it's about 50-something percent or even 60%. So I like this choice right over here. And then last but not least, there were more total no responses from students than from staff and parents combined. So let's look at the no responses from students. No, that's that darker blue color."}, {"video_title": "Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "So I like this choice right over here. And then last but not least, there were more total no responses from students than from staff and parents combined. So let's look at the no responses from students. No, that's that darker blue color. The no responses for students is this area right over here. And then the no, so I can, maybe I'll shade all of that in. And then the no responses from staff and parents combined, that is this area right over here."}, {"video_title": "Significance test for a proportion free response (part 2 with correction) Khan Academy (2).mp3", "Sentence": "In a previous video, I had worked through this AP Statistics example problem right over here, and then I had it looked over by folks who are familiar with how it is graded, including some people who had graded the AP Statistics exam themselves. And they pointed out a few problems with how I actually wrote things down. And so I thought in this video, I would correct those problems. And instead of just redoing the whole example again, the way that would get maximum points on the AP test, I thought it would be even more instructive to show where I went wrong. So the math in this problem was not incorrect, but if I were actually taking the AP Statistics exam, I was told that, say, for the conditions for inference, where we have this random condition here, in the example, I just pointed to the part of the problem where they tell us that we are dealing with a random sample. Assume that the 65 boxes purchased by the students are random samples. I am told that the AP graders do not like this."}, {"video_title": "Significance test for a proportion free response (part 2 with correction) Khan Academy (2).mp3", "Sentence": "And instead of just redoing the whole example again, the way that would get maximum points on the AP test, I thought it would be even more instructive to show where I went wrong. So the math in this problem was not incorrect, but if I were actually taking the AP Statistics exam, I was told that, say, for the conditions for inference, where we have this random condition here, in the example, I just pointed to the part of the problem where they tell us that we are dealing with a random sample. Assume that the 65 boxes purchased by the students are random samples. I am told that the AP graders do not like this. They do not want you to just point to a part of the passage that says that it's a random sample. Instead, what you could say, and they are functionally equivalent, but you wanna make sure that you're doing what people are looking for. Instead, what we could do is, it is stated, over here we could write, stated in problem, problem, that random sample that we're dealing with, that random sample taken."}, {"video_title": "Significance test for a proportion free response (part 2 with correction) Khan Academy (2).mp3", "Sentence": "I am told that the AP graders do not like this. They do not want you to just point to a part of the passage that says that it's a random sample. Instead, what you could say, and they are functionally equivalent, but you wanna make sure that you're doing what people are looking for. Instead, what we could do is, it is stated, over here we could write, stated in problem, problem, that random sample that we're dealing with, that random sample taken. And that would be sufficient, instead of drawing the arrow to that part over there. Then I could check it off. Now, the other thing that I didn't do, I talked about it in the previous video, but I didn't write it down."}, {"video_title": "Significance test for a proportion free response (part 2 with correction) Khan Academy (2).mp3", "Sentence": "Instead, what we could do is, it is stated, over here we could write, stated in problem, problem, that random sample that we're dealing with, that random sample taken. And that would be sufficient, instead of drawing the arrow to that part over there. Then I could check it off. Now, the other thing that I didn't do, I talked about it in the previous video, but I didn't write it down. And I really did wanna model what you would need to do on the actual AP exam is, I came, I said that we failed to reject the null hypothesis, and therefore, there's not enough evidence to suggest the alternative hypothesis. But if you're actually taking the AP exam, you wanna go one step further. You wanna really talk about the conclusion you're making."}, {"video_title": "Significance test for a proportion free response (part 2 with correction) Khan Academy (2).mp3", "Sentence": "Now, the other thing that I didn't do, I talked about it in the previous video, but I didn't write it down. And I really did wanna model what you would need to do on the actual AP exam is, I came, I said that we failed to reject the null hypothesis, and therefore, there's not enough evidence to suggest the alternative hypothesis. But if you're actually taking the AP exam, you wanna go one step further. You wanna really talk about the conclusion you're making. Because they asked this question, based on this sample, is there support for the student's belief that the proportion of boxes with vouchers is less than 0.2? And so now, I can just draw it back to that question. There is not, there is not support, support for student belief, student belief, student belief, that the proportion, proportion of boxes with vouchers, with vouchers, is less than, less than, is less than 0.2."}, {"video_title": "Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3", "Sentence": "The Olympic running team of Freedonia has always used Zeppo's running shoes, but their manager suspects Harpo's shoes can produce better results, which would be lower times. The manager has six runners each run two laps, one lap wearing Zeppo's and another lap wearing Harpo's. Each runner flips a coin to determine which shoes they wear first. The manager wants to test if their times when wearing Harpo's are significantly lower than their times when they wear Zeppo's. Assume that all conditions for inference were met. Which of these is the most appropriate test, an alternative hypothesis? So they're just asking us about the alternative, not even the null, but we can talk about that."}, {"video_title": "Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3", "Sentence": "The manager wants to test if their times when wearing Harpo's are significantly lower than their times when they wear Zeppo's. Assume that all conditions for inference were met. Which of these is the most appropriate test, an alternative hypothesis? So they're just asking us about the alternative, not even the null, but we can talk about that. So pause this video and see if you can figure it out. So before I even go into this particular example, let's just make sure we understand the difference between a two-sample t-test and a paired t-test. So when we're talking about either a two-sample t-test or a two-sample t-interval for the difference between the means, what we're doing is we're considering two populations, and you take two independent samples from those populations."}, {"video_title": "Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3", "Sentence": "So they're just asking us about the alternative, not even the null, but we can talk about that. So pause this video and see if you can figure it out. So before I even go into this particular example, let's just make sure we understand the difference between a two-sample t-test and a paired t-test. So when we're talking about either a two-sample t-test or a two-sample t-interval for the difference between the means, what we're doing is we're considering two populations, and you take two independent samples from those populations. And what you're trying to do is you get statistics off of these samples, and you're trying to estimate the difference between the means of these populations. So it might be the difference mu one minus mu two. That's what you're trying to figure out, mu one minus mu two."}, {"video_title": "Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3", "Sentence": "So when we're talking about either a two-sample t-test or a two-sample t-interval for the difference between the means, what we're doing is we're considering two populations, and you take two independent samples from those populations. And what you're trying to do is you get statistics off of these samples, and you're trying to estimate the difference between the means of these populations. So it might be the difference mu one minus mu two. That's what you're trying to figure out, mu one minus mu two. A paired situation is quite different, even though they might sound the same at first. Here we're looking at just one population. And that's exactly what's happening in this situation right over here."}, {"video_title": "Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3", "Sentence": "That's what you're trying to figure out, mu one minus mu two. A paired situation is quite different, even though they might sound the same at first. Here we're looking at just one population. And that's exactly what's happening in this situation right over here. We're trying to figure out what is the mean difference between using Zeppo's and Harpo's shoes. So this is what we're trying to get at. The mean difference, so we could call that mu sub Zeppo's minus Harpo's."}, {"video_title": "Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3", "Sentence": "And that's exactly what's happening in this situation right over here. We're trying to figure out what is the mean difference between using Zeppo's and Harpo's shoes. So this is what we're trying to get at. The mean difference, so we could call that mu sub Zeppo's minus Harpo's. And the way that we go about doing that is we take a sample, and then for each subject in the sample we perform two measurements, one where they run with the Zeppo's and one when they run with the Harpo's. And then for that sample, you can calculate a mean difference between the Zeppo's and the Harpo's. Zeppo's and the Harpo's."}, {"video_title": "Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3", "Sentence": "The mean difference, so we could call that mu sub Zeppo's minus Harpo's. And the way that we go about doing that is we take a sample, and then for each subject in the sample we perform two measurements, one where they run with the Zeppo's and one when they run with the Harpo's. And then for that sample, you can calculate a mean difference between the Zeppo's and the Harpo's. Zeppo's and the Harpo's. You're gonna calculate this difference for each member of your sample, and then you're gonna take the mean of all of those. So hopefully you notice that this is quite different. And so as you can imagine, here in this example we are dealing with a paired t-test."}, {"video_title": "Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3", "Sentence": "Zeppo's and the Harpo's. You're gonna calculate this difference for each member of your sample, and then you're gonna take the mean of all of those. So hopefully you notice that this is quite different. And so as you can imagine, here in this example we are dealing with a paired t-test. We aren't looking at two independent groups or two independent samples like you would with a two-sample t-test. And so we are in a paired t-test, and the manager wants to test if their times when wearing Harpo's are significantly lower than their times when wearing Zeppo's. So our null hypothesis, even though that they're not asking that, our null hypothesis would be that there's no difference, that the mean time, that the mean difference, that the mean difference between wearing Zeppo's, Zeppo's and Harpo's, and Harpo's, would be equal to zero."}, {"video_title": "Example of hypotheses for paired and two-sample t tests AP Statistics Khan Academy.mp3", "Sentence": "And so as you can imagine, here in this example we are dealing with a paired t-test. We aren't looking at two independent groups or two independent samples like you would with a two-sample t-test. And so we are in a paired t-test, and the manager wants to test if their times when wearing Harpo's are significantly lower than their times when wearing Zeppo's. So our null hypothesis, even though that they're not asking that, our null hypothesis would be that there's no difference, that the mean time, that the mean difference, that the mean difference between wearing Zeppo's, Zeppo's and Harpo's, and Harpo's, would be equal to zero. And the alternative hypothesis, so if they were just saying, hey, is there a difference, then we would say that this would not be equal to zero in the alternative. But the manager explicitly wants to see if Harpo's times are lower than Zeppo's times. So what we would wanna see is if the mean difference, so the mean difference of Zeppo's, Zeppo's minus Harpo's, we're trying to find if we can see if we can have evidence to suggest that this is actually greater than zero."}, {"video_title": "Comparing P value to significance level for test involving difference of proportions Khan Academy.mp3", "Sentence": "They obtain a random sample of records from 500 cats. They find 24 of the 259 male cats have the disease, while 14 of 241 female cats have the disease. The veterinarian uses these results to test their null hypothesis that the true proportion is the same amongst the male and female cats versus the alternative hypothesis that the proportion of males who get the disease is actually higher than the proportion of females. The test statistic was Z is equal to 1.46 and the P value was approximately 0.07. That's useful, they've done a lot of work for us. Assume that all conditions for inference were met. At the alpha equals 0.10 level of significance, is there sufficient evidence to conclude that a larger proportion of male cats have the disease?"}, {"video_title": "Comparing P value to significance level for test involving difference of proportions Khan Academy.mp3", "Sentence": "The test statistic was Z is equal to 1.46 and the P value was approximately 0.07. That's useful, they've done a lot of work for us. Assume that all conditions for inference were met. At the alpha equals 0.10 level of significance, is there sufficient evidence to conclude that a larger proportion of male cats have the disease? Pause this video and see if you can answer that. All right, remember what we do with the significance test. We assume, we assume our null hypothesis and then with that assumption, we look at our data and we calculate a P value."}, {"video_title": "Comparing P value to significance level for test involving difference of proportions Khan Academy.mp3", "Sentence": "At the alpha equals 0.10 level of significance, is there sufficient evidence to conclude that a larger proportion of male cats have the disease? Pause this video and see if you can answer that. All right, remember what we do with the significance test. We assume, we assume our null hypothesis and then with that assumption, we look at our data and we calculate a P value. We say, hey, what is the probability of getting the data that we did assuming our null hypothesis is true? And if that is lower than our significance level, then we reject the null hypothesis because we say, hey, assuming the null hypothesis, we got a pretty unlikely event, we reject it, which would suggest the alternative. And so here, our P value is 0.07, which is indeed less than our alpha, which is less than our significance level."}, {"video_title": "Comparing P value to significance level for test involving difference of proportions Khan Academy.mp3", "Sentence": "We assume, we assume our null hypothesis and then with that assumption, we look at our data and we calculate a P value. We say, hey, what is the probability of getting the data that we did assuming our null hypothesis is true? And if that is lower than our significance level, then we reject the null hypothesis because we say, hey, assuming the null hypothesis, we got a pretty unlikely event, we reject it, which would suggest the alternative. And so here, our P value is 0.07, which is indeed less than our alpha, which is less than our significance level. This thing right over here is 0.10. And so because of that, we would reject the null hypothesis, reject null hypothesis, which is, we could say there is sufficient evidence to conclude that a larger proportion of male cats have the disease. So we could say suggests the alternative hypothesis."}, {"video_title": "Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3", "Sentence": "The graphs below show the test grades of the students in Dexter's class. The first graph shows the relationship between test grades and the amount of time the student spent studying. So this is study time on this axis, and this is the test grade on this axis. And the second graph shows the relationship between test grades and shoe size. So shoe size on this axis, and then test grade. Choose the best description of the relationship between the graphs. So first, before even looking at these, let's just look at these."}, {"video_title": "Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3", "Sentence": "And the second graph shows the relationship between test grades and shoe size. So shoe size on this axis, and then test grade. Choose the best description of the relationship between the graphs. So first, before even looking at these, let's just look at these. Before looking at the explanations, let's look at the actual graphs. So this one on the left right over here, it looks like there is a positive linear relationship right over here. I could almost fit a line that would go just like that."}, {"video_title": "Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3", "Sentence": "So first, before even looking at these, let's just look at these. Before looking at the explanations, let's look at the actual graphs. So this one on the left right over here, it looks like there is a positive linear relationship right over here. I could almost fit a line that would go just like that. And it makes sense that there would be. That the more time that you spend studying, that the better score that you would get. Now, for a certain amount of time studying, some people might do better than others, but it does seem like there's this relationship."}, {"video_title": "Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3", "Sentence": "I could almost fit a line that would go just like that. And it makes sense that there would be. That the more time that you spend studying, that the better score that you would get. Now, for a certain amount of time studying, some people might do better than others, but it does seem like there's this relationship. Here, it doesn't seem like there's really much of a relationship. You see the shoe sizes. For a given shoe size, some people do not so well, and some people do very well."}, {"video_title": "Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3", "Sentence": "Now, for a certain amount of time studying, some people might do better than others, but it does seem like there's this relationship. Here, it doesn't seem like there's really much of a relationship. You see the shoe sizes. For a given shoe size, some people do not so well, and some people do very well. For some people, someone with a size 10 and 1 half, it looks like. Yeah, a size 10 and 1 half, someone, it looks like they flunked the exam. Someone else looks like they got an A minus or a B plus on the exam."}, {"video_title": "Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3", "Sentence": "For a given shoe size, some people do not so well, and some people do very well. For some people, someone with a size 10 and 1 half, it looks like. Yeah, a size 10 and 1 half, someone, it looks like they flunked the exam. Someone else looks like they got an A minus or a B plus on the exam. And it really would be hard to somehow fit a line here. No matter how you draw a line, it doesn't seem like it would really fit some type of a, these dots don't seem to form a trend. So let's see which of these choices apply."}, {"video_title": "Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3", "Sentence": "Someone else looks like they got an A minus or a B plus on the exam. And it really would be hard to somehow fit a line here. No matter how you draw a line, it doesn't seem like it would really fit some type of a, these dots don't seem to form a trend. So let's see which of these choices apply. There's a negative linear relationship between study time and score. No, that's not true. It looks like there's a positive linear relationship."}, {"video_title": "Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3", "Sentence": "So let's see which of these choices apply. There's a negative linear relationship between study time and score. No, that's not true. It looks like there's a positive linear relationship. The more you study, the better your score would be. A negative linear relationship would trend downwards like that. There is a nonlinear relationship between study time and score, and a negative linear relationship between shoe size and score."}, {"video_title": "Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3", "Sentence": "It looks like there's a positive linear relationship. The more you study, the better your score would be. A negative linear relationship would trend downwards like that. There is a nonlinear relationship between study time and score, and a negative linear relationship between shoe size and score. Well, that doesn't seem right either. A nonlinear relationship, it would not be easy to fit a line to it. And this one seems like a line would be very reasonable."}, {"video_title": "Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3", "Sentence": "There is a nonlinear relationship between study time and score, and a negative linear relationship between shoe size and score. Well, that doesn't seem right either. A nonlinear relationship, it would not be easy to fit a line to it. And this one seems like a line would be very reasonable. And there isn't any type of, it doesn't seem like there's any type of relationship between shoe size and score. So I wouldn't pick this one either. There is a positive linear relationship between study time and score."}, {"video_title": "Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3", "Sentence": "And this one seems like a line would be very reasonable. And there isn't any type of, it doesn't seem like there's any type of relationship between shoe size and score. So I wouldn't pick this one either. There is a positive linear relationship between study time and score. That's right. And no relationship between shoe size and score. Well, I'm going to go with that one."}, {"video_title": "Studying, shoe size, and test scores scatter plots Probability and Statistics Khan Academy.mp3", "Sentence": "There is a positive linear relationship between study time and score. That's right. And no relationship between shoe size and score. Well, I'm going to go with that one. Both graphs show positive linear trends of approximately equal strength. No, not at all. This one doesn't show a linear relationship of really any strength."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "What is the range of tree ages that he surveyed? What is the median age of a tree in the forest? So first of all, let's make sure we understand what this box and whiskers plot is even about. This is really a way of seeing the spread of all of the different data points, which are the age of the trees, and to also give other information, like what is the median and where do most of the ages of the trees sit. So this whisker part, so you can see this black part is a whisker, this is the box, and then this is another whisker right over here. The whiskers tell us essentially the spread of all of the data. So it says the lowest data point in this sample is an eight-year-old tree."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "This is really a way of seeing the spread of all of the different data points, which are the age of the trees, and to also give other information, like what is the median and where do most of the ages of the trees sit. So this whisker part, so you can see this black part is a whisker, this is the box, and then this is another whisker right over here. The whiskers tell us essentially the spread of all of the data. So it says the lowest data point in this sample is an eight-year-old tree. I'm assuming that this axis down here is in years. And it says that the highest, the oldest tree right over here is 50 years. So if we want the range, and when we think of range in a statistics point of view, we're thinking of the highest data point minus the lowest data point."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So it says the lowest data point in this sample is an eight-year-old tree. I'm assuming that this axis down here is in years. And it says that the highest, the oldest tree right over here is 50 years. So if we want the range, and when we think of range in a statistics point of view, we're thinking of the highest data point minus the lowest data point. So it's going to be 50 minus 8. So we have a range of 42. So that's what the whiskers tell us."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So if we want the range, and when we think of range in a statistics point of view, we're thinking of the highest data point minus the lowest data point. So it's going to be 50 minus 8. So we have a range of 42. So that's what the whiskers tell us. It tells us that everything falls between 8 and 50 years, including 8 years and 50 years. Now what the box does, the box starts at, well, let me explain it to you this way. This line right over here, this is the median."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So that's what the whiskers tell us. It tells us that everything falls between 8 and 50 years, including 8 years and 50 years. Now what the box does, the box starts at, well, let me explain it to you this way. This line right over here, this is the median. This right over here is the median. And so half of the ages are going to be less than this median. We see right over here the median is 21."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "This line right over here, this is the median. This right over here is the median. And so half of the ages are going to be less than this median. We see right over here the median is 21. So this box and whiskers plot tells us that half of the ages of the trees are less than 21, and half are older than 21. And then these endpoints right over here, these are the medians for each of those sections. So this is the median for all of the trees that are less than the real median, or less than the main median."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "We see right over here the median is 21. So this box and whiskers plot tells us that half of the ages of the trees are less than 21, and half are older than 21. And then these endpoints right over here, these are the medians for each of those sections. So this is the median for all of the trees that are less than the real median, or less than the main median. So this is the middle of all of the ages of trees that are less than 21. This is the middle age for all of the trees that are greater than 21, or older than 21. And so these essentially are splitting, we're actually splitting all of the data into four groups."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So this is the median for all of the trees that are less than the real median, or less than the main median. So this is the middle of all of the ages of trees that are less than 21. This is the middle age for all of the trees that are greater than 21, or older than 21. And so these essentially are splitting, we're actually splitting all of the data into four groups. This we would call the first quartile. So I'll call it Q1 for first quartile. Maybe I'll do 1Q."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And so these essentially are splitting, we're actually splitting all of the data into four groups. This we would call the first quartile. So I'll call it Q1 for first quartile. Maybe I'll do 1Q. This is the first quartile. Roughly, a fourth of the trees, because the way you calculate it, sometimes a tree ends up in one point or another. About a fourth of the trees end up here."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Maybe I'll do 1Q. This is the first quartile. Roughly, a fourth of the trees, because the way you calculate it, sometimes a tree ends up in one point or another. About a fourth of the trees end up here. A fourth of the trees are between 14 and 21. A fourth are between 21 and it looks like 33. And then a fourth are in this quartile."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "About a fourth of the trees end up here. A fourth of the trees are between 14 and 21. A fourth are between 21 and it looks like 33. And then a fourth are in this quartile. So we call this the first quartile, the second quartile, the third quartile, and the fourth quartile. So to answer the question, we already did the range. There's a 42-year spread between the oldest and the youngest tree."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And then a fourth are in this quartile. So we call this the first quartile, the second quartile, the third quartile, and the fourth quartile. So to answer the question, we already did the range. There's a 42-year spread between the oldest and the youngest tree. And then the median age of a tree in the forest is at 21. So even though you might have trees that are as old as 50, the median of the forest is actually closer to the lower end of our entire spectrum of all of the ages. So if you view median as your central tendency measurement, it's only at 21 years."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "Regulations require that product labels on containers of food that are available for sale to the public accurately state the amount of food in those containers. Specifically, if milk containers are labeled to have 128 fluid ounces, and the mean number of fluid ounces of milk in the containers is at least 128, the milk processor is considered to be in compliance with the regulations. The filling machines can be set to the labeled amount. Variability in the filling process causes the actual contents of milk containers to be normally distributed. A random sample of 12 containers of milk was drawn from the milk processing line in a plant, and the amount of milk in each container was recorded. The sample mean and standard deviation of this sample of 12 containers of milk were 127.2 ounces and 2.1 ounces, respectively. Is there sufficient evidence to conclude that the packaging plant is not in compliance with the regulations?"}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "Variability in the filling process causes the actual contents of milk containers to be normally distributed. A random sample of 12 containers of milk was drawn from the milk processing line in a plant, and the amount of milk in each container was recorded. The sample mean and standard deviation of this sample of 12 containers of milk were 127.2 ounces and 2.1 ounces, respectively. Is there sufficient evidence to conclude that the packaging plant is not in compliance with the regulations? Provide statistical justification for your answer. So pause this video and see if you can have a go at it. All right, now let's do this together."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "Is there sufficient evidence to conclude that the packaging plant is not in compliance with the regulations? Provide statistical justification for your answer. So pause this video and see if you can have a go at it. All right, now let's do this together. So first, let's say what we're talking about. So let me define mu, and this is going to be the mean amount amount of milk in population, population of containers, containers at the plant that we care about. And so then we can set up our hypotheses."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "All right, now let's do this together. So first, let's say what we're talking about. So let me define mu, and this is going to be the mean amount amount of milk in population, population of containers, containers at the plant that we care about. And so then we can set up our hypotheses. Our null hypothesis over here is that we are in compliance. We could say that the mean for our population of containers is actually 128. That's our minimum we need to be in compliance."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "And so then we can set up our hypotheses. Our null hypothesis over here is that we are in compliance. We could say that the mean for our population of containers is actually 128. That's our minimum we need to be in compliance. And that our alternative hypothesis is that we are not in compliance. So that's that our mean, the true population mean, is less than 128 fluid ounces. And so this is a situation where we are not in compliance, not in compliance, compliance in the alternative hypothesis."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "That's our minimum we need to be in compliance. And that our alternative hypothesis is that we are not in compliance. So that's that our mean, the true population mean, is less than 128 fluid ounces. And so this is a situation where we are not in compliance, not in compliance, compliance in the alternative hypothesis. Now, if you're going to do a significance test, you need to set a significance level. So let's do that over here. Significance level."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "And so this is a situation where we are not in compliance, not in compliance, compliance in the alternative hypothesis. Now, if you're going to do a significance test, you need to set a significance level. So let's do that over here. Significance level. And if you haven't noticed, I'm doing, I'm trying to do in this video what would be expected of you on a test. And this is an actual question from an AP exam. So our significance level here, I'll just pick it to be 0.05 because, well, that's a fairly typical one."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "Significance level. And if you haven't noticed, I'm doing, I'm trying to do in this video what would be expected of you on a test. And this is an actual question from an AP exam. So our significance level here, I'll just pick it to be 0.05 because, well, that's a fairly typical one. And since they didn't give it one to us, it's important to set one ahead of time. And now we wanna check our conditions for inference. So let me do that over here."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "So our significance level here, I'll just pick it to be 0.05 because, well, that's a fairly typical one. And since they didn't give it one to us, it's important to set one ahead of time. And now we wanna check our conditions for inference. So let me do that over here. Conditions, conditions for inference. And this is to feel good that the sample that we're using to make our inference, to do our significance test, that it's a reasonable one to make inferences from. And so the first one is the random condition."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "So let me do that over here. Conditions, conditions for inference. And this is to feel good that the sample that we're using to make our inference, to do our significance test, that it's a reasonable one to make inferences from. And so the first one is the random condition. And do we meet that? Well, they tell us here it's a random sample of 12 containers of milk. If I was doing this on the AP exam, I would write it out here."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "And so the first one is the random condition. And do we meet that? Well, they tell us here it's a random sample of 12 containers of milk. If I was doing this on the AP exam, I would write it out here. So I would say in the passage or in question, in the question, they say, they say a random, a random sample of 12. And then they go on to say more things. And so I would say that meets condition, meets condition."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "If I was doing this on the AP exam, I would write it out here. So I would say in the passage or in question, in the question, they say, they say a random, a random sample of 12. And then they go on to say more things. And so I would say that meets condition, meets condition. Now the next one we wanna care about is our normal condition and this is to feel good that our sampling distribution is roughly normal. Now there's a couple of ways that we could do that. One is if our sample size is greater than 30 or greater than or equal to 30, then we say, okay, our sampling distribution is going to be roughly normal."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "And so I would say that meets condition, meets condition. Now the next one we wanna care about is our normal condition and this is to feel good that our sampling distribution is roughly normal. Now there's a couple of ways that we could do that. One is if our sample size is greater than 30 or greater than or equal to 30, then we say, okay, our sampling distribution is going to be roughly normal. But in this situation, our sample size n, so sample size, sample size is less than 30, but, but there's another way to meet the normal condition and that's if the underlying parent data is normally distributed. And they actually say it right over here. Variability in the filling process causes the actual contents of milk to be normally distributed."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "One is if our sample size is greater than 30 or greater than or equal to 30, then we say, okay, our sampling distribution is going to be roughly normal. But in this situation, our sample size n, so sample size, sample size is less than 30, but, but there's another way to meet the normal condition and that's if the underlying parent data is normally distributed. And they actually say it right over here. Variability in the filling process causes the actual contents of milk to be normally distributed. So we could say in passage, in passage says, and let's see, I could quote part of this. So actual contents, actual contents, and then dot, dot, dot, normally distributed, normally distributed. So that meets condition, meets condition."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "Variability in the filling process causes the actual contents of milk to be normally distributed. So we could say in passage, in passage says, and let's see, I could quote part of this. So actual contents, actual contents, and then dot, dot, dot, normally distributed, normally distributed. So that meets condition, meets condition. And then the last condition we wanna think about is the independence condition, independence. And this is to feel good that the observations, the individual observations in our sample can be considered to be roughly independent. Now one way is if they were sampling with replacement, which they're not doing here."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "So that meets condition, meets condition. And then the last condition we wanna think about is the independence condition, independence. And this is to feel good that the observations, the individual observations in our sample can be considered to be roughly independent. Now one way is if they were sampling with replacement, which they're not doing here. It looks like they took all 12 containers at once. But another way is if this is less than 10% of the overall population, then you could say, okay, they're gonna, you can view them as roughly independent. And so you can say didn't, didn't sample with replacement, with replacement, but, but assume, assume that 12 is less than 10% of the population."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "Now one way is if they were sampling with replacement, which they're not doing here. It looks like they took all 12 containers at once. But another way is if this is less than 10% of the overall population, then you could say, okay, they're gonna, you can view them as roughly independent. And so you can say didn't, didn't sample with replacement, with replacement, but, but assume, assume that 12 is less than 10% of the population. And in that case, you would meet condition, meet this condition as well. So it looks like we are, we've met these three conditions that we need to make for inference, or we can assume we've done it. They haven't given us any information to the contrary."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "And so you can say didn't, didn't sample with replacement, with replacement, but, but assume, assume that 12 is less than 10% of the population. And in that case, you would meet condition, meet this condition as well. So it looks like we are, we've met these three conditions that we need to make for inference, or we can assume we've done it. They haven't given us any information to the contrary. And so now what we can do is calculate a t-statistic, and then from that, calculate our p-value, compare our p-value to our significance level, and see what kind of conclusions we can make. And so our t-statistic, right over here, and once again, if at any point you're inspired, and if you haven't done so already, try to do it on your own. Our t-statistic is going to be our sample mean minus the assumed mean from the null hypothesis."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "They haven't given us any information to the contrary. And so now what we can do is calculate a t-statistic, and then from that, calculate our p-value, compare our p-value to our significance level, and see what kind of conclusions we can make. And so our t-statistic, right over here, and once again, if at any point you're inspired, and if you haven't done so already, try to do it on your own. Our t-statistic is going to be our sample mean minus the assumed mean from the null hypothesis. And let me, since I'm introducing this notation, this little sub zero, I'll say that's the assumed mean from my null hypothesis. So I'll do that, and then I'll divide. Ideally, if I was doing a z-statistic, I would divide by the standard deviation of the sampling distribution of the sample mean, which is often known as the standard error of the mean."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "Our t-statistic is going to be our sample mean minus the assumed mean from the null hypothesis. And let me, since I'm introducing this notation, this little sub zero, I'll say that's the assumed mean from my null hypothesis. So I'll do that, and then I'll divide. Ideally, if I was doing a z-statistic, I would divide by the standard deviation of the sampling distribution of the sample mean, which is often known as the standard error of the mean. But the whole reason why I'm doing a t-statistic is, well, I don't know exactly what that is, but I could estimate the standard deviation of the sampling distribution of the sample mean using the sample standard deviation divided by the square root of n. And once again, it's always good, if you're doing this on a test, to explain what n is or what some of these things are. If you're using standard notation, people might assume what they are, but if you have time on these tests, you can always explain more of what these actual variables are. But in this case, this is going to be 127.2, that is our sample mean, minus our assumed mean from our null hypothesis, minus 128, all of that over, our sample standard deviation is 2.1, divided by the square root of 12."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "Ideally, if I was doing a z-statistic, I would divide by the standard deviation of the sampling distribution of the sample mean, which is often known as the standard error of the mean. But the whole reason why I'm doing a t-statistic is, well, I don't know exactly what that is, but I could estimate the standard deviation of the sampling distribution of the sample mean using the sample standard deviation divided by the square root of n. And once again, it's always good, if you're doing this on a test, to explain what n is or what some of these things are. If you're using standard notation, people might assume what they are, but if you have time on these tests, you can always explain more of what these actual variables are. But in this case, this is going to be 127.2, that is our sample mean, minus our assumed mean from our null hypothesis, minus 128, all of that over, our sample standard deviation is 2.1, divided by the square root of 12. And so this is going to be approximately equal to, got a calculator out here, and so we have, see in the numerator we have 127.2 minus 128, and then we're gonna divide that by, I'll do another parentheses, 2.1 divided by the square root of 12, and then let me close my parentheses. Did I type that in correctly? Yeah, that looks right."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "But in this case, this is going to be 127.2, that is our sample mean, minus our assumed mean from our null hypothesis, minus 128, all of that over, our sample standard deviation is 2.1, divided by the square root of 12. And so this is going to be approximately equal to, got a calculator out here, and so we have, see in the numerator we have 127.2 minus 128, and then we're gonna divide that by, I'll do another parentheses, 2.1 divided by the square root of 12, and then let me close my parentheses. Did I type that in correctly? Yeah, that looks right. Click Enter. And so this is negative, I'll say it's approximately negative 1.32. So negative 1.32."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "Yeah, that looks right. Click Enter. And so this is negative, I'll say it's approximately negative 1.32. So negative 1.32. And now we can figure out our p-value, our p-value, which is the same thing as the probability of getting a t-statistic this low or lower. So we could say t is less than or equal to negative 1.32 is equal to, so I'll get my calculator back out. And so here, what I would use is I would use the cumulative distribution function for t-statistic, so that's that right over there."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "So negative 1.32. And now we can figure out our p-value, our p-value, which is the same thing as the probability of getting a t-statistic this low or lower. So we could say t is less than or equal to negative 1.32 is equal to, so I'll get my calculator back out. And so here, what I would use is I would use the cumulative distribution function for t-statistic, so that's that right over there. And so I do care about the left tail, so I care about the area under the curve from negative infinity up to and including negative 1.32. So let's do negative 1.32. And then my degrees of freedom, well, it's gonna be my sample size minus one."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "And so here, what I would use is I would use the cumulative distribution function for t-statistic, so that's that right over there. And so I do care about the left tail, so I care about the area under the curve from negative infinity up to and including negative 1.32. So let's do negative 1.32. And then my degrees of freedom, well, it's gonna be my sample size minus one. My sample size was 12, so that minus one is 11. And then I do paste. And so I have this tcdf from negative E99 to negative 1.32 comma 11."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "And then my degrees of freedom, well, it's gonna be my sample size minus one. My sample size was 12, so that minus one is 11. And then I do paste. And so I have this tcdf from negative E99 to negative 1.32 comma 11. And actually, you'd wanna write this down on your exam if you were doing it, just so they know where you got that from. And so this is, this is equal to 0.107. So let me write it."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "And so I have this tcdf from negative E99 to negative 1.32 comma 11. And actually, you'd wanna write this down on your exam if you were doing it, just so they know where you got that from. And so this is, this is equal to 0.107. So let me write it. This is approximately 0.107. And it's important to say how you calculated this. So used, used tcdf, and we went from negative one times 10 to the 99th power, and we went up to negative 1.32, and then we had 11 degrees of freedom to get this result right over here."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "So let me write it. This is approximately 0.107. And it's important to say how you calculated this. So used, used tcdf, and we went from negative one times 10 to the 99th power, and we went up to negative 1.32, and then we had 11 degrees of freedom to get this result right over here. And it also might be good practice to draw your t-distribution right over here. So that's our t-distribution. That's the mean of our t-distribution."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "So used, used tcdf, and we went from negative one times 10 to the 99th power, and we went up to negative 1.32, and then we had 11 degrees of freedom to get this result right over here. And it also might be good practice to draw your t-distribution right over here. So that's our t-distribution. That's the mean of our t-distribution. So we say that this is the area that we care about. So that is that right over there, just to make sure people know what we're talking about. And so here, now we're ready to make a conclusion."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "That's the mean of our t-distribution. So we say that this is the area that we care about. So that is that right over there, just to make sure people know what we're talking about. And so here, now we're ready to make a conclusion. We can compare this to our significance level. And so we can say since, since 0.107 is greater than, our significance level is greater than 0.05, which is alpha, we fail, we fail to reject, reject the null hypothesis. And so let's just make sure we read their question right."}, {"video_title": "Free response example Significance test for a mean AP Statistics Khan Academy.mp3", "Sentence": "And so here, now we're ready to make a conclusion. We can compare this to our significance level. And so we can say since, since 0.107 is greater than, our significance level is greater than 0.05, which is alpha, we fail, we fail to reject, reject the null hypothesis. And so let's just make sure we read their question right. Is there sufficient evidence to conclude that the packaging plant is not in compliance with the regulations? And so another way of saying this is there is not, there is not sufficient, sufficient, I'm gonna have to scroll down a little bit. I'm trying to squeeze it on the page, but I'm gonna have to go down."}, {"video_title": "Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So I've got a binomial variable x, and I'm gonna describe it in very general terms. It is the number of successes after n trials, after n trials, where the probability of success, the probability of success, success for each trial is p. And this is a safe, this is a reasonable way to describe really any random, any binomial variable. We're assuming that each of these trials are independent. The probability stays constant. We have a finite number of trials right over here. Each trial results in either a very clear success or failure. So what we're gonna focus on in this video is, well, what would be the expected value of this binomial variable?"}, {"video_title": "Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "The probability stays constant. We have a finite number of trials right over here. Each trial results in either a very clear success or failure. So what we're gonna focus on in this video is, well, what would be the expected value of this binomial variable? What would the expected value, expected value of x be equal to? And I will just cut to the chase and tell you the answer, and then later in this video, we'll prove it to ourselves a little bit more mathematically. The expected value of x, it turns out, is just going to be equal to the number of trials times the probability of success for each of those trials."}, {"video_title": "Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So what we're gonna focus on in this video is, well, what would be the expected value of this binomial variable? What would the expected value, expected value of x be equal to? And I will just cut to the chase and tell you the answer, and then later in this video, we'll prove it to ourselves a little bit more mathematically. The expected value of x, it turns out, is just going to be equal to the number of trials times the probability of success for each of those trials. And so if you wanted to make that a little bit more concrete, imagine if a trial is a free throw, taking a shot from the free throw line. Success, success is made shot, so you actually make the shot, the ball went in the basket. Your probability is, let me do this yellow color, your probability, this would be your free throw percentage, so let's say it's 30% or 0.3."}, {"video_title": "Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "The expected value of x, it turns out, is just going to be equal to the number of trials times the probability of success for each of those trials. And so if you wanted to make that a little bit more concrete, imagine if a trial is a free throw, taking a shot from the free throw line. Success, success is made shot, so you actually make the shot, the ball went in the basket. Your probability is, let me do this yellow color, your probability, this would be your free throw percentage, so let's say it's 30% or 0.3. And let's say, for the sake of argument, that we're taking 10 free throws, so n is equal to 10. So this is making it all a lot more concrete. So in this particular scenario, your expected value, your expected value, if x is the number of made free throws after taking 10 free throws with a free throw percentage of 30%, well, based on what I just told you, it'd be n times b."}, {"video_title": "Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "Your probability is, let me do this yellow color, your probability, this would be your free throw percentage, so let's say it's 30% or 0.3. And let's say, for the sake of argument, that we're taking 10 free throws, so n is equal to 10. So this is making it all a lot more concrete. So in this particular scenario, your expected value, your expected value, if x is the number of made free throws after taking 10 free throws with a free throw percentage of 30%, well, based on what I just told you, it'd be n times b. It would be the number of trials times the probability of success in any one of those trials times 0.3, which is just going to be, of course, equal to three. Now, does that make intuitive sense? Well, if you're taking 10 shots with a 30% free throw percentage, it actually does feel natural that I would expect to make three shots."}, {"video_title": "Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So in this particular scenario, your expected value, your expected value, if x is the number of made free throws after taking 10 free throws with a free throw percentage of 30%, well, based on what I just told you, it'd be n times b. It would be the number of trials times the probability of success in any one of those trials times 0.3, which is just going to be, of course, equal to three. Now, does that make intuitive sense? Well, if you're taking 10 shots with a 30% free throw percentage, it actually does feel natural that I would expect to make three shots. Now, with that out of the way, let's make ourselves feel good about this mathematically, and we're gonna leverage some of our expected value properties. In particular, we're gonna leverage the fact that if I have the expected value of the sum of two independent random variables, let's say x plus y, it's going to be equal to the expected value of x plus the expected value of y that we talk about in other videos. And so, assuming this right over here, let's construct a new random variable."}, {"video_title": "Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, if you're taking 10 shots with a 30% free throw percentage, it actually does feel natural that I would expect to make three shots. Now, with that out of the way, let's make ourselves feel good about this mathematically, and we're gonna leverage some of our expected value properties. In particular, we're gonna leverage the fact that if I have the expected value of the sum of two independent random variables, let's say x plus y, it's going to be equal to the expected value of x plus the expected value of y that we talk about in other videos. And so, assuming this right over here, let's construct a new random variable. Let's call our random variable y. And we know the following things about y. The probability that y is equal to one is equal to p, and the probability that y is equal to zero is equal to one minus p. And these are the only two outcomes for this random variable."}, {"video_title": "Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so, assuming this right over here, let's construct a new random variable. Let's call our random variable y. And we know the following things about y. The probability that y is equal to one is equal to p, and the probability that y is equal to zero is equal to one minus p. And these are the only two outcomes for this random variable. And so, you might be seeing where this is going. You could view this random variable, it's really representing one trial. It becomes one in a success, zero when you don't have a success."}, {"video_title": "Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "The probability that y is equal to one is equal to p, and the probability that y is equal to zero is equal to one minus p. And these are the only two outcomes for this random variable. And so, you might be seeing where this is going. You could view this random variable, it's really representing one trial. It becomes one in a success, zero when you don't have a success. And so, you could view our original random variable x right over here as being equal to y plus y, and we're gonna have 10 of these. So, we're gonna have 10 y's. In the concrete sense, you could view the random variable y as equaling one if you make a free throw, and equaling zero if you don't make a free throw."}, {"video_title": "Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "It becomes one in a success, zero when you don't have a success. And so, you could view our original random variable x right over here as being equal to y plus y, and we're gonna have 10 of these. So, we're gonna have 10 y's. In the concrete sense, you could view the random variable y as equaling one if you make a free throw, and equaling zero if you don't make a free throw. It's really just representing one of those trials, and you can view x as the sum of n of those trials. Well, actually, let me be very clear here. I immediately went to the concrete, but I really should be saying n y's, because I wanna stay general right over here."}, {"video_title": "Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "In the concrete sense, you could view the random variable y as equaling one if you make a free throw, and equaling zero if you don't make a free throw. It's really just representing one of those trials, and you can view x as the sum of n of those trials. Well, actually, let me be very clear here. I immediately went to the concrete, but I really should be saying n y's, because I wanna stay general right over here. So, there are n n y's right over here. This was just a particular example, but I am going to try to stay general for the rest of the video, because now we are really trying to prove this result right over here. So, let's just take the expected value of both sides."}, {"video_title": "Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "I immediately went to the concrete, but I really should be saying n y's, because I wanna stay general right over here. So, there are n n y's right over here. This was just a particular example, but I am going to try to stay general for the rest of the video, because now we are really trying to prove this result right over here. So, let's just take the expected value of both sides. So, what is it going to be? So, we get the expected value of x is equal to, well, it's the expected value of all of this thing, but by that property right over here, this is going to be the expected value of y plus the expected value of y, plus, and we're gonna do this n times, plus the expected value of y, and we're gonna have n of these. So, we have n. And so, you could rewrite this as being equal to, so this is our n right over here."}, {"video_title": "Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So, let's just take the expected value of both sides. So, what is it going to be? So, we get the expected value of x is equal to, well, it's the expected value of all of this thing, but by that property right over here, this is going to be the expected value of y plus the expected value of y, plus, and we're gonna do this n times, plus the expected value of y, and we're gonna have n of these. So, we have n. And so, you could rewrite this as being equal to, so this is our n right over here. This is n times the expected value of y. Now, what is the expected value of y? Well, this is pretty straightforward."}, {"video_title": "Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So, we have n. And so, you could rewrite this as being equal to, so this is our n right over here. This is n times the expected value of y. Now, what is the expected value of y? Well, this is pretty straightforward. We can actually just do it directly. The expected value of y, let me just write it over here. The expected value of y is just the probability weighted outcomes, and since there's only two discrete outcomes here, it's pretty easy to calculate."}, {"video_title": "Expected value of a binomial variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, this is pretty straightforward. We can actually just do it directly. The expected value of y, let me just write it over here. The expected value of y is just the probability weighted outcomes, and since there's only two discrete outcomes here, it's pretty easy to calculate. We have a probability of p of getting a one, so it's p times one, plus we have a probability of one minus p of getting a zero. Well, what does this simplify to? Well, zero times anything, that's zero, and then you have one times p. This is just equal to p. So, expected value of y is just equal to p, and so there you have it."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "we've seen in the last several videos. You start off with any crazy distribution. It doesn't have to be crazy. It could be a nice normal distribution, but to really make the point that you don't have to have a normal distribution, I like to use crazy ones. So let's say you have some kind of crazy distribution that looks something like that. It could look like anything. So we've seen multiple times you take samples from this crazy distribution."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "It could be a nice normal distribution, but to really make the point that you don't have to have a normal distribution, I like to use crazy ones. So let's say you have some kind of crazy distribution that looks something like that. It could look like anything. So we've seen multiple times you take samples from this crazy distribution. So let's say you were to take samples of, let's say n is equal to 10. So we take 10 instances of this random variable, average them out, and then plot our average. We plot our average."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So we've seen multiple times you take samples from this crazy distribution. So let's say you were to take samples of, let's say n is equal to 10. So we take 10 instances of this random variable, average them out, and then plot our average. We plot our average. We get one instance there. We keep doing that. We do that again."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "We plot our average. We get one instance there. We keep doing that. We do that again. We take 10 samples from this random variable, average them, plot them again. You plot it again, and eventually you do this a gazillion times, in theory infinite number of times, and you're going to approach the sampling distribution of the sample mean. And n equal 10, it's not gonna be a perfect normal distribution, but it's gonna be close."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "We do that again. We take 10 samples from this random variable, average them, plot them again. You plot it again, and eventually you do this a gazillion times, in theory infinite number of times, and you're going to approach the sampling distribution of the sample mean. And n equal 10, it's not gonna be a perfect normal distribution, but it's gonna be close. It'd be perfect only if n was infinity. But let's say we eventually, all of our samples, you know, we get a lot of averages that are there, that stacks up there, that stacks up there, and eventually we'll approach something that looks something like that. And we've seen from the last video that, one, if, let's say we were to do it again, and this time let's say that n is equal to 20."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And n equal 10, it's not gonna be a perfect normal distribution, but it's gonna be close. It'd be perfect only if n was infinity. But let's say we eventually, all of our samples, you know, we get a lot of averages that are there, that stacks up there, that stacks up there, and eventually we'll approach something that looks something like that. And we've seen from the last video that, one, if, let's say we were to do it again, and this time let's say that n is equal to 20. One, the distribution that we get is going to be more normal, and maybe in future videos we'll delve even deeper into things like kurtosis and skew. But it's gonna be more normal, but even more important, or I guess even more obviously to us, and we saw that in the experiment, it's gonna have a lower standard deviation. So they're all gonna have the same mean."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And we've seen from the last video that, one, if, let's say we were to do it again, and this time let's say that n is equal to 20. One, the distribution that we get is going to be more normal, and maybe in future videos we'll delve even deeper into things like kurtosis and skew. But it's gonna be more normal, but even more important, or I guess even more obviously to us, and we saw that in the experiment, it's gonna have a lower standard deviation. So they're all gonna have the same mean. Let's say the mean here is, you know, I don't know, let's say the mean here is five. Then the mean here is also gonna be five. The mean of our sampling distribution of the sample mean is gonna be five."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So they're all gonna have the same mean. Let's say the mean here is, you know, I don't know, let's say the mean here is five. Then the mean here is also gonna be five. The mean of our sampling distribution of the sample mean is gonna be five. And it doesn't matter what our n is. If our n is 20, it's still gonna be five, but our standard deviation is gonna be less in either of these scenarios. And we saw that just by experimenting."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "The mean of our sampling distribution of the sample mean is gonna be five. And it doesn't matter what our n is. If our n is 20, it's still gonna be five, but our standard deviation is gonna be less in either of these scenarios. And we saw that just by experimenting. It might look like this. It's gonna be more normal, but it's gonna have a tighter standard deviation. So maybe it'll look like that."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And we saw that just by experimenting. It might look like this. It's gonna be more normal, but it's gonna have a tighter standard deviation. So maybe it'll look like that. And if we did it with a even larger sample size, let me do that in a different color. If we did that with an even larger sample size, n is equal to 100, what we're gonna get is something that fits the normal distribution even better. We take 100 instances of this random variable, average them, plot it."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So maybe it'll look like that. And if we did it with a even larger sample size, let me do that in a different color. If we did that with an even larger sample size, n is equal to 100, what we're gonna get is something that fits the normal distribution even better. We take 100 instances of this random variable, average them, plot it. 100 instances of this random variable, average them, plot it. We just keep doing that. If we keep doing that, what we're gonna have is something that's even more normal than either of these."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "We take 100 instances of this random variable, average them, plot it. 100 instances of this random variable, average them, plot it. We just keep doing that. If we keep doing that, what we're gonna have is something that's even more normal than either of these. So it's gonna be a much closer fit to a true normal distribution. But even more obvious to the human eye, it's gonna be even tighter. So it's going to be a very low standard deviation."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "If we keep doing that, what we're gonna have is something that's even more normal than either of these. So it's gonna be a much closer fit to a true normal distribution. But even more obvious to the human eye, it's gonna be even tighter. So it's going to be a very low standard deviation. It's gonna look something like that. And I'll show you that on the simulation app in the next, or probably later in this video. So two things happen."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So it's going to be a very low standard deviation. It's gonna look something like that. And I'll show you that on the simulation app in the next, or probably later in this video. So two things happen. As you increase your sample size for every time you do the average, two things are happening. You're becoming more normal, and your standard deviation is getting smaller. So the question might arise, is there a formula?"}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So two things happen. As you increase your sample size for every time you do the average, two things are happening. You're becoming more normal, and your standard deviation is getting smaller. So the question might arise, is there a formula? So if I know the standard deviation, so this is my standard deviation of just my original probability density function. This is the mean of my original probability density function. So if I know the standard deviation, and I know n, n is gonna change depending on how many samples I'm taking every time I do a sample mean."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So the question might arise, is there a formula? So if I know the standard deviation, so this is my standard deviation of just my original probability density function. This is the mean of my original probability density function. So if I know the standard deviation, and I know n, n is gonna change depending on how many samples I'm taking every time I do a sample mean. If I know my standard deviation, or maybe if I know my variance, the variance is just the standard deviation squared. If you don't remember that, you might wanna review those videos. But if I know the variance of my original distribution, and if I know what my n is, how many samples I'm gonna take every time before I average them in order to plot one thing in my sampling distribution of my sample mean, is there a way to predict what the mean of these distributions are?"}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So if I know the standard deviation, and I know n, n is gonna change depending on how many samples I'm taking every time I do a sample mean. If I know my standard deviation, or maybe if I know my variance, the variance is just the standard deviation squared. If you don't remember that, you might wanna review those videos. But if I know the variance of my original distribution, and if I know what my n is, how many samples I'm gonna take every time before I average them in order to plot one thing in my sampling distribution of my sample mean, is there a way to predict what the mean of these distributions are? And so this, sorry, the standard deviation of these distributions, and to make, so you don't get confused between that and that, and let me say the variance. If you know the variance, you can figure out the standard deviation. One is just the square root of the other."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But if I know the variance of my original distribution, and if I know what my n is, how many samples I'm gonna take every time before I average them in order to plot one thing in my sampling distribution of my sample mean, is there a way to predict what the mean of these distributions are? And so this, sorry, the standard deviation of these distributions, and to make, so you don't get confused between that and that, and let me say the variance. If you know the variance, you can figure out the standard deviation. One is just the square root of the other. So this is the variance of our original distribution. Now, to show that this is the variance of our sampling distribution of our sample mean, we'll write it right here. This is the variance of our mean, of our sample mean."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "One is just the square root of the other. So this is the variance of our original distribution. Now, to show that this is the variance of our sampling distribution of our sample mean, we'll write it right here. This is the variance of our mean, of our sample mean. Remember, the sample, our true mean is this, that the Greek letter mu is your true mean. This is equal to the mean, while an X with a line over it means sample mean. Sample mean."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "This is the variance of our mean, of our sample mean. Remember, the sample, our true mean is this, that the Greek letter mu is your true mean. This is equal to the mean, while an X with a line over it means sample mean. Sample mean. So here, what we're saying is this is the variance of our sample means. Now, this is gonna be a true distribution. This isn't an estimate."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Sample mean. So here, what we're saying is this is the variance of our sample means. Now, this is gonna be a true distribution. This isn't an estimate. This is, there's some, you know, if we magically knew this distribution, there's some true variance here. And of course, the mean, so this has a mean. This right here, we can just get our notation right."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "This isn't an estimate. This is, there's some, you know, if we magically knew this distribution, there's some true variance here. And of course, the mean, so this has a mean. This right here, we can just get our notation right. This is the mean of the sampling distribution of the sampling mean. So this is the mean of our means. It just happens to be the same thing."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "This right here, we can just get our notation right. This is the mean of the sampling distribution of the sampling mean. So this is the mean of our means. It just happens to be the same thing. This is the mean of our sample means. It's gonna be the same thing as that, especially if we do the trial over and over again. But anyway, the point of this video, is there any way to figure out this variance, given the variance of the original distribution and your N?"}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "It just happens to be the same thing. This is the mean of our sample means. It's gonna be the same thing as that, especially if we do the trial over and over again. But anyway, the point of this video, is there any way to figure out this variance, given the variance of the original distribution and your N? And it turns out there is. And I'm not gonna do a proof here. I really wanna give you the intuition of it."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But anyway, the point of this video, is there any way to figure out this variance, given the variance of the original distribution and your N? And it turns out there is. And I'm not gonna do a proof here. I really wanna give you the intuition of it. And I think you already do have the sense that every trial you take, if you take 100, you're much more likely, when you average those out, to get close to the true mean than if you took an N of two or an N of five. You're just very unlikely to be far away, right, if you took 100 trials as opposed to taking five. So I think you know that, in some way, it should be inversely proportional to N. The larger your N, the smaller standard deviation."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "I really wanna give you the intuition of it. And I think you already do have the sense that every trial you take, if you take 100, you're much more likely, when you average those out, to get close to the true mean than if you took an N of two or an N of five. You're just very unlikely to be far away, right, if you took 100 trials as opposed to taking five. So I think you know that, in some way, it should be inversely proportional to N. The larger your N, the smaller standard deviation. And actually, it turns out it's about as simple as possible. It's one of those magical things about mathematics. And I'll prove it to you one day."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So I think you know that, in some way, it should be inversely proportional to N. The larger your N, the smaller standard deviation. And actually, it turns out it's about as simple as possible. It's one of those magical things about mathematics. And I'll prove it to you one day. I want to give you a working knowledge first. In statistics, I'm always struggling whether I should be formal and giving you rigorous proofs, but I've kind of come to the conclusion that it's more important to get the working knowledge first in statistics. And then later, once you've gotten all of that down, we can get into the real deep math of it and prove it to you."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And I'll prove it to you one day. I want to give you a working knowledge first. In statistics, I'm always struggling whether I should be formal and giving you rigorous proofs, but I've kind of come to the conclusion that it's more important to get the working knowledge first in statistics. And then later, once you've gotten all of that down, we can get into the real deep math of it and prove it to you. But I think experimental proofs are kind of all you need for right now, using those simulations to show that they're really true. So it turns out that the variance of your sampling distribution of your sample mean is equal to the variance of your original distribution, that guy right there, divided by N. That's all it is. So if this up here has a variance of, let's say this up here has a variance of 20, I'm just making that number up, then, and then let's say your N is 20, then the variance of your sampling distribution of your sample mean for N of 20, well, you're just gonna take that, the variance up here, your variance is 20, divided by your N, 20."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And then later, once you've gotten all of that down, we can get into the real deep math of it and prove it to you. But I think experimental proofs are kind of all you need for right now, using those simulations to show that they're really true. So it turns out that the variance of your sampling distribution of your sample mean is equal to the variance of your original distribution, that guy right there, divided by N. That's all it is. So if this up here has a variance of, let's say this up here has a variance of 20, I'm just making that number up, then, and then let's say your N is 20, then the variance of your sampling distribution of your sample mean for N of 20, well, you're just gonna take that, the variance up here, your variance is 20, divided by your N, 20. So here, your variance is going to be 20 divided by 20, which is equal to one. This is the variance of your original probability distribution, and this is your N. What's your standard deviation gonna be? What's gonna be the square root of that?"}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So if this up here has a variance of, let's say this up here has a variance of 20, I'm just making that number up, then, and then let's say your N is 20, then the variance of your sampling distribution of your sample mean for N of 20, well, you're just gonna take that, the variance up here, your variance is 20, divided by your N, 20. So here, your variance is going to be 20 divided by 20, which is equal to one. This is the variance of your original probability distribution, and this is your N. What's your standard deviation gonna be? What's gonna be the square root of that? Standard deviation is gonna be the square root of one, well, that's also going to be one. So we could also write this. We could take the square root of both sides of this and say the standard deviation of the sampling distribution of the, of the standard deviation of the sampling distribution of the sample mean, it's often called the standard deviation of the mean, and it's also called, I'm gonna write this down, the standard error of the mean, standard error of the mean."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "What's gonna be the square root of that? Standard deviation is gonna be the square root of one, well, that's also going to be one. So we could also write this. We could take the square root of both sides of this and say the standard deviation of the sampling distribution of the, of the standard deviation of the sampling distribution of the sample mean, it's often called the standard deviation of the mean, and it's also called, I'm gonna write this down, the standard error of the mean, standard error of the mean. All of these things that I just mentioned, these all just mean the standard deviation of the sampling distribution of the sample mean. That's why this is confusing, because you use the word mean and sample over and over again, and if it confuses you, let me know, I'll do another video or pause and repeat it, whatever. But if we just take the square root of both sides, the standard error of the mean, or the standard deviation of the sampling distribution of the sample mean is equal to the standard deviation of your original, of your original function, of your original probability density function, which could be very non-normal, divided by the square root of n. I just took the square root of both sides of this equation."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "We could take the square root of both sides of this and say the standard deviation of the sampling distribution of the, of the standard deviation of the sampling distribution of the sample mean, it's often called the standard deviation of the mean, and it's also called, I'm gonna write this down, the standard error of the mean, standard error of the mean. All of these things that I just mentioned, these all just mean the standard deviation of the sampling distribution of the sample mean. That's why this is confusing, because you use the word mean and sample over and over again, and if it confuses you, let me know, I'll do another video or pause and repeat it, whatever. But if we just take the square root of both sides, the standard error of the mean, or the standard deviation of the sampling distribution of the sample mean is equal to the standard deviation of your original, of your original function, of your original probability density function, which could be very non-normal, divided by the square root of n. I just took the square root of both sides of this equation. I personally, I like to remember this, that the variance is just inversely proportional to n, and then I like to go back to this, because this is very simple in my head. You just take the variance divided by n. Oh, and if I want the standard deviation, I just take the square roots of both sides, and I get this formula. So here, the standard deviation, when n is 20, the standard deviation of the sampling distribution of the sample mean is gonna be one."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But if we just take the square root of both sides, the standard error of the mean, or the standard deviation of the sampling distribution of the sample mean is equal to the standard deviation of your original, of your original function, of your original probability density function, which could be very non-normal, divided by the square root of n. I just took the square root of both sides of this equation. I personally, I like to remember this, that the variance is just inversely proportional to n, and then I like to go back to this, because this is very simple in my head. You just take the variance divided by n. Oh, and if I want the standard deviation, I just take the square roots of both sides, and I get this formula. So here, the standard deviation, when n is 20, the standard deviation of the sampling distribution of the sample mean is gonna be one. Here, when n is 100, well, our variance, our variance here, when n is equal to 100, so our variance of the sampling mean of the sample distribution, or our variance of the mean, of the sample mean, we could say, is going to be equal to 20. This guy's variance divided by n. So it equals, n is 100, so it equals 1 5th. Now, this guy's standard deviation, or the standard deviation of the sampling distribution of the sample mean, or the standard error of the mean, is gonna be the square root of that."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So here, the standard deviation, when n is 20, the standard deviation of the sampling distribution of the sample mean is gonna be one. Here, when n is 100, well, our variance, our variance here, when n is equal to 100, so our variance of the sampling mean of the sample distribution, or our variance of the mean, of the sample mean, we could say, is going to be equal to 20. This guy's variance divided by n. So it equals, n is 100, so it equals 1 5th. Now, this guy's standard deviation, or the standard deviation of the sampling distribution of the sample mean, or the standard error of the mean, is gonna be the square root of that. So one over the square root of five. And so, this guy's a little bit under 1 1 2 standard deviation while this guy had a standard deviation of one. So you see, it's definitely thinner."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Now, this guy's standard deviation, or the standard deviation of the sampling distribution of the sample mean, or the standard error of the mean, is gonna be the square root of that. So one over the square root of five. And so, this guy's a little bit under 1 1 2 standard deviation while this guy had a standard deviation of one. So you see, it's definitely thinner. Now, I know what you're saying. Well, Sal, you just gave a formula. I don't necessarily believe you."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So you see, it's definitely thinner. Now, I know what you're saying. Well, Sal, you just gave a formula. I don't necessarily believe you. Well, let's see if we can prove it to ourselves using the simulation. So I'll, just for fun, let me make a, I'll just mess with this distribution a little bit. So that's my new distribution."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "I don't necessarily believe you. Well, let's see if we can prove it to ourselves using the simulation. So I'll, just for fun, let me make a, I'll just mess with this distribution a little bit. So that's my new distribution. And let me take an n of, let me take two things that's easy to take the square root of, because if we're looking at standard deviation. So let's take, we'll take an n of 16, and an n of 25. And let's, well, I'll do a, let's do 10,000 trials."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So that's my new distribution. And let me take an n of, let me take two things that's easy to take the square root of, because if we're looking at standard deviation. So let's take, we'll take an n of 16, and an n of 25. And let's, well, I'll do a, let's do 10,000 trials. So in this case, every one of the trials, we're gonna take 16 samples from here, average them, plot it here, and then do a frequency plot. Here, we're gonna do 25 at a time, and then average them. I'll do it once animated, just to remember."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And let's, well, I'll do a, let's do 10,000 trials. So in this case, every one of the trials, we're gonna take 16 samples from here, average them, plot it here, and then do a frequency plot. Here, we're gonna do 25 at a time, and then average them. I'll do it once animated, just to remember. So I'm taking 16 samples, plot it there. I take 16 samples, as described by this probability density function, or 25 now, plot it down here. Now, if I do that 10,000 times, what do I get?"}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "I'll do it once animated, just to remember. So I'm taking 16 samples, plot it there. I take 16 samples, as described by this probability density function, or 25 now, plot it down here. Now, if I do that 10,000 times, what do I get? What do I get? All right, so here, you know, just visually, you can tell just when n was larger, the standard deviation here is smaller. This is more squeezed together."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Now, if I do that 10,000 times, what do I get? What do I get? All right, so here, you know, just visually, you can tell just when n was larger, the standard deviation here is smaller. This is more squeezed together. But actually, let's write, let's write this stuff down. Let's see if I can remember it here. Here, n is, so in this random distribution I made, my standard deviation was 9.3."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "This is more squeezed together. But actually, let's write, let's write this stuff down. Let's see if I can remember it here. Here, n is, so in this random distribution I made, my standard deviation was 9.3. I'm gonna remember these. Our standard deviation for the original thing was 9.3. And so, standard deviation here was 2.3, and the standard deviation here is 1.87."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Here, n is, so in this random distribution I made, my standard deviation was 9.3. I'm gonna remember these. Our standard deviation for the original thing was 9.3. And so, standard deviation here was 2.3, and the standard deviation here is 1.87. Let's see if it, if it conforms to our formula. So I'm gonna take this offscreen for a second, and I'm gonna go back and do some mathematics. So I have this on my other screen, so I can remember those numbers."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And so, standard deviation here was 2.3, and the standard deviation here is 1.87. Let's see if it, if it conforms to our formula. So I'm gonna take this offscreen for a second, and I'm gonna go back and do some mathematics. So I have this on my other screen, so I can remember those numbers. So in the trial we just did, my wacky distribution had a standard deviation of 9.3. When n is equal to, let me do this in another color. When n was equal to 16, just doing the experiment, doing a bunch of trials, and averaging, and doing all the things, we got the standard deviation of the sampling distribution of the sample mean, or the standard error of the mean."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So I have this on my other screen, so I can remember those numbers. So in the trial we just did, my wacky distribution had a standard deviation of 9.3. When n is equal to, let me do this in another color. When n was equal to 16, just doing the experiment, doing a bunch of trials, and averaging, and doing all the things, we got the standard deviation of the sampling distribution of the sample mean, or the standard error of the mean. We experimentally determined it to be 2.33. And then when n is equal to 25, when n is equal to 25, we got the standard error of the mean being equal to 1.87. Let's see if it conforms to our formulas."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "When n was equal to 16, just doing the experiment, doing a bunch of trials, and averaging, and doing all the things, we got the standard deviation of the sampling distribution of the sample mean, or the standard error of the mean. We experimentally determined it to be 2.33. And then when n is equal to 25, when n is equal to 25, we got the standard error of the mean being equal to 1.87. Let's see if it conforms to our formulas. So we know that the variance, or we could almost say the variance of the mean, or the standard error, well, you know, the variance of the sampling distribution of the sample mean is equal to the variance of our original distribution divided by n. Take the square roots of both sides, and then you get standard error of the mean is equal to standard deviation of your original distribution divided by the square root of n. So let's see if this works out for these two things. So if I were to take 9.3, so let me do this case. So 9.3 divided by the square root of 16, right?"}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Let's see if it conforms to our formulas. So we know that the variance, or we could almost say the variance of the mean, or the standard error, well, you know, the variance of the sampling distribution of the sample mean is equal to the variance of our original distribution divided by n. Take the square roots of both sides, and then you get standard error of the mean is equal to standard deviation of your original distribution divided by the square root of n. So let's see if this works out for these two things. So if I were to take 9.3, so let me do this case. So 9.3 divided by the square root of 16, right? N is 16. So divided by the square root of 16, which is four, what do I get? So 9.3 divided by four."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So 9.3 divided by the square root of 16, right? N is 16. So divided by the square root of 16, which is four, what do I get? So 9.3 divided by four. Let me get a little calculator out here. Let's see, we have, let me clear it out. We wanted to divide 9.3 divided by four."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So 9.3 divided by four. Let me get a little calculator out here. Let's see, we have, let me clear it out. We wanted to divide 9.3 divided by four. 9.3 divided by our square root of n, n was 16, so divided by four is equal to 2.32. 2.32. So this is equal to, this is equal to 2.32, which is pretty darn close to 2.33."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "We wanted to divide 9.3 divided by four. 9.3 divided by our square root of n, n was 16, so divided by four is equal to 2.32. 2.32. So this is equal to, this is equal to 2.32, which is pretty darn close to 2.33. This was after 10,000 trials. Maybe right after this, I'll see what happens if we did 20,000 or 30,000 trials where we take samples of 16 and average them. Now let's look at this."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So this is equal to, this is equal to 2.32, which is pretty darn close to 2.33. This was after 10,000 trials. Maybe right after this, I'll see what happens if we did 20,000 or 30,000 trials where we take samples of 16 and average them. Now let's look at this. Here, we would take 9.3. So let me draw a little line here. Maybe scroll over, that might be better."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Now let's look at this. Here, we would take 9.3. So let me draw a little line here. Maybe scroll over, that might be better. So we take our standard deviation of our original distribution. So just that formula that we derived right here would tell us that our standard error should be equal to the standard deviation of our original distribution, 9.3, divided by the square root of n, divided by the square root of 25, right? Four was just the square root of 16."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Maybe scroll over, that might be better. So we take our standard deviation of our original distribution. So just that formula that we derived right here would tell us that our standard error should be equal to the standard deviation of our original distribution, 9.3, divided by the square root of n, divided by the square root of 25, right? Four was just the square root of 16. So this is equal to 9.3 divided by five, and let's see if it's 1.87. So let me get my calculator back. So if I take 9.3 divided by five, what do I get?"}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Four was just the square root of 16. So this is equal to 9.3 divided by five, and let's see if it's 1.87. So let me get my calculator back. So if I take 9.3 divided by five, what do I get? 1.86, which is very close to 1.87. So we got, we got in this case, 1.86. 1.86."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So if I take 9.3 divided by five, what do I get? 1.86, which is very close to 1.87. So we got, we got in this case, 1.86. 1.86. So as you can see, what we got experimentally was almost exactly, and this was after 10,000 trials, of what you would expect. Let's do another 10,000. So you got another 10,000 trials."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "1.86. So as you can see, what we got experimentally was almost exactly, and this was after 10,000 trials, of what you would expect. Let's do another 10,000. So you got another 10,000 trials. Well, we're still in the ballpark. We're not gonna, maybe I can't hope to get the exact number, you know, rounded or whatever. But as you can see, hopefully that'll be pretty satisfying to you, that the variance of the sampling distribution of the sample mean, the variance of the sampling distribution sampling mean, is just going to be equal to the variance of your original distribution, no matter how wacky that distribution might be, divided by your sample size."}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So you got another 10,000 trials. Well, we're still in the ballpark. We're not gonna, maybe I can't hope to get the exact number, you know, rounded or whatever. But as you can see, hopefully that'll be pretty satisfying to you, that the variance of the sampling distribution of the sample mean, the variance of the sampling distribution sampling mean, is just going to be equal to the variance of your original distribution, no matter how wacky that distribution might be, divided by your sample size. By the number of samples you take for when, for every basket that you average, I guess is the best way to think about it. And you know, sometimes this can get confusing because you are taking samples of averages based on samples. So when someone says sample size, you're like, is sample size the number of times I took averages or the number of things I'm taking averages of each time?"}, {"video_title": "Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But as you can see, hopefully that'll be pretty satisfying to you, that the variance of the sampling distribution of the sample mean, the variance of the sampling distribution sampling mean, is just going to be equal to the variance of your original distribution, no matter how wacky that distribution might be, divided by your sample size. By the number of samples you take for when, for every basket that you average, I guess is the best way to think about it. And you know, sometimes this can get confusing because you are taking samples of averages based on samples. So when someone says sample size, you're like, is sample size the number of times I took averages or the number of things I'm taking averages of each time? And you know, it doesn't hurt to clarify that. Normally when they talk about sample size, they're talking about n. And at least in my head, when I think of the trials as you take a sample size of 16, you average it, that's one trial and you plot it. Then you do it again and you do another trial and you do it over and over again."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "So let's say I have a population and I care about some proportion. Let's say I care about the proportion of folks that are left-handed. I don't know what that is, and so I take a sample of size n, and then from that sample, I can calculate a sample proportion. That's why I put that little hat on top of it. It's a sample proportion that's estimating our true proportion. Now I wanna construct a confidence interval, but before I go down the path, I need to actually set up my conditions for inference, make sure that I meet them, and we've done this many times. So the first condition for inference is the random condition."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "That's why I put that little hat on top of it. It's a sample proportion that's estimating our true proportion. Now I wanna construct a confidence interval, but before I go down the path, I need to actually set up my conditions for inference, make sure that I meet them, and we've done this many times. So the first condition for inference is the random condition. I need to feel good that this is truly a random sample from the population. The second one is often known as the normal condition, and that's the condition that, hey, in order to feel like the sampling distribution for the sample proportions is roughly normal, n times our sample proportion should be greater than or equal to 10, and n times one minus our sample proportion should be greater than or equal to 10. We've seen that multiple times before."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "So the first condition for inference is the random condition. I need to feel good that this is truly a random sample from the population. The second one is often known as the normal condition, and that's the condition that, hey, in order to feel like the sampling distribution for the sample proportions is roughly normal, n times our sample proportion should be greater than or equal to 10, and n times one minus our sample proportion should be greater than or equal to 10. We've seen that multiple times before. And then the third one is the independence condition, and there's two ways to meet this. Either the individual observations in our sample should be done with replacement, or if it's not done with replacement, we can feel pretty confident about this if our sample size is no more than 10% of the size of the entire population. But let's say that we meet these conditions for inference."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "We've seen that multiple times before. And then the third one is the independence condition, and there's two ways to meet this. Either the individual observations in our sample should be done with replacement, or if it's not done with replacement, we can feel pretty confident about this if our sample size is no more than 10% of the size of the entire population. But let's say that we meet these conditions for inference. What do we do? Well, we set up a confidence level, confidence level, for our confidence interval that we're about to construct. And let's say we said it was a 95% confidence level."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "But let's say that we meet these conditions for inference. What do we do? Well, we set up a confidence level, confidence level, for our confidence interval that we're about to construct. And let's say we said it was a 95% confidence level. That would mean that 95% of the time that we went through this exercise, the confidence interval that we get would actually overlap with the true population proportion. And 95% is actually a fairly typical one. But from that confidence level, you can calculate a critical value."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "And let's say we said it was a 95% confidence level. That would mean that 95% of the time that we went through this exercise, the confidence interval that we get would actually overlap with the true population proportion. And 95% is actually a fairly typical one. But from that confidence level, you can calculate a critical value. And the way that you do that, you just look up in a z-table, and once again, all of this is review. You would say, hey, how many standard deviations above and below the mean of a normal distribution would you need to go in order to get, say, 95%, that confidence level of the distribution? And now we're ready to calculate the confidence interval."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "But from that confidence level, you can calculate a critical value. And the way that you do that, you just look up in a z-table, and once again, all of this is review. You would say, hey, how many standard deviations above and below the mean of a normal distribution would you need to go in order to get, say, 95%, that confidence level of the distribution? And now we're ready to calculate the confidence interval. Confidence interval, it is going to be equal to our sample proportion plus or minus our critical value, our critical value, times the standard deviation of the sampling distribution of the sample proportion. Now, there is a way to calculate this exactly if we knew what p is. If we knew what p is, this would be the square root of p times one minus p over n. But if we knew what p is, then we wouldn't even have to do this business of constructing confidence intervals."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "And now we're ready to calculate the confidence interval. Confidence interval, it is going to be equal to our sample proportion plus or minus our critical value, our critical value, times the standard deviation of the sampling distribution of the sample proportion. Now, there is a way to calculate this exactly if we knew what p is. If we knew what p is, this would be the square root of p times one minus p over n. But if we knew what p is, then we wouldn't even have to do this business of constructing confidence intervals. So instead, we estimate this. We say, look, an estimate of the standard deviation of the sampling distribution, often known as the standard error, an estimate of this is going to be the square root of, instead of the true population parameter, we could use the sample proportion. So p hat times one minus p hat, all of that over n. Now, the whole reason why I did this, this is covered in much more detail and much slower in other videos, is to see the parallels between this and a situation when we're constructing a two-sample confidence interval, or z-interval, for a difference between proportions."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "If we knew what p is, this would be the square root of p times one minus p over n. But if we knew what p is, then we wouldn't even have to do this business of constructing confidence intervals. So instead, we estimate this. We say, look, an estimate of the standard deviation of the sampling distribution, often known as the standard error, an estimate of this is going to be the square root of, instead of the true population parameter, we could use the sample proportion. So p hat times one minus p hat, all of that over n. Now, the whole reason why I did this, this is covered in much more detail and much slower in other videos, is to see the parallels between this and a situation when we're constructing a two-sample confidence interval, or z-interval, for a difference between proportions. What am I talking about? Well, let's say that you have two different populations. So this is the first population, and it has some true proportion of the folks that, let's say, are left-handed."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "So p hat times one minus p hat, all of that over n. Now, the whole reason why I did this, this is covered in much more detail and much slower in other videos, is to see the parallels between this and a situation when we're constructing a two-sample confidence interval, or z-interval, for a difference between proportions. What am I talking about? Well, let's say that you have two different populations. So this is the first population, and it has some true proportion of the folks that, let's say, are left-handed. And then there's another population. So let's call that p two. Maybe this is a freshman in your high school or college, and maybe this is sophomores."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "So this is the first population, and it has some true proportion of the folks that, let's say, are left-handed. And then there's another population. So let's call that p two. Maybe this is a freshman in your high school or college, and maybe this is sophomores. So two different populations. And you wanna see if there's a difference between the proportion that are left-handed, say. And so what you could do, just like we've done here, is for each of these populations, you will take a sample here, we'll call that n one, and then from that sample, you calculate a sample proportion, let's call that p one."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "Maybe this is a freshman in your high school or college, and maybe this is sophomores. So two different populations. And you wanna see if there's a difference between the proportion that are left-handed, say. And so what you could do, just like we've done here, is for each of these populations, you will take a sample here, we'll call that n one, and then from that sample, you calculate a sample proportion, let's call that p one. And then from this second population, we do the same thing. This is n two. Notice, n one and n two do not have to be the same sample size."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "And so what you could do, just like we've done here, is for each of these populations, you will take a sample here, we'll call that n one, and then from that sample, you calculate a sample proportion, let's call that p one. And then from this second population, we do the same thing. This is n two. Notice, n one and n two do not have to be the same sample size. That's a common misconception when doing these things. These could be different sample sizes. And then from that sample, you calculate the sample proportion."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "Notice, n one and n two do not have to be the same sample size. That's a common misconception when doing these things. These could be different sample sizes. And then from that sample, you calculate the sample proportion. Now, after you do that, you would wanna check your conditions for inference. And it turns out that the conditions for inference would be exactly the same. Do both of these samples meet the random condition?"}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "And then from that sample, you calculate the sample proportion. Now, after you do that, you would wanna check your conditions for inference. And it turns out that the conditions for inference would be exactly the same. Do both of these samples meet the random condition? Do both of these samples meet the normal condition? And do both of these samples meet the independence condition? And if both samples meet these conditions for inference, then we would have to calculate our critical value."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "Do both of these samples meet the random condition? Do both of these samples meet the normal condition? And do both of these samples meet the independence condition? And if both samples meet these conditions for inference, then we would have to calculate our critical value. And you would do it the exact same way. I'll just write it down again. So first, you need to check all of these."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "And if both samples meet these conditions for inference, then we would have to calculate our critical value. And you would do it the exact same way. I'll just write it down again. So first, you need to check all of these. Then you would take your confidence level, confidence level, and from that, get a critical z. And then you're ready to say what your confidence interval's going to be. So your confidence, confidence interval, interval for p one minus p two."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "So first, you need to check all of these. Then you would take your confidence level, confidence level, and from that, get a critical z. And then you're ready to say what your confidence interval's going to be. So your confidence, confidence interval, interval for p one minus p two. So it's the confidence interval for the difference between these true population proportions. That is going to be equal to the difference between your sample proportions. So p hat one minus p hat two, p hat two, plus or minus, plus or minus your critical value right over here, times the standard deviation of the sampling distribution of the difference between the sample proportions."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "So your confidence, confidence interval, interval for p one minus p two. So it's the confidence interval for the difference between these true population proportions. That is going to be equal to the difference between your sample proportions. So p hat one minus p hat two, p hat two, plus or minus, plus or minus your critical value right over here, times the standard deviation of the sampling distribution of the difference between the sample proportions. So it would be p hat one minus p hat two. And so we already know how to calculate this, how to calculate this, how to calculate this. How do we calculate that?"}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "So p hat one minus p hat two, p hat two, plus or minus, plus or minus your critical value right over here, times the standard deviation of the sampling distribution of the difference between the sample proportions. So it would be p hat one minus p hat two. And so we already know how to calculate this, how to calculate this, how to calculate this. How do we calculate that? Well, I will just give you the formula first. But then we just have to appreciate that this just comes out of the properties of standard deviations and variances that we have studied in the past. So the standard deviation of the sampling distribution of the difference between the sample proportions, it is a mouthful."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "How do we calculate that? Well, I will just give you the formula first. But then we just have to appreciate that this just comes out of the properties of standard deviations and variances that we have studied in the past. So the standard deviation of the sampling distribution of the difference between the sample proportions, it is a mouthful. This is going to be approximately equal to, approximately equal to the square root of p hat one times one minus p hat one over n one, over n one, plus, plus p hat two times one minus p hat two over n two. And then you put that there, you have constructed your confidence interval. And once again, how would you interpret that?"}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "So the standard deviation of the sampling distribution of the difference between the sample proportions, it is a mouthful. This is going to be approximately equal to, approximately equal to the square root of p hat one times one minus p hat one over n one, over n one, plus, plus p hat two times one minus p hat two over n two. And then you put that there, you have constructed your confidence interval. And once again, how would you interpret that? Well, let's say your confidence level is 90%. And from that, you're able to construct this confidence interval. That would mean that 90% of the time that you go through this exercise, your confidence interval would overlap with the true difference between these population parameters, the true difference between these population proportions."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "And once again, how would you interpret that? Well, let's say your confidence level is 90%. And from that, you're able to construct this confidence interval. That would mean that 90% of the time that you go through this exercise, your confidence interval would overlap with the true difference between these population parameters, the true difference between these population proportions. Now, where did this thing come from? Well, you might notice some similarities here. This part over here is an estimate, or it's approximately equal to the variance of the sampling distribution of the sample proportion for our first population."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "That would mean that 90% of the time that you go through this exercise, your confidence interval would overlap with the true difference between these population parameters, the true difference between these population proportions. Now, where did this thing come from? Well, you might notice some similarities here. This part over here is an estimate, or it's approximately equal to the variance of the sampling distribution of the sample proportion for our first population. And then this right over here, once again, is approximately going to be equal to the variance of the sampling distribution for the sample proportions for this population, for p two. How did I know that? Well, look, if this is approximately the standard deviation, you square that, you approximately get the variance."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "This part over here is an estimate, or it's approximately equal to the variance of the sampling distribution of the sample proportion for our first population. And then this right over here, once again, is approximately going to be equal to the variance of the sampling distribution for the sample proportions for this population, for p two. How did I know that? Well, look, if this is approximately the standard deviation, you square that, you approximately get the variance. And so the big takeaway is is that the variance for the sampling distribution of the difference is just the sum of the variances of each of those sampling distributions. That's a lot of big mouthful, I know it can get confusing. But hopefully that makes sense."}, {"video_title": "Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3", "Sentence": "Well, look, if this is approximately the standard deviation, you square that, you approximately get the variance. And so the big takeaway is is that the variance for the sampling distribution of the difference is just the sum of the variances of each of those sampling distributions. That's a lot of big mouthful, I know it can get confusing. But hopefully that makes sense. And that's where this formula comes from. And so it's really not that much more to remember. In the next few videos, we're gonna do many more examples, both looking at these conditions and calculating confidence intervals and critical values."}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So here we have a scenario. I keep picking cards from a standard deck until I get a king. So this is a classic geometric random variable here, and it's important that in this parentheses, it says I replace the cards if they are not a king. And this is important, as we talk about in other videos, because the probability of success each time can't change. And so we could define some random variable, X, this is a geometric random variable, as being equal to the number of picks until we get a king, until we get a king, when we replace the cards if they are not a king. And for this geometric random variable, what's the probability of success on each trial? And remember, one of the conditions for a geometric random variable is that probability of success does not change on each trial."}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "And this is important, as we talk about in other videos, because the probability of success each time can't change. And so we could define some random variable, X, this is a geometric random variable, as being equal to the number of picks until we get a king, until we get a king, when we replace the cards if they are not a king. And for this geometric random variable, what's the probability of success on each trial? And remember, one of the conditions for a geometric random variable is that probability of success does not change on each trial. Well, the probability of success is going to be equal to, there's four kings in a standard deck of 52, and this is the same thing as one over 13. So this first question is, what is the probability that I need to pick five cards? Well, this would be the probability that our geometric random variable, X, is equal to five."}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "And remember, one of the conditions for a geometric random variable is that probability of success does not change on each trial. Well, the probability of success is going to be equal to, there's four kings in a standard deck of 52, and this is the same thing as one over 13. So this first question is, what is the probability that I need to pick five cards? Well, this would be the probability that our geometric random variable, X, is equal to five. And you could actually figure this out by hand, but the whole point here is to think about how to use a calculator. And there is a function called geometpdf, which stands for geometric probability distribution function where what you have to pass it is the probability of success on any given trial, one out of 13, and then the particular value of that random variable that you wanna figure out the probability for. So five right over there."}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, this would be the probability that our geometric random variable, X, is equal to five. And you could actually figure this out by hand, but the whole point here is to think about how to use a calculator. And there is a function called geometpdf, which stands for geometric probability distribution function where what you have to pass it is the probability of success on any given trial, one out of 13, and then the particular value of that random variable that you wanna figure out the probability for. So five right over there. Now, just to be clear, if you're doing this on an AP exam, and this is one of the reasons why a calculator is useful, you actually can use this on an AP exam, AP statistics exam, it's important to tell the graders, if you're doing it on the free response, that this right over here is your P, and that this right over here is your five, just so it's very clear that where you actually got this information from or why you're actually typing it in. But let's just see how it works, what this probability is actually going to amount to. All right, so I have my calculator now, and I just need to type in geometpdf and then those parameters."}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So five right over there. Now, just to be clear, if you're doing this on an AP exam, and this is one of the reasons why a calculator is useful, you actually can use this on an AP exam, AP statistics exam, it's important to tell the graders, if you're doing it on the free response, that this right over here is your P, and that this right over here is your five, just so it's very clear that where you actually got this information from or why you're actually typing it in. But let's just see how it works, what this probability is actually going to amount to. All right, so I have my calculator now, and I just need to type in geometpdf and then those parameters. And so the place where I find that function, I press second, distribution right over here, it's this little blue above the VARS button, and then I can click up, I can scroll down, or I could just go to the bottom of the list, and you can see the second from the bottom is geometpdf, click Enter there. My P value, my probability of success on each trial is one out of 13, and I wanna figure out the probability that I have to pick five cards. And so then click Enter, click Enter again, and there you have it."}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "All right, so I have my calculator now, and I just need to type in geometpdf and then those parameters. And so the place where I find that function, I press second, distribution right over here, it's this little blue above the VARS button, and then I can click up, I can scroll down, or I could just go to the bottom of the list, and you can see the second from the bottom is geometpdf, click Enter there. My P value, my probability of success on each trial is one out of 13, and I wanna figure out the probability that I have to pick five cards. And so then click Enter, click Enter again, and there you have it. It's about 0.056. So this is approximately, approximately 0.056. Now let's answer another question."}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so then click Enter, click Enter again, and there you have it. It's about 0.056. So this is approximately, approximately 0.056. Now let's answer another question. So here they say, what is the probability that I need to pick less than 10 cards? So this is the probability that X is less than 10, or I could say this is equal to the probability that X is less than or equal to nine. And I could say, well, this is the probability that X is equal to one, plus the probability that X is equal to two, all the way to the probability that X is equal to nine, but that would take a while, even if I used this function right over here."}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "Now let's answer another question. So here they say, what is the probability that I need to pick less than 10 cards? So this is the probability that X is less than 10, or I could say this is equal to the probability that X is less than or equal to nine. And I could say, well, this is the probability that X is equal to one, plus the probability that X is equal to two, all the way to the probability that X is equal to nine, but that would take a while, even if I used this function right over here. But lucky for us, there's a cumulative distribution function. Take some space from the next question. This is going to be equal to geometpdf, cumulative distribution function."}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "And I could say, well, this is the probability that X is equal to one, plus the probability that X is equal to two, all the way to the probability that X is equal to nine, but that would take a while, even if I used this function right over here. But lucky for us, there's a cumulative distribution function. Take some space from the next question. This is going to be equal to geometpdf, cumulative distribution function. And once again, I pass the probability of success on any trial, and then up to and including nine. And so let's get the calculator out again. So we go to second, distribution."}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "This is going to be equal to geometpdf, cumulative distribution function. And once again, I pass the probability of success on any trial, and then up to and including nine. And so let's get the calculator out again. So we go to second, distribution. I click up, and there we have geomet cumulative distribution function. Press Enter. One out of 13 chance of success on any trial, up to and including nine, and then Enter."}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So we go to second, distribution. I click up, and there we have geomet cumulative distribution function. Press Enter. One out of 13 chance of success on any trial, up to and including nine, and then Enter. And there you have it. It's approximately 51.3%, or 0.513. So this is approximately 0.513."}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "One out of 13 chance of success on any trial, up to and including nine, and then Enter. And there you have it. It's approximately 51.3%, or 0.513. So this is approximately 0.513. Now let's do one more. What is the probability that I need to pick more than 12 cards? And like, I'll pause the video and see if you can figure this one out."}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So this is approximately 0.513. Now let's do one more. What is the probability that I need to pick more than 12 cards? And like, I'll pause the video and see if you can figure this one out. What function would I use on my calculator? How would I set it up? Well, the probability, this is the probability that X is going to be greater than 12, which is equal to one minus the probability that X is less than or equal to 12."}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "And like, I'll pause the video and see if you can figure this one out. What function would I use on my calculator? How would I set it up? Well, the probability, this is the probability that X is going to be greater than 12, which is equal to one minus the probability that X is less than or equal to 12. And now this, we can just use the cumulative distribution function again. So this is one minus geomet CDF, cumulative distribution function, CDF of one over 13, and up to and including 12. And so what is this going to be equal to?"}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, the probability, this is the probability that X is going to be greater than 12, which is equal to one minus the probability that X is less than or equal to 12. And now this, we can just use the cumulative distribution function again. So this is one minus geomet CDF, cumulative distribution function, CDF of one over 13, and up to and including 12. And so what is this going to be equal to? So second distribution, I click up, I get to the function, click Enter. And so I already have that first, the probability of success on every trial is one over 13, and then cumulative up to 12. And so I click Enter."}, {"video_title": "TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so what is this going to be equal to? So second distribution, I click up, I get to the function, click Enter. And so I already have that first, the probability of success on every trial is one over 13, and then cumulative up to 12. And so I click Enter. And then, well, I could click Enter there, but I really wanna get one minus this value. And so I can do one minus second answer, which would be just one minus that value, which will be equal to, there you have it. It's about 38.3% or 0.383."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "What we're going to do in this video is think about how to visualize distributions of data, then to analyze those visualizations, and we will eventually get to something known as a density curve. But let's start with a simple example, just to review some concepts. Let's say I go to 16 students, and I ask them to measure how many glasses of water they drink per day for the last 30 days, and then to average it. And so this data point right over here tells us one student drank an average of 0.5 glasses of water per day. That person is probably very dehydrated. This person drank 8.1 glasses of water per day on average for the last 30 days. They are better hydrated."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And so this data point right over here tells us one student drank an average of 0.5 glasses of water per day. That person is probably very dehydrated. This person drank 8.1 glasses of water per day on average for the last 30 days. They are better hydrated. If we want to visualize that, we can set up a frequency histogram, where we can create some categories. So this first category would be for data points that are greater than or equal to zero and less than one. And we can see that two data points fall into that category, and that's why the bar right over here for that category is up to two."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "They are better hydrated. If we want to visualize that, we can set up a frequency histogram, where we can create some categories. So this first category would be for data points that are greater than or equal to zero and less than one. And we can see that two data points fall into that category, and that's why the bar right over here for that category is up to two. This category right over here is greater than or equal to three and less than four. Notice there are four data points in that category, and on this frequency histogram, the height of the bar is indeed four. So this is a nice way of looking at a distribution, but you might be more concerned with what percentage of my data falls into each of these categories."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And we can see that two data points fall into that category, and that's why the bar right over here for that category is up to two. This category right over here is greater than or equal to three and less than four. Notice there are four data points in that category, and on this frequency histogram, the height of the bar is indeed four. So this is a nice way of looking at a distribution, but you might be more concerned with what percentage of my data falls into each of these categories. And that becomes especially interesting if we have many, many, many, many data points. And if we had 1,600,432,507 data points, well, just knowing the absolute number that fit into each category isn't so useful. The percent that fits into each category is a lot more useful."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So this is a nice way of looking at a distribution, but you might be more concerned with what percentage of my data falls into each of these categories. And that becomes especially interesting if we have many, many, many, many data points. And if we had 1,600,432,507 data points, well, just knowing the absolute number that fit into each category isn't so useful. The percent that fits into each category is a lot more useful. And so for that, we could set up a relative frequency histogram. So notice, this is representing the same data. But in that first category, instead of the bar height being two, the bar height is now 12.5%."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "The percent that fits into each category is a lot more useful. And so for that, we could set up a relative frequency histogram. So notice, this is representing the same data. But in that first category, instead of the bar height being two, the bar height is now 12.5%. Why is that? Because two of the 16 data points fall into this category. 2 16ths is 1 8th, which is 12.5%."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "But in that first category, instead of the bar height being two, the bar height is now 12.5%. Why is that? Because two of the 16 data points fall into this category. 2 16ths is 1 8th, which is 12.5%. And this one right over here, notice, instead of the height being four, for four data points, it's now 25%. But these are saying the same thing. Four out of the 16 data points fall into this category."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "2 16ths is 1 8th, which is 12.5%. And this one right over here, notice, instead of the height being four, for four data points, it's now 25%. But these are saying the same thing. Four out of the 16 data points fall into this category. 4 16ths is 1 4th, which is 25%. So both of these types of histograms are really useful, and you will see them used all of the time. But there are also cases where you have many, many, many more data points, and you want more granular categories."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Four out of the 16 data points fall into this category. 4 16ths is 1 4th, which is 25%. So both of these types of histograms are really useful, and you will see them used all of the time. But there are also cases where you have many, many, many more data points, and you want more granular categories. So what you could do is, well, let's just make our categories a little more granular. So for example, instead of them being one glass of water wide, maybe you make them half a glass of water wide. So this first category could be greater than or equal to zero, and less than 0.5."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "But there are also cases where you have many, many, many more data points, and you want more granular categories. So what you could do is, well, let's just make our categories a little more granular. So for example, instead of them being one glass of water wide, maybe you make them half a glass of water wide. So this first category could be greater than or equal to zero, and less than 0.5. And that will give you a clearer picture. And I'm now assuming a world where we have more than 16 data points. Maybe we have 16 million data points."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So this first category could be greater than or equal to zero, and less than 0.5. And that will give you a clearer picture. And I'm now assuming a world where we have more than 16 data points. Maybe we have 16 million data points. This would be percentages on the left-hand side. But maybe that isn't good enough for you. Maybe you wanna get even more granular."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Maybe we have 16 million data points. This would be percentages on the left-hand side. But maybe that isn't good enough for you. Maybe you wanna get even more granular. So you make everything, each category, a quarter of a glass. But maybe that doesn't satisfy you. You wanna get more and more and more granular."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Maybe you wanna get even more granular. So you make everything, each category, a quarter of a glass. But maybe that doesn't satisfy you. You wanna get more and more and more granular. Well, you could imagine where this is going. You could get to a point where you're approaching an infinite number of categories, and each category is infinitely thin, is super, super thin, to a point that if you just connect the tops of the bars, that you will actually get a curve. And this type of curve is something that we actually use in the statistics."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "You wanna get more and more and more granular. Well, you could imagine where this is going. You could get to a point where you're approaching an infinite number of categories, and each category is infinitely thin, is super, super thin, to a point that if you just connect the tops of the bars, that you will actually get a curve. And this type of curve is something that we actually use in the statistics. And as promised at the beginning of the video, this is the density curve we talk about. And what's valuable about a density curve, it is a visualization of a distribution where the data points can take on any value in a continuum. They're not just thrown into these coarse buckets."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And this type of curve is something that we actually use in the statistics. And as promised at the beginning of the video, this is the density curve we talk about. And what's valuable about a density curve, it is a visualization of a distribution where the data points can take on any value in a continuum. They're not just thrown into these coarse buckets. So how would you interpret something like this? If you look over the entire interval from zero, let's say, to nine, assuming no one drank more than an average of nine glasses per day, even in our 16 million data points, well then the area under the curve over that interval is going to be 100%, or 1.0. This is going to be true for any density curve, that the entire area of the curve is 100%."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "They're not just thrown into these coarse buckets. So how would you interpret something like this? If you look over the entire interval from zero, let's say, to nine, assuming no one drank more than an average of nine glasses per day, even in our 16 million data points, well then the area under the curve over that interval is going to be 100%, or 1.0. This is going to be true for any density curve, that the entire area of the curve is 100%. It represents all of the data points. A density curve will also never take on a negative value. You won't see the curve dip down and do something strange like that."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "This is going to be true for any density curve, that the entire area of the curve is 100%. It represents all of the data points. A density curve will also never take on a negative value. You won't see the curve dip down and do something strange like that. Now with that out of the way, let's think about how we would make use of it. If I wanted to know what percentage of my data falls between two and four glasses, well I would look at that interval. I'd go from two to four."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "You won't see the curve dip down and do something strange like that. Now with that out of the way, let's think about how we would make use of it. If I wanted to know what percentage of my data falls between two and four glasses, well I would look at that interval. I'd go from two to four. I would look at this interval right over here, and I would try to figure out the area under the curve here. And this area is going to be greater than or equal to zero and less than or equal to 100%. When I eyeball it right over here, it looks like it's about 40% of the entire area under the curve."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "I'd go from two to four. I would look at this interval right over here, and I would try to figure out the area under the curve here. And this area is going to be greater than or equal to zero and less than or equal to 100%. When I eyeball it right over here, it looks like it's about 40% of the entire area under the curve. So just eyeballing it, I would say roughly 40% of my data falls into this interval. If I were to ask you what percentage of the data is greater than three, well then you would be looking at this area, and it looks like it is about 50%, but once again, I am estimating it. But you can start to see how even with estimation, a density curve could be useful."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "When I eyeball it right over here, it looks like it's about 40% of the entire area under the curve. So just eyeballing it, I would say roughly 40% of my data falls into this interval. If I were to ask you what percentage of the data is greater than three, well then you would be looking at this area, and it looks like it is about 50%, but once again, I am estimating it. But you can start to see how even with estimation, a density curve could be useful. In the real world, statisticians will often have tables that might represent the information for the density curve. They might have computer programs or some type of automated tool, and there are also well-known density curves, the famous bell curve that we will study later on where there's a lot of precise data and a lot of tools to exactly figure out the areas. The last thing I'd like to cover is a key misconception for density curves."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "But you can start to see how even with estimation, a density curve could be useful. In the real world, statisticians will often have tables that might represent the information for the density curve. They might have computer programs or some type of automated tool, and there are also well-known density curves, the famous bell curve that we will study later on where there's a lot of precise data and a lot of tools to exactly figure out the areas. The last thing I'd like to cover is a key misconception for density curves. If I were to ask you approximately what percentage of my data is exactly three glasses of water per day, and when I say exactly, I mean exactly the number 3.000 with zeros just going on and on forever, the exact number three. So you might be tempted to just say, okay, this is three. Let me see the corresponding point on the curve."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "The last thing I'd like to cover is a key misconception for density curves. If I were to ask you approximately what percentage of my data is exactly three glasses of water per day, and when I say exactly, I mean exactly the number 3.000 with zeros just going on and on forever, the exact number three. So you might be tempted to just say, okay, this is three. Let me see the corresponding point on the curve. It looks like it is about 0.2 or a little higher than that, so maybe you would say a little bit more than 20% or approximately 20%. And what I would say to you is this is wrong. Remember, the percentage of the data in an interval is not the height of the curve."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Let me see the corresponding point on the curve. It looks like it is about 0.2 or a little higher than that, so maybe you would say a little bit more than 20% or approximately 20%. And what I would say to you is this is wrong. Remember, the percentage of the data in an interval is not the height of the curve. It is the area under the curve in that interval. And if we're just talking about one precise value, like exactly the number three, there is no area under the curve. This vertical line that I just drew over the number of three has no width, and this actually makes sense in the real world."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Remember, the percentage of the data in an interval is not the height of the curve. It is the area under the curve in that interval. And if we're just talking about one precise value, like exactly the number three, there is no area under the curve. This vertical line that I just drew over the number of three has no width, and this actually makes sense in the real world. Even if you were to look at 16 million people, it is very unlikely that even anyone would drink exactly three glasses of water per day. I'm talking about not one atom more or one atom less than three glasses. There might be many people between 2.9 and 3.1, but no one is exactly three glasses a day."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "This vertical line that I just drew over the number of three has no width, and this actually makes sense in the real world. Even if you were to look at 16 million people, it is very unlikely that even anyone would drink exactly three glasses of water per day. I'm talking about not one atom more or one atom less than three glasses. There might be many people between 2.9 and 3.1, but no one is exactly three glasses a day. When someone says I'm drinking three glasses of water per day, that'd be a rough estimate. They're probably 3.0001 or 2.99999 or 3.15 or whatever else. And so instead, you could say what percentage falls in the interval maybe that is greater than or equal to 2.9 and less than or equal to 3.1."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "There might be many people between 2.9 and 3.1, but no one is exactly three glasses a day. When someone says I'm drinking three glasses of water per day, that'd be a rough estimate. They're probably 3.0001 or 2.99999 or 3.15 or whatever else. And so instead, you could say what percentage falls in the interval maybe that is greater than or equal to 2.9 and less than or equal to 3.1. And so once you have an interval, then you actually can look at the area. So we're gonna go from 2.9 to 3.1. So now we have an interval that actually has width, and so it'd be roughly the size of this yellow area that I'm shading in right over here."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And so instead, you could say what percentage falls in the interval maybe that is greater than or equal to 2.9 and less than or equal to 3.1. And so once you have an interval, then you actually can look at the area. So we're gonna go from 2.9 to 3.1. So now we have an interval that actually has width, and so it'd be roughly the size of this yellow area that I'm shading in right over here. And we can approximate it with a rectangle, even though the top of this curve isn't flat, so we could say, look, it's approximately like a rectangle that is 0.2 high. And what's the width? The width here, if we're going from 2.9 to 3.1, the width is going to be 0.2 wide."}, {"video_title": "Density Curves Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So now we have an interval that actually has width, and so it'd be roughly the size of this yellow area that I'm shading in right over here. And we can approximate it with a rectangle, even though the top of this curve isn't flat, so we could say, look, it's approximately like a rectangle that is 0.2 high. And what's the width? The width here, if we're going from 2.9 to 3.1, the width is going to be 0.2 wide. And so we could approximate this area by approximating this rectangle, the area of the rectangle. 0.2 times 0.2, that would give us an area of 0.04. Or we could say approximately 4% of the data falls in this interval."}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So for example, one driver drives one hour a day. Two drivers drive two hours a day. One driver drives three hours a day. It looks like there's five drivers that drive seven hours a day. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? And then they give us some choices. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours?"}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "It looks like there's five drivers that drive seven hours a day. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? And then they give us some choices. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? So pause the video and see if you can figure out which of these percentiles is the closest estimate to the percentile rank of a driver with a daily driving time of six hours, looking at this data right over here. All right, now let's work through this together. So when you think about percentile, you really wanna think about, so let me write this down."}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? So pause the video and see if you can figure out which of these percentiles is the closest estimate to the percentile rank of a driver with a daily driving time of six hours, looking at this data right over here. All right, now let's work through this together. So when you think about percentile, you really wanna think about, so let me write this down. When we're talking about percentile, we're really saying the percentage of the data that, and there's actually two ways that you could compute it. One is the percentage of the data that is below the amount in question, amount in question. The other possibility is the percent of the data that is at or below, that is at or below the amount, the amount in question."}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So when you think about percentile, you really wanna think about, so let me write this down. When we're talking about percentile, we're really saying the percentage of the data that, and there's actually two ways that you could compute it. One is the percentage of the data that is below the amount in question, amount in question. The other possibility is the percent of the data that is at or below, that is at or below the amount, the amount in question. So if we look at this right over here, let's just figure out how many data points, what percentage of the data points are below six hours per day? So let's see, there are, I'm just gonna count them. One, two, three, four, five, six, seven."}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "The other possibility is the percent of the data that is at or below, that is at or below the amount, the amount in question. So if we look at this right over here, let's just figure out how many data points, what percentage of the data points are below six hours per day? So let's see, there are, I'm just gonna count them. One, two, three, four, five, six, seven. So seven of the 14 are below six hours. So we could just say seven, if we use this first technique, we would have seven of the 14 are below six hours per day, and so that would get us a number of 50%, that six hours is at the 50th percentile. If we wanna say what percentage is at that number or below, then we would also count this one."}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "One, two, three, four, five, six, seven. So seven of the 14 are below six hours. So we could just say seven, if we use this first technique, we would have seven of the 14 are below six hours per day, and so that would get us a number of 50%, that six hours is at the 50th percentile. If we wanna say what percentage is at that number or below, then we would also count this one. So we would say eight or eight out of 14, eight out of 14, which is the same thing as four out of seven and if we wanna write that as a decimal, let's see, seven goes into 4.000, we just need to estimate. So seven goes into 45 times 35, we subtract, we get a five, bring down a zero, goes five times, I guess it's just gonna be.5 repeating. So 55.5555%."}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "If we wanna say what percentage is at that number or below, then we would also count this one. So we would say eight or eight out of 14, eight out of 14, which is the same thing as four out of seven and if we wanna write that as a decimal, let's see, seven goes into 4.000, we just need to estimate. So seven goes into 45 times 35, we subtract, we get a five, bring down a zero, goes five times, I guess it's just gonna be.5 repeating. So 55.5555%. So either of these would actually be a legitimate response to the percentile rank for the driver with a daily driving time of six hours. It depends on whether you include the six hours or not. So you could say either the 50th percentile or the roughly the 55th, well actually the 56th percentile if you wanted to round to the nearest percentile."}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So 55.5555%. So either of these would actually be a legitimate response to the percentile rank for the driver with a daily driving time of six hours. It depends on whether you include the six hours or not. So you could say either the 50th percentile or the roughly the 55th, well actually the 56th percentile if you wanted to round to the nearest percentile. Now if you look at these choices here, lucky for us, there's only one choice that's reasonably close to either one of those and that's the 55th percentile. And it looks like the people who wrote this question went with the calculation of percentile where they include the data point in question. So everything at six hours or less, what percentage of the total data is that?"}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "So now I'm going to think about, I'm going to take a fair coin, and I'm going to flip it three times. And I want to find the probability of at least one head out of the three flips. So the easiest way to think about this is how many equally likely possibilities there are. In the last video we saw, if we flip a coin three times, there's eight possibilities. For the first flip, there's two possibilities. Second flip, there's two possibilities. And in the third flip, there are two possibilities."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "In the last video we saw, if we flip a coin three times, there's eight possibilities. For the first flip, there's two possibilities. Second flip, there's two possibilities. And in the third flip, there are two possibilities. So 2 times 2 times 2, there are eight equally likely possibilities if I'm flipping a coin three times. Now how many of those possibilities have at least one head? Well, we drew all of the possibilities over here, so we just have to count how many of these have at least one head."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "And in the third flip, there are two possibilities. So 2 times 2 times 2, there are eight equally likely possibilities if I'm flipping a coin three times. Now how many of those possibilities have at least one head? Well, we drew all of the possibilities over here, so we just have to count how many of these have at least one head. So that's 1, 2, 3, 4, 5, 6, 7. So 7 of these have at least one head in them, and this last one does not. So 7 of the 8 have at least one head."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "Well, we drew all of the possibilities over here, so we just have to count how many of these have at least one head. So that's 1, 2, 3, 4, 5, 6, 7. So 7 of these have at least one head in them, and this last one does not. So 7 of the 8 have at least one head. Now you're probably thinking, OK, Sal, you were able to do it by writing out all of the possibilities, but that would be really hard if I said at least one head out of 20 flips. This would work well because I only had three flips. So let me make it clear."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "So 7 of the 8 have at least one head. Now you're probably thinking, OK, Sal, you were able to do it by writing out all of the possibilities, but that would be really hard if I said at least one head out of 20 flips. This would work well because I only had three flips. So let me make it clear. This is in three flips. This would have been a lot harder to do or more time-consuming to do if I had 20 flips. Is there some shortcut here?"}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "So let me make it clear. This is in three flips. This would have been a lot harder to do or more time-consuming to do if I had 20 flips. Is there some shortcut here? Is there some other way to think about it? And you couldn't just do it in some simple way. You can't just say, oh, probability of heads times probability of heads, because if you got heads the first time, then now you don't have to get heads anymore."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "Is there some shortcut here? Is there some other way to think about it? And you couldn't just do it in some simple way. You can't just say, oh, probability of heads times probability of heads, because if you got heads the first time, then now you don't have to get heads anymore. Or you could get heads again, but you don't have to. So it becomes a little bit more complicated. But there is an easy way to think about it where you could use this methodology right over here."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "You can't just say, oh, probability of heads times probability of heads, because if you got heads the first time, then now you don't have to get heads anymore. Or you could get heads again, but you don't have to. So it becomes a little bit more complicated. But there is an easy way to think about it where you could use this methodology right over here. You'll actually see this on a lot of exams where they make it seem like a harder problem, but if you just think about it the right way, all of a sudden it becomes simpler. One way to think about it is the probability of at least one heads in three flips is the same thing as the probability of not getting all tails. If we got all tails, then we don't have at least one head."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "But there is an easy way to think about it where you could use this methodology right over here. You'll actually see this on a lot of exams where they make it seem like a harder problem, but if you just think about it the right way, all of a sudden it becomes simpler. One way to think about it is the probability of at least one heads in three flips is the same thing as the probability of not getting all tails. If we got all tails, then we don't have at least one head. So these two things are equivalent. The probability of getting at least one head in three flips is the same thing as the probability of not getting all tails in three flips. Let me write in three flips."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "If we got all tails, then we don't have at least one head. So these two things are equivalent. The probability of getting at least one head in three flips is the same thing as the probability of not getting all tails in three flips. Let me write in three flips. So what's the probability of not getting all tails? Well, that's going to be 1 minus the probability of getting all tails. And since it's three flips, it's the probability of tails, tails, and tails."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "Let me write in three flips. So what's the probability of not getting all tails? Well, that's going to be 1 minus the probability of getting all tails. And since it's three flips, it's the probability of tails, tails, and tails. Because any of the other situations are going to have at least one head in them. And that's all of the other possibilities. And this is the only other leftover possibility."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "And since it's three flips, it's the probability of tails, tails, and tails. Because any of the other situations are going to have at least one head in them. And that's all of the other possibilities. And this is the only other leftover possibility. If you add them together, you're going to get one. Let me write it this way. The probability of not all tails plus the probability of all tails, well, this is essentially exhaustive."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "And this is the only other leftover possibility. If you add them together, you're going to get one. Let me write it this way. The probability of not all tails plus the probability of all tails, well, this is essentially exhaustive. This is all of the possible circumstances. So your chances of getting either not all tails or all tails, and these are mutual exclusives, so we can add them, so the probability of not all tails or the probability of all tails is going to be equal to 1. These are mutual exclusives."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "The probability of not all tails plus the probability of all tails, well, this is essentially exhaustive. This is all of the possible circumstances. So your chances of getting either not all tails or all tails, and these are mutual exclusives, so we can add them, so the probability of not all tails or the probability of all tails is going to be equal to 1. These are mutual exclusives. You're either going to have not all tails, which means a head shows up, or you're going to have all tails, but you can't have both of these things happening. And since they're mutual exclusives, and you're saying the probability of this or this happening, you can add their probabilities. And this is essentially all of the possible events."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "These are mutual exclusives. You're either going to have not all tails, which means a head shows up, or you're going to have all tails, but you can't have both of these things happening. And since they're mutual exclusives, and you're saying the probability of this or this happening, you can add their probabilities. And this is essentially all of the possible events. So this is essentially, if you combine these, this is the probability of any of the events happening, and that's going to be a 1 or 100% chance. So another way to think about it is the probability of not all tails is going to be 1 minus the probability of all tails. So that's what we did right over here."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "And this is essentially all of the possible events. So this is essentially, if you combine these, this is the probability of any of the events happening, and that's going to be a 1 or 100% chance. So another way to think about it is the probability of not all tails is going to be 1 minus the probability of all tails. So that's what we did right over here. And the probability of all tails is pretty straightforward. That's the probability of it's going to be 1 half, because you have a 1 half chance of getting a tails on the first flip, times, let me write it here so it becomes a little clearer, so this is going to be 1 minus the probability of getting all tails, well, you have a 1 half chance of getting tails on the first flip, and then you're going to have to get another tails on the second flip, and then you're going to have to get another tails on the third flip. And then 1 half times 1 half times 1 half, this is going to be 1 eighth."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "So that's what we did right over here. And the probability of all tails is pretty straightforward. That's the probability of it's going to be 1 half, because you have a 1 half chance of getting a tails on the first flip, times, let me write it here so it becomes a little clearer, so this is going to be 1 minus the probability of getting all tails, well, you have a 1 half chance of getting tails on the first flip, and then you're going to have to get another tails on the second flip, and then you're going to have to get another tails on the third flip. And then 1 half times 1 half times 1 half, this is going to be 1 eighth. And then 1 minus 1 eighth, or 8 eighths minus 1 eighth, is going to be equal to 7 eighths. So we can apply that to a problem that is harder to do than writing all of the scenarios like we did in the first problem, we can say, let's say we have 10 flips, the probability of at least 1 head in 10 flips. Well, we use the same idea, this is the same thing, this is going to be equal to the probability of not all tails in 10 flips."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "And then 1 half times 1 half times 1 half, this is going to be 1 eighth. And then 1 minus 1 eighth, or 8 eighths minus 1 eighth, is going to be equal to 7 eighths. So we can apply that to a problem that is harder to do than writing all of the scenarios like we did in the first problem, we can say, let's say we have 10 flips, the probability of at least 1 head in 10 flips. Well, we use the same idea, this is the same thing, this is going to be equal to the probability of not all tails in 10 flips. So we're just saying the probability of not getting all of the flips going to be tails, all of the flips is tails, not all tails in 10 flips. And this is going to be 1 minus the probability of flipping tails 10 times. So it's 1 minus 10 tails in a row."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "Well, we use the same idea, this is the same thing, this is going to be equal to the probability of not all tails in 10 flips. So we're just saying the probability of not getting all of the flips going to be tails, all of the flips is tails, not all tails in 10 flips. And this is going to be 1 minus the probability of flipping tails 10 times. So it's 1 minus 10 tails in a row. And so this is going to be equal to, this part right over here, let me write this. So this is going to be this 1, let me just rewrite it, this is equal to 1 minus, and this part is going to be, well, 1 tail, another tail, so it's 1 half times 1 half, and I'm going to do this 10 times. Let me write this a little neater, because I need a 1 half, so that's 5, 6, 7, 8, 9, and 10."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "So it's 1 minus 10 tails in a row. And so this is going to be equal to, this part right over here, let me write this. So this is going to be this 1, let me just rewrite it, this is equal to 1 minus, and this part is going to be, well, 1 tail, another tail, so it's 1 half times 1 half, and I'm going to do this 10 times. Let me write this a little neater, because I need a 1 half, so that's 5, 6, 7, 8, 9, and 10. And so we really just have to, the numerator is going to be 1. So this is going to be 1, this is going to be equal to 1, let me do it in that same color of green. This is going to be equal to 1 minus, our numerator, you just have 1 times itself 10 times, so that's 1."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "Let me write this a little neater, because I need a 1 half, so that's 5, 6, 7, 8, 9, and 10. And so we really just have to, the numerator is going to be 1. So this is going to be 1, this is going to be equal to 1, let me do it in that same color of green. This is going to be equal to 1 minus, our numerator, you just have 1 times itself 10 times, so that's 1. And then on the denominator, you have 2 times 2 is 4, 4 times 2 is 8, 16, 32, 64, 128, 256, 512, 1,024. This is the same exact same thing as 1 is 1024 over 1024 minus 1 over 1024, which is equal to 1023, or 1,023, over 1,024, we have a common denominator here, so 1,000, doing that same blue, over 1,000 and 1,024. So if you flip a coin 10 times in a row, a fair coin, you're probability of getting at least one heads in that 10 flips is pretty high."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "This is going to be equal to 1 minus, our numerator, you just have 1 times itself 10 times, so that's 1. And then on the denominator, you have 2 times 2 is 4, 4 times 2 is 8, 16, 32, 64, 128, 256, 512, 1,024. This is the same exact same thing as 1 is 1024 over 1024 minus 1 over 1024, which is equal to 1023, or 1,023, over 1,024, we have a common denominator here, so 1,000, doing that same blue, over 1,000 and 1,024. So if you flip a coin 10 times in a row, a fair coin, you're probability of getting at least one heads in that 10 flips is pretty high. It's 1,023 over 1,024, and you can get a calculator out to figure that out in terms of a percentage. Actually, let me just do that just for fun. So if we have 1,023 divided by 1,024, that gives us, you have a 99.9% chance that you're going to have at least one heads."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "So if you flip a coin 10 times in a row, a fair coin, you're probability of getting at least one heads in that 10 flips is pretty high. It's 1,023 over 1,024, and you can get a calculator out to figure that out in terms of a percentage. Actually, let me just do that just for fun. So if we have 1,023 divided by 1,024, that gives us, you have a 99.9% chance that you're going to have at least one heads. So this is, if we round, this is equal to 99.9% chance. And I rounded a little bit. It's actually even slightly higher than that."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "So if we have 1,023 divided by 1,024, that gives us, you have a 99.9% chance that you're going to have at least one heads. So this is, if we round, this is equal to 99.9% chance. And I rounded a little bit. It's actually even slightly higher than that. And this is a pretty powerful tool, or a pretty powerful way to think about it, because it would have taken you forever to write all of the scenarios down. In fact, there would have been 1,024 scenarios to write down. So doing this exercise for 10 flips would have taken up all of our time."}, {"video_title": "Coin flipping probability Probability and Statistics Khan Academy.mp3", "Sentence": "It's actually even slightly higher than that. And this is a pretty powerful tool, or a pretty powerful way to think about it, because it would have taken you forever to write all of the scenarios down. In fact, there would have been 1,024 scenarios to write down. So doing this exercise for 10 flips would have taken up all of our time. But when you think about it in a slightly different way, when you just say, look, the probability of getting at least one heads in 10 flips is the same thing as the probability of not getting all tails. That's 1 minus the probability of getting all tails. And this is actually a pretty easy thing to think about."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "What proportion of exam scores are higher than Ludwig's score? Give your answer correct to four decimal places. So let's just visualize what's going on here. So the scores are normally distributed. So it would look like this. So the distribution would look something like that. Trying to make that pretty symmetric looking."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So the scores are normally distributed. So it would look like this. So the distribution would look something like that. Trying to make that pretty symmetric looking. The mean is 40 points. So that would be 40 points right over there. Standard deviation is three points."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "Trying to make that pretty symmetric looking. The mean is 40 points. So that would be 40 points right over there. Standard deviation is three points. So this could be one standard deviation above the mean. That would be one standard deviation below the mean. And once again, this is just very rough."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "Standard deviation is three points. So this could be one standard deviation above the mean. That would be one standard deviation below the mean. And once again, this is just very rough. And so this would be 43. This would be 37 right over here. And they say Ludwig got a score of 47.5 points on the exam."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "And once again, this is just very rough. And so this would be 43. This would be 37 right over here. And they say Ludwig got a score of 47.5 points on the exam. So Ludwig's score is going to be someplace around here. So Ludwig got a 47.5 on the exam. And they're saying what proportion of exam scores are higher than Ludwig's score?"}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "And they say Ludwig got a score of 47.5 points on the exam. So Ludwig's score is going to be someplace around here. So Ludwig got a 47.5 on the exam. And they're saying what proportion of exam scores are higher than Ludwig's score? So what we need to do is figure out what is the area under the normal distribution curve that is above 47.5? So the way we will tackle this is we will figure out the z-score for 47.5. How many standard deviations above the mean is that?"}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "And they're saying what proportion of exam scores are higher than Ludwig's score? So what we need to do is figure out what is the area under the normal distribution curve that is above 47.5? So the way we will tackle this is we will figure out the z-score for 47.5. How many standard deviations above the mean is that? Then we will look at a z-table to figure out what proportion is below that. Because that's what z-tables give us. They give us the proportion that is below a certain z-score."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "How many standard deviations above the mean is that? Then we will look at a z-table to figure out what proportion is below that. Because that's what z-tables give us. They give us the proportion that is below a certain z-score. And then we could take one minus that to figure out the proportion that is above. Remember, the entire area under the curve is one. So if we can figure out this orange area and take one minus that, we're gonna get the red area."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "They give us the proportion that is below a certain z-score. And then we could take one minus that to figure out the proportion that is above. Remember, the entire area under the curve is one. So if we can figure out this orange area and take one minus that, we're gonna get the red area. So let's do that. So first of all, let's figure out the z-score for 47.5. So let's see."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So if we can figure out this orange area and take one minus that, we're gonna get the red area. So let's do that. So first of all, let's figure out the z-score for 47.5. So let's see. We would take 47.5 and we would subtract the mean. So this is his score. We'll subtract the mean minus 40."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So let's see. We would take 47.5 and we would subtract the mean. So this is his score. We'll subtract the mean minus 40. We know what that's gonna be. That's 7.5. So that's how much more above the mean."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "We'll subtract the mean minus 40. We know what that's gonna be. That's 7.5. So that's how much more above the mean. But how many standard deviations is that? Well, each standard deviation is three. So what's 7.5 divided by three?"}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So that's how much more above the mean. But how many standard deviations is that? Well, each standard deviation is three. So what's 7.5 divided by three? This just means the previous answer divided by three. So he has 2.5 standard deviations above the mean. So the z-score here, z-score here is a positive 2.5."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So what's 7.5 divided by three? This just means the previous answer divided by three. So he has 2.5 standard deviations above the mean. So the z-score here, z-score here is a positive 2.5. If he was below the mean, it would be a negative. So now we can look at a z-table to figure out what proportion is less than 2.5 standard deviations above the mean. So that'll give us that orange and then we'll subtract that from one."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So the z-score here, z-score here is a positive 2.5. If he was below the mean, it would be a negative. So now we can look at a z-table to figure out what proportion is less than 2.5 standard deviations above the mean. So that'll give us that orange and then we'll subtract that from one. So let's get our z-table. So here we go. And we've already done this in previous videos."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So that'll give us that orange and then we'll subtract that from one. So let's get our z-table. So here we go. And we've already done this in previous videos. But what's going on here is this left column gives us our z-score up to the tenths place. And then these other columns give us the hundredths place. So what we wanna do is find 2.5 right over here on the left."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "And we've already done this in previous videos. But what's going on here is this left column gives us our z-score up to the tenths place. And then these other columns give us the hundredths place. So what we wanna do is find 2.5 right over here on the left. And it's actually gonna be 2.50. There's no, there's zero hundredths here. So we're gonna, we wanna look up 2.50."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So what we wanna do is find 2.5 right over here on the left. And it's actually gonna be 2.50. There's no, there's zero hundredths here. So we're gonna, we wanna look up 2.50. So let me scroll my z-table. So I'm gonna go down to 2.5. Alright, I think I am there."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So we're gonna, we wanna look up 2.50. So let me scroll my z-table. So I'm gonna go down to 2.5. Alright, I think I am there. So what I have here, so I have 2.5. So I am going to be in this row. And it's now scrolled off, but this first column we saw, this is the hundredths place and this is zero hundredths."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "Alright, I think I am there. So what I have here, so I have 2.5. So I am going to be in this row. And it's now scrolled off, but this first column we saw, this is the hundredths place and this is zero hundredths. And so 2.50 puts us right over here. Now you might be tempted to say, okay, that's the proportion that scores higher than Ludwig, but you'd be wrong. This is the proportion that scores lower than Ludwig."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "And it's now scrolled off, but this first column we saw, this is the hundredths place and this is zero hundredths. And so 2.50 puts us right over here. Now you might be tempted to say, okay, that's the proportion that scores higher than Ludwig, but you'd be wrong. This is the proportion that scores lower than Ludwig. So what we wanna do is take one minus this value. So let me get my calculator out again. So what I'm going to do is I'm going to take one minus this."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "This is the proportion that scores lower than Ludwig. So what we wanna do is take one minus this value. So let me get my calculator out again. So what I'm going to do is I'm going to take one minus this. One minus 0.9938 is equal to, now this is, so this is the proportion that scores less than Ludwig. One minus that is gonna be the proportion that scores more than him. The reason why we have to do this is because the z-table gives us the proportion less than a certain z-score."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So what I'm going to do is I'm going to take one minus this. One minus 0.9938 is equal to, now this is, so this is the proportion that scores less than Ludwig. One minus that is gonna be the proportion that scores more than him. The reason why we have to do this is because the z-table gives us the proportion less than a certain z-score. So this gives us right over here, 0.0062. So that's the proportion. If you thought of it in percent, it would be.62% scores higher than Ludwig."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "The reason why we have to do this is because the z-table gives us the proportion less than a certain z-score. So this gives us right over here, 0.0062. So that's the proportion. If you thought of it in percent, it would be.62% scores higher than Ludwig. And that makes sense, because Ludwig scored over two standard deviations, two and a half standard deviations above the mean. So our answer here is 0.0062. So this is going to be 0.0062."}, {"video_title": "Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And we talked about different scenarios. We could use a z-table plus the true population standard deviation, and that actually would construct pretty valid confidence intervals. But the problem is you don't know the population standard deviation. And so you might try to use a z-table to find your critical values, plus the sample standard deviation. But what we talked about is that this doesn't actually do a good job of calculating our confidence intervals. And we're going to see that experimentally in a few seconds. And so instead, we have something called a t-statistic, where if we want our critical value, we use a t-table instead of a z-table."}, {"video_title": "Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And so you might try to use a z-table to find your critical values, plus the sample standard deviation. But what we talked about is that this doesn't actually do a good job of calculating our confidence intervals. And we're going to see that experimentally in a few seconds. And so instead, we have something called a t-statistic, where if we want our critical value, we use a t-table instead of a z-table. And then we use that in conjunction with our sample standard deviation. And then all of a sudden, we are actually going to have pretty good confidence intervals. To make this a little bit more real, let's look at a simulation."}, {"video_title": "Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And so instead, we have something called a t-statistic, where if we want our critical value, we use a t-table instead of a z-table. And then we use that in conjunction with our sample standard deviation. And then all of a sudden, we are actually going to have pretty good confidence intervals. To make this a little bit more real, let's look at a simulation. So this is a scratch pad on Khan Academy made by Khan Academy user Charlotte Allen. And the whole point there is to see what our confidence intervals look like with these different scenarios. So let's say we have a true population mean of 2.0 for some, let's say it's the average number, the mean number of apples people eat a day."}, {"video_title": "Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "To make this a little bit more real, let's look at a simulation. So this is a scratch pad on Khan Academy made by Khan Academy user Charlotte Allen. And the whole point there is to see what our confidence intervals look like with these different scenarios. So let's say we have a true population mean of 2.0 for some, let's say it's the average number, the mean number of apples people eat a day. The true population mean is two. That seems high, but maybe it's in a certain country that has a lot of apples. And let's say we know that the population standard deviation is 0.5."}, {"video_title": "Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So let's say we have a true population mean of 2.0 for some, let's say it's the average number, the mean number of apples people eat a day. The true population mean is two. That seems high, but maybe it's in a certain country that has a lot of apples. And let's say we know that the population standard deviation is 0.5. And we're gonna create confidence intervals with the goal of having a 95% confidence level. And we're gonna take sample sizes of 12. So first, we can construct our confidence intervals using z and sigma, which is a legitimate way to do it."}, {"video_title": "Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "And let's say we know that the population standard deviation is 0.5. And we're gonna create confidence intervals with the goal of having a 95% confidence level. And we're gonna take sample sizes of 12. So first, we can construct our confidence intervals using z and sigma, which is a legitimate way to do it. And so let's just draw a bunch of samples here. And so we do see that it looks like it is roughly 95% when we keep making these samples and constructing these confidence intervals, that 95% of the time, these confidence intervals contain our true population mean. So these look like good confidence intervals."}, {"video_title": "Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So first, we can construct our confidence intervals using z and sigma, which is a legitimate way to do it. And so let's just draw a bunch of samples here. And so we do see that it looks like it is roughly 95% when we keep making these samples and constructing these confidence intervals, that 95% of the time, these confidence intervals contain our true population mean. So these look like good confidence intervals. But what we've talked about is, normally when you're doing this type of thing, this type of inferential statistics, you don't know the population standard deviation. You don't know sigma. So instead, you might be tempted to use z with our sample standard deviations."}, {"video_title": "Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So these look like good confidence intervals. But what we've talked about is, normally when you're doing this type of thing, this type of inferential statistics, you don't know the population standard deviation. You don't know sigma. So instead, you might be tempted to use z with our sample standard deviations. But if you look at that for these exact same samples we just calculated, notice now when we did it over and over again, we've done this 625 times, in this scenario where we keep calculating the confidence intervals with z and s, the true population mean is contained in the intervals only 92.2% of the time. And we could keep going. So we have a much lower hit rate than we would hope to have if we were actually using z and sigma."}, {"video_title": "Simulation showing value of t statistic Confidence intervals AP Statistics Khan Academy.mp3", "Sentence": "So instead, you might be tempted to use z with our sample standard deviations. But if you look at that for these exact same samples we just calculated, notice now when we did it over and over again, we've done this 625 times, in this scenario where we keep calculating the confidence intervals with z and s, the true population mean is contained in the intervals only 92.2% of the time. And we could keep going. So we have a much lower hit rate than we would hope to have if we were actually using z and sigma. Now what's neat is if we use t, use a t-table, notice this is getting much closer. And this is neat because with a t-table and something that we can actually get from the sample, the sample standard deviation, we're actually able to have a pretty close hit rate to what we would have if we actually knew the population standard deviation. So that's the value of t and t-statistics."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "So let's say that we have two random variables, X and Y, and they are completely independent. They are independent random variables. And I'm just going to go over a little bit of notation here. If we wanted to know the expected, or if we talked about the expected value of this random variable X, that is the same thing as the mean value of this random variable X. If we talk about the expected value of Y, that is the same thing as the mean of Y. If we talk about the variance of the random variable X, that is the same thing as the expected value of the squared distances between our random variable X and its mean. And that right there, squared."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "If we wanted to know the expected, or if we talked about the expected value of this random variable X, that is the same thing as the mean value of this random variable X. If we talk about the expected value of Y, that is the same thing as the mean of Y. If we talk about the variance of the random variable X, that is the same thing as the expected value of the squared distances between our random variable X and its mean. And that right there, squared. So the expected value of these squared differences, and that is, you can also use the notation, sigma squared for the random variable X. This is just a review of things we already know, but I just want to reintroduce it, because I'll use this to build up some of our tools. So you do the same thing with this with random variable Y."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "And that right there, squared. So the expected value of these squared differences, and that is, you can also use the notation, sigma squared for the random variable X. This is just a review of things we already know, but I just want to reintroduce it, because I'll use this to build up some of our tools. So you do the same thing with this with random variable Y. The variance of random variable Y is the expected value of the squared difference between our random variable Y and the mean of Y, or the expected value of Y, squared. And that's the same thing as sigma squared of Y. There's a variance of Y."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "So you do the same thing with this with random variable Y. The variance of random variable Y is the expected value of the squared difference between our random variable Y and the mean of Y, or the expected value of Y, squared. And that's the same thing as sigma squared of Y. There's a variance of Y. Now, you may or may not already know these properties of expected values and variances, but I will reintroduce them to you, and I won't go into some rigorous proof. Actually, I think they're fairly easy to digest. So one is that if I have some third random variable, let's say I have some third random variable that is defined as being the random variable X plus the random variable Y."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "There's a variance of Y. Now, you may or may not already know these properties of expected values and variances, but I will reintroduce them to you, and I won't go into some rigorous proof. Actually, I think they're fairly easy to digest. So one is that if I have some third random variable, let's say I have some third random variable that is defined as being the random variable X plus the random variable Y. Let me stay with my colors just so everything becomes clear. The random variable X plus the random variable Y. What is the expected value of Z going to be?"}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "So one is that if I have some third random variable, let's say I have some third random variable that is defined as being the random variable X plus the random variable Y. Let me stay with my colors just so everything becomes clear. The random variable X plus the random variable Y. What is the expected value of Z going to be? The expected value of Z is going to be equal to the expected value of X plus Y. And this is a property of expected values. I'm not going to prove it rigorously right here, but it's the expected value of X plus the expected value of Y."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "What is the expected value of Z going to be? The expected value of Z is going to be equal to the expected value of X plus Y. And this is a property of expected values. I'm not going to prove it rigorously right here, but it's the expected value of X plus the expected value of Y. Or another way to think about this is that the mean of Z is going to be the mean of X plus the mean of Y. Or another way to view it is if I wanted to take, let's say I have some other random variable. I'm running out of letters here."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "I'm not going to prove it rigorously right here, but it's the expected value of X plus the expected value of Y. Or another way to think about this is that the mean of Z is going to be the mean of X plus the mean of Y. Or another way to view it is if I wanted to take, let's say I have some other random variable. I'm running out of letters here. Let's say I have the random variable A, and I define random variable A to be X minus Y. So what's its expected value going to be? The expected value of A is going to be equal to the expected value of X minus Y, which is equal to, you can either view it as the expected value of X plus the expected value of negative Y, or the expected value of X minus the expected value of Y, which is the same thing as the mean of X minus the mean of Y."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "I'm running out of letters here. Let's say I have the random variable A, and I define random variable A to be X minus Y. So what's its expected value going to be? The expected value of A is going to be equal to the expected value of X minus Y, which is equal to, you can either view it as the expected value of X plus the expected value of negative Y, or the expected value of X minus the expected value of Y, which is the same thing as the mean of X minus the mean of Y. So this is what the mean of our random variable A would be equal to. All of this is review, and I'm going to use this when we start talking about distributions that are sums and differences of other distributions. Now let's think about what the variance of random variable Z is and what the variance of random variable A is."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "The expected value of A is going to be equal to the expected value of X minus Y, which is equal to, you can either view it as the expected value of X plus the expected value of negative Y, or the expected value of X minus the expected value of Y, which is the same thing as the mean of X minus the mean of Y. So this is what the mean of our random variable A would be equal to. All of this is review, and I'm going to use this when we start talking about distributions that are sums and differences of other distributions. Now let's think about what the variance of random variable Z is and what the variance of random variable A is. So the variance of Z, and just to kind of always focus back on the intuition, it makes sense. If X is completely independent of Y, and if I have some random variable that is the sum of the two, then the expected value of that variable, of that new variable, is going to be the sum of the expected values of the other two, because they are unrelated. If I think, if my expected value here is 5 and my expected value here is 7, it's completely reasonable that my expected value here is 12, assuming that they are completely independent."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "Now let's think about what the variance of random variable Z is and what the variance of random variable A is. So the variance of Z, and just to kind of always focus back on the intuition, it makes sense. If X is completely independent of Y, and if I have some random variable that is the sum of the two, then the expected value of that variable, of that new variable, is going to be the sum of the expected values of the other two, because they are unrelated. If I think, if my expected value here is 5 and my expected value here is 7, it's completely reasonable that my expected value here is 12, assuming that they are completely independent. Now, if we have a situation, so what is the variance of my random variable Z? And once again, I'm not going to do a rigorous proof here, this is really just a property of variances, but I'm going to use this to establish what the variance of our random variable A is. So if this squared distance on average is some variance, and this one is completely independent, and its squared distance on average is some distance, then the variance of their sum is actually going to be the sum of their variances."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "If I think, if my expected value here is 5 and my expected value here is 7, it's completely reasonable that my expected value here is 12, assuming that they are completely independent. Now, if we have a situation, so what is the variance of my random variable Z? And once again, I'm not going to do a rigorous proof here, this is really just a property of variances, but I'm going to use this to establish what the variance of our random variable A is. So if this squared distance on average is some variance, and this one is completely independent, and its squared distance on average is some distance, then the variance of their sum is actually going to be the sum of their variances. So this is going to be equal to the variance of random variable X plus the variance of random variable Y. So another way of thinking about it is that the variance of Z, which is the same thing as the variance of X plus Y, is equal to the variance of X plus the variance of random variable Y. And hopefully that makes some sense, I'm not proving it too rigorously, and you'll see this in a lot of statistics books."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "So if this squared distance on average is some variance, and this one is completely independent, and its squared distance on average is some distance, then the variance of their sum is actually going to be the sum of their variances. So this is going to be equal to the variance of random variable X plus the variance of random variable Y. So another way of thinking about it is that the variance of Z, which is the same thing as the variance of X plus Y, is equal to the variance of X plus the variance of random variable Y. And hopefully that makes some sense, I'm not proving it too rigorously, and you'll see this in a lot of statistics books. Now, what I want to show you is that the variance of random variable A is actually this exact same thing. And that's the interesting thing, because you might say, hey, why wouldn't it be the difference? We had the differences over here, so let's experiment with this a little bit."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "And hopefully that makes some sense, I'm not proving it too rigorously, and you'll see this in a lot of statistics books. Now, what I want to show you is that the variance of random variable A is actually this exact same thing. And that's the interesting thing, because you might say, hey, why wouldn't it be the difference? We had the differences over here, so let's experiment with this a little bit. The variance of random variable A is the same thing as the variance of X minus Y, which is equal to the variance of X plus negative Y. These are equivalent statements. So you could view this as being equal to, just using this over here, the sum of these two variances."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "We had the differences over here, so let's experiment with this a little bit. The variance of random variable A is the same thing as the variance of X minus Y, which is equal to the variance of X plus negative Y. These are equivalent statements. So you could view this as being equal to, just using this over here, the sum of these two variances. So it's going to be equal to the sum of the variance of X plus the variance of negative Y. And what I need to show you is that the variance of negative Y, the negative of that random variable, is going to be the same thing as the variance of Y. So what is the variance of negative Y?"}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "So you could view this as being equal to, just using this over here, the sum of these two variances. So it's going to be equal to the sum of the variance of X plus the variance of negative Y. And what I need to show you is that the variance of negative Y, the negative of that random variable, is going to be the same thing as the variance of Y. So what is the variance of negative Y? The variance of negative Y is the same thing as the variance of negative Y, which is equal to the expected value of the distance between negative Y and the expected value of negative Y squared. That's all the variance actually is. Now, what is the expected value of negative Y right over here?"}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "So what is the variance of negative Y? The variance of negative Y is the same thing as the variance of negative Y, which is equal to the expected value of the distance between negative Y and the expected value of negative Y squared. That's all the variance actually is. Now, what is the expected value of negative Y right over here? Actually, even better, let me factor out a negative 1. So what's in the parentheses right here, this is the exact same thing as negative 1 squared times Y plus the expected value of negative Y. So that's the exact same thing in the parentheses squared."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "Now, what is the expected value of negative Y right over here? Actually, even better, let me factor out a negative 1. So what's in the parentheses right here, this is the exact same thing as negative 1 squared times Y plus the expected value of negative Y. So that's the exact same thing in the parentheses squared. Everything in magenta is everything in magenta here, and it is the expected value of that thing. Now, what is the expected value of negative Y? The expected value of negative Y, I'll do it over here, the expected value of the negative of a random variable is just the negative of the expected value of that random variable."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "So that's the exact same thing in the parentheses squared. Everything in magenta is everything in magenta here, and it is the expected value of that thing. Now, what is the expected value of negative Y? The expected value of negative Y, I'll do it over here, the expected value of the negative of a random variable is just the negative of the expected value of that random variable. So if you look at this, we can rewrite this. I'll give myself a little bit more space. We can rewrite this as the expected value, the variance of negative Y is the expected value."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "The expected value of negative Y, I'll do it over here, the expected value of the negative of a random variable is just the negative of the expected value of that random variable. So if you look at this, we can rewrite this. I'll give myself a little bit more space. We can rewrite this as the expected value, the variance of negative Y is the expected value. This is just 1. Negative 1 squared is just 1. And over here you have Y."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "We can rewrite this as the expected value, the variance of negative Y is the expected value. This is just 1. Negative 1 squared is just 1. And over here you have Y. And instead of just writing plus the expected value of negative Y, that's the same thing as minus the expected value of Y. So you have that and then all of that squared. Now notice, this is the exact same thing by definition as the variance of Y."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "And over here you have Y. And instead of just writing plus the expected value of negative Y, that's the same thing as minus the expected value of Y. So you have that and then all of that squared. Now notice, this is the exact same thing by definition as the variance of Y. So we just showed you just now, so this is the variance of Y. So we just showed you that the variance of the difference of two independent random variables is equal to the sum of the variances. You could definitely believe this."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "Now notice, this is the exact same thing by definition as the variance of Y. So we just showed you just now, so this is the variance of Y. So we just showed you that the variance of the difference of two independent random variables is equal to the sum of the variances. You could definitely believe this. It's equal to the sum of the variance of the first one plus the variance of the negative of the second one. And we just showed that that variance is the same thing as the variance of the positive version of that variable, which makes sense. Your distance from the mean is going to be, it doesn't matter whether you're taking the positive or the negative of the variable."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "You could definitely believe this. It's equal to the sum of the variance of the first one plus the variance of the negative of the second one. And we just showed that that variance is the same thing as the variance of the positive version of that variable, which makes sense. Your distance from the mean is going to be, it doesn't matter whether you're taking the positive or the negative of the variable. You just care about absolute distance. So it makes complete sense that that quantity and that quantity is going to be the same thing. So the whole reason why I went through this exercise, the important takeaways here, is that the mean of differences right over here, so I could rewrite it as the mean of the differences of the random variable is the same thing as the differences of their means."}, {"video_title": "Variance of differences of random variables Probability and Statistics Khan Academy.mp3", "Sentence": "Your distance from the mean is going to be, it doesn't matter whether you're taking the positive or the negative of the variable. You just care about absolute distance. So it makes complete sense that that quantity and that quantity is going to be the same thing. So the whole reason why I went through this exercise, the important takeaways here, is that the mean of differences right over here, so I could rewrite it as the mean of the differences of the random variable is the same thing as the differences of their means. And then the other important takeaway, and I'm going to build on this in the next few videos, is that the variance of the difference, if I define a new random variable as the difference of two other random variables, the variance of that random variable is actually the sum of the variances of the two random variables. So these are the two important takeaways that we'll use to build on in future videos. Anyway, hopefully that wasn't too confusing."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "A cereal company is giving away a prize in each box of cereal, and they advertise, collect all six prizes. Each box of cereal has one prize, and each prize is equally likely to appear in any given box. Amanda wonders how many boxes it takes, on average, to get all six prizes. So there's several ways to approach this for Amanda. She could try to figure out a mathematical way to determine what is the expected number of boxes she would need to collect, on average, to get all six prizes. Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes. So for example, she could say, alright, each box is gonna have one of six prizes."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So there's several ways to approach this for Amanda. She could try to figure out a mathematical way to determine what is the expected number of boxes she would need to collect, on average, to get all six prizes. Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes. So for example, she could say, alright, each box is gonna have one of six prizes. So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. And then she could have a computer generate a random string of numbers, maybe something that looks like this. And the general method, she could start at the left here, and each new number she gets, she can say, hey, this is like getting a cereal box, and then it's going to tell me which prize I got."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So for example, she could say, alright, each box is gonna have one of six prizes. So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. And then she could have a computer generate a random string of numbers, maybe something that looks like this. And the general method, she could start at the left here, and each new number she gets, she can say, hey, this is like getting a cereal box, and then it's going to tell me which prize I got. So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes. You might say, well look, there are numbers here that aren't one through six."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "And the general method, she could start at the left here, and each new number she gets, she can say, hey, this is like getting a cereal box, and then it's going to tell me which prize I got. So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes. You might say, well look, there are numbers here that aren't one through six. There's zero, there's seven, there's eight or nine. Well, for those numbers, she could just ignore them. She could just pretend like they aren't there, and just keep going past them."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "You might say, well look, there are numbers here that aren't one through six. There's zero, there's seven, there's eight or nine. Well, for those numbers, she could just ignore them. She could just pretend like they aren't there, and just keep going past them. So why don't you pause this video, and do it for the first experiment. On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes? So let's, let me make a table here."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "She could just pretend like they aren't there, and just keep going past them. So why don't you pause this video, and do it for the first experiment. On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes? So let's, let me make a table here. So this is the experiment. And then in the second column, I'm gonna say number of boxes. Number of boxes you would have to get in that simulation."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So let's, let me make a table here. So this is the experiment. And then in the second column, I'm gonna say number of boxes. Number of boxes you would have to get in that simulation. So maybe I'll do the first one in this blue color. So we're in the first simulation. So one box, we got the one."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Number of boxes you would have to get in that simulation. So maybe I'll do the first one in this blue color. So we're in the first simulation. So one box, we got the one. Actually, maybe I'll check things off. So we have to get a one, a two, a three, a four, a five, and a six. So let's see, we have a one."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So one box, we got the one. Actually, maybe I'll check things off. So we have to get a one, a two, a three, a four, a five, and a six. So let's see, we have a one. I'll check that off. We have a five. I'll check that off."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So let's see, we have a one. I'll check that off. We have a five. I'll check that off. We get a six. I'll check that off. Well, the next box, we got another six."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "I'll check that off. We get a six. I'll check that off. Well, the next box, we got another six. We've already have that prize, but we're gonna keep getting boxes. Then the next box, we get a two. Then the next box, we get a four."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Well, the next box, we got another six. We've already have that prize, but we're gonna keep getting boxes. Then the next box, we get a two. Then the next box, we get a four. Then the next box, the number's a seven. So we will just ignore this right over here. The box after that, we get a six, but we already have that prize."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Then the next box, we get a four. Then the next box, the number's a seven. So we will just ignore this right over here. The box after that, we get a six, but we already have that prize. Then we ignore the next box, a zero. That doesn't give us a prize. We assume that that didn't even happen."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "The box after that, we get a six, but we already have that prize. Then we ignore the next box, a zero. That doesn't give us a prize. We assume that that didn't even happen. And then we would go to the number three, which is the last prize that we need. So how many boxes did we have to go through? Well, we would only count the valid ones, the ones that gave a valid prize between the numbers one through six, including one and six."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "We assume that that didn't even happen. And then we would go to the number three, which is the last prize that we need. So how many boxes did we have to go through? Well, we would only count the valid ones, the ones that gave a valid prize between the numbers one through six, including one and six. So let's see. We went through one, two, three, four, five, six, seven, eight boxes in the first experiment. So experiment number one, it took us eight boxes to get all six prizes."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Well, we would only count the valid ones, the ones that gave a valid prize between the numbers one through six, including one and six. So let's see. We went through one, two, three, four, five, six, seven, eight boxes in the first experiment. So experiment number one, it took us eight boxes to get all six prizes. Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. This just meant that on this first experiment, it took eight boxes. If you wanted to figure out, on average, you wanna do many experiments."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So experiment number one, it took us eight boxes to get all six prizes. Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. This just meant that on this first experiment, it took eight boxes. If you wanted to figure out, on average, you wanna do many experiments. And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. So now let's do our second experiment. And remember, it's important that these are truly random numbers."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "If you wanted to figure out, on average, you wanna do many experiments. And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. So now let's do our second experiment. And remember, it's important that these are truly random numbers. And so we will now start at the first valid number. So we have a two. So this is our second experiment."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "And remember, it's important that these are truly random numbers. And so we will now start at the first valid number. So we have a two. So this is our second experiment. We got a two. We got a one. We can ignore this eight."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So this is our second experiment. We got a two. We got a one. We can ignore this eight. Then we get a two again. We already have that prize. Ignore the nine."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "We can ignore this eight. Then we get a two again. We already have that prize. Ignore the nine. Five, that's a prize we need in this experiment. Nine, we can ignore. And then four, haven't gotten that prize yet in this experiment."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Ignore the nine. Five, that's a prize we need in this experiment. Nine, we can ignore. And then four, haven't gotten that prize yet in this experiment. Three, haven't gotten that prize yet in this experiment. One, we already got that prize. Three, we already got that prize."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "And then four, haven't gotten that prize yet in this experiment. Three, haven't gotten that prize yet in this experiment. One, we already got that prize. Three, we already got that prize. Three, already got that prize. Two, two, already got those prizes. Zero, we already got all of these prizes over here."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Three, we already got that prize. Three, already got that prize. Two, two, already got those prizes. Zero, we already got all of these prizes over here. We can ignore the zero. Already got that prize. And finally, we get prize number six."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Zero, we already got all of these prizes over here. We can ignore the zero. Already got that prize. And finally, we get prize number six. So how many boxes did we need in that second experiment? Well, let's see. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17 boxes."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "And finally, we get prize number six. So how many boxes did we need in that second experiment? Well, let's see. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17 boxes. So in experiment two, I needed 17, or Amanda needed 17 boxes. And she can keep going. Let's do this one more time."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17 boxes. So in experiment two, I needed 17, or Amanda needed 17 boxes. And she can keep going. Let's do this one more time. This is strangely fun. So experiment three. Now remember, we only wanna look at the valid numbers."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Let's do this one more time. This is strangely fun. So experiment three. Now remember, we only wanna look at the valid numbers. We'll ignore the invalid numbers, the ones that don't give us a valid prize number. So four, we get that prize. These are all invalid."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Now remember, we only wanna look at the valid numbers. We'll ignore the invalid numbers, the ones that don't give us a valid prize number. So four, we get that prize. These are all invalid. In fact, and then we go to five, we get that prize. Five, we already have it. We get the two prize."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "These are all invalid. In fact, and then we go to five, we get that prize. Five, we already have it. We get the two prize. Seven and eight are invalid. Seven's invalid. Six, we get that prize."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "We get the two prize. Seven and eight are invalid. Seven's invalid. Six, we get that prize. Seven's invalid. One, we got that prize. One, we already got it."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Six, we get that prize. Seven's invalid. One, we got that prize. One, we already got it. Nine's invalid. Two, we already got it. Nine is invalid."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "One, we already got it. Nine's invalid. Two, we already got it. Nine is invalid. One, we already got the one prize. And then finally, we get prize number three, which was the missing prize. So how many boxes, valid boxes, did we have to go through?"}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Nine is invalid. One, we already got the one prize. And then finally, we get prize number three, which was the missing prize. So how many boxes, valid boxes, did we have to go through? Let's see. One, two, three, four, five, six, seven, eight, nine, 10. 10."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So how many boxes, valid boxes, did we have to go through? Let's see. One, two, three, four, five, six, seven, eight, nine, 10. 10. So with only three experiments, what was our average? Well, with these three experiments, our average is going to be eight plus 17 plus 10 over three. So let's see, this is 2535 over three, which is equal to 11 2 3rds."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "10. So with only three experiments, what was our average? Well, with these three experiments, our average is going to be eight plus 17 plus 10 over three. So let's see, this is 2535 over three, which is equal to 11 2 3rds. Now, do we know that this is the true theoretical expected number of boxes that you would need to get? No, we don't know that. But the more experiments we run, the closer our averages likely get to the true theoretical average."}, {"video_title": "Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3", "Sentence": "Estella oversees 12 of these residence halls. Her department surveyed a large, simple, random sample of first-year students who live in those 12 residence halls about their overall satisfaction with campus living. Estella can safely generalize the results of the survey to which population. So pause this video and see if you can figure it out. Alright, so let's do this together. So Estella has done, it's a large, simple, random sample of first-year students. So let's see."}, {"video_title": "Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3", "Sentence": "So pause this video and see if you can figure it out. Alright, so let's do this together. So Estella has done, it's a large, simple, random sample of first-year students. So let's see. Choice A is only those students who were surveyed. Well no, this was a simple, random sample and it was a large sample, so it's meant to be indicative of all first-year students. You can generalize more than just making statements about just the students who were surveyed."}, {"video_title": "Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3", "Sentence": "So let's see. Choice A is only those students who were surveyed. Well no, this was a simple, random sample and it was a large sample, so it's meant to be indicative of all first-year students. You can generalize more than just making statements about just the students who were surveyed. All first-year students, but only those who live in these 12 residence halls. Yeah, I think this one looks fair, because you can't generalize to people who don't live in those residence halls. Maybe Estella oversees the 12 best residence halls or the 12 worst residence halls."}, {"video_title": "Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3", "Sentence": "You can generalize more than just making statements about just the students who were surveyed. All first-year students, but only those who live in these 12 residence halls. Yeah, I think this one looks fair, because you can't generalize to people who don't live in those residence halls. Maybe Estella oversees the 12 best residence halls or the 12 worst residence halls. And so you wouldn't get, if that were the case, you would not be able to generalize beyond that. Or these might be the 12 that are closest to campus or the 12 that are furthest from campus. So you can only generalize to people who live in those residence halls."}, {"video_title": "Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3", "Sentence": "Maybe Estella oversees the 12 best residence halls or the 12 worst residence halls. And so you wouldn't get, if that were the case, you would not be able to generalize beyond that. Or these might be the 12 that are closest to campus or the 12 that are furthest from campus. So you can only generalize to people who live in those residence halls. All students, first-year or not, but only those who live in these 12 residence halls. Well the issue here is that a second-year student or third-year student might just have a different perspective, even if they live in that same building, and we did a large, random sample of first-year students. We didn't do a large, random sample of all people in those 12 residence halls, so we'll rule that out."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "A statistician for a basketball team tracked the number of points that each of the 12 players on the team had in one game, and then made a stem and the leaf plot to show the data. Sometimes it's called a stem plot. How many points did the team score? And when you first look at this plot right over here, it seems a little hard to understand. Under stem you have zero, one, two. Under leaf you have all of these digits here. How does this relate to the number of points each student or each player actually scored?"}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "And when you first look at this plot right over here, it seems a little hard to understand. Under stem you have zero, one, two. Under leaf you have all of these digits here. How does this relate to the number of points each student or each player actually scored? And the way to interpret a stem and leaf plot is the leaf contain, at least the way that this statistician used it, the leaf contains the smallest digit, or the ones digit, in the number of points that each player scored, and the stem contains the tens digits. And usually the leaf will contain the rightmost digit, or the ones digit, and then the stem will contain all of the other digits. And what's useful about this is it gives kind of a distribution of where the players were."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "How does this relate to the number of points each student or each player actually scored? And the way to interpret a stem and leaf plot is the leaf contain, at least the way that this statistician used it, the leaf contains the smallest digit, or the ones digit, in the number of points that each player scored, and the stem contains the tens digits. And usually the leaf will contain the rightmost digit, or the ones digit, and then the stem will contain all of the other digits. And what's useful about this is it gives kind of a distribution of where the players were. You see that most of the players scored points that started with a zero, then a few more scored points that started with a one, and then only one score scored points that started with a two, and it was actually 20 points. So let me actually write down all of this data in a way that maybe you're a little bit more used to understanding it. So I'm gonna write the zeros in purple."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "And what's useful about this is it gives kind of a distribution of where the players were. You see that most of the players scored points that started with a zero, then a few more scored points that started with a one, and then only one score scored points that started with a two, and it was actually 20 points. So let me actually write down all of this data in a way that maybe you're a little bit more used to understanding it. So I'm gonna write the zeros in purple. So there's, let's see, one, two, three, four, five, six, seven players had zero as the first digit. So one, two, three, four, five, six, seven. I wrote seven zeros, and then this player also had a zero in his ones digit."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "So I'm gonna write the zeros in purple. So there's, let's see, one, two, three, four, five, six, seven players had zero as the first digit. So one, two, three, four, five, six, seven. I wrote seven zeros, and then this player also had a zero in his ones digit. This player, I'm gonna try to do all the colors, this player also had a zero in his ones digit. This player right here had a two in his ones digit, so he scored a total of two points. This player, let me do orange, this player had four for his ones digit."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "I wrote seven zeros, and then this player also had a zero in his ones digit. This player, I'm gonna try to do all the colors, this player also had a zero in his ones digit. This player right here had a two in his ones digit, so he scored a total of two points. This player, let me do orange, this player had four for his ones digit. This player had seven for his ones digit. Then this player had seven for his ones digit. And then, let me see, I'm almost using all the colors, this player had nine for his ones digit."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "This player, let me do orange, this player had four for his ones digit. This player had seven for his ones digit. Then this player had seven for his ones digit. And then, let me see, I'm almost using all the colors, this player had nine for his ones digit. So the way to read this is you had one player with zero points, zero, two, four, seven, nine, and nine. But you can see, I'm just kind of silly saying the zero was the tens digit. You could have even put a blank there, but the zero lets us know that they didn't score, they didn't score anything in the tens place."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "And then, let me see, I'm almost using all the colors, this player had nine for his ones digit. So the way to read this is you had one player with zero points, zero, two, four, seven, nine, and nine. But you can see, I'm just kind of silly saying the zero was the tens digit. You could have even put a blank there, but the zero lets us know that they didn't score, they didn't score anything in the tens place. But these are the actual scores for those seven players. Now let's go to the next row in the stem and leaf plot. So over here, all of the digits start with, or all of the points start with one for each of the players, and there's four of them."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "You could have even put a blank there, but the zero lets us know that they didn't score, they didn't score anything in the tens place. But these are the actual scores for those seven players. Now let's go to the next row in the stem and leaf plot. So over here, all of the digits start with, or all of the points start with one for each of the players, and there's four of them. So one, one, one, and one. And then we have this player over here, it's one, his ones digit, or her ones digit is a one. So this player, this represents 11."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "So over here, all of the digits start with, or all of the points start with one for each of the players, and there's four of them. So one, one, one, and one. And then we have this player over here, it's one, his ones digit, or her ones digit is a one. So this player, this represents 11. One in the tens place, one in the ones place. This player over here also got 11. One in the tens place, one in the ones place."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "So this player, this represents 11. One in the tens place, one in the ones place. This player over here also got 11. One in the tens place, one in the ones place. This player, let me do orange. This player has three in the ones place. So he or she scored 13 points."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "One in the tens place, one in the ones place. This player, let me do orange. This player has three in the ones place. So he or she scored 13 points. One in the tens place, three in the ones place, 13 points. And then, I will do this in purple, this player has eight in their ones place. So he or she scored 18 points."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "So he or she scored 13 points. One in the tens place, three in the ones place, 13 points. And then, I will do this in purple, this player has eight in their ones place. So he or she scored 18 points. One in the tens place, eight in the ones place, 18 points. And then finally, you have this player that has two, the tens digit is a two, and then the ones digit is a zero. Is a zero, I'll circle that in yellow."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "So he or she scored 18 points. One in the tens place, eight in the ones place, 18 points. And then finally, you have this player that has two, the tens digit is a two, and then the ones digit is a zero. Is a zero, I'll circle that in yellow. It is a zero. So he or she scored 20 points. So using, looking at the stem and leaf plot, we're able to extract out all of the number of points that all of the players scored."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "Is a zero, I'll circle that in yellow. It is a zero. So he or she scored 20 points. So using, looking at the stem and leaf plot, we're able to extract out all of the number of points that all of the players scored. And once again, what was useful about this is you see how many players scored between zero and nine points, including nine points. How many scored between 10 and 19 points, and then how many scored 20 points or over. And you see the distribution right over here."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "So using, looking at the stem and leaf plot, we're able to extract out all of the number of points that all of the players scored. And once again, what was useful about this is you see how many players scored between zero and nine points, including nine points. How many scored between 10 and 19 points, and then how many scored 20 points or over. And you see the distribution right over here. But let's actually answer the question that they're asking us to answer. How many points did the team score? So here, we just have to add up all of these numbers right over here."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "And you see the distribution right over here. But let's actually answer the question that they're asking us to answer. How many points did the team score? So here, we just have to add up all of these numbers right over here. So we're going to add up, I'll start with the largest. So 20 plus 18 plus 13 plus 11 plus 11, 13, 11, 11, plus nine plus seven, plus seven again, plus four, plus two. Did I do that right?"}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "So here, we just have to add up all of these numbers right over here. So we're going to add up, I'll start with the largest. So 20 plus 18 plus 13 plus 11 plus 11, 13, 11, 11, plus nine plus seven, plus seven again, plus four, plus two. Did I do that right? We have two 11s, then a nine, then two sevens, then a four, then a two, and then these two characters didn't score anything. So let's add up all of these together. Let's add them all up."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "Did I do that right? We have two 11s, then a nine, then two sevens, then a four, then a two, and then these two characters didn't score anything. So let's add up all of these together. Let's add them all up. So zero plus eight is eight, plus three is 11, plus one is 12, plus one is 13, plus nine is 22, plus seven is 27, 34, 38, 30, or 40. So that gets us to 40. Let me do that one more time."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "Let's add them all up. So zero plus eight is eight, plus three is 11, plus one is 12, plus one is 13, plus nine is 22, plus seven is 27, 34, 38, 30, or 40. So that gets us to 40. Let me do that one more time. Eight, 11, 11, 12, 13, 22, 29, 29, and then, 29, 36, 40, and 42. Looks like I actually might have messed, let me do it one more time. This is the hardest part, adding these up."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "Let me do that one more time. Eight, 11, 11, 12, 13, 22, 29, 29, and then, 29, 36, 40, and 42. Looks like I actually might have messed, let me do it one more time. This is the hardest part, adding these up. So let me try that one last time. I'm just going to state where my sum is. So zero, eight, add three, 11, 12, 13, 22, 29, 36, 40, 42."}, {"video_title": "Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "This is the hardest part, adding these up. So let me try that one last time. I'm just going to state where my sum is. So zero, eight, add three, 11, 12, 13, 22, 29, 36, 40, 42. So it's a good thing that I double-checked that, I made a mistake the first time. Four plus two is six, seven, eight, nine, 10. So we get to 102 points."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "And so we're gonna get an example of doing that right over here. Sometimes in life, say on an exam, especially on something like an AP exam, you're asked to describe or compare a distribution. And what we're going to do in this video is do exactly that. In fact, this one we're going to describe, and then in a future video, we're going to compare distributions. Now before we even read about this distribution or look at this distribution, if you're asked to describe a distribution, there's four things that you should be thinking about. You should be thinking about the shape of the distribution. And when we're talking about shape, it's going to be, there could be left skew, there could be right skew, and we'll see examples of these."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "In fact, this one we're going to describe, and then in a future video, we're going to compare distributions. Now before we even read about this distribution or look at this distribution, if you're asked to describe a distribution, there's four things that you should be thinking about. You should be thinking about the shape of the distribution. And when we're talking about shape, it's going to be, there could be left skew, there could be right skew, and we'll see examples of these. We've talked about them in detail in other videos. They could be symmetric. These are the ones that we typically see, although there might be other types of shapes."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "And when we're talking about shape, it's going to be, there could be left skew, there could be right skew, and we'll see examples of these. We've talked about them in detail in other videos. They could be symmetric. These are the ones that we typically see, although there might be other types of shapes. You will have your center of distribution. And there's multiple ways of thinking about the center of distribution. We've talked about this before."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "These are the ones that we typically see, although there might be other types of shapes. You will have your center of distribution. And there's multiple ways of thinking about the center of distribution. We've talked about this before. You have your mean, you have your median. These are the two most typical ones. You have a notion of spread."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "We've talked about this before. You have your mean, you have your median. These are the two most typical ones. You have a notion of spread. And for spread, you could use range, you could use interquartile range, you could use something like a mean absolute deviation, you could use a, you could use the standard deviation. These are all measures of spread. And then you probably should at least comment about outliers, even if you don't see them."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "You have a notion of spread. And for spread, you could use range, you could use interquartile range, you could use something like a mean absolute deviation, you could use a, you could use the standard deviation. These are all measures of spread. And then you probably should at least comment about outliers, even if you don't see them. It's a good idea to comment, just to make sure that you are being relatively comprehensive. So now, given that, let's do, let's describe the distribution right over here. It says, in the state of Connecticut, the Department of Motor Vehicles, the DMV, requires 16 and 17 year olds to take a 25 question knowledge test in order to obtain a learner's permit."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "And then you probably should at least comment about outliers, even if you don't see them. It's a good idea to comment, just to make sure that you are being relatively comprehensive. So now, given that, let's do, let's describe the distribution right over here. It says, in the state of Connecticut, the Department of Motor Vehicles, the DMV, requires 16 and 17 year olds to take a 25 question knowledge test in order to obtain a learner's permit. To pass, prospective drivers must correctly answer at least 20 questions. On one Monday, 22 teenagers took the test. The dot plot below shows their scores."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "It says, in the state of Connecticut, the Department of Motor Vehicles, the DMV, requires 16 and 17 year olds to take a 25 question knowledge test in order to obtain a learner's permit. To pass, prospective drivers must correctly answer at least 20 questions. On one Monday, 22 teenagers took the test. The dot plot below shows their scores. So why don't you pause this video and see if you can take a shot at describing the shape, the center, the spread, and the outliers. Some of these you might be able to come with the actual numbers. You might be able to calculate some of these."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "The dot plot below shows their scores. So why don't you pause this video and see if you can take a shot at describing the shape, the center, the spread, and the outliers. Some of these you might be able to come with the actual numbers. You might be able to calculate some of these. But really, just to get a sense of it, why don't you take a shot at it? All right, now let's do this together. So first, on the shape."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "You might be able to calculate some of these. But really, just to get a sense of it, why don't you take a shot at it? All right, now let's do this together. So first, on the shape. So what we see is we have, most of the distribution is in this part between 20 and 25. But then we have this fairly long tail to the left. And so this tells us that we have a left skew, or it is a left skewed distribution right over here."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "So first, on the shape. So what we see is we have, most of the distribution is in this part between 20 and 25. But then we have this fairly long tail to the left. And so this tells us that we have a left skew, or it is a left skewed distribution right over here. So we have done the shape. It's a left skewed distribution because the tail goes to the left. Now what about the center of this distribution?"}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "And so this tells us that we have a left skew, or it is a left skewed distribution right over here. So we have done the shape. It's a left skewed distribution because the tail goes to the left. Now what about the center of this distribution? So there's a few ways to measure center, mean or median. Just for the sake of simplicity, I'll think about the median here. I can eyeball that to some degree."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "Now what about the center of this distribution? So there's a few ways to measure center, mean or median. Just for the sake of simplicity, I'll think about the median here. I can eyeball that to some degree. You could also calculate the mean. It would take a little bit more time. I would guess that it's someplace, not even calculating it, I would guess that it's someplace in this range right over there."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "I can eyeball that to some degree. You could also calculate the mean. It would take a little bit more time. I would guess that it's someplace, not even calculating it, I would guess that it's someplace in this range right over there. But let me actually calculate it. So the median, there's 22 data points. So the median is whatever number has 11 on to the right of it and 11 to the left, half of 22."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "I would guess that it's someplace, not even calculating it, I would guess that it's someplace in this range right over there. But let me actually calculate it. So the median, there's 22 data points. So the median is whatever number has 11 on to the right of it and 11 to the left, half of 22. So let's see. We have one, two, three, four, five, six, seven, eight, nine, 10, 11. So the median here is going to be, let's see, this is 23 because we have a bunch of 23s."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "So the median is whatever number has 11 on to the right of it and 11 to the left, half of 22. So let's see. We have one, two, three, four, five, six, seven, eight, nine, 10, 11. So the median here is going to be, let's see, this is 23 because we have a bunch of 23s. One, two, three, four, five, six 23s. And if we were to just order all of the data points, 11 of the data points would be 23 or less, and then 11 would be 23 or more. So our median here, so I could say our center is 23 if we use the median."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "So the median here is going to be, let's see, this is 23 because we have a bunch of 23s. One, two, three, four, five, six 23s. And if we were to just order all of the data points, 11 of the data points would be 23 or less, and then 11 would be 23 or more. So our median here, so I could say our center is 23 if we use the median. And actually, let me write that down. So our median is 23. That's the measure of center that I decided to use."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "So our median here, so I could say our center is 23 if we use the median. And actually, let me write that down. So our median is 23. That's the measure of center that I decided to use. Now, what about spread? Well, the simplest measure of spread is just the range, which is the highest value minus the lowest value. And so our range here would be 25 minus four."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "That's the measure of center that I decided to use. Now, what about spread? Well, the simplest measure of spread is just the range, which is the highest value minus the lowest value. And so our range here would be 25 minus four. 25 minus four is equal to 21. So that is a measure of range. You could have others, but this one is very easy to calculate."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "And so our range here would be 25 minus four. 25 minus four is equal to 21. So that is a measure of range. You could have others, but this one is very easy to calculate. And then if we think about outliers, well, there are a few outliers I would consider, and it's very subjective. People can debate if there's a dot right over there, is that an outlier or not? But I would say that these four right over here, I would consider outliers."}, {"video_title": "Example Describing a distribution AP Statistics Khan Academy.mp3", "Sentence": "You could have others, but this one is very easy to calculate. And then if we think about outliers, well, there are a few outliers I would consider, and it's very subjective. People can debate if there's a dot right over there, is that an outlier or not? But I would say that these four right over here, I would consider outliers. So I would say approximately four outliers, but once again, this is subjective. The main point of this exercise is to just get in the habit of thinking about these things. And statistics is all about creating engineering, one could say, different measurements for center, for spread, and different ways to describe the shape."}, {"video_title": "Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "And if you don't know what antibodies are, these are things that your immune system keeps around so it's very easy to recognize future infection. But you don't have to worry too much about that for this video. In this video, we're just trying to think about how we can visualize data to understand if there's a relationship between having antibodies and the age of the individual. So let's say we go out and collect a bunch of data. So we test 120 adults and 114 have antibodies, six don't. We test 60 children, 54 have antibodies, six don't. We test 20 infants and then eight have antibodies and 12 don't."}, {"video_title": "Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "So let's say we go out and collect a bunch of data. So we test 120 adults and 114 have antibodies, six don't. We test 60 children, 54 have antibodies, six don't. We test 20 infants and then eight have antibodies and 12 don't. So we can just look at this data but this really still doesn't give us a visual representation of what's going on. One step we can take, it still doesn't give us a fully visual representation, is to just think about percentages that might help us think about the likelihood of having antibodies. So if we calculate the percentages, we might see something like this."}, {"video_title": "Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "We test 20 infants and then eight have antibodies and 12 don't. So we can just look at this data but this really still doesn't give us a visual representation of what's going on. One step we can take, it still doesn't give us a fully visual representation, is to just think about percentages that might help us think about the likelihood of having antibodies. So if we calculate the percentages, we might see something like this. For example, 114 over 120 is 95% or 95% have antibodies. That 114 over 120 is 95%. And then the number that don't have antibodies, this six right over here, that is 5%, six over 120."}, {"video_title": "Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "So if we calculate the percentages, we might see something like this. For example, 114 over 120 is 95% or 95% have antibodies. That 114 over 120 is 95%. And then the number that don't have antibodies, this six right over here, that is 5%, six over 120. And you can do that for each of the categories. 54 over 60 is 90% while six over 60, you can do that math in your head, is 10%. And we could do the same thing for the infants."}, {"video_title": "Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "And then the number that don't have antibodies, this six right over here, that is 5%, six over 120. And you can do that for each of the categories. 54 over 60 is 90% while six over 60, you can do that math in your head, is 10%. And we could do the same thing for the infants. Eight out of 20 is 40% while 12 out of 20 is 60%. So that helps us a little bit. It helps us think about, well, what's the percentage of adults that have the antibody or children or infants?"}, {"video_title": "Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "And we could do the same thing for the infants. Eight out of 20 is 40% while 12 out of 20 is 60%. So that helps us a little bit. It helps us think about, well, what's the percentage of adults that have the antibody or children or infants? But if we really wanna visualize it, we can look at two different types of visualizations. One, we can call a segmented bar chart. And I will show a segmented bar chart for this data right over here."}, {"video_title": "Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "It helps us think about, well, what's the percentage of adults that have the antibody or children or infants? But if we really wanna visualize it, we can look at two different types of visualizations. One, we can call a segmented bar chart. And I will show a segmented bar chart for this data right over here. Now in a segmented bar chart, we group, we have a bar for each category here, and we're making adults, children, and infants the different categories, because we're thinking maybe that has something to do with the likelihood of having antibodies. And then for each bar, for example, this adult bar, you can see the percentage that have the antibodies and the percentage that don't. So 95% of the adult bar is filled in blue."}, {"video_title": "Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "And I will show a segmented bar chart for this data right over here. Now in a segmented bar chart, we group, we have a bar for each category here, and we're making adults, children, and infants the different categories, because we're thinking maybe that has something to do with the likelihood of having antibodies. And then for each bar, for example, this adult bar, you can see the percentage that have the antibodies and the percentage that don't. So 95% of the adult bar is filled in blue. That's for yes, they have the antibodies, and 5% is filled in red. And then for children, you can see that 90% is filled in blue and 10% is filled in red because 10% don't have the antibodies. And then for infants, you can see that 40% is filled in blue and 60% don't have the antibodies."}, {"video_title": "Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "So 95% of the adult bar is filled in blue. That's for yes, they have the antibodies, and 5% is filled in red. And then for children, you can see that 90% is filled in blue and 10% is filled in red because 10% don't have the antibodies. And then for infants, you can see that 40% is filled in blue and 60% don't have the antibodies. Now this by itself is pretty useful to visually see, all right, it looks like adults are much more likely to have the antibodies than children, and children are far more likely to have the antibodies than infants. And so it looks like this idea of making a bar for each of adults, children, or infants was a good way to start to understand the likelihood of having antibodies. You could have done it other ways."}, {"video_title": "Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "And then for infants, you can see that 40% is filled in blue and 60% don't have the antibodies. Now this by itself is pretty useful to visually see, all right, it looks like adults are much more likely to have the antibodies than children, and children are far more likely to have the antibodies than infants. And so it looks like this idea of making a bar for each of adults, children, or infants was a good way to start to understand the likelihood of having antibodies. You could have done it other ways. You could have had a bar for have antibodies and another bar for not have antibodies, and then you could have segmented the bar chart by whether they are adults, children, or infants. But if you did that, that would have been trying to understand whether having antibodies or not having antibodies is predictive of whether you're an adult, child, or infant, while this one makes, at least to me, a little bit more sense that whether you're an adult, child, or infant might be predictive of whether or not you have antibodies. But there is some information lost in this segmented bar chart."}, {"video_title": "Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "You could have done it other ways. You could have had a bar for have antibodies and another bar for not have antibodies, and then you could have segmented the bar chart by whether they are adults, children, or infants. But if you did that, that would have been trying to understand whether having antibodies or not having antibodies is predictive of whether you're an adult, child, or infant, while this one makes, at least to me, a little bit more sense that whether you're an adult, child, or infant might be predictive of whether or not you have antibodies. But there is some information lost in this segmented bar chart. For example, we have lost the fact that we have sampled or we have tested a lot more adults than children and far more children than infants. So one way to incorporate that data back into a visualization to essentially show how many people you sampled in each of these categories, we can generate what's known as a mosaic plot. So this is a mosaic plot right over here."}, {"video_title": "Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "But there is some information lost in this segmented bar chart. For example, we have lost the fact that we have sampled or we have tested a lot more adults than children and far more children than infants. So one way to incorporate that data back into a visualization to essentially show how many people you sampled in each of these categories, we can generate what's known as a mosaic plot. So this is a mosaic plot right over here. And one way to think about it is we have just adjusted the width of each of these bars based on how many people we tested. So we tested 200 people. And so you can view this width right over here as being 200."}, {"video_title": "Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "So this is a mosaic plot right over here. And one way to think about it is we have just adjusted the width of each of these bars based on how many people we tested. So we tested 200 people. And so you can view this width right over here as being 200. And you can see that we tested 120 adults. So the width of this first bar, I guess you could say, although now we're dealing with a mosaic plot, this width right over here would be 60% of this entire width, which you can see that it is. And then the children are 60 of the 200 that we tested."}, {"video_title": "Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "And so you can view this width right over here as being 200. And you can see that we tested 120 adults. So the width of this first bar, I guess you could say, although now we're dealing with a mosaic plot, this width right over here would be 60% of this entire width, which you can see that it is. And then the children are 60 of the 200 that we tested. And so this width right over here would be 60 over the entire 200, or it would be about 30% of the entire width. And we can see that we tested the fewest number of infants. And so this 20 right over here represents the 20 infants we tested."}, {"video_title": "Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3", "Sentence": "And then the children are 60 of the 200 that we tested. And so this width right over here would be 60 over the entire 200, or it would be about 30% of the entire width. And we can see that we tested the fewest number of infants. And so this 20 right over here represents the 20 infants we tested. And the reason why this mosaic plot conveys more information, it conveys all the same information that our segmented bar chart does, but it also gives us a sense that we tested more adults than children and far more children than infants. And it's also easy to then look at it and say, okay, of the total number of people who don't have the antibodies, so that would be the red area right over here, even though we tested the fewest number of infants, it looks like infants make up a large chunk of the total number of folks who don't have antibodies. So I'll leave you there."}, {"video_title": "Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3", "Sentence": "In a previous video, we began to think about how we can use a regression line, and in particular, the slope of a regression line based on sample data, how we can use that in order to make inference about the slope of the true population regression line. In this video, we're going to think about what are the conditions for inference when we're dealing with regression lines? And these are going to be in some ways similar to the conditions for inference that we thought about when we were doing hypothesis testing and confidence intervals for means and for proportions, but there's also going to be a few new conditions. So to help us remember these conditions, you might want to think about the LINER acronym, L-I-N-E-R, and if it isn't obvious to you, this almost is linear. Liner, with an A, it would be linear, and this is valuable because remember, we're thinking about linear regression. So the L right over here actually does stand for linear, and here, the condition is is that the actual relationship in the population between your X and Y variables actually is a linear relationship. So actual linear relationship between X and Y."}, {"video_title": "Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3", "Sentence": "So to help us remember these conditions, you might want to think about the LINER acronym, L-I-N-E-R, and if it isn't obvious to you, this almost is linear. Liner, with an A, it would be linear, and this is valuable because remember, we're thinking about linear regression. So the L right over here actually does stand for linear, and here, the condition is is that the actual relationship in the population between your X and Y variables actually is a linear relationship. So actual linear relationship between X and Y. Now, in a lot of cases, you might just have to assume that this is going to be the case when you see it on an exam, like an AP exam, for example. They might say, hey, assume this condition is met. Oftentimes, they'll say, assume all of these conditions are met."}, {"video_title": "Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3", "Sentence": "So actual linear relationship between X and Y. Now, in a lot of cases, you might just have to assume that this is going to be the case when you see it on an exam, like an AP exam, for example. They might say, hey, assume this condition is met. Oftentimes, they'll say, assume all of these conditions are met. They just want you to maybe know about these conditions, but this is something to think about. If the underlying relationship is nonlinear, well, then maybe some of your inferences might not be as robust. Now, the next one is one we have seen before when we're talking about general conditions for inference, and this is the independence, independence condition, and there's a couple of ways to think about it."}, {"video_title": "Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3", "Sentence": "Oftentimes, they'll say, assume all of these conditions are met. They just want you to maybe know about these conditions, but this is something to think about. If the underlying relationship is nonlinear, well, then maybe some of your inferences might not be as robust. Now, the next one is one we have seen before when we're talking about general conditions for inference, and this is the independence, independence condition, and there's a couple of ways to think about it. Either individual observations are independent of each other, so you could be sampling with replacement, or you could be thinking about your 10% rule that we have done when we thought about the independence condition for proportions and for means, where we would need to feel confident that the size of our sample is no more than 10% of the size of the population. Now, the next one is the normal condition, which we have talked about when we were doing inference for proportions and for means, although it means something a little bit more sophisticated when we're dealing with a regression. The normal condition, and once again, many times people will just say, assume it's been met, but let me actually draw a regression line, but do it with a little perspective, and I'm gonna add a third dimension."}, {"video_title": "Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3", "Sentence": "Now, the next one is one we have seen before when we're talking about general conditions for inference, and this is the independence, independence condition, and there's a couple of ways to think about it. Either individual observations are independent of each other, so you could be sampling with replacement, or you could be thinking about your 10% rule that we have done when we thought about the independence condition for proportions and for means, where we would need to feel confident that the size of our sample is no more than 10% of the size of the population. Now, the next one is the normal condition, which we have talked about when we were doing inference for proportions and for means, although it means something a little bit more sophisticated when we're dealing with a regression. The normal condition, and once again, many times people will just say, assume it's been met, but let me actually draw a regression line, but do it with a little perspective, and I'm gonna add a third dimension. Let's say that's the x-axis, and let's say this is the y-axis, and the true population regression line looks like this. And so the normal condition tells us that for any given x in the true population, the distribution of y's that you would expect is normal, is normal, so let me see if I can draw a normal distribution for the y's given that x. So that would be that normal distribution there, and then let's say for this x right over here, you would expect a normal distribution as well, so just like, just like this."}, {"video_title": "Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3", "Sentence": "The normal condition, and once again, many times people will just say, assume it's been met, but let me actually draw a regression line, but do it with a little perspective, and I'm gonna add a third dimension. Let's say that's the x-axis, and let's say this is the y-axis, and the true population regression line looks like this. And so the normal condition tells us that for any given x in the true population, the distribution of y's that you would expect is normal, is normal, so let me see if I can draw a normal distribution for the y's given that x. So that would be that normal distribution there, and then let's say for this x right over here, you would expect a normal distribution as well, so just like, just like this. So for given x, the distribution of y's should be normal. Once again, many times you'll just be told to assume that that has been met because it might, at least in an introductory statistics class, be a little bit hard to figure this out on your own. Now the next condition is related to that, and this is the idea of having equal variance, equal variance, and that's just saying that each of these normal distributions should have the same spread for a given x."}, {"video_title": "Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3", "Sentence": "So that would be that normal distribution there, and then let's say for this x right over here, you would expect a normal distribution as well, so just like, just like this. So for given x, the distribution of y's should be normal. Once again, many times you'll just be told to assume that that has been met because it might, at least in an introductory statistics class, be a little bit hard to figure this out on your own. Now the next condition is related to that, and this is the idea of having equal variance, equal variance, and that's just saying that each of these normal distributions should have the same spread for a given x. And so you could say equal variance, or you could even think about them having the equal standard deviation. So for example, if for a given x, let's say for this x, all of a sudden you had a much lower variance, maybe it looked like this, then you would no longer meet your conditions for inference. Last but not least, and this is one we've seen many times, this is the random condition."}, {"video_title": "Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3", "Sentence": "Now the next condition is related to that, and this is the idea of having equal variance, equal variance, and that's just saying that each of these normal distributions should have the same spread for a given x. And so you could say equal variance, or you could even think about them having the equal standard deviation. So for example, if for a given x, let's say for this x, all of a sudden you had a much lower variance, maybe it looked like this, then you would no longer meet your conditions for inference. Last but not least, and this is one we've seen many times, this is the random condition. And this is that the data comes from a well-designed random sample or some type of randomized experiment. And this condition we have seen in every type of condition for inference that we have looked at so far. So I'll leave you there."}, {"video_title": "Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3", "Sentence": "Last but not least, and this is one we've seen many times, this is the random condition. And this is that the data comes from a well-designed random sample or some type of randomized experiment. And this condition we have seen in every type of condition for inference that we have looked at so far. So I'll leave you there. It's good to know it will show up on some exams, but many times when it comes to problem solving, in an introductory statistics class, they will tell you, hey, just assume the conditions for inference have been met, or what are the conditions for inference? But they're not going to actually make you prove, for example, the normal or the equal variance condition. That might be a bit much for an introductory statistics class."}, {"video_title": "Conditions for a z test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Management claimed that 10% of employees earned this rating but Jules suspected it was actually less common. She obtained an anonymous random sample of 10 ratings for employees on her team. She wants to use the sample data to test her null hypothesis that the true proportion is 10% versus her alternative hypothesis that the true proportion is less than 10% where P is the proportion of all employees on her team who earned exceeds expectations. Which conditions for performing this type of test did Jules' sample meet? And when they're saying which conditions, they are talking about the three conditions, the random condition, the normal condition, and we've seen these before, and the independence condition. So I will let you pause the video now and try to figure this out on your own, and then we will review each of these conditions and think about whether Jules' sample meets the conditions that we need to feel good about some of our significance testing. All right, now let's work through this together."}, {"video_title": "Conditions for a z test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Which conditions for performing this type of test did Jules' sample meet? And when they're saying which conditions, they are talking about the three conditions, the random condition, the normal condition, and we've seen these before, and the independence condition. So I will let you pause the video now and try to figure this out on your own, and then we will review each of these conditions and think about whether Jules' sample meets the conditions that we need to feel good about some of our significance testing. All right, now let's work through this together. So let's just remind ourselves what we're going to do in a significance test. We have our null hypothesis, we have our alternative hypothesis. What we do is we look at the population, the population size, there's 40 employees on staff at this company."}, {"video_title": "Conditions for a z test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "All right, now let's work through this together. So let's just remind ourselves what we're going to do in a significance test. We have our null hypothesis, we have our alternative hypothesis. What we do is we look at the population, the population size, there's 40 employees on staff at this company. We take a sample, in Jules' case, she took a sample size of 10, and then we calculate a sample statistic, in this case it is the sample proportion, which is equal to, let's just call it p hat sub one. And then we want to calculate a p value. And just as a bit of review, a p value is the probability of getting a result at least as extreme as this one if we assume our null hypothesis is true."}, {"video_title": "Conditions for a z test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "What we do is we look at the population, the population size, there's 40 employees on staff at this company. We take a sample, in Jules' case, she took a sample size of 10, and then we calculate a sample statistic, in this case it is the sample proportion, which is equal to, let's just call it p hat sub one. And then we want to calculate a p value. And just as a bit of review, a p value is the probability of getting a result at least as extreme as this one if we assume our null hypothesis is true. And in this particular case, because she suspects that not 10% are getting the exceeds expectations, this would be the probability of your sample statistic being less than or equal to the one that you just calculated for a sample size of n equals 10 given that your null hypothesis is true. And if this p value is less than your predetermined significance level, maybe that's 5% or 10%, but you'd want to decide it ahead of time, then you would reject, you would reject your null hypothesis because the probability of getting this result seems pretty low, in which case it would suggest the alternative. But then if the p value is not less than this, then you wouldn't be able to reject the null hypothesis."}, {"video_title": "Conditions for a z test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "And just as a bit of review, a p value is the probability of getting a result at least as extreme as this one if we assume our null hypothesis is true. And in this particular case, because she suspects that not 10% are getting the exceeds expectations, this would be the probability of your sample statistic being less than or equal to the one that you just calculated for a sample size of n equals 10 given that your null hypothesis is true. And if this p value is less than your predetermined significance level, maybe that's 5% or 10%, but you'd want to decide it ahead of time, then you would reject, you would reject your null hypothesis because the probability of getting this result seems pretty low, in which case it would suggest the alternative. But then if the p value is not less than this, then you wouldn't be able to reject the null hypothesis. But the key thing, and this is what this question is all about, in order to feel good about this calculation, we need to make some assumptions about the sampling distribution. We have to assume that it's reasonably normal, that it can actually be used to calculate this probability, and that's where these conditions come into play. The first is the random condition, and that's that the data points in this sample were truly randomly selected."}, {"video_title": "Conditions for a z test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "But then if the p value is not less than this, then you wouldn't be able to reject the null hypothesis. But the key thing, and this is what this question is all about, in order to feel good about this calculation, we need to make some assumptions about the sampling distribution. We have to assume that it's reasonably normal, that it can actually be used to calculate this probability, and that's where these conditions come into play. The first is the random condition, and that's that the data points in this sample were truly randomly selected. So pause this video. Did she meet the random condition? Well, it says she obtained an anonymous random sample of 10 ratings of employees on her team."}, {"video_title": "Conditions for a z test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "The first is the random condition, and that's that the data points in this sample were truly randomly selected. So pause this video. Did she meet the random condition? Well, it says she obtained an anonymous random sample of 10 ratings of employees on her team. They don't say how she did it, but we'll have to take their word for it that it was an anonymous random sample, so she meets the random condition. Now, what about the normal condition? The normal condition tells us that the expected number of successes, which would be our sample size times the true proportion, and the number of failures, sample size times one minus p, need to be at least equal to 10."}, {"video_title": "Conditions for a z test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Well, it says she obtained an anonymous random sample of 10 ratings of employees on her team. They don't say how she did it, but we'll have to take their word for it that it was an anonymous random sample, so she meets the random condition. Now, what about the normal condition? The normal condition tells us that the expected number of successes, which would be our sample size times the true proportion, and the number of failures, sample size times one minus p, need to be at least equal to 10. So they need to be greater than or equal to 10. Now, what are they for this particular scenario? Well, n is equal to 10, n is equal to 10, and our true proportion, remember, we're going to assume, when we do the significance test, we assume the null hypothesis is true, and the null hypothesis tells us that our true proportion is 0.1."}, {"video_title": "Conditions for a z test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "The normal condition tells us that the expected number of successes, which would be our sample size times the true proportion, and the number of failures, sample size times one minus p, need to be at least equal to 10. So they need to be greater than or equal to 10. Now, what are they for this particular scenario? Well, n is equal to 10, n is equal to 10, and our true proportion, remember, we're going to assume, when we do the significance test, we assume the null hypothesis is true, and the null hypothesis tells us that our true proportion is 0.1. So this is 0.1. This is one minus 0.1, which is 0.9. Well, 10 times 0.1 is one, so that's not greater than or equal to 10, so just off of that, we don't meet the normal condition, but even the second one, 10 times 0.9 is nine."}, {"video_title": "Conditions for a z test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Well, n is equal to 10, n is equal to 10, and our true proportion, remember, we're going to assume, when we do the significance test, we assume the null hypothesis is true, and the null hypothesis tells us that our true proportion is 0.1. So this is 0.1. This is one minus 0.1, which is 0.9. Well, 10 times 0.1 is one, so that's not greater than or equal to 10, so just off of that, we don't meet the normal condition, but even the second one, 10 times 0.9 is nine. That's also not greater than or equal to 10, so we don't meet this normal condition. We can't feel good that the sampling distribution is roughly normal, which we normally assume when we're trying to make this type of calculation. And then last but not least, independence."}, {"video_title": "Conditions for a z test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Well, 10 times 0.1 is one, so that's not greater than or equal to 10, so just off of that, we don't meet the normal condition, but even the second one, 10 times 0.9 is nine. That's also not greater than or equal to 10, so we don't meet this normal condition. We can't feel good that the sampling distribution is roughly normal, which we normally assume when we're trying to make this type of calculation. And then last but not least, independence. Independence is to feel good that each of the data points in your sample are independent. The results of whether they are a success or a failure is independent of each other. Now, if she was surveying these people with replacement, if each data point was with replacement, you would definitely meet this independence condition, but she didn't do it with replacement, but there's another way to go about it."}, {"video_title": "Conditions for a z test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "And then last but not least, independence. Independence is to feel good that each of the data points in your sample are independent. The results of whether they are a success or a failure is independent of each other. Now, if she was surveying these people with replacement, if each data point was with replacement, you would definitely meet this independence condition, but she didn't do it with replacement, but there's another way to go about it. You could use your 10% rule. If your sample size is less than 10% of the population size, then it's okay. It's considered roughly okay that you didn't do it with replacement, but her sample size here is 25%, clearly greater than 10%, and so she does not meet the independence condition either."}, {"video_title": "Conditions for a t test about a mean AP Statistics Khan Academy.mp3", "Sentence": "He suspects that on average, they send each other more than 100 messages per day. Sunil takes a random sample of seven days from their chat history and records how many messages were sent on those days. The sample data are strongly skewed to the right with a mean of 125 messages and a standard deviation of 44 messages. He wants to use these sample data to conduct a t-test about the mean. Which conditions for performing this type of significance test have been met? So let's just think about what's going on here. Sunil might have some type of a null hypothesis."}, {"video_title": "Conditions for a t test about a mean AP Statistics Khan Academy.mp3", "Sentence": "He wants to use these sample data to conduct a t-test about the mean. Which conditions for performing this type of significance test have been met? So let's just think about what's going on here. Sunil might have some type of a null hypothesis. Maybe he got this 100, maybe he read a magazine article that says that on average, the average teenager sends 100 text messages per day. And so maybe the null hypothesis is that the mean amount of messages per day that he and his friends send, which was signified by mu, maybe the null is 100, that they're no different than all other teenagers, and maybe he suspects, and actually they say it right over here, his alternative hypothesis would be what he suspects, that they'd send more than 100 text messages per day. And so what he does is he takes a sample from the population of days, and there's over 365."}, {"video_title": "Conditions for a t test about a mean AP Statistics Khan Academy.mp3", "Sentence": "Sunil might have some type of a null hypothesis. Maybe he got this 100, maybe he read a magazine article that says that on average, the average teenager sends 100 text messages per day. And so maybe the null hypothesis is that the mean amount of messages per day that he and his friends send, which was signified by mu, maybe the null is 100, that they're no different than all other teenagers, and maybe he suspects, and actually they say it right over here, his alternative hypothesis would be what he suspects, that they'd send more than 100 text messages per day. And so what he does is he takes a sample from the population of days, and there's over 365. They say they've been using the group messaging app for over a year. And he takes seven of those days. So n is equal to seven."}, {"video_title": "Conditions for a t test about a mean AP Statistics Khan Academy.mp3", "Sentence": "And so what he does is he takes a sample from the population of days, and there's over 365. They say they've been using the group messaging app for over a year. And he takes seven of those days. So n is equal to seven. And from that, he calculates sample statistics. He calculates the sample mean, which is trying to estimate the true population mean right over here. And he also is able to calculate a sample standard deviation."}, {"video_title": "Conditions for a t test about a mean AP Statistics Khan Academy.mp3", "Sentence": "So n is equal to seven. And from that, he calculates sample statistics. He calculates the sample mean, which is trying to estimate the true population mean right over here. And he also is able to calculate a sample standard deviation. And what you do in a significance test is you say, well, what is the probability of getting this sample mean or something even more extreme, assuming the null hypothesis? And if that probability is below a preset threshold, then you would reject the null hypothesis, and it would suggest the alternative. But in order to feel good about that significance test and be able to even calculate that p-value with confidence, there are conditions for performing this type of significance test."}, {"video_title": "Conditions for a t test about a mean AP Statistics Khan Academy.mp3", "Sentence": "And he also is able to calculate a sample standard deviation. And what you do in a significance test is you say, well, what is the probability of getting this sample mean or something even more extreme, assuming the null hypothesis? And if that probability is below a preset threshold, then you would reject the null hypothesis, and it would suggest the alternative. But in order to feel good about that significance test and be able to even calculate that p-value with confidence, there are conditions for performing this type of significance test. The first is is that this is truly a random sample, and that's known as the random condition. And you have seen this before when we did significance tests with proportions. Here we're doing it with means, population mean, sample mean."}, {"video_title": "Conditions for a t test about a mean AP Statistics Khan Academy.mp3", "Sentence": "But in order to feel good about that significance test and be able to even calculate that p-value with confidence, there are conditions for performing this type of significance test. The first is is that this is truly a random sample, and that's known as the random condition. And you have seen this before when we did significance tests with proportions. Here we're doing it with means, population mean, sample mean. In the past, we did it with population proportion and sample proportion. Well, the random condition, it says it right here, Sunil takes a random sample of seven days from their chat history. They don't say how he did it, but we'll just take their word for it that it was a random sample."}, {"video_title": "Conditions for a t test about a mean AP Statistics Khan Academy.mp3", "Sentence": "Here we're doing it with means, population mean, sample mean. In the past, we did it with population proportion and sample proportion. Well, the random condition, it says it right here, Sunil takes a random sample of seven days from their chat history. They don't say how he did it, but we'll just take their word for it that it was a random sample. The next condition is sometimes known as the independence, independence condition. And that's that the individual observations in our sample are roughly independent. One way that they would be independent for sure is if Sunil is sampling with replacement."}, {"video_title": "Conditions for a t test about a mean AP Statistics Khan Academy.mp3", "Sentence": "They don't say how he did it, but we'll just take their word for it that it was a random sample. The next condition is sometimes known as the independence, independence condition. And that's that the individual observations in our sample are roughly independent. One way that they would be independent for sure is if Sunil is sampling with replacement. They don't say that, but another condition, so you either could have replacement, sampling with replacement, or another way where you could feel that it's roughly independent is if your sample size is less than or equal to 10% of the population. Now in this situation, he took seven, he took a sample size of seven, and then the population of days, it says that they've been using the group messaging app for over a year, so they've been using it for over 365 days. So seven is for sure less than or equal to 10% of 365, which would be 36.5."}, {"video_title": "Conditions for a t test about a mean AP Statistics Khan Academy.mp3", "Sentence": "One way that they would be independent for sure is if Sunil is sampling with replacement. They don't say that, but another condition, so you either could have replacement, sampling with replacement, or another way where you could feel that it's roughly independent is if your sample size is less than or equal to 10% of the population. Now in this situation, he took seven, he took a sample size of seven, and then the population of days, it says that they've been using the group messaging app for over a year, so they've been using it for over 365 days. So seven is for sure less than or equal to 10% of 365, which would be 36.5. So we meet this condition, which allows us to meet the independence condition. Now the last condition is often known as the normal condition, and this is to feel good that the sampling distribution of the sample means right over here is approximately normal. And this is going to be a little bit different than what we saw with significance tests when we dealt with proportions."}, {"video_title": "Conditions for a t test about a mean AP Statistics Khan Academy.mp3", "Sentence": "So seven is for sure less than or equal to 10% of 365, which would be 36.5. So we meet this condition, which allows us to meet the independence condition. Now the last condition is often known as the normal condition, and this is to feel good that the sampling distribution of the sample means right over here is approximately normal. And this is going to be a little bit different than what we saw with significance tests when we dealt with proportions. There's a few ways to feel good that the sampling distribution of the sample means is normal. One is is if the underlying parent population normal. So parent, parent population normal."}, {"video_title": "Conditions for a t test about a mean AP Statistics Khan Academy.mp3", "Sentence": "And this is going to be a little bit different than what we saw with significance tests when we dealt with proportions. There's a few ways to feel good that the sampling distribution of the sample means is normal. One is is if the underlying parent population normal. So parent, parent population normal. Now they don't tell us anything that there's actually a normal distribution for the amount of time that they spend on a given day. So we don't know this one for sure, but sometimes you might. Another way is to feel good that our sample size is greater than or equal to 30."}, {"video_title": "Conditions for a t test about a mean AP Statistics Khan Academy.mp3", "Sentence": "So parent, parent population normal. Now they don't tell us anything that there's actually a normal distribution for the amount of time that they spend on a given day. So we don't know this one for sure, but sometimes you might. Another way is to feel good that our sample size is greater than or equal to 30. And this comes from the central limit theorem that then our sampling distribution is going to be roughly normal. But we see very clearly our sample size is not greater than or equal to 30, so we don't meet that constraint either. Now the third way that we could feel good that our sampling distribution of our sample mean is roughly normal is if our sample, is if our sample is symmetric, symmetric, and there are no outliers, or maybe even you could say no significant outliers."}, {"video_title": "Conditions for a t test about a mean AP Statistics Khan Academy.mp3", "Sentence": "Another way is to feel good that our sample size is greater than or equal to 30. And this comes from the central limit theorem that then our sampling distribution is going to be roughly normal. But we see very clearly our sample size is not greater than or equal to 30, so we don't meet that constraint either. Now the third way that we could feel good that our sampling distribution of our sample mean is roughly normal is if our sample, is if our sample is symmetric, symmetric, and there are no outliers, or maybe even you could say no significant outliers. Now is this the case? Well it says right over here, the sample data are strongly skewed to the right with a mean of 125 messages and a standard deviation of 44 messages. So this strongly skewed to the right, it's clearly not a symmetric sample data."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "Darnell is a middle school student with a height of 161.4 centimeters. What proportion of student heights are lower than Darnell's height? So let's think about what they are asking. So they're saying that heights are normally distributed. So it would have a shape that looks something like that. That's my hand-drawn version of it. There's a mean of 150 centimeters."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "So they're saying that heights are normally distributed. So it would have a shape that looks something like that. That's my hand-drawn version of it. There's a mean of 150 centimeters. So right over here, that would be 150 centimeters. They tell us that there's a standard deviation of 20 centimeters, and Darnell has a height of 161.4 centimeters. So Darnell is above the mean."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "There's a mean of 150 centimeters. So right over here, that would be 150 centimeters. They tell us that there's a standard deviation of 20 centimeters, and Darnell has a height of 161.4 centimeters. So Darnell is above the mean. So let's say he is right over here, and I'm not drawing it exactly, but you get the idea. That is 161.4 centimeters. And we want to figure out what proportion of students' heights are lower than Darnell's height."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "So Darnell is above the mean. So let's say he is right over here, and I'm not drawing it exactly, but you get the idea. That is 161.4 centimeters. And we want to figure out what proportion of students' heights are lower than Darnell's height. So we want to figure out what is the area under the normal curve right over here. That will give us the proportion that are lower than Darnell's height. And I'll give you a hint on how to do this."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "And we want to figure out what proportion of students' heights are lower than Darnell's height. So we want to figure out what is the area under the normal curve right over here. That will give us the proportion that are lower than Darnell's height. And I'll give you a hint on how to do this. We need to think about how many standard deviations above the mean is Darnell. And we can do that because they tell us what the standard deviation is, and we know the difference between Darnell's height and the mean height. And then once we know how many standard deviations he is above the mean, that's our z-score, we can look at a z-table that tell us what proportion is less than that amount in a normal distribution."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "And I'll give you a hint on how to do this. We need to think about how many standard deviations above the mean is Darnell. And we can do that because they tell us what the standard deviation is, and we know the difference between Darnell's height and the mean height. And then once we know how many standard deviations he is above the mean, that's our z-score, we can look at a z-table that tell us what proportion is less than that amount in a normal distribution. So let's do that. So I have my TI-84 emulator right over here. And let's see, Darnell is 161.4 centimeters, 161.4."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "And then once we know how many standard deviations he is above the mean, that's our z-score, we can look at a z-table that tell us what proportion is less than that amount in a normal distribution. So let's do that. So I have my TI-84 emulator right over here. And let's see, Darnell is 161.4 centimeters, 161.4. Now the mean is 150, minus 150, is equal to, we could have done that in our head, 11.4 centimeters. Now how many standard deviations is that above the mean? Well they tell us that a standard deviation in this case for this distribution is 20 centimeters."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "And let's see, Darnell is 161.4 centimeters, 161.4. Now the mean is 150, minus 150, is equal to, we could have done that in our head, 11.4 centimeters. Now how many standard deviations is that above the mean? Well they tell us that a standard deviation in this case for this distribution is 20 centimeters. So we'll take 11.4 divided by 20. So we will just take our previous answer. So this just means our previous answer divided by 20 centimeters."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "Well they tell us that a standard deviation in this case for this distribution is 20 centimeters. So we'll take 11.4 divided by 20. So we will just take our previous answer. So this just means our previous answer divided by 20 centimeters. And that gets us 0.57. So we can say that this is 0.57 standard deviations 0.57 standard deviations, deviations above the mean. Now why is that useful?"}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "So this just means our previous answer divided by 20 centimeters. And that gets us 0.57. So we can say that this is 0.57 standard deviations 0.57 standard deviations, deviations above the mean. Now why is that useful? Well you could take this z-score right over here and look at a z-table to figure out what proportion is less than 0.57 standard deviations above the mean. So let's get a z-table over here. So what we're going to do is we're gonna look up this z-score on this table."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "Now why is that useful? Well you could take this z-score right over here and look at a z-table to figure out what proportion is less than 0.57 standard deviations above the mean. So let's get a z-table over here. So what we're going to do is we're gonna look up this z-score on this table. And the way that you do it, this first column, each row tells us our z-score up until the tenths place. And then each of these columns after that tell us which hundredths we're in. So 0.57, the tenths place is right over here."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "So what we're going to do is we're gonna look up this z-score on this table. And the way that you do it, this first column, each row tells us our z-score up until the tenths place. And then each of these columns after that tell us which hundredths we're in. So 0.57, the tenths place is right over here. So we're going to be in this row. And then our hundredths place is this seven. So we'll look right over here."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "So 0.57, the tenths place is right over here. So we're going to be in this row. And then our hundredths place is this seven. So we'll look right over here. So 0.57, this tells us the proportion that is lower than 0.57 standard deviations above the mean. And so it is 0.7157, or another way to think about it is, if the heights are truly normally distributed, 71.57% of the students would have a height less than Darnell's. But the answer to this question, what proportion of students' heights are lower than Darnell's height?"}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "It is election season and there is a runoff between candidate A versus candidate B. And we are pollsters and we're interested in figuring out well what's the likelihood that candidate A wins this election? Well ideally, we would go to the entire population of likely voters right over here. Let's say there's 100,000 likely voters and we would ask every one of them, who do you support? And from that, we would be able to get the population proportion. Which would be, this is the proportion that support candidate A. But it might not be realistic."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "Let's say there's 100,000 likely voters and we would ask every one of them, who do you support? And from that, we would be able to get the population proportion. Which would be, this is the proportion that support candidate A. But it might not be realistic. In fact, it definitely will not be realistic to ask all 100,000 people. So instead, we do the thing that we tend to do in statistics is that we sample this population and we calculate a statistic from that sample in order to estimate this parameter. So let's say we take a sample right over here."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "But it might not be realistic. In fact, it definitely will not be realistic to ask all 100,000 people. So instead, we do the thing that we tend to do in statistics is that we sample this population and we calculate a statistic from that sample in order to estimate this parameter. So let's say we take a sample right over here. So this sample size, let's say n equals 100 and we calculate the sample proportion that support candidate A. So out of the 100, let's say that 54 say that they're going to support candidate A. So the sample proportion here is 0.54."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "So let's say we take a sample right over here. So this sample size, let's say n equals 100 and we calculate the sample proportion that support candidate A. So out of the 100, let's say that 54 say that they're going to support candidate A. So the sample proportion here is 0.54. And just to appreciate that we're not always going to get 0.54, there could have been a situation where we sampled a different 100 and we would have maybe gotten a different sample proportion, maybe in that one we got 0.58. And we already have the tools in statistics to think about this, the distribution of the possible sample proportions we could get. We've talked about it when we thought about sampling distributions."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "So the sample proportion here is 0.54. And just to appreciate that we're not always going to get 0.54, there could have been a situation where we sampled a different 100 and we would have maybe gotten a different sample proportion, maybe in that one we got 0.58. And we already have the tools in statistics to think about this, the distribution of the possible sample proportions we could get. We've talked about it when we thought about sampling distributions. So you could have the sampling distribution of the sample proportions, of the sample proportions, and it's going to, this distribution is going to be specific to what our sample size is for n is equal to 100. And so we can describe the possible sample proportions we could get and their likelihoods with this sampling distribution. So let me do that."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "We've talked about it when we thought about sampling distributions. So you could have the sampling distribution of the sample proportions, of the sample proportions, and it's going to, this distribution is going to be specific to what our sample size is for n is equal to 100. And so we can describe the possible sample proportions we could get and their likelihoods with this sampling distribution. So let me do that. So it would look something like this. Because our sample size is so much smaller than the population, it's way less than 10%, we can assume that each person we're asking that it's approximately independent. Also, if we make the assumption that the true proportion isn't too close to zero or not too close to one, then we can say that, well look, this sampling distribution is roughly going to be normal."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "So let me do that. So it would look something like this. Because our sample size is so much smaller than the population, it's way less than 10%, we can assume that each person we're asking that it's approximately independent. Also, if we make the assumption that the true proportion isn't too close to zero or not too close to one, then we can say that, well look, this sampling distribution is roughly going to be normal. So we'll have a normal, this kind of bell curve shape. And we know a lot about the sampling distribution of the sample proportions. We know already, for example, and if this is foreign to you, I encourage you to watch the videos on this on Khan Academy, that the mean of this sampling distribution is going to be the actual population proportion."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "Also, if we make the assumption that the true proportion isn't too close to zero or not too close to one, then we can say that, well look, this sampling distribution is roughly going to be normal. So we'll have a normal, this kind of bell curve shape. And we know a lot about the sampling distribution of the sample proportions. We know already, for example, and if this is foreign to you, I encourage you to watch the videos on this on Khan Academy, that the mean of this sampling distribution is going to be the actual population proportion. And we also know what the standard deviation of this is going to be. So let me, that's maybe, that's one standard deviation, this is two standard deviations, that's three standard deviations. Above the mean, that's one standard deviation, two standard deviations, three standard deviations below the mean."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "We know already, for example, and if this is foreign to you, I encourage you to watch the videos on this on Khan Academy, that the mean of this sampling distribution is going to be the actual population proportion. And we also know what the standard deviation of this is going to be. So let me, that's maybe, that's one standard deviation, this is two standard deviations, that's three standard deviations. Above the mean, that's one standard deviation, two standard deviations, three standard deviations below the mean. So this distance, let me do this in a different color. This standard deviation right over here, which we denote as the standard deviation of the sample proportions for this sampling distribution. This is, we've already seen the formula there."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "Above the mean, that's one standard deviation, two standard deviations, three standard deviations below the mean. So this distance, let me do this in a different color. This standard deviation right over here, which we denote as the standard deviation of the sample proportions for this sampling distribution. This is, we've already seen the formula there. It's the square root of p times one minus p, where p is once again our population proportion, divided by our sample size. That's why it's specific for n equals 100 here. And so in this first scenario, and let's just focus on this one right over here."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "This is, we've already seen the formula there. It's the square root of p times one minus p, where p is once again our population proportion, divided by our sample size. That's why it's specific for n equals 100 here. And so in this first scenario, and let's just focus on this one right over here. When we took a sample size of n equals 100 and we got the sample proportion of 0.54, we could have gotten all sorts of outcomes here. Maybe 0.54 is right over here. Maybe 0.54 is right over here."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "And so in this first scenario, and let's just focus on this one right over here. When we took a sample size of n equals 100 and we got the sample proportion of 0.54, we could have gotten all sorts of outcomes here. Maybe 0.54 is right over here. Maybe 0.54 is right over here. And the reason why I have this uncertainty is we actually don't know what the real population parameter is, what the real population proportion is. But let me ask you maybe a slightly easier question. What is, what is the probability, probability that our sample proportion of 0.54 is within, is within two times two standard deviations of p?"}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "Maybe 0.54 is right over here. And the reason why I have this uncertainty is we actually don't know what the real population parameter is, what the real population proportion is. But let me ask you maybe a slightly easier question. What is, what is the probability, probability that our sample proportion of 0.54 is within, is within two times two standard deviations of p? Pause the video and think about that. Well, that's just saying, look, if I'm gonna take a sample and calculate the sample proportion right over here, what's the probability that I'm within two standard deviations of the mean? Well, that's essentially going to be this area right over here."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "What is, what is the probability, probability that our sample proportion of 0.54 is within, is within two times two standard deviations of p? Pause the video and think about that. Well, that's just saying, look, if I'm gonna take a sample and calculate the sample proportion right over here, what's the probability that I'm within two standard deviations of the mean? Well, that's essentially going to be this area right over here. And we know from studying normal curves that approximately 95% of the area is within two standard deviations. So this is approximately 95%. 95% of the time that I take a sample size of 100 and I calculate this sample proportion, 95% of the time I'm going to be within two standard deviations."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "Well, that's essentially going to be this area right over here. And we know from studying normal curves that approximately 95% of the area is within two standard deviations. So this is approximately 95%. 95% of the time that I take a sample size of 100 and I calculate this sample proportion, 95% of the time I'm going to be within two standard deviations. But if you take this statement, you can actually construct another statement that starts to feel a little bit more, I guess we could say, inferential. We could say there, there is a 95% probability that the population proportion, p, is within two standard deviations, two standard deviations of p hat, which is equal to 0.54. Pause this video."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "95% of the time that I take a sample size of 100 and I calculate this sample proportion, 95% of the time I'm going to be within two standard deviations. But if you take this statement, you can actually construct another statement that starts to feel a little bit more, I guess we could say, inferential. We could say there, there is a 95% probability that the population proportion, p, is within two standard deviations, two standard deviations of p hat, which is equal to 0.54. Pause this video. Appreciate that these two are equivalent statements. If there's a 95% chance that our sample proportion is within two standard deviations of the true proportion, well, that's equivalent to saying that there's a 95% chance that our true proportion is within two standard deviations of our sample proportion. And this is really, really interesting because if we were able to figure out what this value is, well, then we would be able to create what you could call a confidence interval."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "Pause this video. Appreciate that these two are equivalent statements. If there's a 95% chance that our sample proportion is within two standard deviations of the true proportion, well, that's equivalent to saying that there's a 95% chance that our true proportion is within two standard deviations of our sample proportion. And this is really, really interesting because if we were able to figure out what this value is, well, then we would be able to create what you could call a confidence interval. Now, you immediately might be seeing a problem here. In order to calculate this, our standard deviation of this distribution, we have to know our population parameter. So pause this video and think about what we would do instead if we don't know what p is here, if we don't know our population proportion, do we have something that we could use as an estimate for our population proportion?"}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "And this is really, really interesting because if we were able to figure out what this value is, well, then we would be able to create what you could call a confidence interval. Now, you immediately might be seeing a problem here. In order to calculate this, our standard deviation of this distribution, we have to know our population parameter. So pause this video and think about what we would do instead if we don't know what p is here, if we don't know our population proportion, do we have something that we could use as an estimate for our population proportion? Well, yes, we calculated p hat already. We calculated our sample proportion. And so a new statistic that we could define is the standard error."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "So pause this video and think about what we would do instead if we don't know what p is here, if we don't know our population proportion, do we have something that we could use as an estimate for our population proportion? Well, yes, we calculated p hat already. We calculated our sample proportion. And so a new statistic that we could define is the standard error. The standard error of our sample proportions. And we can define that as being equal to, since we don't know the population proportion, we're gonna use a sample proportion. P hat times one minus p hat, all of that over n. In this case, of course, n is 100."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "And so a new statistic that we could define is the standard error. The standard error of our sample proportions. And we can define that as being equal to, since we don't know the population proportion, we're gonna use a sample proportion. P hat times one minus p hat, all of that over n. In this case, of course, n is 100. We do know that. And it actually turns out, I'm not gonna prove it in this video, that this actually is an unbiased estimator for this right over here. So this is going to be equal to 0.54 times one minus 0.54."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "P hat times one minus p hat, all of that over n. In this case, of course, n is 100. We do know that. And it actually turns out, I'm not gonna prove it in this video, that this actually is an unbiased estimator for this right over here. So this is going to be equal to 0.54 times one minus 0.54. So it's 0.46. All of that over 100. So we have the square root of 0.54 times 0.46 divided by 100, close my parentheses, Enter."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "So this is going to be equal to 0.54 times one minus 0.54. So it's 0.46. All of that over 100. So we have the square root of 0.54 times 0.46 divided by 100, close my parentheses, Enter. So if we're around to the nearest hundredth, it's going to be actually, even around to the nearest thousandth, it's gonna be approximately five hundredths. So this is going to be, this is approximately 0.05. So another way to say all of these things is, instead, we don't know exactly this, but now we have an estimate for it."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "So we have the square root of 0.54 times 0.46 divided by 100, close my parentheses, Enter. So if we're around to the nearest hundredth, it's going to be actually, even around to the nearest thousandth, it's gonna be approximately five hundredths. So this is going to be, this is approximately 0.05. So another way to say all of these things is, instead, we don't know exactly this, but now we have an estimate for it. So we can now say, with 95% confidence, and that will often be known as our confidence level right over here, with 95% confidence, between, between, and so we'd wanna go two standard errors below our sample proportion that we just happened to calculate. So that would be 0.54 minus two times five hundredths. So that would be 0.54 minus 10 hundredths, which would be 0.44."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "So another way to say all of these things is, instead, we don't know exactly this, but now we have an estimate for it. So we can now say, with 95% confidence, and that will often be known as our confidence level right over here, with 95% confidence, between, between, and so we'd wanna go two standard errors below our sample proportion that we just happened to calculate. So that would be 0.54 minus two times five hundredths. So that would be 0.54 minus 10 hundredths, which would be 0.44. And we'd also wanna go two standard errors above the sample proportion. So that would be that plus 10 hundredths, and 0.64 of voters, of voters, support, support A. And so this interval that we have right over here, from 0.44 to 0.64, this will be known as our confidence interval."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "So that would be 0.54 minus 10 hundredths, which would be 0.44. And we'd also wanna go two standard errors above the sample proportion. So that would be that plus 10 hundredths, and 0.64 of voters, of voters, support, support A. And so this interval that we have right over here, from 0.44 to 0.64, this will be known as our confidence interval. Confidence interval. And this will change, not just in the starting point and the end point, but it will change the actual length of our confidence interval, will change depending on what sample proportion we happen to pick for that sample of 100. A related idea to the confidence interval is this notion of margin of error."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "And so this interval that we have right over here, from 0.44 to 0.64, this will be known as our confidence interval. Confidence interval. And this will change, not just in the starting point and the end point, but it will change the actual length of our confidence interval, will change depending on what sample proportion we happen to pick for that sample of 100. A related idea to the confidence interval is this notion of margin of error. Margin of error. And for this particular case, for this particular sample, our margin of error, because we care about 95% confidence, so that would be two standard errors. So our margin of error here is two times our standard error, which is be 0.1 or 0.10."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "A related idea to the confidence interval is this notion of margin of error. Margin of error. And for this particular case, for this particular sample, our margin of error, because we care about 95% confidence, so that would be two standard errors. So our margin of error here is two times our standard error, which is be 0.1 or 0.10. And so we're going one margin of error above our sample proportion, right over here, and one margin of error below our sample proportion, right over here, to define our confidence interval. And as I mentioned, this margin of error is not going to be fixed every time we take a sample. Depending on what our sample proportion is, it's going to affect our margin of error, because that is calculated essentially with the standard error."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "So our margin of error here is two times our standard error, which is be 0.1 or 0.10. And so we're going one margin of error above our sample proportion, right over here, and one margin of error below our sample proportion, right over here, to define our confidence interval. And as I mentioned, this margin of error is not going to be fixed every time we take a sample. Depending on what our sample proportion is, it's going to affect our margin of error, because that is calculated essentially with the standard error. Another interpretation of this is that the method that we used to get this interval right over here, the method that we used to get this confidence interval, when we use it over and over, it will produce intervals, and the intervals won't always be the same, it's gonna be dependent on our sample proportion, but it will produce intervals which include the true proportion, which we might not know and often don't know. It'll include the true proportion 95% of the time. I'll cover that intuition more in future videos."}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "Depending on what our sample proportion is, it's going to affect our margin of error, because that is calculated essentially with the standard error. Another interpretation of this is that the method that we used to get this interval right over here, the method that we used to get this confidence interval, when we use it over and over, it will produce intervals, and the intervals won't always be the same, it's gonna be dependent on our sample proportion, but it will produce intervals which include the true proportion, which we might not know and often don't know. It'll include the true proportion 95% of the time. I'll cover that intuition more in future videos. We'll see how the interval changes, how the margin of error changes, but when you do this calculation over and over and over again, 95% of the time, your true proportion is going to be contained in whatever interval you happen to calculate that time. Now another interesting question is, well, what if you wanted to tighten up the intervals on average? How would you do that?"}, {"video_title": "Confidence intervals and margin of error AP Statistics Khan Academy.mp3", "Sentence": "I'll cover that intuition more in future videos. We'll see how the interval changes, how the margin of error changes, but when you do this calculation over and over and over again, 95% of the time, your true proportion is going to be contained in whatever interval you happen to calculate that time. Now another interesting question is, well, what if you wanted to tighten up the intervals on average? How would you do that? Well, if you wanted to lower your margin of error, the best way to lower the margin of error is if you increase this denominator right over here, and increasing that denominator means increasing the sample size. And so one thing that you will often see when people are talking about election coverage is, well, we need to sample more people in order to get a lower margin of error. But I'll leave you there, and I'll see you in future videos."}, {"video_title": "Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3", "Sentence": "His sample mean was four years and his sample standard deviation was two years. Rory wants to use these sample data to conduct a t-test on the mean. Assume that all conditions for inference have been met. Calculate the test statistic for Rory's test. So I always just like to remind ourselves what's going on. So you have your null hypothesis here that the mean number of years of experience for teachers in the district is five and then the alternative hypothesis is that the mean years of experience is less than five for teachers in the district. So if this represents all the teachers in the district, the population, then what he did is he took a sample and said he used a sample of 25 teachers."}, {"video_title": "Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3", "Sentence": "Calculate the test statistic for Rory's test. So I always just like to remind ourselves what's going on. So you have your null hypothesis here that the mean number of years of experience for teachers in the district is five and then the alternative hypothesis is that the mean years of experience is less than five for teachers in the district. So if this represents all the teachers in the district, the population, then what he did is he took a sample and said he used a sample of 25 teachers. So n here is equal to 25. And then from that sample, he was able to calculate some statistics. He was able to calculate the sample mean."}, {"video_title": "Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3", "Sentence": "So if this represents all the teachers in the district, the population, then what he did is he took a sample and said he used a sample of 25 teachers. So n here is equal to 25. And then from that sample, he was able to calculate some statistics. He was able to calculate the sample mean. So that sample mean was four years. The sample mean was four years. And then he was also able to calculate the sample standard deviation."}, {"video_title": "Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3", "Sentence": "He was able to calculate the sample mean. So that sample mean was four years. The sample mean was four years. And then he was also able to calculate the sample standard deviation. The sample standard deviation was equal to two years. Now, the whole point that we do or the main thing we do when we do significance tests is we say, all right, if we assume the null hypothesis is true, what's the probability of getting a sample mean this low or lower? And if that probability is below a preset significance level, then we reject the null hypothesis and it suggests the alternative."}, {"video_title": "Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3", "Sentence": "And then he was also able to calculate the sample standard deviation. The sample standard deviation was equal to two years. Now, the whole point that we do or the main thing we do when we do significance tests is we say, all right, if we assume the null hypothesis is true, what's the probability of getting a sample mean this low or lower? And if that probability is below a preset significance level, then we reject the null hypothesis and it suggests the alternative. But in order to figure out that probability, we need to figure out a test statistic. Sometimes we use a z-test. If we're dealing with proportions."}, {"video_title": "Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3", "Sentence": "And if that probability is below a preset significance level, then we reject the null hypothesis and it suggests the alternative. But in order to figure out that probability, we need to figure out a test statistic. Sometimes we use a z-test. If we're dealing with proportions. But when we deal with means, we tend to use a t-test. And the reason why is if you wanted to figure out a z-statistic, what you would do is you would take your sample mean, subtract from that the assumed mean from the null hypothesis. So mu, and I'll just put a little zero, sub zero there."}, {"video_title": "Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3", "Sentence": "If we're dealing with proportions. But when we deal with means, we tend to use a t-test. And the reason why is if you wanted to figure out a z-statistic, what you would do is you would take your sample mean, subtract from that the assumed mean from the null hypothesis. So mu, and I'll just put a little zero, sub zero there. So this is the assumed mean from the null hypothesis. And then you would want to divide by the standard deviation of the sampling distribution of the sample mean. So you'd wanna divide by that."}, {"video_title": "Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3", "Sentence": "So mu, and I'll just put a little zero, sub zero there. So this is the assumed mean from the null hypothesis. And then you would want to divide by the standard deviation of the sampling distribution of the sample mean. So you'd wanna divide by that. But this, we don't know. And so that's why instead we do a t-statistic, in which case we take the difference between our sample mean and our assumed population mean, the population parameter, and we try to estimate this. And we estimate that with our sample standard deviation divided by the square root of our sample size."}, {"video_title": "Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3", "Sentence": "So you'd wanna divide by that. But this, we don't know. And so that's why instead we do a t-statistic, in which case we take the difference between our sample mean and our assumed population mean, the population parameter, and we try to estimate this. And we estimate that with our sample standard deviation divided by the square root of our sample size. And so if you're inspired, I encourage you, even if you're not inspired, I encourage you to pause this video and try to calculate this t-statistic. Well, this is going to be equal to, let's see, our sample mean is four minus our assumed mean is five, our assumed population mean is five. Our sample standard deviation is two."}, {"video_title": "Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3", "Sentence": "And we estimate that with our sample standard deviation divided by the square root of our sample size. And so if you're inspired, I encourage you, even if you're not inspired, I encourage you to pause this video and try to calculate this t-statistic. Well, this is going to be equal to, let's see, our sample mean is four minus our assumed mean is five, our assumed population mean is five. Our sample standard deviation is two. All of that over the square root of the sample size, all of that over the square root of 25. So this is going to be equal, our numerator is negative one. So it's negative one divided by two over five, which is equal to negative one times five over two."}, {"video_title": "Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3", "Sentence": "Our sample standard deviation is two. All of that over the square root of the sample size, all of that over the square root of 25. So this is going to be equal, our numerator is negative one. So it's negative one divided by two over five, which is equal to negative one times five over two. And so this is going to be equal to, equal to negative five over two, or negative 2.5. And then what we would do in this, what Rory would do, is then look this t-value up on a t-table and say, so if you look at a distribution of a t-statistic, something like that, and say, okay, we are negative 2.5 below the mean. So negative, negative 2.5."}, {"video_title": "Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3", "Sentence": "So it's negative one divided by two over five, which is equal to negative one times five over two. And so this is going to be equal to, equal to negative five over two, or negative 2.5. And then what we would do in this, what Rory would do, is then look this t-value up on a t-table and say, so if you look at a distribution of a t-statistic, something like that, and say, okay, we are negative 2.5 below the mean. So negative, negative 2.5. And so what he would wanna do is figure out this area here, because this would be the probability of being that far below the mean or even further below the mean. And so that would give us our p-value. And then if that p-value is below some preset significance level that Rory should have set, maybe 5% or 1%, then he'll reject the null hypothesis, which would suggest his suspicion that the true mean of years of experience for the teachers in his district is less than five."}, {"video_title": "Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3", "Sentence": "So negative, negative 2.5. And so what he would wanna do is figure out this area here, because this would be the probability of being that far below the mean or even further below the mean. And so that would give us our p-value. And then if that p-value is below some preset significance level that Rory should have set, maybe 5% or 1%, then he'll reject the null hypothesis, which would suggest his suspicion that the true mean of years of experience for the teachers in his district is less than five. Now another really important thing to keep in mind is, they told us that assume all conditions for inference have been met. And so that's the, assuming that this was truly a random sample, that each, the individual observations are either truly independent or roughly independent, that maybe he observed either with replacement or it's less than 10% of the population, and he feels good that the sampling distribution is going to be roughly normal. And we've talked about that in other videos."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "What is the probability that you will run out of water? So let's think about what's happening here. So there's some distribution of how many liters the average man needs when they're active outdoors. And let me just draw an example. It might look something like this. So they're all going to need at least more than 0 liters. So this would be 0 liters over here."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "And let me just draw an example. It might look something like this. So they're all going to need at least more than 0 liters. So this would be 0 liters over here. The average male, so the mean of the amount of water a man needs when active outdoors, is 2 liters. So 2 liters would be right over here. So the mean is equal to 2 liters."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So this would be 0 liters over here. The average male, so the mean of the amount of water a man needs when active outdoors, is 2 liters. So 2 liters would be right over here. So the mean is equal to 2 liters. It has a standard deviation of 0.7 liters, or 0.7 liters. So the standard deviation, maybe I'll draw it this way. So this distribution, once again, we don't know whether it's a normal distribution or not."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So the mean is equal to 2 liters. It has a standard deviation of 0.7 liters, or 0.7 liters. So the standard deviation, maybe I'll draw it this way. So this distribution, once again, we don't know whether it's a normal distribution or not. It could just be some type of crazy distribution. So maybe some people need almost close to, well, everyone needs a little bit of water, but maybe some people need very, very little water. Then you have a lot of people who need that, maybe some people who need more, and maybe no one can drink more than maybe this is like 4 liters of water."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So this distribution, once again, we don't know whether it's a normal distribution or not. It could just be some type of crazy distribution. So maybe some people need almost close to, well, everyone needs a little bit of water, but maybe some people need very, very little water. Then you have a lot of people who need that, maybe some people who need more, and maybe no one can drink more than maybe this is like 4 liters of water. So maybe this is the actual distribution. And then one standard deviation is going to be 0.7 liters away. So this is 1, 0.7 liters is, so this would be 1 liter, 2 liters, 3 liters."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "Then you have a lot of people who need that, maybe some people who need more, and maybe no one can drink more than maybe this is like 4 liters of water. So maybe this is the actual distribution. And then one standard deviation is going to be 0.7 liters away. So this is 1, 0.7 liters is, so this would be 1 liter, 2 liters, 3 liters. So one standard deviation is going to be about that far away from the mean. If you go above it, it'll be about that far if you go below it. So let me draw."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So this is 1, 0.7 liters is, so this would be 1 liter, 2 liters, 3 liters. So one standard deviation is going to be about that far away from the mean. If you go above it, it'll be about that far if you go below it. So let me draw. This is the standard deviation. That right there is the standard deviation to the right. That's the standard deviation to the left."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So let me draw. This is the standard deviation. That right there is the standard deviation to the right. That's the standard deviation to the left. And we know that the standard deviation is equal to, I'll write the 0 in front, 0.7 liters. So that's the actual distribution of how much water the average man needs when active. Now what's interesting about this problem, we are planning a full day nature trip for 50 men and we'll bring 110 liters of water."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "That's the standard deviation to the left. And we know that the standard deviation is equal to, I'll write the 0 in front, 0.7 liters. So that's the actual distribution of how much water the average man needs when active. Now what's interesting about this problem, we are planning a full day nature trip for 50 men and we'll bring 110 liters of water. What is the probability that you will run out? So the probability that you will run out, let me write this down. The probability that I will or that you will run out is equal or is the same thing as the probability that we use more than 110 liters on our outdoor nature day, whatever we're doing, which is the same thing as the probability."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "Now what's interesting about this problem, we are planning a full day nature trip for 50 men and we'll bring 110 liters of water. What is the probability that you will run out? So the probability that you will run out, let me write this down. The probability that I will or that you will run out is equal or is the same thing as the probability that we use more than 110 liters on our outdoor nature day, whatever we're doing, which is the same thing as the probability. If we use more than 110 liters, that means that on average, because we have 50 men, so 110 divided by 50 is what? That's 2. Let me get the calculator out just so we don't make any mistakes here."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "The probability that I will or that you will run out is equal or is the same thing as the probability that we use more than 110 liters on our outdoor nature day, whatever we're doing, which is the same thing as the probability. If we use more than 110 liters, that means that on average, because we have 50 men, so 110 divided by 50 is what? That's 2. Let me get the calculator out just so we don't make any mistakes here. So this is going to be the calculator out. So on average, if we have 110 liters and it's going to be drunk by 50 men, including ourselves, I guess, that means that it's the problem. So we would run out if on average more than 2.2 liters is used per man."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "Let me get the calculator out just so we don't make any mistakes here. So this is going to be the calculator out. So on average, if we have 110 liters and it's going to be drunk by 50 men, including ourselves, I guess, that means that it's the problem. So we would run out if on average more than 2.2 liters is used per man. So this is the same thing as the probability of the average, or maybe we should say the sample mean, or let me write it this way, that the average water use per man of our 50 men is greater than, or we could say greater than or equal to, let me say greater than, well, I'll say greater than, because if we write on the money, then we won't run out of water, is greater than 2.2 liters per man. So let's think about this. We are essentially taking 50 men out of a universal sample."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So we would run out if on average more than 2.2 liters is used per man. So this is the same thing as the probability of the average, or maybe we should say the sample mean, or let me write it this way, that the average water use per man of our 50 men is greater than, or we could say greater than or equal to, let me say greater than, well, I'll say greater than, because if we write on the money, then we won't run out of water, is greater than 2.2 liters per man. So let's think about this. We are essentially taking 50 men out of a universal sample. And we got this data. Who knows where we got this data from, that the average man drinks 2 liters, and that the standard deviation is this. Maybe there's some huge study, and this was the best estimate of what the population parameters are, that this is the mean and this is the standard deviation."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "We are essentially taking 50 men out of a universal sample. And we got this data. Who knows where we got this data from, that the average man drinks 2 liters, and that the standard deviation is this. Maybe there's some huge study, and this was the best estimate of what the population parameters are, that this is the mean and this is the standard deviation. Now we're sampling 50 men. And what we need to do is figure out, essentially, what is the probability that the mean of this sample, that the sample mean is going to be greater than 2.2 liters. And to do that, we have to figure out the distribution of the sampling mean."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "Maybe there's some huge study, and this was the best estimate of what the population parameters are, that this is the mean and this is the standard deviation. Now we're sampling 50 men. And what we need to do is figure out, essentially, what is the probability that the mean of this sample, that the sample mean is going to be greater than 2.2 liters. And to do that, we have to figure out the distribution of the sampling mean. And we know what that's called. It's the sampling distribution of the sample means. And we know that that is going to be a normal distribution."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "And to do that, we have to figure out the distribution of the sampling mean. And we know what that's called. It's the sampling distribution of the sample means. And we know that that is going to be a normal distribution. We know a few of the properties of that normal distribution. So this is the distribution of just all men. And then if you take samples of, say, 50 men."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "And we know that that is going to be a normal distribution. We know a few of the properties of that normal distribution. So this is the distribution of just all men. And then if you take samples of, say, 50 men. So this will be, let me write this down. So down here, I'm going to draw the sampling distribution of the sample mean when n. So when our sample size is equal to 50. So this is essentially telling us the likelihood of the different means when we are sampling 50 men from this population and taking their average water use."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "And then if you take samples of, say, 50 men. So this will be, let me write this down. So down here, I'm going to draw the sampling distribution of the sample mean when n. So when our sample size is equal to 50. So this is essentially telling us the likelihood of the different means when we are sampling 50 men from this population and taking their average water use. So let me draw that. So let's say this is the frequency, and then here are the different values. Now, the mean value of this, the mean of the sampling distribution of the sample mean, this x bar, that's really just the sample mean right over there, is equal to, it's going to be equal to, if we were to do this millions and millions of times, if we were to plot all of the means when we keep taking samples of 50 and then we were to plot them all out, we would show that this mean of distribution is actually going to be the mean of our actual population."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So this is essentially telling us the likelihood of the different means when we are sampling 50 men from this population and taking their average water use. So let me draw that. So let's say this is the frequency, and then here are the different values. Now, the mean value of this, the mean of the sampling distribution of the sample mean, this x bar, that's really just the sample mean right over there, is equal to, it's going to be equal to, if we were to do this millions and millions of times, if we were to plot all of the means when we keep taking samples of 50 and then we were to plot them all out, we would show that this mean of distribution is actually going to be the mean of our actual population. So it's going to be the same value. I want to do it in that same blue. It's going to be the same value as this population over here."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "Now, the mean value of this, the mean of the sampling distribution of the sample mean, this x bar, that's really just the sample mean right over there, is equal to, it's going to be equal to, if we were to do this millions and millions of times, if we were to plot all of the means when we keep taking samples of 50 and then we were to plot them all out, we would show that this mean of distribution is actually going to be the mean of our actual population. So it's going to be the same value. I want to do it in that same blue. It's going to be the same value as this population over here. So that is going to be 2 liters. So we're still centered at 2 liters. But what's neat about this is that the sampling distribution of the sample mean, so you take 50 people, find their mean, plot the frequency."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "It's going to be the same value as this population over here. So that is going to be 2 liters. So we're still centered at 2 liters. But what's neat about this is that the sampling distribution of the sample mean, so you take 50 people, find their mean, plot the frequency. 50 people, find the mean. This is actually going to be a normal distribution regardless of, this one just has a well-defined standard deviation. It's not normal."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "But what's neat about this is that the sampling distribution of the sample mean, so you take 50 people, find their mean, plot the frequency. 50 people, find the mean. This is actually going to be a normal distribution regardless of, this one just has a well-defined standard deviation. It's not normal. Even though this one isn't normal, this one over here will be. And we've seen it in multiple videos already. So this is going to be a normal distribution."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "It's not normal. Even though this one isn't normal, this one over here will be. And we've seen it in multiple videos already. So this is going to be a normal distribution. And the standard deviation, and we saw this in the last video, and hopefully we've got a little bit of intuition for why this is true. The standard deviation, actually maybe put it a better way. The variance of the sample mean is going to be the variance."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So this is going to be a normal distribution. And the standard deviation, and we saw this in the last video, and hopefully we've got a little bit of intuition for why this is true. The standard deviation, actually maybe put it a better way. The variance of the sample mean is going to be the variance. So remember, this is standard deviation. So it's going to be the variance of the population divided by n. And if you wanted the standard deviation of this distribution right here, you just take the square root of both sides. You take the square root of both sides of that."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "The variance of the sample mean is going to be the variance. So remember, this is standard deviation. So it's going to be the variance of the population divided by n. And if you wanted the standard deviation of this distribution right here, you just take the square root of both sides. You take the square root of both sides of that. We have the standard deviation of the sample mean is going to be equal to, the square root of this side over here, is going to be equal to the standard deviation of the population divided by the square root of n. And what's this going to be in our case? We know what the standard deviation of the population is. It is 0.7."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "You take the square root of both sides of that. We have the standard deviation of the sample mean is going to be equal to, the square root of this side over here, is going to be equal to the standard deviation of the population divided by the square root of n. And what's this going to be in our case? We know what the standard deviation of the population is. It is 0.7. And what is n? We have 50 men. So 0.7 over the square root of 50."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "It is 0.7. And what is n? We have 50 men. So 0.7 over the square root of 50. Now let's figure out what that is with the calculator. So we have 0.7 divided by the square root of 50. And we have 0.098."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So 0.7 over the square root of 50. Now let's figure out what that is with the calculator. So we have 0.7 divided by the square root of 50. And we have 0.098. It was pretty close to 0.99. So I'll just write that down. So this is equal to 0.099."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "And we have 0.098. It was pretty close to 0.99. So I'll just write that down. So this is equal to 0.099. That's going to be the standard deviation of this. So it's going to have a lower standard deviation. So it's going to look, the distribution is going to be normal."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So this is equal to 0.099. That's going to be the standard deviation of this. So it's going to have a lower standard deviation. So it's going to look, the distribution is going to be normal. It's going to look something like this. So this is 3 liters over here. This is 1 liter."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So it's going to look, the distribution is going to be normal. It's going to look something like this. So this is 3 liters over here. This is 1 liter. The standard deviation is almost a tenth. So it's going to be a much narrower distribution. It's going to look something."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "This is 1 liter. The standard deviation is almost a tenth. So it's going to be a much narrower distribution. It's going to look something. I'm trying my best to draw it. It's going to look something like this. You get the idea."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "It's going to look something. I'm trying my best to draw it. It's going to look something like this. You get the idea. Where the standard deviation right now is almost 0.1. So it's 0.09, almost a tenth. So it's going to be one standard deviation away."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "You get the idea. Where the standard deviation right now is almost 0.1. So it's 0.09, almost a tenth. So it's going to be one standard deviation away. It's going to look something like that. So we have our distribution. It's a normal distribution."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So it's going to be one standard deviation away. It's going to look something like that. So we have our distribution. It's a normal distribution. And now let's go back to our question that we're asking. We want to know the probability that our sample will have an average greater than 2.2. So this is the distribution of all of the possible samples, the means of all of the possible samples."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "It's a normal distribution. And now let's go back to our question that we're asking. We want to know the probability that our sample will have an average greater than 2.2. So this is the distribution of all of the possible samples, the means of all of the possible samples. Now to be greater than 2.2, 2.2 is going to be right around here. 2.2 is going to be right around here. So we are essentially asking, we will run out if our sample mean falls into this bucket over here."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So this is the distribution of all of the possible samples, the means of all of the possible samples. Now to be greater than 2.2, 2.2 is going to be right around here. 2.2 is going to be right around here. So we are essentially asking, we will run out if our sample mean falls into this bucket over here. So we essentially need to figure out what is, you can even view it as, what's this area under this curve there. And to figure that out, we just have to figure out how many standard deviations above the mean we are, which is going to be our z-score. And then we could use a z-table to figure out what this area right over here is."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So we are essentially asking, we will run out if our sample mean falls into this bucket over here. So we essentially need to figure out what is, you can even view it as, what's this area under this curve there. And to figure that out, we just have to figure out how many standard deviations above the mean we are, which is going to be our z-score. And then we could use a z-table to figure out what this area right over here is. So we want to know, when we are above 2.2 liters, so 2.2 liters, we could even do it in our head, 2.2 liters is what we care about. That's right over here. That is, our mean is 2, so we are 0.2 above the mean."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "And then we could use a z-table to figure out what this area right over here is. So we want to know, when we are above 2.2 liters, so 2.2 liters, we could even do it in our head, 2.2 liters is what we care about. That's right over here. That is, our mean is 2, so we are 0.2 above the mean. And if we want that in terms of standard deviations, we just divide this by the standard deviation of this distribution over here. And we figured out what that is. The standard deviation of this distribution is 0.099."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "That is, our mean is 2, so we are 0.2 above the mean. And if we want that in terms of standard deviations, we just divide this by the standard deviation of this distribution over here. And we figured out what that is. The standard deviation of this distribution is 0.099. So if we take, and you'll see a formula where you take this value minus the mean and divide it by the standard deviation, that's all we're doing. We're just figuring out how many standard deviations above the mean we are. So you just take this number right over here, divided by the standard deviation, so 0.099, or 0.099."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "The standard deviation of this distribution is 0.099. So if we take, and you'll see a formula where you take this value minus the mean and divide it by the standard deviation, that's all we're doing. We're just figuring out how many standard deviations above the mean we are. So you just take this number right over here, divided by the standard deviation, so 0.099, or 0.099. And then we get, let's get our calculator. And actually, we had the exact number over here. So we could just take 0.2, we could just take this 0.2, 0.2, divided by this value over here."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So you just take this number right over here, divided by the standard deviation, so 0.099, or 0.099. And then we get, let's get our calculator. And actually, we had the exact number over here. So we could just take 0.2, we could just take this 0.2, 0.2, divided by this value over here. And on this calculator, when I press second answer, it just means the last answer. So I'm taking 0.2 divided by this value over there. And I get 2.020."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So we could just take 0.2, we could just take this 0.2, 0.2, divided by this value over here. And on this calculator, when I press second answer, it just means the last answer. So I'm taking 0.2 divided by this value over there. And I get 2.020. So this means, so that means that this value, or I should write this probability is the same probability of being 2.02 standard deviations. Maybe I should write it this way. More than, let me write it down here where I have more space."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "And I get 2.020. So this means, so that means that this value, or I should write this probability is the same probability of being 2.02 standard deviations. Maybe I should write it this way. More than, let me write it down here where I have more space. So this all boils down to the probability of running out of water is the probability that the sample mean will be more than just the 50 that we happen to select. Remember, if we take a bunch of samples of 50 and plot all of them, we'll get this whole distribution. But the 150, the group of 50 that we happen to select, the probability of running out of water is the same thing as the probability of the mean of those people will be more than 2.020 standard deviations above the mean."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "More than, let me write it down here where I have more space. So this all boils down to the probability of running out of water is the probability that the sample mean will be more than just the 50 that we happen to select. Remember, if we take a bunch of samples of 50 and plot all of them, we'll get this whole distribution. But the 150, the group of 50 that we happen to select, the probability of running out of water is the same thing as the probability of the mean of those people will be more than 2.020 standard deviations above the mean. Above the mean of this distribution, which is actually the same distribution. So what is that going to be? And here we just have to look up our z table."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "But the 150, the group of 50 that we happen to select, the probability of running out of water is the same thing as the probability of the mean of those people will be more than 2.020 standard deviations above the mean. Above the mean of this distribution, which is actually the same distribution. So what is that going to be? And here we just have to look up our z table. Remember, this 2.02 is just this value right here. 0.2 divided by 0.09. I just had to pause the video because there's some type of fighter jet outside or something."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "And here we just have to look up our z table. Remember, this 2.02 is just this value right here. 0.2 divided by 0.09. I just had to pause the video because there's some type of fighter jet outside or something. Anyway, hopefully they won't come back. But anyway, so we need to figure out the probability that the sample mean will be more than 2.02 standard deviations above the mean. And to figure that out, we go to a z table."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "I just had to pause the video because there's some type of fighter jet outside or something. Anyway, hopefully they won't come back. But anyway, so we need to figure out the probability that the sample mean will be more than 2.02 standard deviations above the mean. And to figure that out, we go to a z table. And you can find this pretty much anywhere. Usually it's in any stat book or on the internet, wherever. And so essentially, we want to know the probability."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "And to figure that out, we go to a z table. And you can find this pretty much anywhere. Usually it's in any stat book or on the internet, wherever. And so essentially, we want to know the probability. The z table will tell you how much area is below this value. So if you go to z of 2.02, that was the value that we were dealing with, right? Yeah, 2.02."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "And so essentially, we want to know the probability. The z table will tell you how much area is below this value. So if you go to z of 2.02, that was the value that we were dealing with, right? Yeah, 2.02. It was, so you go for the first digit. We go to 2.0, and it was 2.02. 2.02 is right over there, right?"}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "Yeah, 2.02. It was, so you go for the first digit. We go to 2.0, and it was 2.02. 2.02 is right over there, right? So we had 2.0, and then the next digit you go up here. So it's 2.02 is right over there. So this 0.9783, let me write it down over here."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "2.02 is right over there, right? So we had 2.0, and then the next digit you go up here. So it's 2.02 is right over there. So this 0.9783, let me write it down over here. This 0.9783, I want to be very careful, 0.9783. That z table, that's not this value over here. This 0.9783 on the z table, that is giving us this whole area over here."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So this 0.9783, let me write it down over here. This 0.9783, I want to be very careful, 0.9783. That z table, that's not this value over here. This 0.9783 on the z table, that is giving us this whole area over here. It's giving us the probability that we are below that value, that we are less than 2.02 standard deviations above the mean. So it's giving us that value over here. So to answer our question, to answer this probability, we just have to subtract this from 1, because these will all add up to 1."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "This 0.9783 on the z table, that is giving us this whole area over here. It's giving us the probability that we are below that value, that we are less than 2.02 standard deviations above the mean. So it's giving us that value over here. So to answer our question, to answer this probability, we just have to subtract this from 1, because these will all add up to 1. So we just take our calculator back out, and we just take 1 minus 0.9783 is equal to 0.0217. So this right here is 0.0217. Or another way you could say it, it is a 2.17% probability that we will run out of water."}, {"video_title": "Sampling distribution example problem Probability and Statistics Khan Academy.mp3", "Sentence": "So to answer our question, to answer this probability, we just have to subtract this from 1, because these will all add up to 1. So we just take our calculator back out, and we just take 1 minus 0.9783 is equal to 0.0217. So this right here is 0.0217. Or another way you could say it, it is a 2.17% probability that we will run out of water. And we are done. Let me make sure I got that number right. So that number, it was 0.2017."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "You have a 99 out of 100 chance that any time you apply this test, that it is going to be accurate. Now let's say that you, and you just magically know that, we're just assuming that. Now let's just say that you get 100 folks into this room, and you apply this test to all 100 of them. So apply, apply test 100 times. So what are some of the possible outcomes here? Is it for sure that 99, exactly 99 out of 100 are going to be accurate, and that one out of 100 is going to be inaccurate? Well that's definitely a likely possibility, but it's also possible you get a little lucky and all 100 are accurate, or you get a little unlucky, and that 98 are accurate, and that two are inaccurate."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "So apply, apply test 100 times. So what are some of the possible outcomes here? Is it for sure that 99, exactly 99 out of 100 are going to be accurate, and that one out of 100 is going to be inaccurate? Well that's definitely a likely possibility, but it's also possible you get a little lucky and all 100 are accurate, or you get a little unlucky, and that 98 are accurate, and that two are inaccurate. And actually I calculated the probabilities ahead of time, and the goal of this video isn't to go into the probability and combinatorics of it, but if you're curious about it, there's a lot of good videos on probability and combinatorics on Khan Academy. But I calculated it ahead of time, and the probability, if you have something that has a 99% chance of being accurate, and you apply it 100 times, the probability that it is accurate, that it is accurate 100 out of the 100 times, is approximately equal to, approximately equal to 36.6%. I rounded to the nearest tenth of a percent."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "Well that's definitely a likely possibility, but it's also possible you get a little lucky and all 100 are accurate, or you get a little unlucky, and that 98 are accurate, and that two are inaccurate. And actually I calculated the probabilities ahead of time, and the goal of this video isn't to go into the probability and combinatorics of it, but if you're curious about it, there's a lot of good videos on probability and combinatorics on Khan Academy. But I calculated it ahead of time, and the probability, if you have something that has a 99% chance of being accurate, and you apply it 100 times, the probability that it is accurate, that it is accurate 100 out of the 100 times, is approximately equal to, approximately equal to 36.6%. I rounded to the nearest tenth of a percent. So it's a little better than a third chance that you'll actually get all of, all of the people are going to get an accurate result, even though for any one of them, there's a 99% chance that it's accurate. Now we could keep going. The probability that it is accurate, I'm just going to put these quotes here so I don't have to rewrite accurate over and over again."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "I rounded to the nearest tenth of a percent. So it's a little better than a third chance that you'll actually get all of, all of the people are going to get an accurate result, even though for any one of them, there's a 99% chance that it's accurate. Now we could keep going. The probability that it is accurate, I'm just going to put these quotes here so I don't have to rewrite accurate over and over again. The probability that it is accurate 99 out of 100 times, I calculated it ahead of time, it is approximately 37.0%. So this is what you would expect. Getting 100 out of 100 doesn't seem that unlikely if each of the times you apply it has a 99% chance of being accurate."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "The probability that it is accurate, I'm just going to put these quotes here so I don't have to rewrite accurate over and over again. The probability that it is accurate 99 out of 100 times, I calculated it ahead of time, it is approximately 37.0%. So this is what you would expect. Getting 100 out of 100 doesn't seem that unlikely if each of the times you apply it has a 99% chance of being accurate. But it makes sense that you, that you would expect 99 out of 100 to be even more likely, slightly more likely. And we can of course keep going. The probability that it is accurate 98 out of 100 times is approximately 18.5%."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "Getting 100 out of 100 doesn't seem that unlikely if each of the times you apply it has a 99% chance of being accurate. But it makes sense that you, that you would expect 99 out of 100 to be even more likely, slightly more likely. And we can of course keep going. The probability that it is accurate 98 out of 100 times is approximately 18.5%. And I'm just going to do a few more. The probability that it is accurate 97 out of 100 times, and once again I calculated all of these ahead of time, is 6%. So it's definitely in the realm of possibility, but it's, the probability is much lower than getting, having 99 out of 100 or 100 out of 100 being accurate."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "The probability that it is accurate 98 out of 100 times is approximately 18.5%. And I'm just going to do a few more. The probability that it is accurate 97 out of 100 times, and once again I calculated all of these ahead of time, is 6%. So it's definitely in the realm of possibility, but it's, the probability is much lower than getting, having 99 out of 100 or 100 out of 100 being accurate. And then the probability, let me put the double quotes here, the probability that it's accurate 96 out of 100 times is approximately 1.5%. And then the probability, and I'll just do one more, and I could keep going, the probability, there's some probability that even though each test has a 99% chance, you just get super unlucky and that very few of them are accurate. But I'll just, and you see, you see what's happening to the probabilities as we have fewer and fewer of them being accurate."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "So it's definitely in the realm of possibility, but it's, the probability is much lower than getting, having 99 out of 100 or 100 out of 100 being accurate. And then the probability, let me put the double quotes here, the probability that it's accurate 96 out of 100 times is approximately 1.5%. And then the probability, and I'll just do one more, and I could keep going, the probability, there's some probability that even though each test has a 99% chance, you just get super unlucky and that very few of them are accurate. But I'll just, and you see, you see what's happening to the probabilities as we have fewer and fewer of them being accurate. It becomes less and less probable. So the probability that 95 out of the 100 are accurate is approximately 0.3%. So this was just kind of a, I guess you could say a thought experiment."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "But I'll just, and you see, you see what's happening to the probabilities as we have fewer and fewer of them being accurate. It becomes less and less probable. So the probability that 95 out of the 100 are accurate is approximately 0.3%. So this was just kind of a, I guess you could say a thought experiment. If we had a test that we know for sure that every time you administer it the probability that it's accurate is 99%, then these are the probabilities that if you administer it 100 times that you get 100 out of 100 accurate, the probability that you get 99 out of 100 accurate, and so on and so forth. So let's just keep that in mind and then think a little bit about hypothesis testing and how we can use this framework. So let's put all that in the back of our minds."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "So this was just kind of a, I guess you could say a thought experiment. If we had a test that we know for sure that every time you administer it the probability that it's accurate is 99%, then these are the probabilities that if you administer it 100 times that you get 100 out of 100 accurate, the probability that you get 99 out of 100 accurate, and so on and so forth. So let's just keep that in mind and then think a little bit about hypothesis testing and how we can use this framework. So let's put all that in the back of our minds. And let's say that you have devised a new test. You have a new test and you don't know how accurate it is. You have a new cholesterol test."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "So let's put all that in the back of our minds. And let's say that you have devised a new test. You have a new test and you don't know how accurate it is. You have a new cholesterol test. You don't know how accurate it is. You know that in order for it to be approved by whatever governing body, it has to be accurate 99, the probability of it being accurate has to be 99%. So needs to have probability of accurate equal to 99%."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "You have a new cholesterol test. You don't know how accurate it is. You know that in order for it to be approved by whatever governing body, it has to be accurate 99, the probability of it being accurate has to be 99%. So needs to have probability of accurate equal to 99%. You don't know if this is true. You just know that that's what it needs to be. And so you have your test and let's say you set up a hypothesis and your hypothesis could be a lot of things and once you get deeper into statistics, there's null hypothesis and alternate hypotheses, but let's just start with just a simple hypothesis."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "So needs to have probability of accurate equal to 99%. You don't know if this is true. You just know that that's what it needs to be. And so you have your test and let's say you set up a hypothesis and your hypothesis could be a lot of things and once you get deeper into statistics, there's null hypothesis and alternate hypotheses, but let's just start with just a simple hypothesis. You're hopeful your hypothesis is that the probability that your new test is accurate is this is your hypothesis because you want that to be your hypothesis because if you feel good about it, then you're like, okay, maybe I'll get approved by the appropriate governing body. So you say, hey, my hypothesis is that my new test is accurate 99, the probability of it being accurate is 99%. So then you go off and you apply it 100 times."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "And so you have your test and let's say you set up a hypothesis and your hypothesis could be a lot of things and once you get deeper into statistics, there's null hypothesis and alternate hypotheses, but let's just start with just a simple hypothesis. You're hopeful your hypothesis is that the probability that your new test is accurate is this is your hypothesis because you want that to be your hypothesis because if you feel good about it, then you're like, okay, maybe I'll get approved by the appropriate governing body. So you say, hey, my hypothesis is that my new test is accurate 99, the probability of it being accurate is 99%. So then you go off and you apply it 100 times. So you apply your new test. You don't know the actual probability of it being accurate. You apply the test 100 times."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "So then you go off and you apply it 100 times. So you apply your new test. You don't know the actual probability of it being accurate. You apply the test 100 times. And let's say out of those 100 times, you get that they are accurate, you get that it is accurate and you're able to use some other test that you know, some for sure test, some super accurate test to verify your own test results and you see that it is accurate 95 out of the 100 times. So the question you have is, well, does the hypothesis make sense to you? Will you accept this hypothesis?"}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "You apply the test 100 times. And let's say out of those 100 times, you get that they are accurate, you get that it is accurate and you're able to use some other test that you know, some for sure test, some super accurate test to verify your own test results and you see that it is accurate 95 out of the 100 times. So the question you have is, well, does the hypothesis make sense to you? Will you accept this hypothesis? Well, what you say is, well, if my hypothesis was true, if my test were accurate 99, if the probability of my test being accurate is 99%, what's the probability of me getting this outcome? Well, we figured that out. If it really was accurate 99% of the time, then the probability of getting this outcome is only 0.3%."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "Will you accept this hypothesis? Well, what you say is, well, if my hypothesis was true, if my test were accurate 99, if the probability of my test being accurate is 99%, what's the probability of me getting this outcome? Well, we figured that out. If it really was accurate 99% of the time, then the probability of getting this outcome is only 0.3%. So if you assume true, if you assume hypothesis, I'll just write hype. If you assume the hypothesis is true, the probability of the outcome you got, probability of observed outcome is approximately 0.3%. And so you say, look, you know, maybe it's definitely possible that I just got very, very, very, very unlikely, but based on this, I probably should reject my hypothesis because the probability of me getting this outcome, if the hypothesis was true, is very, very, very, very low."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "If it really was accurate 99% of the time, then the probability of getting this outcome is only 0.3%. So if you assume true, if you assume hypothesis, I'll just write hype. If you assume the hypothesis is true, the probability of the outcome you got, probability of observed outcome is approximately 0.3%. And so you say, look, you know, maybe it's definitely possible that I just got very, very, very, very unlikely, but based on this, I probably should reject my hypothesis because the probability of me getting this outcome, if the hypothesis was true, is very, very, very, very low. And as we go deeper into statistics, you'll see that there are thresholds that people often set for, you know, if the probability of something happening or not happening is above or below some threshold, then we might reject a certain hypothesis. But in this world, you could see that, look, if my test really was accurate 99% of the time, for me to get, when I apply it to 100 people, it's only accurate 95 out of 100. If my hypothesis is true, that would have only, there's only a 0.3% chance that I would have seen this observation."}, {"video_title": "Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3", "Sentence": "And so you say, look, you know, maybe it's definitely possible that I just got very, very, very, very unlikely, but based on this, I probably should reject my hypothesis because the probability of me getting this outcome, if the hypothesis was true, is very, very, very, very low. And as we go deeper into statistics, you'll see that there are thresholds that people often set for, you know, if the probability of something happening or not happening is above or below some threshold, then we might reject a certain hypothesis. But in this world, you could see that, look, if my test really was accurate 99% of the time, for me to get, when I apply it to 100 people, it's only accurate 95 out of 100. If my hypothesis is true, that would have only, there's only a 0.3% chance that I would have seen this observation. So based on that, it might be completely reasonable to say, you know what, I might reject my hypothesis. Look for a new test. I don't feel good about this new cholesterol test that I constructed."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "The manager looks at the difference between the proportions of cars with the defect in each sample. So they're looking at the difference of sample proportions every month. Describe the distribution of the difference of sample proportions in terms of its mean, standard deviation, and shape. So let's take these step by step. So first, let's think about the mean of the difference of our sample proportions. Pause this video and try to figure out what that's going to be. Well, we have seen this in previous videos, that if we have the mean of the difference of two random variables, that's the same as the difference of the means."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "So let's take these step by step. So first, let's think about the mean of the difference of our sample proportions. Pause this video and try to figure out what that's going to be. Well, we have seen this in previous videos, that if we have the mean of the difference of two random variables, that's the same as the difference of the means. Or another way to think about it is, if we wanna figure out the mean of this, so sample proportion from plant A minus sample proportion from plant B, this is just going to be equal to the mean of the sample proportion from plant A minus the mean of the sample proportion from plant B. Now, what are these going to be equal to? Well, what's the mean of the sample proportion of plant A?"}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "Well, we have seen this in previous videos, that if we have the mean of the difference of two random variables, that's the same as the difference of the means. Or another way to think about it is, if we wanna figure out the mean of this, so sample proportion from plant A minus sample proportion from plant B, this is just going to be equal to the mean of the sample proportion from plant A minus the mean of the sample proportion from plant B. Now, what are these going to be equal to? Well, what's the mean of the sample proportion of plant A? It's just going to be the true population proportion for plant A, and they tell us that. They tell us that 8% of all cars produced at plant A have a certain defect. So this could be 8%, or we could write it as 0.08."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "Well, what's the mean of the sample proportion of plant A? It's just going to be the true population proportion for plant A, and they tell us that. They tell us that 8% of all cars produced at plant A have a certain defect. So this could be 8%, or we could write it as 0.08. And then from that, we are going to subtract the mean of the sample proportion from plant B. And we know what that mean's going to be. The mean of a sample proportion is going to be the population proportion."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "So this could be 8%, or we could write it as 0.08. And then from that, we are going to subtract the mean of the sample proportion from plant B. And we know what that mean's going to be. The mean of a sample proportion is going to be the population proportion. The parameter of the population, which we know for plant B is 6%, 0.06. And then that gets us a mean of the difference of 0.02, or 2% having a, or 2% difference in defect rate would be the mean. Now let's think about the standard deviation."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "The mean of a sample proportion is going to be the population proportion. The parameter of the population, which we know for plant B is 6%, 0.06. And then that gets us a mean of the difference of 0.02, or 2% having a, or 2% difference in defect rate would be the mean. Now let's think about the standard deviation. So instead of thinking in terms of standard deviation, let's think about the square of the standard deviation, which is variance. And from there, we can go back to standard deviation by taking a square root. So if we're looking at the variance, and let me write it this way."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "Now let's think about the standard deviation. So instead of thinking in terms of standard deviation, let's think about the square of the standard deviation, which is variance. And from there, we can go back to standard deviation by taking a square root. So if we're looking at the variance, and let me write it this way. If we're looking at the variance of the difference of the sample proportions, so the sample proportion from plant A minus the sample proportion from plant B, but just as a review, if you assume that we're sampling independently from each of the plants, so what we're sampling from plant A does not affect what we're sampling from plant B or vice versa, then we can add the variances. So this is going to be equal to the variance of the sample proportion from plant A plus the variance of the sample proportion from plant B. Some of you might be saying, wait, aren't we taking the difference of sample proportions here?"}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "So if we're looking at the variance, and let me write it this way. If we're looking at the variance of the difference of the sample proportions, so the sample proportion from plant A minus the sample proportion from plant B, but just as a review, if you assume that we're sampling independently from each of the plants, so what we're sampling from plant A does not affect what we're sampling from plant B or vice versa, then we can add the variances. So this is going to be equal to the variance of the sample proportion from plant A plus the variance of the sample proportion from plant B. Some of you might be saying, wait, aren't we taking the difference of sample proportions here? Why are we adding? And the reminder is, remember, variance is a measure of spread. And whether you're now taking the difference of random variables or you're taking the sum of them, when you have more variables, you're going to have more spread."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "Some of you might be saying, wait, aren't we taking the difference of sample proportions here? Why are we adding? And the reminder is, remember, variance is a measure of spread. And whether you're now taking the difference of random variables or you're taking the sum of them, when you have more variables, you're going to have more spread. So regardless of whether this is a negative or a positive over here, this is going to be a positive. So what is this going to be equal to? Well, we could take each of these terms."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "And whether you're now taking the difference of random variables or you're taking the sum of them, when you have more variables, you're going to have more spread. So regardless of whether this is a negative or a positive over here, this is going to be a positive. So what is this going to be equal to? Well, we could take each of these terms. What's going to be the variance of the sample proportion from plant A? Well, if every time we looked at one of the cars, we looked at it and then we put it back into the mix. So if we were sampling with replacement, which means that each of our observations are independent of the other ones, we have a formula."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "Well, we could take each of these terms. What's going to be the variance of the sample proportion from plant A? Well, if every time we looked at one of the cars, we looked at it and then we put it back into the mix. So if we were sampling with replacement, which means that each of our observations are independent of the other ones, we have a formula. We know that this variance would be the population proportion of plant A times one minus the population proportion of plant A divided by the number that we sampled from plant A. Now, in the scenario that we are talking about, we didn't sample with replacement. We just took 200 at a time and looked at them."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "So if we were sampling with replacement, which means that each of our observations are independent of the other ones, we have a formula. We know that this variance would be the population proportion of plant A times one minus the population proportion of plant A divided by the number that we sampled from plant A. Now, in the scenario that we are talking about, we didn't sample with replacement. We just took 200 at a time and looked at them. We didn't take one at a time and replace it and do that 200 times. But we also know that this is a pretty good approximation even when you are not sampling with replacement. If your sample is less than 10% of the population and 200 is less than 10% of 3000."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "We just took 200 at a time and looked at them. We didn't take one at a time and replace it and do that 200 times. But we also know that this is a pretty good approximation even when you are not sampling with replacement. If your sample is less than 10% of the population and 200 is less than 10% of 3000. So this is a pretty good approximation, what you would use in a first year statistics class. And of course, we can use the same logic. This is going to be equal to the population proportion of plant B times one minus the population proportion in plant B, all of that over your sample size from plant B."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "If your sample is less than 10% of the population and 200 is less than 10% of 3000. So this is a pretty good approximation, what you would use in a first year statistics class. And of course, we can use the same logic. This is going to be equal to the population proportion of plant B times one minus the population proportion in plant B, all of that over your sample size from plant B. And we know all of these things. We know that your population proportion in plant A is 8% or 0.08. One minus that is 0.92."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "This is going to be equal to the population proportion of plant B times one minus the population proportion in plant B, all of that over your sample size from plant B. And we know all of these things. We know that your population proportion in plant A is 8% or 0.08. One minus that is 0.92. We're taking samples of 200 at a time from plant A. And then in plant B, we know the population proportion they told us is 6% or 0.06. One minus that is 0.94."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "One minus that is 0.92. We're taking samples of 200 at a time from plant A. And then in plant B, we know the population proportion they told us is 6% or 0.06. One minus that is 0.94. And then the sample size from plant B is also going to be 200. It's going to be 200. We get 0.08 times 0.92 divided by 200."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "One minus that is 0.94. And then the sample size from plant B is also going to be 200. It's going to be 200. We get 0.08 times 0.92 divided by 200. And then plus, let's open parentheses here. We get 0.06 times 0.94 divided by 200. And then actually let me close the parentheses."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "We get 0.08 times 0.92 divided by 200. And then plus, let's open parentheses here. We get 0.06 times 0.94 divided by 200. And then actually let me close the parentheses. And that equals this business. So 0.00065. So 0.00065."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "And then actually let me close the parentheses. And that equals this business. So 0.00065. So 0.00065. And then from this, we can figure out what the standard deviation is going to be. The standard deviation of the difference between our sample proportions is going to be just the square root of this. It's going to be the square root of 0.00065."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "So 0.00065. And then from this, we can figure out what the standard deviation is going to be. The standard deviation of the difference between our sample proportions is going to be just the square root of this. It's going to be the square root of 0.00065. And that is approximately equal to, let's just take the square root, and we get this 0.025. 0.025. And there you have it."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "It's going to be the square root of 0.00065. And that is approximately equal to, let's just take the square root, and we get this 0.025. 0.025. And there you have it. We have thought about the standard deviation. And then last but not least, let's think about the shape. So just as a review, we just have to remind ourselves that the distribution of each sample proportion is going to be normal as long as we expect at least 10 successes and 10 failures."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "And there you have it. We have thought about the standard deviation. And then last but not least, let's think about the shape. So just as a review, we just have to remind ourselves that the distribution of each sample proportion is going to be normal as long as we expect at least 10 successes and 10 failures. Well, let's look at each of these. How many successes do you expect? Or where success would actually be a defect."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "So just as a review, we just have to remind ourselves that the distribution of each sample proportion is going to be normal as long as we expect at least 10 successes and 10 failures. Well, let's look at each of these. How many successes do you expect? Or where success would actually be a defect. But let's think about this. 8% of, in each case, of a sample of 200, that's going to be 16. So you would expect 16 defects."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "Or where success would actually be a defect. But let's think about this. 8% of, in each case, of a sample of 200, that's going to be 16. So you would expect 16 defects. And then you would expect 200 minus 16, which is a lot larger than 10, of no defects. So both of those are greater than or equal to 10. And then if you did the same thing for plant B, you get the same idea."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "So you would expect 16 defects. And then you would expect 200 minus 16, which is a lot larger than 10, of no defects. So both of those are greater than or equal to 10. And then if you did the same thing for plant B, you get the same idea. 6% of 200 is 12. And then if you say the ones that have no defects, that's 200 minus 12, which is way more than 10, especially in that latter case. But in every situation, we expect to have at least 10 successes and 10 failures."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "And then if you did the same thing for plant B, you get the same idea. 6% of 200 is 12. And then if you say the ones that have no defects, that's 200 minus 12, which is way more than 10, especially in that latter case. But in every situation, we expect to have at least 10 successes and 10 failures. And so we can assume that the distributions of each of these are going to be normal. And we also know that the difference of two normally distributed variables is also normal so long as they pass that large count condition that we just talked about. And so let's draw what this distribution might look like."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "But in every situation, we expect to have at least 10 successes and 10 failures. And so we can assume that the distributions of each of these are going to be normal. And we also know that the difference of two normally distributed variables is also normal so long as they pass that large count condition that we just talked about. And so let's draw what this distribution might look like. It might look something like this. It's going to be a normal distribution where you have a mean right over here of, I'll do that in that same color, a mean of 0.02. You can definitely take on negative values because there's some situations in which your sample proportion from plant B actually could be larger just by random chance than it is from plant A."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "And so let's draw what this distribution might look like. It might look something like this. It's going to be a normal distribution where you have a mean right over here of, I'll do that in that same color, a mean of 0.02. You can definitely take on negative values because there's some situations in which your sample proportion from plant B actually could be larger just by random chance than it is from plant A. So you can definitely take on negative values. But if I wanted to show where zero is, maybe zero is right over here. So we could draw an axis right over here."}, {"video_title": "Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3", "Sentence": "You can definitely take on negative values because there's some situations in which your sample proportion from plant B actually could be larger just by random chance than it is from plant A. So you can definitely take on negative values. But if I wanted to show where zero is, maybe zero is right over here. So we could draw an axis right over here. And then we know what the standard deviation is. It's 0.025 or it's approximately that. So if we were to go one standard deviation down, we would go right about there."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "Administrators at Riverview High School surveyed a random sample of 100 of their seniors to see how they felt about the lunch offerings at the school's cafeteria. So you have all of the seniors, I'm assuming there's more than 100 of them, and then they sampled 100 of them. So this is the sample. So the population is all of the seniors at the school. That's the population, all of the seniors. And they sampled 100 of them. So the 100 seniors that they talked to, that is the sample."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "So the population is all of the seniors at the school. That's the population, all of the seniors. And they sampled 100 of them. So the 100 seniors that they talked to, that is the sample. That is the sample. So they tell us, identify the population and the sample in the setting. So let's just see which of these choices actually match up to what I just said."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "So the 100 seniors that they talked to, that is the sample. That is the sample. So they tell us, identify the population and the sample in the setting. So let's just see which of these choices actually match up to what I just said. And like always, I encourage you to pause the video and see if you can work through it on your own. So the population is all high school seniors in the world. The sample is all of the seniors at Riverview High."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "So let's just see which of these choices actually match up to what I just said. And like always, I encourage you to pause the video and see if you can work through it on your own. So the population is all high school seniors in the world. The sample is all of the seniors at Riverview High. No, this is not right. We're not trying to figure out, we're not trying to get an indication of how all of the high school seniors in the world feel about the food at Riverview High School. We're trying to get an indication of how the seniors at Riverview High School feel about the lunch at the school's cafeteria."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "The sample is all of the seniors at Riverview High. No, this is not right. We're not trying to figure out, we're not trying to get an indication of how all of the high school seniors in the world feel about the food at Riverview High School. We're trying to get an indication of how the seniors at Riverview High School feel about the lunch at the school's cafeteria. So they did a sample of 100 of them. So this is definitely not going to be, let me cross this one out. The population is all students at Riverview High."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "We're trying to get an indication of how the seniors at Riverview High School feel about the lunch at the school's cafeteria. So they did a sample of 100 of them. So this is definitely not going to be, let me cross this one out. The population is all students at Riverview High. The sample is all of the seniors at Riverview High. Well, they clearly didn't sample all of the seniors. They sample 100 of the seniors."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "The population is all students at Riverview High. The sample is all of the seniors at Riverview High. Well, they clearly didn't sample all of the seniors. They sample 100 of the seniors. So this isn't gonna be right either. Let's hope that the third choice is working out. The population is all seniors at Riverview High."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "They sample 100 of the seniors. So this isn't gonna be right either. Let's hope that the third choice is working out. The population is all seniors at Riverview High. The sample is the 100 seniors surveyed. Yep, that's exactly what we talked here. We're trying to get an indication about how all of the seniors at Riverview High feel about the food, the lunch offerings."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "The population is all seniors at Riverview High. The sample is the 100 seniors surveyed. Yep, that's exactly what we talked here. We're trying to get an indication about how all of the seniors at Riverview High feel about the food, the lunch offerings. We probably think it's impractical, or the administrators feel it's impractical to talk to everyone so they get exactly what the population thinks. So instead, they're gonna do a random sample of 100 of them. So the sample is 100 seniors who are actually surveyed."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "He asks his listeners to visit his website and participate in the poll. The poll shows that 89% of about 200 respondents love his show. What is the most concerning source of bias in this scenario? And like always, pause this video and see if you can figure it out on your own and then we'll work through it together. Let's think about what's going on. He has this population of listeners right over here. I'll assume that the number of listeners is more than 200."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "And like always, pause this video and see if you can figure it out on your own and then we'll work through it together. Let's think about what's going on. He has this population of listeners right over here. I'll assume that the number of listeners is more than 200. And he says, hey, I wanna find a sample and I can't ask all of my listeners. Who knows, maybe he has 10,000 listeners. They don't tell us that."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "I'll assume that the number of listeners is more than 200. And he says, hey, I wanna find a sample and I can't ask all of my listeners. Who knows, maybe he has 10,000 listeners. They don't tell us that. But let's say there's 10,000 listeners here and he says, well, I wanna get an indication of what percent like my show. So I need a sample. But instead of taking a truly random sample, he asks them to volunteer."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "They don't tell us that. But let's say there's 10,000 listeners here and he says, well, I wanna get an indication of what percent like my show. So I need a sample. But instead of taking a truly random sample, he asks them to volunteer. He asks his listeners to visit his website. So that's classic volunteer response sampling. This is non-random because who decides to go to his website and listen to what he just said and maybe even has access to a computer?"}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "But instead of taking a truly random sample, he asks them to volunteer. He asks his listeners to visit his website. So that's classic volunteer response sampling. This is non-random because who decides to go to his website and listen to what he just said and maybe even has access to a computer? That's not random. In fact, the people more likely to do that, so these are the people out of the 10,000. So these are the 200 responses here who decide to do it."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "This is non-random because who decides to go to his website and listen to what he just said and maybe even has access to a computer? That's not random. In fact, the people more likely to do that, so these are the people out of the 10,000. So these are the 200 responses here who decide to do it. These are more likely to be the people who already like David or like to listen to what he tells them to do. The people, the listeners who are not into David or don't wanna do what he tells them to do, well, they're unlikely to be in the say, oh, I'm not really into David and I don't like him telling me what to do, but hey, I'm gonna go to his website anyway. I'm gonna fill out that poll."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "So these are the 200 responses here who decide to do it. These are more likely to be the people who already like David or like to listen to what he tells them to do. The people, the listeners who are not into David or don't wanna do what he tells them to do, well, they're unlikely to be in the say, oh, I'm not really into David and I don't like him telling me what to do, but hey, I'm gonna go to his website anyway. I'm gonna fill out that poll. That's less likely. Or you might get extreme. So people who really don't like him might say, I'm gonna definitely go there."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "I'm gonna fill out that poll. That's less likely. Or you might get extreme. So people who really don't like him might say, I'm gonna definitely go there. But in this case, I would say that it's more likely your fans are gonna do what you ask them to do and go to your website and spend time on your website. And because of that, that 89% is probably an overestimate. 89% is probably an overestimate of the number of listeners who really love his show, because you're more likely to get the ones who love him to show up and fill out that actual survey."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "So people who really don't like him might say, I'm gonna definitely go there. But in this case, I would say that it's more likely your fans are gonna do what you ask them to do and go to your website and spend time on your website. And because of that, that 89% is probably an overestimate. 89% is probably an overestimate of the number of listeners who really love his show, because you're more likely to get the ones who love him to show up and fill out that actual survey. Now, these other forms of bias, response bias. This is when you're asking something that people don't necessarily wanna answer truthfully or the way that it's phrased, it might make someone respond, you say, in a biased way. Classic examples of this are like, have you lied to your parents in the past week?"}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "89% is probably an overestimate of the number of listeners who really love his show, because you're more likely to get the ones who love him to show up and fill out that actual survey. Now, these other forms of bias, response bias. This is when you're asking something that people don't necessarily wanna answer truthfully or the way that it's phrased, it might make someone respond, you say, in a biased way. Classic examples of this are like, have you lied to your parents in the past week? Or have you ever cheated on your spouse? Or something like that. Or have you smoked?"}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "Classic examples of this are like, have you lied to your parents in the past week? Or have you ever cheated on your spouse? Or something like that. Or have you smoked? Any of these things that people might not wanna answer completely truthfully or they might be hiding from the world and might not just wanna answer that truthfully on a survey. And so you're gonna have response bias. But that's not the case right over here."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "Or have you smoked? Any of these things that people might not wanna answer completely truthfully or they might be hiding from the world and might not just wanna answer that truthfully on a survey. And so you're gonna have response bias. But that's not the case right over here. And undercoverage is when the way that you're sampling, you're definitely missing out on an important constituency. You know, voluntary response, we're likely missing out on some important constituencies, on some people who might not be into going to your website. But undercoverage is where it's a little bit more clear that that is happening."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "But that's not the case right over here. And undercoverage is when the way that you're sampling, you're definitely missing out on an important constituency. You know, voluntary response, we're likely missing out on some important constituencies, on some people who might not be into going to your website. But undercoverage is where it's a little bit more clear that that is happening. Now let's do another case. Let's do another case, maybe an alternate reality, where David's trying to figure this out again. He's still hosting a podcast, and he's still curious how much his listeners like his show, but he tries to take a different sample."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "But undercoverage is where it's a little bit more clear that that is happening. Now let's do another case. Let's do another case, maybe an alternate reality, where David's trying to figure this out again. He's still hosting a podcast, and he's still curious how much his listeners like his show, but he tries to take a different sample. He decides, in this case, to poll the next 100 listeners who send him fan emails. They don't all respond, but 94 out of the 97 listeners polled say they loved his show. What is the most concerning source of bias in this scenario?"}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "He's still hosting a podcast, and he's still curious how much his listeners like his show, but he tries to take a different sample. He decides, in this case, to poll the next 100 listeners who send him fan emails. They don't all respond, but 94 out of the 97 listeners polled say they loved his show. What is the most concerning source of bias in this scenario? Well, this is a classic, hey, I have a group, I have a sample sitting in front of me, it's in my inbox on my email, let me just go to them. Isn't that convenient? So this is classic convenience sample."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "What is the most concerning source of bias in this scenario? Well, this is a classic, hey, I have a group, I have a sample sitting in front of me, it's in my inbox on my email, let me just go to them. Isn't that convenient? So this is classic convenience sample. And this isn't just like, hey, you know, these are the first 100 people to walk through the door, and there's, you know, a lot of times you could argue why that might be not so random, but these are the next 100 listeners who sent him fan emails. So this is convenience sampling, and the sample that you happen to use out of convenience is one that's going to be very skewed to liking you. So once again, this is overestimating, overestimating the percent, the percent that love his show."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "So this is classic convenience sample. And this isn't just like, hey, you know, these are the first 100 people to walk through the door, and there's, you know, a lot of times you could argue why that might be not so random, but these are the next 100 listeners who sent him fan emails. So this is convenience sampling, and the sample that you happen to use out of convenience is one that's going to be very skewed to liking you. So once again, this is overestimating, overestimating the percent, the percent that love his show. Now, nonresponse is when you ask a certain number of people to fill out a survey, or to answer a questionnaire, and for some reason, some percent do not fill it out, and you're like, well, who were those people? Maybe they would have said something important, and maybe their viewpoint is not properly represented in the overall number that actually did fill it out. And there is some nonresponse going on here."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "So once again, this is overestimating, overestimating the percent, the percent that love his show. Now, nonresponse is when you ask a certain number of people to fill out a survey, or to answer a questionnaire, and for some reason, some percent do not fill it out, and you're like, well, who were those people? Maybe they would have said something important, and maybe their viewpoint is not properly represented in the overall number that actually did fill it out. And there is some nonresponse going on here. He asks 100 people who sent fan emails to fill out the survey, to say whether they love it or not, 97 fill it out, so there were three people who did not fill out the survey. So there is some nonresponse going on that would be a source of bias, but it's not the most concerning. You know, right over here, they're asking us, fill out the most concerning source of bias, and the convenience sampling is definitely the biggest deal here."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "And there is some nonresponse going on here. He asks 100 people who sent fan emails to fill out the survey, to say whether they love it or not, 97 fill it out, so there were three people who did not fill out the survey. So there is some nonresponse going on that would be a source of bias, but it's not the most concerning. You know, right over here, they're asking us, fill out the most concerning source of bias, and the convenience sampling is definitely the biggest deal here. There were three people who didn't respond, but that's not as big of a deal. Voluntary response sampling, well, he didn't ask people, like in the last example, like, hey, if you can go here and fill it out, I guess there is actually, actually, no, take that back. There is a little bit of voluntary response here, where he goes to these 100 people and he asks them to respond, and so you have the 97 people who choose to respond."}, {"video_title": "Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "She randomly assigned people to exercise for 30 or 60 minutes, then measured their temperatures. The 18 people who exercised for 30 minutes had a mean temperature, so this is the sample mean for that sample of 18 folks, of 38.3 degrees Celsius with a standard deviation, this is a sample standard deviation for those 18 folks, of 0.27 degrees Celsius. The 24 people who exercised 60 minutes had a mean temperature of 38.9 degrees Celsius with a standard deviation, this is once again, these are both sample means and standard deviations, of 0.29 degrees Celsius. Assume that the conditions for inference have been met and that Kylie will use the conservative degrees of freedom from the smaller sample size. Which of the following is a 90% confidence interval for the difference in mean body temperature after exercising for the two amounts of time? So pause this video and see if you can figure it out. All right, now let's work through this together."}, {"video_title": "Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "Assume that the conditions for inference have been met and that Kylie will use the conservative degrees of freedom from the smaller sample size. Which of the following is a 90% confidence interval for the difference in mean body temperature after exercising for the two amounts of time? So pause this video and see if you can figure it out. All right, now let's work through this together. So in previous videos, we talked about the general form of our confidence interval, our T interval, which we're going to use because we're dealing with means and we're dealing with the differences in means. And so our T interval is going to have the form, our difference between our sample means, so it could be the sample mean for the 60 minute group minus the sample mean for the 30 minute group, plus or minus our critical T value, times our estimate of the sampling distribution of the difference of the sample means. And that is going to be, I think I have enough space here to do it, that is going to be the sample standard deviation of the 60 minute group squared over the sample size of the 60 minute group plus the sample standard deviation of the 30 minute group squared divided by the sample size of the 30 minute group."}, {"video_title": "Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "All right, now let's work through this together. So in previous videos, we talked about the general form of our confidence interval, our T interval, which we're going to use because we're dealing with means and we're dealing with the differences in means. And so our T interval is going to have the form, our difference between our sample means, so it could be the sample mean for the 60 minute group minus the sample mean for the 30 minute group, plus or minus our critical T value, times our estimate of the sampling distribution of the difference of the sample means. And that is going to be, I think I have enough space here to do it, that is going to be the sample standard deviation of the 60 minute group squared over the sample size of the 60 minute group plus the sample standard deviation of the 30 minute group squared divided by the sample size of the 30 minute group. And so we can actually figure out all of these things. So this is going to be equal to the sample mean for the 60 minute group is 38.9, so it's 38.9, minus the sample mean for the 30 minute group, which is 38.3, 38.3, plus or minus our critical T value. Now how do we figure that out?"}, {"video_title": "Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And that is going to be, I think I have enough space here to do it, that is going to be the sample standard deviation of the 60 minute group squared over the sample size of the 60 minute group plus the sample standard deviation of the 30 minute group squared divided by the sample size of the 30 minute group. And so we can actually figure out all of these things. So this is going to be equal to the sample mean for the 60 minute group is 38.9, so it's 38.9, minus the sample mean for the 30 minute group, which is 38.3, 38.3, plus or minus our critical T value. Now how do we figure that out? Well, we can use our 90% confidence level that we care about, this 90% confidence interval, but if we're looking up things on a T table, we also need to know our degrees of freedom. And it says here that Kylie will use the conservative degrees of freedom. And that means that she will look at each of those samples, so one has a sample size of 18, one has a sample size of 24, whichever is lower, she will use one less than that as her degrees of freedom."}, {"video_title": "Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "Now how do we figure that out? Well, we can use our 90% confidence level that we care about, this 90% confidence interval, but if we're looking up things on a T table, we also need to know our degrees of freedom. And it says here that Kylie will use the conservative degrees of freedom. And that means that she will look at each of those samples, so one has a sample size of 18, one has a sample size of 24, whichever is lower, she will use one less than that as her degrees of freedom. 18 is clearly less, lower than 24, so the degrees of freedom in this situation is 18 or are 18 minus one, so 17. And so using that and that, we can now look this up on a T table. So our confidence level, 90%, and then our degrees of freedom, 17, so that is that row."}, {"video_title": "Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And that means that she will look at each of those samples, so one has a sample size of 18, one has a sample size of 24, whichever is lower, she will use one less than that as her degrees of freedom. 18 is clearly less, lower than 24, so the degrees of freedom in this situation is 18 or are 18 minus one, so 17. And so using that and that, we can now look this up on a T table. So our confidence level, 90%, and then our degrees of freedom, 17, so that is that row. The 90% confidence level is this column. And so that gives us our critical T value of 1.74. So going back here, this is going to be plus or minus 1.74 times the square root, times the square root."}, {"video_title": "Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "So our confidence level, 90%, and then our degrees of freedom, 17, so that is that row. The 90% confidence level is this column. And so that gives us our critical T value of 1.74. So going back here, this is going to be plus or minus 1.74 times the square root, times the square root. What's our sample standard deviation for the 60-minute group? Well, they give it right over here, 0.29, and we're gonna have to square that, divided by the sample size for the 60-minute group. So let's see, the 24 people who exercised for 60 minutes, so divided by 24, plus the sample standard deviation for the 30-minute group, so that's 0.27, 0.27 squared, divided by the sample size for the 30-minute group, divided by 18."}, {"video_title": "Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "So going back here, this is going to be plus or minus 1.74 times the square root, times the square root. What's our sample standard deviation for the 60-minute group? Well, they give it right over here, 0.29, and we're gonna have to square that, divided by the sample size for the 60-minute group. So let's see, the 24 people who exercised for 60 minutes, so divided by 24, plus the sample standard deviation for the 30-minute group, so that's 0.27, 0.27 squared, divided by the sample size for the 30-minute group, divided by 18. And we're done, and we can look down at the choices. Let's see, they all got the first part the same, because that's maybe the most straightforward part, 38.9 minus 38.3, plus or minus 1.74, so both of these are looking good. We can rule out these two, because they have a different critical T value."}, {"video_title": "Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "So let's see, the 24 people who exercised for 60 minutes, so divided by 24, plus the sample standard deviation for the 30-minute group, so that's 0.27, 0.27 squared, divided by the sample size for the 30-minute group, divided by 18. And we're done, and we can look down at the choices. Let's see, they all got the first part the same, because that's maybe the most straightforward part, 38.9 minus 38.3, plus or minus 1.74, so both of these are looking good. We can rule out these two, because they have a different critical T value. And let's see, we have 0.29 squared, divided by 24, plus 0.27 squared, divided by 18. This one is looking good. Over here, let's see, they put the 30-minute sample size with the sample standard deviation of the 60-minute group, so that won't work, and so we like choice A."}, {"video_title": "Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3", "Sentence": "The drinks in the sample contained a mean amount of 528 milliliters with a standard deviation of four milliliters. These results produced a test statistic of t is equal to negative 2.236 and a p-value of approximately 0.038. Assuming the conditions for inference were met, what is an appropriate conclusion at the alpha equals 0.05 significance level? And they give us some choices here. And like always, I encourage you to pause this video and see if you can figure it out on your own. All right, so now let's work through this together. So let's just remind ourselves what's going on."}, {"video_title": "Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3", "Sentence": "And they give us some choices here. And like always, I encourage you to pause this video and see if you can figure it out on your own. All right, so now let's work through this together. So let's just remind ourselves what's going on. So you have some population of drinks, and we care about the true population mean. You have a null hypothesis around it that the true mean is 530 milliliters, but then there's the alternative hypothesis that it's not 530 milliliters. So to test your null hypothesis, you take a sample."}, {"video_title": "Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3", "Sentence": "So let's just remind ourselves what's going on. So you have some population of drinks, and we care about the true population mean. You have a null hypothesis around it that the true mean is 530 milliliters, but then there's the alternative hypothesis that it's not 530 milliliters. So to test your null hypothesis, you take a sample. In this case, we had a sample of 20 drinks. And using that sample, you calculate a sample mean, and then you also calculate a sample standard deviation. They tell us these things right over here."}, {"video_title": "Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3", "Sentence": "So to test your null hypothesis, you take a sample. In this case, we had a sample of 20 drinks. And using that sample, you calculate a sample mean, and then you also calculate a sample standard deviation. They tell us these things right over here. And then using this information and actually our sample size, you're able to calculate a t-statistic. You're able to calculate a t-statistic, and then using that t-statistic, you are able to calculate a p-value. And the p-value is what is the probability of getting a result at least this extreme if we assume that the null hypothesis is true?"}, {"video_title": "Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3", "Sentence": "They tell us these things right over here. And then using this information and actually our sample size, you're able to calculate a t-statistic. You're able to calculate a t-statistic, and then using that t-statistic, you are able to calculate a p-value. And the p-value is what is the probability of getting a result at least this extreme if we assume that the null hypothesis is true? And if that probability is lower than our significance level, then we say, hey, that's a very low probability. We're going to reject our null hypothesis, which would suggest our alternative. So the key to this question is just to compare this p-value right over here to our significance level."}, {"video_title": "Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3", "Sentence": "And the p-value is what is the probability of getting a result at least this extreme if we assume that the null hypothesis is true? And if that probability is lower than our significance level, then we say, hey, that's a very low probability. We're going to reject our null hypothesis, which would suggest our alternative. So the key to this question is just to compare this p-value right over here to our significance level. And as we see, the p-value, 0.038, is indeed less than 0.05. And so because of this, we would reject the null hypothesis. We would reject the null hypothesis, which would suggest the alternative, that the true mean is something different than 530 mL."}, {"video_title": "Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3", "Sentence": "So the key to this question is just to compare this p-value right over here to our significance level. And as we see, the p-value, 0.038, is indeed less than 0.05. And so because of this, we would reject the null hypothesis. We would reject the null hypothesis, which would suggest the alternative, that the true mean is something different than 530 mL. And so if we look at our choices here, so the first choice says reject the null hypothesis. This is strong evidence that the mean filling amount is different than 530 mL. Yeah, that one looks good."}, {"video_title": "Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3", "Sentence": "We would reject the null hypothesis, which would suggest the alternative, that the true mean is something different than 530 mL. And so if we look at our choices here, so the first choice says reject the null hypothesis. This is strong evidence that the mean filling amount is different than 530 mL. Yeah, that one looks good. This suggests, this is strong evidence. This suggests the alternative hypothesis, which is that right over there. But let's read the other one, just to make sure that they don't make sense."}, {"video_title": "Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3", "Sentence": "Yeah, that one looks good. This suggests, this is strong evidence. This suggests the alternative hypothesis, which is that right over there. But let's read the other one, just to make sure that they don't make sense. So this is rejecting the null hypothesis. That looks true so far. This isn't enough evidence to conclude that the mean filling amount is different than 530 mL."}, {"video_title": "Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3", "Sentence": "But let's read the other one, just to make sure that they don't make sense. So this is rejecting the null hypothesis. That looks true so far. This isn't enough evidence to conclude that the mean filling amount is different than 530 mL. No, not, the first one is definitely much stronger. Fail to reject the null hypothesis. No, we are rejecting the null hypothesis because our p-value is lower than our significance level."}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "Nutritionists measured the sugar content in grams for 32 drinks at Starbucks. A cumulative relative frequency graph, let me underline that, a cumulative relative frequency graph for the data is shown below. So they have different, on the horizontal axis, different amounts of sugar in grams, and then we have the cumulative relative frequency. So let's just make sure we understand how to read this. This is saying that zero, or 0%, of the drinks have a sugar content, have no sugar content. This right over here, this data point, this looks like it's at the.5 grams, and then this looks like it's at 0.1. This says that 0.1, or I guess we would say 10%, of the drinks that Starbucks offers has five grams of sugar or less."}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "So let's just make sure we understand how to read this. This is saying that zero, or 0%, of the drinks have a sugar content, have no sugar content. This right over here, this data point, this looks like it's at the.5 grams, and then this looks like it's at 0.1. This says that 0.1, or I guess we would say 10%, of the drinks that Starbucks offers has five grams of sugar or less. This data point tells us that 100% of the drinks at Starbucks has 50 grams of sugar or less. The cumulative relative frequency, that's why we, for each of these points, we say this is the frequency that has that much sugar or less, and that's why it just keeps on increasing and increasing. As we add more sugar, we're going to see, we're gonna see a larger proportion or a larger relative frequency has that much sugar or less."}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "This says that 0.1, or I guess we would say 10%, of the drinks that Starbucks offers has five grams of sugar or less. This data point tells us that 100% of the drinks at Starbucks has 50 grams of sugar or less. The cumulative relative frequency, that's why we, for each of these points, we say this is the frequency that has that much sugar or less, and that's why it just keeps on increasing and increasing. As we add more sugar, we're going to see, we're gonna see a larger proportion or a larger relative frequency has that much sugar or less. So let's read the first question. An iced coffee has 15 grams of sugar. Estimate the percentile of this drink to the nearest whole percent."}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "As we add more sugar, we're going to see, we're gonna see a larger proportion or a larger relative frequency has that much sugar or less. So let's read the first question. An iced coffee has 15 grams of sugar. Estimate the percentile of this drink to the nearest whole percent. So iced coffee has 15 grams of sugar, which would be right over here. And so let's estimate the percentile. So we can see they actually have a data point right over here."}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "Estimate the percentile of this drink to the nearest whole percent. So iced coffee has 15 grams of sugar, which would be right over here. And so let's estimate the percentile. So we can see they actually have a data point right over here. And we can see that 20%, or 0.2, 20% of the drinks that Starbucks offers has 15 grams of sugar or less. So the percentile of this drink, if I were to estimate it, it looks like it's the relative frequency, 0.2, has that much sugar or less, and so this percentile would be 20%. Once again, another way to think about it, to read this, you could convert these to percentages."}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "So we can see they actually have a data point right over here. And we can see that 20%, or 0.2, 20% of the drinks that Starbucks offers has 15 grams of sugar or less. So the percentile of this drink, if I were to estimate it, it looks like it's the relative frequency, 0.2, has that much sugar or less, and so this percentile would be 20%. Once again, another way to think about it, to read this, you could convert these to percentages. You could say that 20% has this much sugar or less, 15 grams of sugar or less, so an iced coffee is in the 20th percentile. Let's do another question. So here we are asked to estimate the median of the distribution of drinks."}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "Once again, another way to think about it, to read this, you could convert these to percentages. You could say that 20% has this much sugar or less, 15 grams of sugar or less, so an iced coffee is in the 20th percentile. Let's do another question. So here we are asked to estimate the median of the distribution of drinks. Hint, think about the 50th percentile. So the median, if you were to line up all of the drinks, you would take the middle drink. And so you could view that as, well, what drink is exactly at the 50th percentile?"}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "So here we are asked to estimate the median of the distribution of drinks. Hint, think about the 50th percentile. So the median, if you were to line up all of the drinks, you would take the middle drink. And so you could view that as, well, what drink is exactly at the 50th percentile? So now let's look at the 50th percentile would be a cumulative relative frequency of 0.5, which would be right over here on our vertical axis. Another way to think about it is 0.5, or 50% of the drinks are going, if we go to this point right over here, what has a cumulative relative frequency of 0.5? We see that we are right at, looks like this is 25 grams."}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "And so you could view that as, well, what drink is exactly at the 50th percentile? So now let's look at the 50th percentile would be a cumulative relative frequency of 0.5, which would be right over here on our vertical axis. Another way to think about it is 0.5, or 50% of the drinks are going, if we go to this point right over here, what has a cumulative relative frequency of 0.5? We see that we are right at, looks like this is 25 grams. So one way to interpret this is 50% of the drinks have less than, or have 25 grams of sugar or less. So this looks like a pretty good estimate for the median, for the middle data point. So the median is approximately 25 grams, that half of the drinks have 25 grams or less of sugar."}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "We see that we are right at, looks like this is 25 grams. So one way to interpret this is 50% of the drinks have less than, or have 25 grams of sugar or less. So this looks like a pretty good estimate for the median, for the middle data point. So the median is approximately 25 grams, that half of the drinks have 25 grams or less of sugar. Let's do one more based on the same data set. So here we're asked, what is the best estimate for the interquartile range of the distribution of drinks? So the interquartile range, we wanna figure out, well, what's sitting at the 25th percentile?"}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "So the median is approximately 25 grams, that half of the drinks have 25 grams or less of sugar. Let's do one more based on the same data set. So here we're asked, what is the best estimate for the interquartile range of the distribution of drinks? So the interquartile range, we wanna figure out, well, what's sitting at the 25th percentile? And we wanna think about what's at the 75th percentile. And then we want to take the difference. That's what the interquartile range is."}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "So the interquartile range, we wanna figure out, well, what's sitting at the 25th percentile? And we wanna think about what's at the 75th percentile. And then we want to take the difference. That's what the interquartile range is. So let's do that. So first, the 25th percentile, we'd wanna look at the cumulative relative frequency. So 25th, this would be 30th, so 25th would be right around here."}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "That's what the interquartile range is. So let's do that. So first, the 25th percentile, we'd wanna look at the cumulative relative frequency. So 25th, this would be 30th, so 25th would be right around here. And so it looks like the 25th percentile is, that looks like about, I don't know. And we're estimating here, so that looks like it's about, this would be 15, looks like I would say maybe 18 grams. So approximately 18 grams."}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "So 25th, this would be 30th, so 25th would be right around here. And so it looks like the 25th percentile is, that looks like about, I don't know. And we're estimating here, so that looks like it's about, this would be 15, looks like I would say maybe 18 grams. So approximately 18 grams. Once again, one way to think about it is, 25% of the drinks have 18 grams of sugar or less. And let's look at the 75th percentile. So this is 70th, 75th would be right over there."}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "So approximately 18 grams. Once again, one way to think about it is, 25% of the drinks have 18 grams of sugar or less. And let's look at the 75th percentile. So this is 70th, 75th would be right over there. Actually, I can draw a straighter line than that. I have a line tool here. So 75th percentile would put me right over there."}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "So this is 70th, 75th would be right over there. Actually, I can draw a straighter line than that. I have a line tool here. So 75th percentile would put me right over there. I don't know, that looks like, well, I'll go with 39 grams, roughly 39 grams. And so what's the difference between these two? Well, the difference between these two, it looks like it's about 21 grams."}, {"video_title": "Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3", "Sentence": "So 75th percentile would put me right over there. I don't know, that looks like, well, I'll go with 39 grams, roughly 39 grams. And so what's the difference between these two? Well, the difference between these two, it looks like it's about 21 grams. So our interquartile range, our estimate of our interquartile range, looking at this cumulative relative frequency distribution, because we're saying, hey, look, it looks like the 25th percentile, it looks like 25% of the drinks have 18 grams or less. 75% of the drinks have 39 grams or less. If we take the difference between these two quartiles, this is the first quartile, this is our third quartile, we're gonna get 21 grams."}, {"video_title": "Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "She wants to test whether this holds true for teachers in her state. So she is going to take a random sample of these teachers and see what percent of them are members of a union. Let P represent the proportion of teachers in her state that are members of a union. Write an appropriate set of hypotheses for her significance test. So pause this video and see if you can do that. All right, now let's do it together. So what we wanna do for this significance test is set up a null hypothesis and an alternative hypothesis."}, {"video_title": "Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Write an appropriate set of hypotheses for her significance test. So pause this video and see if you can do that. All right, now let's do it together. So what we wanna do for this significance test is set up a null hypothesis and an alternative hypothesis. Now, your null hypothesis is the hypothesis that, hey, there's no news here. It's what you would expect it to be. And so if you read a report saying that 49% of teachers in the United States were members of labor unions, well, then it would be reasonable to say that the null hypothesis, the no news here, is that the same percentage of teachers in her state are members of a labor union."}, {"video_title": "Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "So what we wanna do for this significance test is set up a null hypothesis and an alternative hypothesis. Now, your null hypothesis is the hypothesis that, hey, there's no news here. It's what you would expect it to be. And so if you read a report saying that 49% of teachers in the United States were members of labor unions, well, then it would be reasonable to say that the null hypothesis, the no news here, is that the same percentage of teachers in her state are members of a labor union. So that percentage, that proportion is P. So this would be the null hypothesis, that the proportion in her state is also 49%. And now what would the alternative be? Well, the alternative is that the proportion in her state is not 49%."}, {"video_title": "Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "And so if you read a report saying that 49% of teachers in the United States were members of labor unions, well, then it would be reasonable to say that the null hypothesis, the no news here, is that the same percentage of teachers in her state are members of a labor union. So that percentage, that proportion is P. So this would be the null hypothesis, that the proportion in her state is also 49%. And now what would the alternative be? Well, the alternative is that the proportion in her state is not 49%. This is the thing that, hey, there would be news here. There'd be something interesting to report. There's something different about her state."}, {"video_title": "Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Well, the alternative is that the proportion in her state is not 49%. This is the thing that, hey, there would be news here. There'd be something interesting to report. There's something different about her state. And how would she use this? Well, she would take a sample of teachers in her state, figure out the sample proportion, figure out the probability of getting that sample proportion if we were to assume that the null hypothesis is true. If that probability is lower than a threshold, which she should have said ahead of time, her significance level, then she would reject the null hypothesis, which would suggest the alternative."}, {"video_title": "Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "There's something different about her state. And how would she use this? Well, she would take a sample of teachers in her state, figure out the sample proportion, figure out the probability of getting that sample proportion if we were to assume that the null hypothesis is true. If that probability is lower than a threshold, which she should have said ahead of time, her significance level, then she would reject the null hypothesis, which would suggest the alternative. Let's do another example here. According to a very large poll in 2015, about 90% of homes in California had access to the internet. Market researchers want to test if that proportion is now higher."}, {"video_title": "Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "If that probability is lower than a threshold, which she should have said ahead of time, her significance level, then she would reject the null hypothesis, which would suggest the alternative. Let's do another example here. According to a very large poll in 2015, about 90% of homes in California had access to the internet. Market researchers want to test if that proportion is now higher. So they take a random sample of 1,000 homes in California and find that 920, or 92% of homes sampled, have access to the internet. Let P represent the proportion of homes in California that have access to the internet. Write an appropriate set of hypotheses for their significance test."}, {"video_title": "Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Market researchers want to test if that proportion is now higher. So they take a random sample of 1,000 homes in California and find that 920, or 92% of homes sampled, have access to the internet. Let P represent the proportion of homes in California that have access to the internet. Write an appropriate set of hypotheses for their significance test. So once again, pause this video and see if you can figure it out. So once again, we want to have a null hypothesis and we want to have an alternative hypothesis. The null hypothesis is the, hey, there's no news here."}, {"video_title": "Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Write an appropriate set of hypotheses for their significance test. So once again, pause this video and see if you can figure it out. So once again, we want to have a null hypothesis and we want to have an alternative hypothesis. The null hypothesis is the, hey, there's no news here. And so that would say that, you know, it's kind of the status quo, that the proportion of people who have internet is still the same as the last study, is still the same at 90%. Or I could write 90%, or I could write 0.9 right over here. Now some of you might have been tempted to put 92% there."}, {"video_title": "Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "The null hypothesis is the, hey, there's no news here. And so that would say that, you know, it's kind of the status quo, that the proportion of people who have internet is still the same as the last study, is still the same at 90%. Or I could write 90%, or I could write 0.9 right over here. Now some of you might have been tempted to put 92% there. But it's very important to realize, 92% is the sample proportion, that's the sample statistic. When we're writing these hypotheses, this is about, these are hypotheses about the true parameter. What is the proportion of, the true proportion of homes in California that now have the internet?"}, {"video_title": "Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Now some of you might have been tempted to put 92% there. But it's very important to realize, 92% is the sample proportion, that's the sample statistic. When we're writing these hypotheses, this is about, these are hypotheses about the true parameter. What is the proportion of, the true proportion of homes in California that now have the internet? And so this is about the true proportion. And so the alternative here is that it's now greater than 90%, or I could say it's greater than 0.9. I could have written 90% or 0.9 here."}, {"video_title": "Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "What is the proportion of, the true proportion of homes in California that now have the internet? And so this is about the true proportion. And so the alternative here is that it's now greater than 90%, or I could say it's greater than 0.9. I could have written 90% or 0.9 here. And so they really, in this question, they wrote this to kind of distract you, to make you think, oh, maybe I have to incorporate this 92% somehow. And once again, how will they use these hypotheses? Well, they will take this sample in which they got 92% of homes samples had access to the internet."}, {"video_title": "Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "I could have written 90% or 0.9 here. And so they really, in this question, they wrote this to kind of distract you, to make you think, oh, maybe I have to incorporate this 92% somehow. And once again, how will they use these hypotheses? Well, they will take this sample in which they got 92% of homes samples had access to the internet. So this right over here is my sample proportion. And then they're gonna figure out, well, what's the probability of getting this sample proportion for this sample size if we were to assume that the null hypothesis is true? If this probability of getting this is below a threshold, it's below alpha, below our significance level, then we'll reject the null hypothesis, which would suggest the alternative."}, {"video_title": "Generalizabilty of survey results example AP Statistics Khan Academy.mp3", "Sentence": "Here is computer output from a least squares regression analysis for using fertility rate to predict life expectancy. Use this model to predict the life expectancy of a country whose fertility rate is two babies per woman. And you can round your answer to the nearest whole number of years. So pause this number and see if you can do it. You might need to use a calculator. All right, now let's do this together. So in general, this computer output is actually giving us a lot of data, more than we need actually to do this prediction."}, {"video_title": "Generalizabilty of survey results example AP Statistics Khan Academy.mp3", "Sentence": "So pause this number and see if you can do it. You might need to use a calculator. All right, now let's do this together. So in general, this computer output is actually giving us a lot of data, more than we need actually to do this prediction. But it's giving us the data we need to know the equation for a regression line. So the general form of a regression line, a linear regression line, would be our estimate, and that little hat means we're estimating our y value, would be equal to our y-intercept plus our slope times our x value. Now in this situation, we're using fertility to predict life expectancy."}, {"video_title": "Generalizabilty of survey results example AP Statistics Khan Academy.mp3", "Sentence": "So in general, this computer output is actually giving us a lot of data, more than we need actually to do this prediction. But it's giving us the data we need to know the equation for a regression line. So the general form of a regression line, a linear regression line, would be our estimate, and that little hat means we're estimating our y value, would be equal to our y-intercept plus our slope times our x value. Now in this situation, we're using fertility to predict life expectancy. Or let me circle all of life expectancy. So the thing that we're trying to predict, that is y, life expectancy, and fertility is the thing that we're using to predict that. So that is going to be our x right over there."}, {"video_title": "Generalizabilty of survey results example AP Statistics Khan Academy.mp3", "Sentence": "Now in this situation, we're using fertility to predict life expectancy. Or let me circle all of life expectancy. So the thing that we're trying to predict, that is y, life expectancy, and fertility is the thing that we're using to predict that. So that is going to be our x right over there. Now what are a and b? Well, our computer output gives us that. It's these numbers right over here."}, {"video_title": "Generalizabilty of survey results example AP Statistics Khan Academy.mp3", "Sentence": "So that is going to be our x right over there. Now what are a and b? Well, our computer output gives us that. It's these numbers right over here. Our constant coefficient right over here, this is a, and our slope is going to be negative 5.97. You could view it as the coefficient on fertility. Remember, this right over here is fertility."}, {"video_title": "Generalizabilty of survey results example AP Statistics Khan Academy.mp3", "Sentence": "It's these numbers right over here. Our constant coefficient right over here, this is a, and our slope is going to be negative 5.97. You could view it as the coefficient on fertility. Remember, this right over here is fertility. You could even rewrite this as our estimated life expectancy, estimated life expectancy, I could put a little hat on it to show this is estimated life expectancy, is going to be equal to 89.70 minus 5.97 times fertility, times fertility rate. I'll just call it, say, Fert and period right over there. Notice, this is the coefficient on fertility and then this is the constant coefficient."}, {"video_title": "Generalizabilty of survey results example AP Statistics Khan Academy.mp3", "Sentence": "Remember, this right over here is fertility. You could even rewrite this as our estimated life expectancy, estimated life expectancy, I could put a little hat on it to show this is estimated life expectancy, is going to be equal to 89.70 minus 5.97 times fertility, times fertility rate. I'll just call it, say, Fert and period right over there. Notice, this is the coefficient on fertility and then this is the constant coefficient. We could view that right over there. And now we can use this to estimate the life expectancy of a country whose fertility rate is two babies per woman. For fertility, you just put a two here and then you get your estimated life expectancy."}, {"video_title": "Generalizabilty of survey results example AP Statistics Khan Academy.mp3", "Sentence": "Notice, this is the coefficient on fertility and then this is the constant coefficient. We could view that right over there. And now we can use this to estimate the life expectancy of a country whose fertility rate is two babies per woman. For fertility, you just put a two here and then you get your estimated life expectancy. So what's that going to be? We can get out a calculator. So we can say 5.97 times two is equal to that and then we wanna subtract that from, so I'll put a negative there, and add that to 89.7 is equal to, and we wanna round to the nearest whole number of years, so that's approximately 78 years."}, {"video_title": "Generalizabilty of survey results example AP Statistics Khan Academy.mp3", "Sentence": "For fertility, you just put a two here and then you get your estimated life expectancy. So what's that going to be? We can get out a calculator. So we can say 5.97 times two is equal to that and then we wanna subtract that from, so I'll put a negative there, and add that to 89.7 is equal to, and we wanna round to the nearest whole number of years, so that's approximately 78 years. So this is approximately 78 years. And we're done. And just to be clear, what even happened here is that Nkechi, she did a regression."}, {"video_title": "Generalizabilty of survey results example AP Statistics Khan Academy.mp3", "Sentence": "So we can say 5.97 times two is equal to that and then we wanna subtract that from, so I'll put a negative there, and add that to 89.7 is equal to, and we wanna round to the nearest whole number of years, so that's approximately 78 years. So this is approximately 78 years. And we're done. And just to be clear, what even happened here is that Nkechi, she did a regression. On the x-axis is fertility. Fertility. On the y-axis is, let's call it L period dot E period."}, {"video_title": "Generalizabilty of survey results example AP Statistics Khan Academy.mp3", "Sentence": "And just to be clear, what even happened here is that Nkechi, she did a regression. On the x-axis is fertility. Fertility. On the y-axis is, let's call it L period dot E period. That's our y-axis. Took 10 data points, one, two, three, four, five, six, seven, eight, nine, 10, put a regression line on, tried to fit a regression line. Saw a negative linear relationship."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3", "Sentence": "Here's a simulation created by Khan Academy user TETF. I can assume that's pronounced Ted-eff. And what it allows us to do is give us an intuition as to why we divide by n minus one when we calculate our sample variance and why that gives us an unbiased estimate of population variance. So the way this starts off, and I encourage you to go try this out yourself, is that you can construct a distribution. It says build a population by clicking in the blue area. So here, we're actually creating a population. So we're creating, every time I click, it increases the population size."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3", "Sentence": "So the way this starts off, and I encourage you to go try this out yourself, is that you can construct a distribution. It says build a population by clicking in the blue area. So here, we're actually creating a population. So we're creating, every time I click, it increases the population size. So let me just, and I'm just randomly doing this, and I encourage you to go onto this scratch pad. It's on the Khan Academy Computer Science, and try to do it yourself. So here, we are, I can stop at some point."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3", "Sentence": "So we're creating, every time I click, it increases the population size. So let me just, and I'm just randomly doing this, and I encourage you to go onto this scratch pad. It's on the Khan Academy Computer Science, and try to do it yourself. So here, we are, I can stop at some point. So I've constructed a population. I can throw out some random points up here. So this is our population."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3", "Sentence": "So here, we are, I can stop at some point. So I've constructed a population. I can throw out some random points up here. So this is our population. And as you saw while I was doing that, I was calculating parameters for the population. It was calculating the population mean at 204.09, and also the population standard deviation, which is derived from the population variance. This is the square root of the population variance, and it's at 63.8."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3", "Sentence": "So this is our population. And as you saw while I was doing that, I was calculating parameters for the population. It was calculating the population mean at 204.09, and also the population standard deviation, which is derived from the population variance. This is the square root of the population variance, and it's at 63.8. It was also plotting the population variance down here. You see it's 63.8, which is the standard deviation. And it's a little harder to see, but it says squared."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3", "Sentence": "This is the square root of the population variance, and it's at 63.8. It was also plotting the population variance down here. You see it's 63.8, which is the standard deviation. And it's a little harder to see, but it says squared. These are these numbers squared. So this is essentially 63.8 is the population, 63.8 squared is the population variance. So that's interesting by itself, but it really doesn't tell us a lot so far about why we divide by n minus one."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3", "Sentence": "And it's a little harder to see, but it says squared. These are these numbers squared. So this is essentially 63.8 is the population, 63.8 squared is the population variance. So that's interesting by itself, but it really doesn't tell us a lot so far about why we divide by n minus one. And this is the interesting part. We can now start to take samples, and we can decide what sample size we wanna do. I'll start with really small samples, so the smallest possible sample that makes any sense."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3", "Sentence": "So that's interesting by itself, but it really doesn't tell us a lot so far about why we divide by n minus one. And this is the interesting part. We can now start to take samples, and we can decide what sample size we wanna do. I'll start with really small samples, so the smallest possible sample that makes any sense. So I'm gonna start with really small samples. And what they're going to do, what the simulation is going to do, is every time I take a sample, it's going to calculate the variance. So the numerator is going to be the sum of each of my data points in my sample minus my sample mean, and I'm gonna square it."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3", "Sentence": "I'll start with really small samples, so the smallest possible sample that makes any sense. So I'm gonna start with really small samples. And what they're going to do, what the simulation is going to do, is every time I take a sample, it's going to calculate the variance. So the numerator is going to be the sum of each of my data points in my sample minus my sample mean, and I'm gonna square it. And then it's going to divide it by n plus a. So, and it's going to vary a. It's going to divide it by anywhere between n plus negative three, so n minus three, all the way to n plus a."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3", "Sentence": "So the numerator is going to be the sum of each of my data points in my sample minus my sample mean, and I'm gonna square it. And then it's going to divide it by n plus a. So, and it's going to vary a. It's going to divide it by anywhere between n plus negative three, so n minus three, all the way to n plus a. And we're gonna do it many, many, many, many times. We're gonna essentially take the mean of those variances for any a, and figure out which gives us the best estimate. So if I just generate one sample right over there, well, we see, when we, when our, we see kind of this curve."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3", "Sentence": "It's going to divide it by anywhere between n plus negative three, so n minus three, all the way to n plus a. And we're gonna do it many, many, many, many times. We're gonna essentially take the mean of those variances for any a, and figure out which gives us the best estimate. So if I just generate one sample right over there, well, we see, when we, when our, we see kind of this curve. When we have high values of a, we are underestimating. When we have lower values of a, we are overestimating the population variance. But that was just for one, that was just for one sample, not really that meaningful."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3", "Sentence": "So if I just generate one sample right over there, well, we see, when we, when our, we see kind of this curve. When we have high values of a, we are underestimating. When we have lower values of a, we are overestimating the population variance. But that was just for one, that was just for one sample, not really that meaningful. It's one sample of size two. Let's generate a bunch of samples, and then average them over many of them. And you see, when you look at many, many, many, many, many samples, something interesting is happening."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3", "Sentence": "But that was just for one, that was just for one sample, not really that meaningful. It's one sample of size two. Let's generate a bunch of samples, and then average them over many of them. And you see, when you look at many, many, many, many, many samples, something interesting is happening. When you look at the mean of those samples, when you average together those curves from all of those samples, you see that our best estimate is when a is pretty close to negative one, is when this is n plus negative one, or n minus one. Anything less than negative one, if we did negative n minus 1.05, or n minus 1.5, we start overestimating the variance. Anything less than negative one, so if we have n plus zero, if we divide by n, or if we have n plus.05, or whatever it might be, we start underestimating, we start underestimating the population variance."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3", "Sentence": "And you see, when you look at many, many, many, many, many samples, something interesting is happening. When you look at the mean of those samples, when you average together those curves from all of those samples, you see that our best estimate is when a is pretty close to negative one, is when this is n plus negative one, or n minus one. Anything less than negative one, if we did negative n minus 1.05, or n minus 1.5, we start overestimating the variance. Anything less than negative one, so if we have n plus zero, if we divide by n, or if we have n plus.05, or whatever it might be, we start underestimating, we start underestimating the population variance. And you can do this for samples of different sizes. Let me try a sample size six. And here you go, once again, as I press, I'm just keeping generate sample pressed down."}, {"video_title": "Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3", "Sentence": "Anything less than negative one, so if we have n plus zero, if we divide by n, or if we have n plus.05, or whatever it might be, we start underestimating, we start underestimating the population variance. And you can do this for samples of different sizes. Let me try a sample size six. And here you go, once again, as I press, I'm just keeping generate sample pressed down. As we generate more and more and more samples, and for all of the a's, we essentially take the average across those samples for the variance, depending on how we calculate it, you'll see that, once again, our best estimate is pretty darn close, is pretty darn close to negative one. And if you were to try this, if you were to get this to millions of samples generated, you will see that your best estimate is when a is negative one, or when you are dividing by, when you're dividing by n minus one. So once again, thanks TETF for this, I think it's a really interesting way to think about why we divide by n minus one."}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "Assume the method of placing each cake order is independent. So C, if we assume a few things, is a classic geometric random variable. What tells us that? Well, the giveaway is that we're gonna keep doing these independent trials where the probability of success is constant and there's a clear success. A telephone order in this case is a success. The probability is 10% of it happening. And we're gonna keep doing it until we get a success."}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "Well, the giveaway is that we're gonna keep doing these independent trials where the probability of success is constant and there's a clear success. A telephone order in this case is a success. The probability is 10% of it happening. And we're gonna keep doing it until we get a success. So classic geometric random variable. Now they ask us, find the probability, the probability that it takes fewer than five orders for Liliana to get her first telephone order of the month. So it's really the probability that C is less than five."}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "And we're gonna keep doing it until we get a success. So classic geometric random variable. Now they ask us, find the probability, the probability that it takes fewer than five orders for Liliana to get her first telephone order of the month. So it's really the probability that C is less than five. So like always, pause this video and have a go at it. And even if you struggle with it, that's even, that's better, your brain will be more primed for the actual solution that we can go through together. All right."}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "So it's really the probability that C is less than five. So like always, pause this video and have a go at it. And even if you struggle with it, that's even, that's better, your brain will be more primed for the actual solution that we can go through together. All right. So I'm assuming you've had a go at it. So there's a couple of ways to approach it. You could say, well look, this is just gonna be the probability that C is equal to one plus the probability that C is equal to two plus the probability that C is equal to three plus the probability that C is equal to four."}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "All right. So I'm assuming you've had a go at it. So there's a couple of ways to approach it. You could say, well look, this is just gonna be the probability that C is equal to one plus the probability that C is equal to two plus the probability that C is equal to three plus the probability that C is equal to four. And we can calculate it this way. What is the probability that C equals one? Well, it's the probability that her very first order is a telephone order."}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "You could say, well look, this is just gonna be the probability that C is equal to one plus the probability that C is equal to two plus the probability that C is equal to three plus the probability that C is equal to four. And we can calculate it this way. What is the probability that C equals one? Well, it's the probability that her very first order is a telephone order. And so we'll have 0.1. What's the probability that C equals two? Well, it's the probability that her first order is not a telephone order."}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "Well, it's the probability that her very first order is a telephone order. And so we'll have 0.1. What's the probability that C equals two? Well, it's the probability that her first order is not a telephone order. So it's one minus 10%. There's a 90% chance it's not a telephone order. And that her second order is a telephone order."}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "Well, it's the probability that her first order is not a telephone order. So it's one minus 10%. There's a 90% chance it's not a telephone order. And that her second order is a telephone order. What about the probability C equals three? Well, her first two orders would not be telephone orders and her third order would be one. And then C equals four?"}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "And that her second order is a telephone order. What about the probability C equals three? Well, her first two orders would not be telephone orders and her third order would be one. And then C equals four? Well, her first three orders would not be telephone orders and her fourth one would. And we could get a calculator maybe and add all of these things up and we would actually get the answer. But you're probably wondering, well, this is kinda hairy to type into a calculator."}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "And then C equals four? Well, her first three orders would not be telephone orders and her fourth one would. And we could get a calculator maybe and add all of these things up and we would actually get the answer. But you're probably wondering, well, this is kinda hairy to type into a calculator. Maybe there is an easier way to tackle this. And indeed, there is. So think about it."}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "But you're probably wondering, well, this is kinda hairy to type into a calculator. Maybe there is an easier way to tackle this. And indeed, there is. So think about it. The probability that C is less than five, that's the same thing as one minus the probability that we don't have a telephone order in the first four. One minus the probability that no telephone order in first four orders. So what's this?"}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "So think about it. The probability that C is less than five, that's the same thing as one minus the probability that we don't have a telephone order in the first four. One minus the probability that no telephone order in first four orders. So what's this? Well, because this is just saying, what's the probability we do have an order in the first four? So it's the same thing as one minus the probability that we don't have an order in the first four. And this is pretty straightforward to calculate."}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "So what's this? Well, because this is just saying, what's the probability we do have an order in the first four? So it's the same thing as one minus the probability that we don't have an order in the first four. And this is pretty straightforward to calculate. So this is going to be equal to one minus, and let me do this in another color so we know what I'm referring to. So what's the probability that we have no telephone orders in the first four orders? Well, the probability on a given order that you don't have a telephone order is 0.9."}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "And this is pretty straightforward to calculate. So this is going to be equal to one minus, and let me do this in another color so we know what I'm referring to. So what's the probability that we have no telephone orders in the first four orders? Well, the probability on a given order that you don't have a telephone order is 0.9. And then if that has to be true for the first four, well, it's gonna be 0.9 times 0.9 times 0.9 times 0.9, or 0.9 to the fourth power. So this is a lot easier to calculate. So let's do that."}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "Well, the probability on a given order that you don't have a telephone order is 0.9. And then if that has to be true for the first four, well, it's gonna be 0.9 times 0.9 times 0.9 times 0.9, or 0.9 to the fourth power. So this is a lot easier to calculate. So let's do that. Let's get a calculator out. All right, so let me just take 0.9 to the fourth power is equal to, and then let me subtract that from one. So let me make that a negative, and then let me add one to it."}, {"video_title": "Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3", "Sentence": "So let's do that. Let's get a calculator out. All right, so let me just take 0.9 to the fourth power is equal to, and then let me subtract that from one. So let me make that a negative, and then let me add one to it. And we get, there you go, 0.3439. So this is equal to 0.3439. And we're done."}, {"video_title": "Generalizing k scores in n attempts Probability and Statistics Khan Academy.mp3", "Sentence": "I have a 70% chance of making any free throw, which is actually higher than my actual free throw probability, which might be a surprise to you. But we said in that circumstance, if it's a 70% chance of making it, well that means that you have a one minus 70%, or 30% chance of missing. And we said if you took six attempts, the probability of you getting exactly, making two of the baskets, exactly two scores, and I call them scores instead of making it just because I wanted making and missing, have different letters in that video, we said, well, there's six choose two different ways of making two, exactly two out of the six free throws, and then the probability of any one of those ways is going to be making it twice, which is 0.7 squared, and missing it four times, so 0.3 to the fourth power. So this was just one particular situation, but we could generalize based on the logic that we had in that video. In fact, let's do that. So if I were to generalize it, if I were to say the probability, the probability, it's the exact same logic, of exactly, now let's say K, let me do this in a color, an interesting color, so let me do it in this orangish brown color, K shots, or exactly K scores, I'll call making a free throw a score, we'll just assume you got a point for it. So exactly two K scores, in N attempts, let's just say in N attempts, in N, and let me go back to that green color, N attempts, is going to be equal to, well, how many ways can you pick K things out of N, or N choose K, N choose K, and then, actually let's just generalize it even more."}, {"video_title": "Generalizing k scores in n attempts Probability and Statistics Khan Academy.mp3", "Sentence": "So this was just one particular situation, but we could generalize based on the logic that we had in that video. In fact, let's do that. So if I were to generalize it, if I were to say the probability, the probability, it's the exact same logic, of exactly, now let's say K, let me do this in a color, an interesting color, so let me do it in this orangish brown color, K shots, or exactly K scores, I'll call making a free throw a score, we'll just assume you got a point for it. So exactly two K scores, in N attempts, let's just say in N attempts, in N, and let me go back to that green color, N attempts, is going to be equal to, well, how many ways can you pick K things out of N, or N choose K, N choose K, and then, actually let's just generalize it even more. Let's just say that you have, your free throw probability is P. So let's say P is, so for this situation right over here, since we generalized it fully, let's say that P is the probability of making a free throw. Actually, since I already have a P here, let me just say F is equal to the probability of making a free throw, or you could say your probability of scoring, if you call a score making a free throw. So if F is your probability of making a free throw, so if you want N scores, then this is going to be, this is going to be, well, it's going to be F to the N power, and then you're going to have, and then you're going to miss the remainder, sorry, F to the K power, because you're making exactly K scores."}, {"video_title": "Generalizing k scores in n attempts Probability and Statistics Khan Academy.mp3", "Sentence": "So exactly two K scores, in N attempts, let's just say in N attempts, in N, and let me go back to that green color, N attempts, is going to be equal to, well, how many ways can you pick K things out of N, or N choose K, N choose K, and then, actually let's just generalize it even more. Let's just say that you have, your free throw probability is P. So let's say P is, so for this situation right over here, since we generalized it fully, let's say that P is the probability of making a free throw. Actually, since I already have a P here, let me just say F is equal to the probability of making a free throw, or you could say your probability of scoring, if you call a score making a free throw. So if F is your probability of making a free throw, so if you want N scores, then this is going to be, this is going to be, well, it's going to be F to the N power, and then you're going to have, and then you're going to miss the remainder, sorry, F to the K power, because you're making exactly K scores. So F to the K power, and then the remainder, so the N minus K attempts, you're going to miss it. So it's going to be that probability of missing, and the probability of missing is going to be one minus F, so it's going to be times one minus F to the N minus K power, to the N minus K power. And just, if you like, or I encourage you, pause the video, and just make sure you understand the parallels between this example where I had a set where, I guess our F was 70%, our F was 70%, one minus F, or our F was.7, and one minus F would be.3, and we were seeing, how do we get two scores in six attempts?"}, {"video_title": "Generalizing k scores in n attempts Probability and Statistics Khan Academy.mp3", "Sentence": "So if F is your probability of making a free throw, so if you want N scores, then this is going to be, this is going to be, well, it's going to be F to the N power, and then you're going to have, and then you're going to miss the remainder, sorry, F to the K power, because you're making exactly K scores. So F to the K power, and then the remainder, so the N minus K attempts, you're going to miss it. So it's going to be that probability of missing, and the probability of missing is going to be one minus F, so it's going to be times one minus F to the N minus K power, to the N minus K power. And just, if you like, or I encourage you, pause the video, and just make sure you understand the parallels between this example where I had a set where, I guess our F was 70%, our F was 70%, one minus F, or our F was.7, and one minus F would be.3, and we were seeing, how do we get two scores in six attempts? And here we're saying K scores in N attempts, and this is just a general way to think about it. And the whole reason why I'm setting this up this way is it's interesting to now think about the probability distribution for a random variable that's defined by the number of scores in your N attempts, or the number of scores in your six attempts. And actually, since I've been pushing the limit, or I've been doing longer videos than I intend to, I will do that in the next video."}, {"video_title": "Constructing t interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And we are going to think about the means of these populations. So let's say this first population is a population of golden retrievers, and this second population is a population of chihuahuas. And the mean that we're going to think about is maybe the mean weight. So mu one would be the true mean weight of the population of golden retrievers, and mu two would be the true mean weight of the population of chihuahuas. And what we wanna think about is what is the difference between these two population means, between these two population parameters? Well, if we don't know this, all we can do is try to estimate it and maybe construct some type of confidence interval. And that's what we're going to talk about in this video."}, {"video_title": "Constructing t interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "So mu one would be the true mean weight of the population of golden retrievers, and mu two would be the true mean weight of the population of chihuahuas. And what we wanna think about is what is the difference between these two population means, between these two population parameters? Well, if we don't know this, all we can do is try to estimate it and maybe construct some type of confidence interval. And that's what we're going to talk about in this video. So how do we go about doing it? Well, we've seen this or similar things before. What you would do is you would take a sample from both populations, so from population one here, I would take a sample of size N sub one, and from that, I can calculate a sample mean."}, {"video_title": "Constructing t interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And that's what we're going to talk about in this video. So how do we go about doing it? Well, we've seen this or similar things before. What you would do is you would take a sample from both populations, so from population one here, I would take a sample of size N sub one, and from that, I can calculate a sample mean. So this is a statistic that is trying to estimate that. And I can also calculate a sample standard deviation. And I can do the same thing in the population of chihuahuas, if that's what our population two is all about."}, {"video_title": "Constructing t interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "What you would do is you would take a sample from both populations, so from population one here, I would take a sample of size N sub one, and from that, I can calculate a sample mean. So this is a statistic that is trying to estimate that. And I can also calculate a sample standard deviation. And I can do the same thing in the population of chihuahuas, if that's what our population two is all about. So I could take a sample, and actually, this sample does not have to be the same as N one, so I'll call it N sub two. It could be, but it doesn't have to be. And from that, I can calculate a sample mean, X bar sub two, and a sample standard deviation."}, {"video_title": "Constructing t interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And I can do the same thing in the population of chihuahuas, if that's what our population two is all about. So I could take a sample, and actually, this sample does not have to be the same as N one, so I'll call it N sub two. It could be, but it doesn't have to be. And from that, I can calculate a sample mean, X bar sub two, and a sample standard deviation. So now, assuming that our conditions for inference are met, and we've talked about those before, we have the random condition, we have the normal condition, we have the independence condition. Assuming those conditions are met, and we talk about those in other videos for means, let's think about how we can construct a confidence interval. And so you might say, all right, well, that would be the difference of my sample means, X bar sub one minus X bar sub two, plus or minus some Z value times my standard deviation, times the standard deviation of the sampling distribution of the difference of the sample means."}, {"video_title": "Constructing t interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And from that, I can calculate a sample mean, X bar sub two, and a sample standard deviation. So now, assuming that our conditions for inference are met, and we've talked about those before, we have the random condition, we have the normal condition, we have the independence condition. Assuming those conditions are met, and we talk about those in other videos for means, let's think about how we can construct a confidence interval. And so you might say, all right, well, that would be the difference of my sample means, X bar sub one minus X bar sub two, plus or minus some Z value times my standard deviation, times the standard deviation of the sampling distribution of the difference of the sample means. So X bar sub one minus X bar sub two. And you might say, well, where do I get my Z from? Well, our confidence level would determine that."}, {"video_title": "Constructing t interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And so you might say, all right, well, that would be the difference of my sample means, X bar sub one minus X bar sub two, plus or minus some Z value times my standard deviation, times the standard deviation of the sampling distribution of the difference of the sample means. So X bar sub one minus X bar sub two. And you might say, well, where do I get my Z from? Well, our confidence level would determine that. Confidence, confidence level. If our confidence level is 95%, that would determine our Z. Now, this would not be incorrect, but we face a problem, because we are going to need to estimate what the standard deviation of the sampling distribution of the difference between our sample means actually is."}, {"video_title": "Constructing t interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "Well, our confidence level would determine that. Confidence, confidence level. If our confidence level is 95%, that would determine our Z. Now, this would not be incorrect, but we face a problem, because we are going to need to estimate what the standard deviation of the sampling distribution of the difference between our sample means actually is. To make that clear, let me write it this way. So the variance of the sampling distribution of the difference of our sample means is going to be equal to the variance of the sampling distribution of sample mean one, plus the variance of the sampling distribution of sample mean two. Now, if we knew the true underlying standard deviations of this population and this population, then we could actually come up with these."}, {"video_title": "Constructing t interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "Now, this would not be incorrect, but we face a problem, because we are going to need to estimate what the standard deviation of the sampling distribution of the difference between our sample means actually is. To make that clear, let me write it this way. So the variance of the sampling distribution of the difference of our sample means is going to be equal to the variance of the sampling distribution of sample mean one, plus the variance of the sampling distribution of sample mean two. Now, if we knew the true underlying standard deviations of this population and this population, then we could actually come up with these. In that case, this right over here would be equal to the variance of the population, of population one, divided by our sample size, N one, plus, plus the variance of the underlying population two divided by this sample size. But we don't know these variances, and so we try to estimate them. So we estimate them with our sample standard deviations."}, {"video_title": "Constructing t interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "Now, if we knew the true underlying standard deviations of this population and this population, then we could actually come up with these. In that case, this right over here would be equal to the variance of the population, of population one, divided by our sample size, N one, plus, plus the variance of the underlying population two divided by this sample size. But we don't know these variances, and so we try to estimate them. So we estimate them with our sample standard deviations. So we say this is going to be approximately equal to our first sample standard deviation squared over N one, plus our second sample standard deviation squared over N two. And so we can say that an estimate of the standard deviation of the sampling distribution of the difference between our sample means, an estimate, is going to be equal to the square root of this. It's going to be approximately equal to the square root of S one squared over N one, plus S two squared over N two."}, {"video_title": "Constructing t interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "So we estimate them with our sample standard deviations. So we say this is going to be approximately equal to our first sample standard deviation squared over N one, plus our second sample standard deviation squared over N two. And so we can say that an estimate of the standard deviation of the sampling distribution of the difference between our sample means, an estimate, is going to be equal to the square root of this. It's going to be approximately equal to the square root of S one squared over N one, plus S two squared over N two. But the problem is, is once we use this estimate that we can figure out, a critical Z value isn't going to be as good as a critical T value. So instead, you would say my confidence interval is going to be X bar sub one minus X bar sub two, plus or minus a critical T value instead of a Z value, because that works better when you are estimating standard deviation of the sampling distribution of the difference between the sample means. And so you have T star times our estimate of this, which is going to be equal to the square root of S sub one squared over N one, plus S sub two squared over N two."}, {"video_title": "Constructing t interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "It's going to be approximately equal to the square root of S one squared over N one, plus S two squared over N two. But the problem is, is once we use this estimate that we can figure out, a critical Z value isn't going to be as good as a critical T value. So instead, you would say my confidence interval is going to be X bar sub one minus X bar sub two, plus or minus a critical T value instead of a Z value, because that works better when you are estimating standard deviation of the sampling distribution of the difference between the sample means. And so you have T star times our estimate of this, which is going to be equal to the square root of S sub one squared over N one, plus S sub two squared over N two. And then you might say, well, what determines our T star? Well, once again, you would look it up on a table using your confidence level, and you might be saying, wait, hold on. When I look up a T value, I don't just care about a confidence level, I also care about degrees of freedom."}, {"video_title": "Constructing t interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And so you have T star times our estimate of this, which is going to be equal to the square root of S sub one squared over N one, plus S sub two squared over N two. And then you might say, well, what determines our T star? Well, once again, you would look it up on a table using your confidence level, and you might be saying, wait, hold on. When I look up a T value, I don't just care about a confidence level, I also care about degrees of freedom. What is going to be the degrees of freedom in this situation? Well, there's a simple answer and a complicated answer. Once we think about the difference of means, there's fairly sophisticated formulas that computers can use to get a more precise degrees of freedom."}, {"video_title": "Constructing t interval for difference of means AP Statistics Khan Academy.mp3", "Sentence": "When I look up a T value, I don't just care about a confidence level, I also care about degrees of freedom. What is going to be the degrees of freedom in this situation? Well, there's a simple answer and a complicated answer. Once we think about the difference of means, there's fairly sophisticated formulas that computers can use to get a more precise degrees of freedom. But what you will typically see in a statistics class is a conservative view of degrees of freedom, where you take the lower of N one and N two, and you subtract one from that. So the degrees of freedom here, so the degrees of freedom here is going to be the lower, lower of N one minus one, or N two minus one. Or you take the lower of N one or N two, and you subtract one from that."}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "And we have four of them right over here, and the first thing I want to think about is if we can approximate what value would be the middle value, or the median for the data set described by these density curves. So just to remind ourselves, if we have a set of numbers and we order them from least to greatest, the median would be the middle value, or the midway between the middle two values. In a case like this, we want to find the value for which half of the values are above that value and half of the values are below. So when you're looking at a density curve, you'd want to look at the area, and you'd want to say, okay, at what value do we have equal area above and below that value? And so for this one, just eyeballing it, this value right over here would be the median. And in general, if you have a symmetric distribution like this, the median will be right along that line of symmetry. Here we have a slightly more unusual distribution."}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "So when you're looking at a density curve, you'd want to look at the area, and you'd want to say, okay, at what value do we have equal area above and below that value? And so for this one, just eyeballing it, this value right over here would be the median. And in general, if you have a symmetric distribution like this, the median will be right along that line of symmetry. Here we have a slightly more unusual distribution. This would be called a bimodal distribution, but you have two major lumps right over here. But it is symmetric, and that point of symmetry is right over here, and so this value, once again, would be the median. Another way to think about it is the area to the left of that value is equal to the area to the right of that value, making it the median."}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "Here we have a slightly more unusual distribution. This would be called a bimodal distribution, but you have two major lumps right over here. But it is symmetric, and that point of symmetry is right over here, and so this value, once again, would be the median. Another way to think about it is the area to the left of that value is equal to the area to the right of that value, making it the median. What if we're dealing with non-symmetric distributions? Well, we'd want to do the same principle. We'd want to think at what value is the area on the right and the area on the left equal?"}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "Another way to think about it is the area to the left of that value is equal to the area to the right of that value, making it the median. What if we're dealing with non-symmetric distributions? Well, we'd want to do the same principle. We'd want to think at what value is the area on the right and the area on the left equal? And once again, this isn't going to be super exact, but I'm going to try to approximate it. You might be tempted to go right at the top of this lump right over here, but if I were to do that, it's pretty clear, even eyeballing it, that the right area right over here is larger than the left area. So that would not be the median."}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "We'd want to think at what value is the area on the right and the area on the left equal? And once again, this isn't going to be super exact, but I'm going to try to approximate it. You might be tempted to go right at the top of this lump right over here, but if I were to do that, it's pretty clear, even eyeballing it, that the right area right over here is larger than the left area. So that would not be the median. If I move the median a little bit over to the right, this may be right around here, this looks a lot closer. Once again, I'm approximating it, but it's reasonable to say that the area here looks pretty close to the area right over there, and if that is the case, then this is going to be the median. Similarly, on this one right over here, maybe right over here."}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "So that would not be the median. If I move the median a little bit over to the right, this may be right around here, this looks a lot closer. Once again, I'm approximating it, but it's reasonable to say that the area here looks pretty close to the area right over there, and if that is the case, then this is going to be the median. Similarly, on this one right over here, maybe right over here. And once again, I'm just approximating it, but that seems reasonable, that this area is equal to that one. Even though this is longer, it's much lower. This part of the curve is much higher, even though it goes on less to the right."}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "Similarly, on this one right over here, maybe right over here. And once again, I'm just approximating it, but that seems reasonable, that this area is equal to that one. Even though this is longer, it's much lower. This part of the curve is much higher, even though it goes on less to the right. So that's the median. For well-behaved continuous distributions like this, it's going to be the value for which the area to the left and the area to the right are equal. But what about the mean?"}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "This part of the curve is much higher, even though it goes on less to the right. So that's the median. For well-behaved continuous distributions like this, it's going to be the value for which the area to the left and the area to the right are equal. But what about the mean? Well, the mean is you take each of the possible values and you weight it by their frequencies, you weight it by their frequencies, and you add all of that up. And so for symmetric distributions, your mean and your median are actually going to be the same. So this is going to be your mean as well, this is going to be your mean as well."}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "But what about the mean? Well, the mean is you take each of the possible values and you weight it by their frequencies, you weight it by their frequencies, and you add all of that up. And so for symmetric distributions, your mean and your median are actually going to be the same. So this is going to be your mean as well, this is going to be your mean as well. If you wanna think about it in terms of physics, the mean would be your balancing point, the point at which you would wanna put a little fulcrum and you would wanna balance the distribution. And so you could put a little fulcrum here and you could imagine that this thing would balance. This thing would balance."}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "So this is going to be your mean as well, this is going to be your mean as well. If you wanna think about it in terms of physics, the mean would be your balancing point, the point at which you would wanna put a little fulcrum and you would wanna balance the distribution. And so you could put a little fulcrum here and you could imagine that this thing would balance. This thing would balance. And that all comes out of this idea of the weighted average of all of these possible values. What about for these less symmetric distributions? Well, let's think about it over here."}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "This thing would balance. And that all comes out of this idea of the weighted average of all of these possible values. What about for these less symmetric distributions? Well, let's think about it over here. Where would I have to put the fulcrum? Or what does our intuition say if we wanted to balance this? Well, we have equal areas on either side, but when you have this long tail to the right, it's going to pull the mean to the right of the median in this case."}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "Well, let's think about it over here. Where would I have to put the fulcrum? Or what does our intuition say if we wanted to balance this? Well, we have equal areas on either side, but when you have this long tail to the right, it's going to pull the mean to the right of the median in this case. And so our balance point is probably going to be something closer to that. And once again, this is me approximating it, but this would roughly be our mean. It would sit in this case to the right of our median."}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "Well, we have equal areas on either side, but when you have this long tail to the right, it's going to pull the mean to the right of the median in this case. And so our balance point is probably going to be something closer to that. And once again, this is me approximating it, but this would roughly be our mean. It would sit in this case to the right of our median. Let me make it clear. This median is referring to that, the mean is referring to this. In this case, because I have this long tail to the left, it's likely that I would have to balance it out right over here."}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "It would sit in this case to the right of our median. Let me make it clear. This median is referring to that, the mean is referring to this. In this case, because I have this long tail to the left, it's likely that I would have to balance it out right over here. So the mean would be this value right over there. And there's actually a term for these non-symmetric distributions where the mean is varying from the median. Distributions like this are referred to as being skewed."}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "In this case, because I have this long tail to the left, it's likely that I would have to balance it out right over here. So the mean would be this value right over there. And there's actually a term for these non-symmetric distributions where the mean is varying from the median. Distributions like this are referred to as being skewed. And this distribution, where you have the mean to the right of the median, where you have this long tail to the right, this is called right skewed. Now, the technical idea of skewness can get quite complicated, but generally speaking, you can spot it out when you have a long tail on one direction. That's the direction in which it will be skewed."}, {"video_title": "Median, mean and skew from density curves AP Statistics Khan Academy.mp3", "Sentence": "Distributions like this are referred to as being skewed. And this distribution, where you have the mean to the right of the median, where you have this long tail to the right, this is called right skewed. Now, the technical idea of skewness can get quite complicated, but generally speaking, you can spot it out when you have a long tail on one direction. That's the direction in which it will be skewed. Or if the mean is to that direction of the median. So the mean is to the right of the median. So generally speaking, that's going to be a right skewed distribution."}, {"video_title": "Influential points in regression AP Statistics Khan Academy.mp3", "Sentence": "I'm pretty sure I just tore my calf muscle this morning while sprinting with my son, but the math must not stop. So I'm here to help us think about what we could call influential points when we're thinking about regressions. And to help us here, I have this tool from BFW Publishing. I encourage you to go here and use this tool yourself. But what it allows us to do is to draw some points. So just like that, let me draw some points, and then fit a least squares line. So that's the least squares line right over there."}, {"video_title": "Influential points in regression AP Statistics Khan Academy.mp3", "Sentence": "I encourage you to go here and use this tool yourself. But what it allows us to do is to draw some points. So just like that, let me draw some points, and then fit a least squares line. So that's the least squares line right over there. And you can not only see the line, we can see our correlation coefficient. It's pretty good, 0.8156. It's pretty close to one."}, {"video_title": "Influential points in regression AP Statistics Khan Academy.mp3", "Sentence": "So that's the least squares line right over there. And you can not only see the line, we can see our correlation coefficient. It's pretty good, 0.8156. It's pretty close to one. So we have a pretty good fit right over here. But what we're gonna think about are points that might influence, or might be overly influential, we could say, to different aspects of this regression line. So one type of influential point is known as an outlier."}, {"video_title": "Influential points in regression AP Statistics Khan Academy.mp3", "Sentence": "It's pretty close to one. So we have a pretty good fit right over here. But what we're gonna think about are points that might influence, or might be overly influential, we could say, to different aspects of this regression line. So one type of influential point is known as an outlier. And a good way of identifying an outlier is it's a very bad fit to the line, or it has a very large residual. And so if I put a point right over here, that is an outlier. So what happens when we have an outlier like that?"}, {"video_title": "Influential points in regression AP Statistics Khan Academy.mp3", "Sentence": "So one type of influential point is known as an outlier. And a good way of identifying an outlier is it's a very bad fit to the line, or it has a very large residual. And so if I put a point right over here, that is an outlier. So what happens when we have an outlier like that? So before we had a correlation coefficient of 0.8 something, you put one outlier like that, it's now one out of 16 points, it dramatically lowered our correlation coefficient because we have a really large residual right over here. So an outlier like this has been very influential on the correlation coefficient. It didn't impact the slope of the line a tremendous amount."}, {"video_title": "Influential points in regression AP Statistics Khan Academy.mp3", "Sentence": "So what happens when we have an outlier like that? So before we had a correlation coefficient of 0.8 something, you put one outlier like that, it's now one out of 16 points, it dramatically lowered our correlation coefficient because we have a really large residual right over here. So an outlier like this has been very influential on the correlation coefficient. It didn't impact the slope of the line a tremendous amount. It did a little bit. Actually, when I put it there, it didn't impact the slope much at all. And it does impact the y-intercept a little bit."}, {"video_title": "Influential points in regression AP Statistics Khan Academy.mp3", "Sentence": "It didn't impact the slope of the line a tremendous amount. It did a little bit. Actually, when I put it there, it didn't impact the slope much at all. And it does impact the y-intercept a little bit. Actually, when I put it out here, it doesn't impact the y-intercept much at all. If I put it a little bit more to the left, it impacts it a little bit. But these outliers that are at least close to the mean x-value, these seem to be most relevant in terms of impacting or most influential in terms of the correlation coefficient."}, {"video_title": "Influential points in regression AP Statistics Khan Academy.mp3", "Sentence": "And it does impact the y-intercept a little bit. Actually, when I put it out here, it doesn't impact the y-intercept much at all. If I put it a little bit more to the left, it impacts it a little bit. But these outliers that are at least close to the mean x-value, these seem to be most relevant in terms of impacting or most influential in terms of the correlation coefficient. Now, what about an outlier that's further away from the mean x-value? And something, a point whose x-value is further away from the mean x-value is considered a high leverage point. And the way you could think about that is if you imagine this as being some type of a seesaw somehow pivoted on the mean x-value, well, if you put a point out here, it looks like it's pivoting down."}, {"video_title": "Influential points in regression AP Statistics Khan Academy.mp3", "Sentence": "But these outliers that are at least close to the mean x-value, these seem to be most relevant in terms of impacting or most influential in terms of the correlation coefficient. Now, what about an outlier that's further away from the mean x-value? And something, a point whose x-value is further away from the mean x-value is considered a high leverage point. And the way you could think about that is if you imagine this as being some type of a seesaw somehow pivoted on the mean x-value, well, if you put a point out here, it looks like it's pivoting down. It's like someone's sitting at this end of the seesaw. And so that's where I think the term leverage comes from. And you can see when I put an outlier, if I put a high leverage outlier out here, that does many things."}, {"video_title": "Influential points in regression AP Statistics Khan Academy.mp3", "Sentence": "And the way you could think about that is if you imagine this as being some type of a seesaw somehow pivoted on the mean x-value, well, if you put a point out here, it looks like it's pivoting down. It's like someone's sitting at this end of the seesaw. And so that's where I think the term leverage comes from. And you can see when I put an outlier, if I put a high leverage outlier out here, that does many things. It definitely drops the correlation coefficient. It changes the slope, and it changes the y-intercept. So it does a lot of things."}, {"video_title": "Influential points in regression AP Statistics Khan Academy.mp3", "Sentence": "And you can see when I put an outlier, if I put a high leverage outlier out here, that does many things. It definitely drops the correlation coefficient. It changes the slope, and it changes the y-intercept. So it does a lot of things. So it's highly influential for everything I just talked about. Now, if I have a high leverage point that's maybe a little bit less of an outlier, something like this, based on the points that I happen to have, it didn't hurt the correlation coefficient. In fact, in that example, it's actually improved it a little bit, but it did change the y-intercept a bit, and it did change the slope a bit, although obviously not as dramatic as when you do something like that."}, {"video_title": "Influential points in regression AP Statistics Khan Academy.mp3", "Sentence": "So it does a lot of things. So it's highly influential for everything I just talked about. Now, if I have a high leverage point that's maybe a little bit less of an outlier, something like this, based on the points that I happen to have, it didn't hurt the correlation coefficient. In fact, in that example, it's actually improved it a little bit, but it did change the y-intercept a bit, and it did change the slope a bit, although obviously not as dramatic as when you do something like that. And that kills the correlation coefficient as well. Let's see what happens if we do things over here. So if I have a high leverage outlier over here, you see the same thing."}, {"video_title": "Influential points in regression AP Statistics Khan Academy.mp3", "Sentence": "In fact, in that example, it's actually improved it a little bit, but it did change the y-intercept a bit, and it did change the slope a bit, although obviously not as dramatic as when you do something like that. And that kills the correlation coefficient as well. Let's see what happens if we do things over here. So if I have a high leverage outlier over here, you see the same thing. A high leverage outlier seems to influence everything. If it is a high leverage point that is less of an outlier, actually, once again, it improved the correlation coefficient. You could say that it's still influential on the correlation coefficient."}, {"video_title": "Influential points in regression AP Statistics Khan Academy.mp3", "Sentence": "So if I have a high leverage outlier over here, you see the same thing. A high leverage outlier seems to influence everything. If it is a high leverage point that is less of an outlier, actually, once again, it improved the correlation coefficient. You could say that it's still influential on the correlation coefficient. In this case, it's improving it. But it's less influential in terms of the slope and the y-intercept, although it is making a difference there. So I encourage you to play with this."}, {"video_title": "When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3", "Sentence": "What I wanna do in this video is give a primer on thinking about when to use a z statistic versus a t statistic when we are doing significance tests. So there's two major scenarios that we will see in an introductory statistics class. One is when we are dealing with proportions. So I'll write that on the left side right over here. And the other is when we are dealing with means. In the proportion case, when we're doing our significance test, we will set up some null hypothesis that usually deals with the population proportion. We might say it is equal to some value."}, {"video_title": "When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So I'll write that on the left side right over here. And the other is when we are dealing with means. In the proportion case, when we're doing our significance test, we will set up some null hypothesis that usually deals with the population proportion. We might say it is equal to some value. Let's just call that P sub one. And then maybe you have an alternative hypothesis that well, no, the population proportion is greater than that or less than that or it's just not equal to that. So let me just go with that one."}, {"video_title": "When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3", "Sentence": "We might say it is equal to some value. Let's just call that P sub one. And then maybe you have an alternative hypothesis that well, no, the population proportion is greater than that or less than that or it's just not equal to that. So let me just go with that one. It's not equal to P sub one. And then what we do to actually test, to actually do the significance test is we take a sample from the population. It's going to have a sample size of N. We need to make sure that we feel good about making the inference."}, {"video_title": "When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So let me just go with that one. It's not equal to P sub one. And then what we do to actually test, to actually do the significance test is we take a sample from the population. It's going to have a sample size of N. We need to make sure that we feel good about making the inference. We've talked about the conditions for inference in previous videos multiple times. But from this, we calculate the sample proportion. And then from this, we calculate the P value."}, {"video_title": "When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3", "Sentence": "It's going to have a sample size of N. We need to make sure that we feel good about making the inference. We've talked about the conditions for inference in previous videos multiple times. But from this, we calculate the sample proportion. And then from this, we calculate the P value. And the way that we do the P value, remember, the P value is the probability of getting a sample proportion at least this extreme. And if it's below some threshold, we reject the null hypothesis and it suggests the alternative. And over here, the way we do that is we find an associated Z value for that P, for that sample proportion."}, {"video_title": "When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3", "Sentence": "And then from this, we calculate the P value. And the way that we do the P value, remember, the P value is the probability of getting a sample proportion at least this extreme. And if it's below some threshold, we reject the null hypothesis and it suggests the alternative. And over here, the way we do that is we find an associated Z value for that P, for that sample proportion. And the way that we calculate it, we say, okay, look, our Z is going to be how many of the sampling distributions, standard deviations, are we away from the mean? And remember, the mean of the sampling distribution is going to be the population proportion. So here, we got this sample statistic, this sample proportion."}, {"video_title": "When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3", "Sentence": "And over here, the way we do that is we find an associated Z value for that P, for that sample proportion. And the way that we calculate it, we say, okay, look, our Z is going to be how many of the sampling distributions, standard deviations, are we away from the mean? And remember, the mean of the sampling distribution is going to be the population proportion. So here, we got this sample statistic, this sample proportion. The difference between that and the assumed proportion, remember, when we do these significance tests, we try to figure out the probability assuming the null hypothesis is true. And so when we see this P sub zero, this is the assumed proportion from the null hypothesis. So that's the difference between these two, the sample proportion and the assumed proportion."}, {"video_title": "When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So here, we got this sample statistic, this sample proportion. The difference between that and the assumed proportion, remember, when we do these significance tests, we try to figure out the probability assuming the null hypothesis is true. And so when we see this P sub zero, this is the assumed proportion from the null hypothesis. So that's the difference between these two, the sample proportion and the assumed proportion. And then you'd wanna divide it by what's often known as the standard error of the statistic, which is just the standard deviation of the sampling distribution of the sample proportion. And this works out well for proportions because in proportions, I can figure out what this is. This is going to be equal to the square root of the assumed population proportion times one minus the assumed population proportion, all of that over N. And then I would use this Z statistic to figure out the P value."}, {"video_title": "When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So that's the difference between these two, the sample proportion and the assumed proportion. And then you'd wanna divide it by what's often known as the standard error of the statistic, which is just the standard deviation of the sampling distribution of the sample proportion. And this works out well for proportions because in proportions, I can figure out what this is. This is going to be equal to the square root of the assumed population proportion times one minus the assumed population proportion, all of that over N. And then I would use this Z statistic to figure out the P value. And in this case, I would look at both tails of the distribution because I care about how far I am either above or below the assumed population proportion. Now with means, there's definitely some similarities here. You will make a null hypothesis."}, {"video_title": "When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3", "Sentence": "This is going to be equal to the square root of the assumed population proportion times one minus the assumed population proportion, all of that over N. And then I would use this Z statistic to figure out the P value. And in this case, I would look at both tails of the distribution because I care about how far I am either above or below the assumed population proportion. Now with means, there's definitely some similarities here. You will make a null hypothesis. Maybe you assume the population mean is equal to mu one. And then there's going to be an alternative hypothesis that maybe your population mean is not equal to mu one. And you're gonna do something very simple."}, {"video_title": "When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3", "Sentence": "You will make a null hypothesis. Maybe you assume the population mean is equal to mu one. And then there's going to be an alternative hypothesis that maybe your population mean is not equal to mu one. And you're gonna do something very simple. You take your population, take a sample of size N. Instead of calculating a sample proportion, you calculate a sample mean. And actually, you can calculate other things like a sample standard deviation. But now you have an issue."}, {"video_title": "When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3", "Sentence": "And you're gonna do something very simple. You take your population, take a sample of size N. Instead of calculating a sample proportion, you calculate a sample mean. And actually, you can calculate other things like a sample standard deviation. But now you have an issue. You say, well, ideally, I would use a Z statistic. And you could if you were able to say, well, I could take the difference between my sample mean and the assumed mean in the null hypothesis. So that would be this right over here."}, {"video_title": "When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3", "Sentence": "But now you have an issue. You say, well, ideally, I would use a Z statistic. And you could if you were able to say, well, I could take the difference between my sample mean and the assumed mean in the null hypothesis. So that would be this right over here. That's what that zero means, the assumed mean from the null hypothesis. And I would then divide by the standard error of the mean, which is another way of saying the standard deviation of the sampling distribution of the sample mean. But this is not so easy to figure out."}, {"video_title": "When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3", "Sentence": "So that would be this right over here. That's what that zero means, the assumed mean from the null hypothesis. And I would then divide by the standard error of the mean, which is another way of saying the standard deviation of the sampling distribution of the sample mean. But this is not so easy to figure out. In order to figure out this, this is going to be the standard deviation of the underlying population divided by the square root of N. We know what N is going to be if we conducted the sample, but we don't know what the standard deviation is. So instead, what we do is we estimate this. And so we'll take the sample mean."}, {"video_title": "When to use z or t statistics in significance tests AP Statistics Khan Academy.mp3", "Sentence": "But this is not so easy to figure out. In order to figure out this, this is going to be the standard deviation of the underlying population divided by the square root of N. We know what N is going to be if we conducted the sample, but we don't know what the standard deviation is. So instead, what we do is we estimate this. And so we'll take the sample mean. We subtract from that the assumed population mean from the null hypothesis. And we divide by an estimate of this, which is going to be our sample standard deviation divided by the square root of N. But because this is an estimate, we actually get a better result. Instead of saying, hey, this is an estimate of our Z statistic, we will call this our T statistic."}, {"video_title": "More solving problems with bar graphs Fractions 3rd grade Khan Academy.mp3", "Sentence": "Alright, so it says that, so it says 20 people said their favorite type of movie was comedy. Six people said their favorite type of movie was scary. 10 people said their favorite type of movie was adventure. 10 people also said their favorite type of movie is cartoons. And then 16 people said their favorite type of movie is a mystery. Alright, so let's read the questions here. So which of the following types of movies were picked by fewer than 14 people?"}, {"video_title": "More solving problems with bar graphs Fractions 3rd grade Khan Academy.mp3", "Sentence": "10 people also said their favorite type of movie is cartoons. And then 16 people said their favorite type of movie is a mystery. Alright, so let's read the questions here. So which of the following types of movies were picked by fewer than 14 people? So fewer than 14 people. So more than 14 picked comedy or mystery. So this line right over here would be 14."}, {"video_title": "More solving problems with bar graphs Fractions 3rd grade Khan Academy.mp3", "Sentence": "So which of the following types of movies were picked by fewer than 14 people? So fewer than 14 people. So more than 14 picked comedy or mystery. So this line right over here would be 14. So fewer than 14 were scary, adventure, or cartoon. This was six people, 10 people, and 10 people. Scary, adventure, and cartoon."}, {"video_title": "More solving problems with bar graphs Fractions 3rd grade Khan Academy.mp3", "Sentence": "So this line right over here would be 14. So fewer than 14 were scary, adventure, or cartoon. This was six people, 10 people, and 10 people. Scary, adventure, and cartoon. So we want to pick scary, adventure, and cartoon. These three had less than 14 people pick them. So let's do a couple more of these."}, {"video_title": "More solving problems with bar graphs Fractions 3rd grade Khan Academy.mp3", "Sentence": "Scary, adventure, and cartoon. So we want to pick scary, adventure, and cartoon. These three had less than 14 people pick them. So let's do a couple more of these. Let's do a couple more. Puppy Party Place graphed the number of dogs that came in each weekday. So let's see, 80 dogs came in on Monday, 160 dogs came in on Tuesday, 80 dogs came in on Wednesday, 140 dogs came in on Thursday, and then Friday was a very popular day."}, {"video_title": "More solving problems with bar graphs Fractions 3rd grade Khan Academy.mp3", "Sentence": "So let's do a couple more of these. Let's do a couple more. Puppy Party Place graphed the number of dogs that came in each weekday. So let's see, 80 dogs came in on Monday, 160 dogs came in on Tuesday, 80 dogs came in on Wednesday, 140 dogs came in on Thursday, and then Friday was a very popular day. 180 dogs came into the Puppy Party Place on Friday. Alright, so what are they asking us? On which day did the same number of dogs come to Puppy Party Place as on Monday and Wednesday combined?"}, {"video_title": "More solving problems with bar graphs Fractions 3rd grade Khan Academy.mp3", "Sentence": "So let's see, 80 dogs came in on Monday, 160 dogs came in on Tuesday, 80 dogs came in on Wednesday, 140 dogs came in on Thursday, and then Friday was a very popular day. 180 dogs came into the Puppy Party Place on Friday. Alright, so what are they asking us? On which day did the same number of dogs come to Puppy Party Place as on Monday and Wednesday combined? Alright, so let's see, Monday and Wednesday combined. So on Monday we had 80 puppies come there, or 80 dogs, and on Wednesday we had 80 dogs. And they're saying which day had the same as Monday and Wednesday combined?"}, {"video_title": "More solving problems with bar graphs Fractions 3rd grade Khan Academy.mp3", "Sentence": "On which day did the same number of dogs come to Puppy Party Place as on Monday and Wednesday combined? Alright, so let's see, Monday and Wednesday combined. So on Monday we had 80 puppies come there, or 80 dogs, and on Wednesday we had 80 dogs. And they're saying which day had the same as Monday and Wednesday combined? So Monday and Wednesday combined would be 80 plus 80, which would be 160. So on which of these days did 160 dogs come? Well, we see it right over here, 160 dogs."}, {"video_title": "More solving problems with bar graphs Fractions 3rd grade Khan Academy.mp3", "Sentence": "And they're saying which day had the same as Monday and Wednesday combined? So Monday and Wednesday combined would be 80 plus 80, which would be 160. So on which of these days did 160 dogs come? Well, we see it right over here, 160 dogs. That was on Tuesday, and only on Tuesday. So Tuesday. Let's do a few more of these."}, {"video_title": "More solving problems with bar graphs Fractions 3rd grade Khan Academy.mp3", "Sentence": "Well, we see it right over here, 160 dogs. That was on Tuesday, and only on Tuesday. So Tuesday. Let's do a few more of these. Desert Zone asked its customers about their favorite ice cream flavor and graphed the results. So 175 customers said vanilla, 225 said chocolate, 75 said strawberry, and 200 said cookie dough. Alright, how many more customers picked the most popular flavor than the least popular flavor?"}, {"video_title": "More solving problems with bar graphs Fractions 3rd grade Khan Academy.mp3", "Sentence": "Let's do a few more of these. Desert Zone asked its customers about their favorite ice cream flavor and graphed the results. So 175 customers said vanilla, 225 said chocolate, 75 said strawberry, and 200 said cookie dough. Alright, how many more customers picked the most popular flavor than the least popular flavor? Alright, so the most popular flavor was chocolate. That was 225. And the least popular flavor, it's late, to the least popular flavor, it's a tongue twister, was 75."}, {"video_title": "More solving problems with bar graphs Fractions 3rd grade Khan Academy.mp3", "Sentence": "Alright, how many more customers picked the most popular flavor than the least popular flavor? Alright, so the most popular flavor was chocolate. That was 225. And the least popular flavor, it's late, to the least popular flavor, it's a tongue twister, was 75. So how many more picked the most popular than the least popular? Well, the most popular is 225, the least popular was 75. So it's 225 minus 75."}, {"video_title": "More solving problems with bar graphs Fractions 3rd grade Khan Academy.mp3", "Sentence": "And the least popular flavor, it's late, to the least popular flavor, it's a tongue twister, was 75. So how many more picked the most popular than the least popular? Well, the most popular is 225, the least popular was 75. So it's 225 minus 75. Let's see, 225 minus 25 would be 200, minus 50 would be 175, minus 75 would be 150. So we get 150 customers. And once again, we're saying that 150 more customers picked chocolate, the most popular flavor, than the least popular."}, {"video_title": "More solving problems with bar graphs Fractions 3rd grade Khan Academy.mp3", "Sentence": "So it's 225 minus 75. Let's see, 225 minus 25 would be 200, minus 50 would be 175, minus 75 would be 150. So we get 150 customers. And once again, we're saying that 150 more customers picked chocolate, the most popular flavor, than the least popular. Another way you could do it is you could look at the least popular and the most popular, and say, well, how many more do I have to add to the least popular to get to the most popular? And you see 25, 50, 75, 100, 125, 150. Each of these lines is 25 more."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "The scatter plot and trend line below show the relationship between how many hours students spent studying and their score on the test. The line fitted to model the data has a slope of 15. So the line that they're talking about is right here. So this is the scatter plot. This shows that some student who spent some time between half an hour and an hour studying got a little bit less than a 45 on the test. The student here who got a little bit higher than a 60 spent a little under two hours studying. This student over here who looks like they got like a 94 or a 95 spent over four hours studying."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "So this is the scatter plot. This shows that some student who spent some time between half an hour and an hour studying got a little bit less than a 45 on the test. The student here who got a little bit higher than a 60 spent a little under two hours studying. This student over here who looks like they got like a 94 or a 95 spent over four hours studying. And so then they fit a line to it and this line has a slope of 15. And before I even read these choices, what's the best interpretation of this slope? Well, if you think this line is indicative of the trend and it does look like that from the scatter plot, that implies that roughly every extra hour that you study is going to improve your score by 15."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "This student over here who looks like they got like a 94 or a 95 spent over four hours studying. And so then they fit a line to it and this line has a slope of 15. And before I even read these choices, what's the best interpretation of this slope? Well, if you think this line is indicative of the trend and it does look like that from the scatter plot, that implies that roughly every extra hour that you study is going to improve your score by 15. You could say on average according to this regression. So if we start over here and we were to increase by one hour our score should improve by 15. And it does indeed look like that."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "Well, if you think this line is indicative of the trend and it does look like that from the scatter plot, that implies that roughly every extra hour that you study is going to improve your score by 15. You could say on average according to this regression. So if we start over here and we were to increase by one hour our score should improve by 15. And it does indeed look like that. We're going from, we're going in the horizontal direction, we're going one hour, and then in the vertical direction we're going from 45 to 60. So that's how I would interpret it. Every hour, based on this regression, you could, it's not unreasonable to expect 15 points improvement, or at least that's what we're seeing, that's what we're seeing from the regression of the data."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "And it does indeed look like that. We're going from, we're going in the horizontal direction, we're going one hour, and then in the vertical direction we're going from 45 to 60. So that's how I would interpret it. Every hour, based on this regression, you could, it's not unreasonable to expect 15 points improvement, or at least that's what we're seeing, that's what we're seeing from the regression of the data. So let's look at which of these choices actually describe something like that. The model predicts that the student who scored zero studied for an average of 15 hours. No, it definitely doesn't say that."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "Every hour, based on this regression, you could, it's not unreasonable to expect 15 points improvement, or at least that's what we're seeing, that's what we're seeing from the regression of the data. So let's look at which of these choices actually describe something like that. The model predicts that the student who scored zero studied for an average of 15 hours. No, it definitely doesn't say that. The model predicts that students who didn't study at all will have an average score of 15 points. No, we didn't see that. Students, if you take this, if you believe this model, someone who doesn't study at all would get close to, would get between 35 and 40 points, so like a 37 or a 38."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "No, it definitely doesn't say that. The model predicts that students who didn't study at all will have an average score of 15 points. No, we didn't see that. Students, if you take this, if you believe this model, someone who doesn't study at all would get close to, would get between 35 and 40 points, so like a 37 or a 38. So don't like that choice. The model predicts that the score will increase 15 points for each additional hour of study time. Yes, that is exactly what we were thinking about when we were looking at the model."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "Students, if you take this, if you believe this model, someone who doesn't study at all would get close to, would get between 35 and 40 points, so like a 37 or a 38. So don't like that choice. The model predicts that the score will increase 15 points for each additional hour of study time. Yes, that is exactly what we were thinking about when we were looking at the model. That's what a slope of 15 tells you. You increase studying time by an hour, it increases score by 15 points. The model predicts that the study time will increase 15 hours for each additional point scored."}, {"video_title": "General multiplication rule example dependent events Probability & combinatorics.mp3", "Sentence": "For the final round, each of them will randomly select a card without replacement that will reveal what the star material must be in their craft. Here are the available cards. So I guess the star material is the primary material they need to use in this competition. Maya and Doug both want to get silk as their star material. Maya will draw first, followed by Doug. What is the probability that neither contestant draws silk? Pause this video and see if you can work through that before we work through this together."}, {"video_title": "General multiplication rule example dependent events Probability & combinatorics.mp3", "Sentence": "Maya and Doug both want to get silk as their star material. Maya will draw first, followed by Doug. What is the probability that neither contestant draws silk? Pause this video and see if you can work through that before we work through this together. All right, now let's work through this together. So the probability that neither contestant draws silk. So that would be, I'll just write it another way, the probability that, I'll write MNS for Maya no silk."}, {"video_title": "General multiplication rule example dependent events Probability & combinatorics.mp3", "Sentence": "Pause this video and see if you can work through that before we work through this together. All right, now let's work through this together. So the probability that neither contestant draws silk. So that would be, I'll just write it another way, the probability that, I'll write MNS for Maya no silk. So Maya no silk and Doug no silk. That's just another way of saying what is the probability that neither contestant draws silk? And so this is going to be equivalent to the probability that Maya does not get silk, Maya no silk, right over here, times the probability that Doug doesn't get silk, given that Maya did not get silk, given Maya no silk."}, {"video_title": "General multiplication rule example dependent events Probability & combinatorics.mp3", "Sentence": "So that would be, I'll just write it another way, the probability that, I'll write MNS for Maya no silk. So Maya no silk and Doug no silk. That's just another way of saying what is the probability that neither contestant draws silk? And so this is going to be equivalent to the probability that Maya does not get silk, Maya no silk, right over here, times the probability that Doug doesn't get silk, given that Maya did not get silk, given Maya no silk. This line right over here, this vertical line, this is shorthand for given. And so let's calculate each of these. So this is going to be equal to the probability that Maya gets no silk, she picked first."}, {"video_title": "General multiplication rule example dependent events Probability & combinatorics.mp3", "Sentence": "And so this is going to be equivalent to the probability that Maya does not get silk, Maya no silk, right over here, times the probability that Doug doesn't get silk, given that Maya did not get silk, given Maya no silk. This line right over here, this vertical line, this is shorthand for given. And so let's calculate each of these. So this is going to be equal to the probability that Maya gets no silk, she picked first. There's six options out of here. Five of them are not silk. So it is five over six."}, {"video_title": "General multiplication rule example dependent events Probability & combinatorics.mp3", "Sentence": "So this is going to be equal to the probability that Maya gets no silk, she picked first. There's six options out of here. Five of them are not silk. So it is five over six. And then the probability that Doug does not get silk, given that Maya did not get silk. So if Maya did not get silk, then that means that silk is still in the mix, but there's only five possibilities left because Maya picked one of them. And four of them are not silk."}, {"video_title": "General multiplication rule example dependent events Probability & combinatorics.mp3", "Sentence": "So it is five over six. And then the probability that Doug does not get silk, given that Maya did not get silk. So if Maya did not get silk, then that means that silk is still in the mix, but there's only five possibilities left because Maya picked one of them. And four of them are not silk. They're still silk as an option. And it's important to recognize that the probability that Doug gets no silk is dependent on whether Maya got silk or not. So it's very important to have this given right over here."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "In the last few videos, we saw that if we had n points, each of them have x and y coordinates. So let me draw n of those points. So let's call this point 1. It has a coordinate x1, y1. You have the second point over here that has a coordinate x2, y2. And then we keep putting points up here and eventually we get to the nth point over here. So we have the nth point that has the coordinates xn, yn."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "It has a coordinate x1, y1. You have the second point over here that has a coordinate x2, y2. And then we keep putting points up here and eventually we get to the nth point over here. So we have the nth point that has the coordinates xn, yn. What we saw is that there is a line that we can find. We can find a line that minimizes the squared distance. So this line right here, I'll call it y is equal to mx plus b."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "So we have the nth point that has the coordinates xn, yn. What we saw is that there is a line that we can find. We can find a line that minimizes the squared distance. So this line right here, I'll call it y is equal to mx plus b. That there is some line that minimizes the squared distance to the point. So let me just review what those squared distances are. Sometimes it's called the squared error."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "So this line right here, I'll call it y is equal to mx plus b. That there is some line that minimizes the squared distance to the point. So let me just review what those squared distances are. Sometimes it's called the squared error. So this is the error between the line and point 1. So I'll call that error 1. This is the error between the line and point 2."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Sometimes it's called the squared error. So this is the error between the line and point 1. So I'll call that error 1. This is the error between the line and point 2. We'll call this error 2. This is the error between the line and point 3. Sorry, and point n. So if you wanted the total error, if you want the total squared error, and this is actually how we started off this whole discussion, the total squared error between the points and the line, you literally just take the y value at each point."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "This is the error between the line and point 2. We'll call this error 2. This is the error between the line and point 3. Sorry, and point n. So if you wanted the total error, if you want the total squared error, and this is actually how we started off this whole discussion, the total squared error between the points and the line, you literally just take the y value at each point. So for example, you would take y1, that's this value right over here. You would take y1 minus the y value at this point in the line. Well, that point in the line is essentially the y value you get when you substitute x1 into this equation."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Sorry, and point n. So if you wanted the total error, if you want the total squared error, and this is actually how we started off this whole discussion, the total squared error between the points and the line, you literally just take the y value at each point. So for example, you would take y1, that's this value right over here. You would take y1 minus the y value at this point in the line. Well, that point in the line is essentially the y value you get when you substitute x1 into this equation. So I'll just substitute x1 into this equation. So minus mx1 plus b. This right here, that is this y value right over here."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Well, that point in the line is essentially the y value you get when you substitute x1 into this equation. So I'll just substitute x1 into this equation. So minus mx1 plus b. This right here, that is this y value right over here. That is mx1 plus b. I don't want to get my graph too cluttered, so I'll just delete that there. That is error 1 right over there. That is error 1, and we want the squared errors between each of the points in the line."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "This right here, that is this y value right over here. That is mx1 plus b. I don't want to get my graph too cluttered, so I'll just delete that there. That is error 1 right over there. That is error 1, and we want the squared errors between each of the points in the line. So that's the first one. Then you do the same thing for the second point. So we started our discussion this way, y2 minus mx2 plus b squared all the way, I'll do dot, dot, dot to show that there are a bunch of these that we have to do until we get to the nth point, all the way to yn minus mxn plus b squared."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "That is error 1, and we want the squared errors between each of the points in the line. So that's the first one. Then you do the same thing for the second point. So we started our discussion this way, y2 minus mx2 plus b squared all the way, I'll do dot, dot, dot to show that there are a bunch of these that we have to do until we get to the nth point, all the way to yn minus mxn plus b squared. Now that we actually know how to find these m's and b's, I showed you the formula, in fact we've proved the formula of how to find these m's and b's. We can find this line, and if we wanted to say, well, you know, how much error is there, we can then calculate it because we now know the m's and the b's. So we can calculate it for a certain set of data."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "So we started our discussion this way, y2 minus mx2 plus b squared all the way, I'll do dot, dot, dot to show that there are a bunch of these that we have to do until we get to the nth point, all the way to yn minus mxn plus b squared. Now that we actually know how to find these m's and b's, I showed you the formula, in fact we've proved the formula of how to find these m's and b's. We can find this line, and if we wanted to say, well, you know, how much error is there, we can then calculate it because we now know the m's and the b's. So we can calculate it for a certain set of data. Now what I want to do is kind of come up with a more meaningful estimate of how good this line is fitting the data points that we have. To do that, we're going to ask ourselves the question, how much, or we could even say what percentage, what percentage of the variation in y is described by the variation in x? Let's think about this."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "So we can calculate it for a certain set of data. Now what I want to do is kind of come up with a more meaningful estimate of how good this line is fitting the data points that we have. To do that, we're going to ask ourselves the question, how much, or we could even say what percentage, what percentage of the variation in y is described by the variation in x? Let's think about this. How much of the total variation in y, there's obviously variation in y. This y value is over here, this point's y value is over here. There's clearly a bunch of variation in the y, but how much of that is essentially described by the variation in x or described by the line?"}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Let's think about this. How much of the total variation in y, there's obviously variation in y. This y value is over here, this point's y value is over here. There's clearly a bunch of variation in the y, but how much of that is essentially described by the variation in x or described by the line? Let's think about that. First let's think about what the total variation is. How much of the, we could even say total variation, how much of the total variation in y?"}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "There's clearly a bunch of variation in the y, but how much of that is essentially described by the variation in x or described by the line? Let's think about that. First let's think about what the total variation is. How much of the, we could even say total variation, how much of the total variation in y? Let's just figure out what the total variation in y is. The total variation, it's really just a tool for measuring, total variation in y, well we care, when we think about variation, and this is even true when we talk about variance, which was the mean variation in y, is we think about the square distance from some central tendency and the best central measure we can have of y is the arithmetic mean. We could just say the total variation in y is just going to be the sum of the distances of each of the y's, so you get y1, let me do this in another color, you get y1, this y1 over here, this is y1 over here, you get y1 minus the mean of all the y's, minus the mean of all the y's squared, plus y2, plus y2, minus the mean of all of the y squared, plus, and you just keep going all the way to the nth y value, to yn minus the mean of all the y's squared."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "How much of the, we could even say total variation, how much of the total variation in y? Let's just figure out what the total variation in y is. The total variation, it's really just a tool for measuring, total variation in y, well we care, when we think about variation, and this is even true when we talk about variance, which was the mean variation in y, is we think about the square distance from some central tendency and the best central measure we can have of y is the arithmetic mean. We could just say the total variation in y is just going to be the sum of the distances of each of the y's, so you get y1, let me do this in another color, you get y1, this y1 over here, this is y1 over here, you get y1 minus the mean of all the y's, minus the mean of all the y's squared, plus y2, plus y2, minus the mean of all of the y squared, plus, and you just keep going all the way to the nth y value, to yn minus the mean of all the y's squared. This gives you the total variation in y. You can just take out all the y values, find their mean, it'll be some value, maybe it's right over here someplace, maybe that is the mean value of all the y's, and so you can even visualize it the same way we visualized the squared error from the line. So if you visualize it, you can imagine a line that's y is equal to the mean of y, which would look just like that, and what we're measuring over here, this error right over here is the square of this distance right over here, between this point vertically and this line."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "We could just say the total variation in y is just going to be the sum of the distances of each of the y's, so you get y1, let me do this in another color, you get y1, this y1 over here, this is y1 over here, you get y1 minus the mean of all the y's, minus the mean of all the y's squared, plus y2, plus y2, minus the mean of all of the y squared, plus, and you just keep going all the way to the nth y value, to yn minus the mean of all the y's squared. This gives you the total variation in y. You can just take out all the y values, find their mean, it'll be some value, maybe it's right over here someplace, maybe that is the mean value of all the y's, and so you can even visualize it the same way we visualized the squared error from the line. So if you visualize it, you can imagine a line that's y is equal to the mean of y, which would look just like that, and what we're measuring over here, this error right over here is the square of this distance right over here, between this point vertically and this line. The second one is going to be this distance, is going to be this distance, just right up to the line. The nth one is going to be the distance from there all the way to the line right over there, and then there are these other points in between. This is the total variation y."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "So if you visualize it, you can imagine a line that's y is equal to the mean of y, which would look just like that, and what we're measuring over here, this error right over here is the square of this distance right over here, between this point vertically and this line. The second one is going to be this distance, is going to be this distance, just right up to the line. The nth one is going to be the distance from there all the way to the line right over there, and then there are these other points in between. This is the total variation y. Makes sense, if you divide this by n, you actually will get the, I should say this is the total variation in y, if you divide this by n, you're going to get what we typically associate as the variance of y, which is kind of the average square distance. Now we have the total square distance. So what we want to do is how much of this, how much of the total variation y is described by the variation in x?"}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "This is the total variation y. Makes sense, if you divide this by n, you actually will get the, I should say this is the total variation in y, if you divide this by n, you're going to get what we typically associate as the variance of y, which is kind of the average square distance. Now we have the total square distance. So what we want to do is how much of this, how much of the total variation y is described by the variation in x? So maybe we can think of it this way, so our denominator, we want what percentage of the total variation in y? So let me write it this way. Let me call this as the squared error from the average."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "So what we want to do is how much of this, how much of the total variation y is described by the variation in x? So maybe we can think of it this way, so our denominator, we want what percentage of the total variation in y? So let me write it this way. Let me call this as the squared error from the average. Let me call this, this is equal to the squared error, maybe I'll call this the squared error from the mean of y. And this is really the total variation in y. So let's put that as the denominator."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Let me call this as the squared error from the average. Let me call this, this is equal to the squared error, maybe I'll call this the squared error from the mean of y. And this is really the total variation in y. So let's put that as the denominator. Let's put that as the denominator, the total variation y, which is the squared error from the mean of the y's. Now we want to know what percentage of this is described by the variation in x. Now what is not described by the variation in x?"}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "So let's put that as the denominator. Let's put that as the denominator, the total variation y, which is the squared error from the mean of the y's. Now we want to know what percentage of this is described by the variation in x. Now what is not described by the variation in x? We want how much is described by the variation in x. But what if we want how much of the total error, how much of the total variation is not described by the line over here, is not described by the regression line. How much of the total data is not?"}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Now what is not described by the variation in x? We want how much is described by the variation in x. But what if we want how much of the total error, how much of the total variation is not described by the line over here, is not described by the regression line. How much of the total data is not? Well, we already have a measure for that. We have the squared error of the line. This tells us the square of the distances from each point to our line."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "How much of the total data is not? Well, we already have a measure for that. We have the squared error of the line. This tells us the square of the distances from each point to our line. So it is exactly this measure. It tells us how much of the total variation is not described by the regression line. So if you want to know what percentage of the total variation is not described by the regression line, you would just say, this is the total, it would just be the squared error, the squared error of the line, because this is the total variation not described by the regression line, divided by the total variation."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "This tells us the square of the distances from each point to our line. So it is exactly this measure. It tells us how much of the total variation is not described by the regression line. So if you want to know what percentage of the total variation is not described by the regression line, you would just say, this is the total, it would just be the squared error, the squared error of the line, because this is the total variation not described by the regression line, divided by the total variation. So let me make it clear. This right over here tells us what percentage of variation, of the total variation, is not described by the variation in x, by the variation in x, or by the line, or by the regression line. Regression, by the regression line."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "So if you want to know what percentage of the total variation is not described by the regression line, you would just say, this is the total, it would just be the squared error, the squared error of the line, because this is the total variation not described by the regression line, divided by the total variation. So let me make it clear. This right over here tells us what percentage of variation, of the total variation, is not described by the variation in x, by the variation in x, or by the line, or by the regression line. Regression, by the regression line. So to answer our question, what percentage is described by the variation, well, the rest of it has to be described by the variation in x. Because our question is, what percentage of the total variation is described by the variation in x? This is the percentage that is not described."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Regression, by the regression line. So to answer our question, what percentage is described by the variation, well, the rest of it has to be described by the variation in x. Because our question is, what percentage of the total variation is described by the variation in x? This is the percentage that is not described. So if this number right here, if this number is, I don't know, 30%, if 30% of the variation in y is not described by the line, then the remainder will be described by the line. So we can essentially just subtract this from 1. So if we take 1 minus the squared error between our data points and the line, over the squared error between the data points, between the y's and the mean y, we have, we now have a percentage, this actually tells us what percentage of total variation, total variation, is described by the line."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "This is the percentage that is not described. So if this number right here, if this number is, I don't know, 30%, if 30% of the variation in y is not described by the line, then the remainder will be described by the line. So we can essentially just subtract this from 1. So if we take 1 minus the squared error between our data points and the line, over the squared error between the data points, between the y's and the mean y, we have, we now have a percentage, this actually tells us what percentage of total variation, total variation, is described by the line. Is described, is described, you can either view it as described by the line, or by the variation in x. Is described by the variation, by the variation in x. And this number right here, this is called the coefficient of determination."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "So if we take 1 minus the squared error between our data points and the line, over the squared error between the data points, between the y's and the mean y, we have, we now have a percentage, this actually tells us what percentage of total variation, total variation, is described by the line. Is described, is described, you can either view it as described by the line, or by the variation in x. Is described by the variation, by the variation in x. And this number right here, this is called the coefficient of determination. This is called the coefficient of determination. It's just what statisticians have decided to name it. Coefficient, coefficient of determination."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "And this number right here, this is called the coefficient of determination. This is called the coefficient of determination. It's just what statisticians have decided to name it. Coefficient, coefficient of determination. Of determination. Determination. And it's also called r squared."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Coefficient, coefficient of determination. Of determination. Determination. And it's also called r squared. You might have even heard that term when people talk about regression. Now, let's think about it. If the standard, if the squared error of the line, if the squared error is really small, if the squared error is really small, what does that mean?"}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "And it's also called r squared. You might have even heard that term when people talk about regression. Now, let's think about it. If the standard, if the squared error of the line, if the squared error is really small, if the squared error is really small, what does that mean? It means that these errors, it means that these errors right over here are really small, are really small, which means that the line is a really good fit. Which means that the line, this line over here, it tells us that the line is a really good fit. So if the, let me write it over here."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "If the standard, if the squared error of the line, if the squared error is really small, if the squared error is really small, what does that mean? It means that these errors, it means that these errors right over here are really small, are really small, which means that the line is a really good fit. Which means that the line, this line over here, it tells us that the line is a really good fit. So if the, let me write it over here. If the squared error of the line is small, is small, it tells us that the line is a good fit. Line is a good, it tells us it's a good fit. Now, what would happen over here?"}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "So if the, let me write it over here. If the squared error of the line is small, is small, it tells us that the line is a good fit. Line is a good, it tells us it's a good fit. Now, what would happen over here? Well, if this number is really small, this is going to be a very small fraction over here. One minus a very small fraction is going to be a pretty large, it's going to be a number close to one. So then, so then we're going to have our r squared will be close, close to one, which tells us that a lot of the variation in y is described by the variation in x, which makes sense because the line is a good fit."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Now, what would happen over here? Well, if this number is really small, this is going to be a very small fraction over here. One minus a very small fraction is going to be a pretty large, it's going to be a number close to one. So then, so then we're going to have our r squared will be close, close to one, which tells us that a lot of the variation in y is described by the variation in x, which makes sense because the line is a good fit. You take the opposite case. If the squared error of the line is huge, if this number over here is huge, if this number over here is huge, then that means there's a lot of error between the data points and the line. And so if this number is huge, then this number over here is going to be huge."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "So then, so then we're going to have our r squared will be close, close to one, which tells us that a lot of the variation in y is described by the variation in x, which makes sense because the line is a good fit. You take the opposite case. If the squared error of the line is huge, if this number over here is huge, if this number over here is huge, then that means there's a lot of error between the data points and the line. And so if this number is huge, then this number over here is going to be huge. One minus, or it's going to be a percentage close to one, and one minus that is going to be close to zero. And so if this, if the squared error of the line is large, is large, is large, if this is large, this whole thing is going to be close to one. And if this whole thing is close to one, the whole coefficient of determination, the whole r squared is going to be close to zero, which makes sense."}, {"video_title": "R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3", "Sentence": "And so if this number is huge, then this number over here is going to be huge. One minus, or it's going to be a percentage close to one, and one minus that is going to be close to zero. And so if this, if the squared error of the line is large, is large, is large, if this is large, this whole thing is going to be close to one. And if this whole thing is close to one, the whole coefficient of determination, the whole r squared is going to be close to zero, which makes sense. R squared will be close to zero, which makes sense. That tells us that very little of the total variation in y is described by the variation in x, or described by the line. Well, anyway, everything I've been dealing with so far has been a little bit in the abstract."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "But he only has enough money to buy at most four packs. Suppose that each pack has probability 0.2 of containing the card Hugo is hoping for. Let the random variable X be the number of packs of cards Hugo buys. Here is the probability distribution for X. So it looks like there is a 0.2 probability that he buys one pack, and that makes sense because that first pack, there is a 0.2 probability that it contains his favorite player's card. And if it does, at that point, he'll just stop. He won't buy any more packs."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "Here is the probability distribution for X. So it looks like there is a 0.2 probability that he buys one pack, and that makes sense because that first pack, there is a 0.2 probability that it contains his favorite player's card. And if it does, at that point, he'll just stop. He won't buy any more packs. Now what about the probability that he buys two packs? Well, over here, they give it a 0.16, and that makes sense. There is a 0.8 probability that he does not get the card he wants on the first one, and then there's another 0.2 that he gets it on the second one."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "He won't buy any more packs. Now what about the probability that he buys two packs? Well, over here, they give it a 0.16, and that makes sense. There is a 0.8 probability that he does not get the card he wants on the first one, and then there's another 0.2 that he gets it on the second one. So 0.8 times 0.2 does indeed equal 0.16. But they're not asking us to calculate that. They give it to us."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "There is a 0.8 probability that he does not get the card he wants on the first one, and then there's another 0.2 that he gets it on the second one. So 0.8 times 0.2 does indeed equal 0.16. But they're not asking us to calculate that. They give it to us. Then the probability that he gets three packs is 0.128, and then they've left blank the probability that he gets four packs. But this is the entire discrete probability distribution because Hugo has to stop at four. Even if he doesn't get the card he wants at four on the fourth pack, he's just going to stop over there."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "They give it to us. Then the probability that he gets three packs is 0.128, and then they've left blank the probability that he gets four packs. But this is the entire discrete probability distribution because Hugo has to stop at four. Even if he doesn't get the card he wants at four on the fourth pack, he's just going to stop over there. So we could actually figure out this question mark by just realizing that these four probabilities have to add up to one. But let's just first answer the question. Find the indicated probability."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "Even if he doesn't get the card he wants at four on the fourth pack, he's just going to stop over there. So we could actually figure out this question mark by just realizing that these four probabilities have to add up to one. But let's just first answer the question. Find the indicated probability. What is the probability that X is greater than or equal to two? What is the probability? Remember, X is the number of packs of cards Hugo buys."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "Find the indicated probability. What is the probability that X is greater than or equal to two? What is the probability? Remember, X is the number of packs of cards Hugo buys. I encourage you to pause the video and try to figure it out. So let's look at the scenarios we're talking about. Probability that our discrete random variable X is greater than or equal to two."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "Remember, X is the number of packs of cards Hugo buys. I encourage you to pause the video and try to figure it out. So let's look at the scenarios we're talking about. Probability that our discrete random variable X is greater than or equal to two. Well, that's these three scenarios right over here. And so what is their combined probability? Well, you might want to say, hey, we need to figure out what the probability of getting exactly four packs are."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "Probability that our discrete random variable X is greater than or equal to two. Well, that's these three scenarios right over here. And so what is their combined probability? Well, you might want to say, hey, we need to figure out what the probability of getting exactly four packs are. But we have to remember that these all add up to 100%. And so this right over here is 0.2. And so this is 0.2, the other three combined have to add up to 0.8."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, you might want to say, hey, we need to figure out what the probability of getting exactly four packs are. But we have to remember that these all add up to 100%. And so this right over here is 0.2. And so this is 0.2, the other three combined have to add up to 0.8. 0.8 plus 0.2 is one, or 100%. So just like that, we know that this is 0.8. If for kicks, we wanted to figure out this question mark right over here, we could just say that, look, have to add up to one."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so this is 0.2, the other three combined have to add up to 0.8. 0.8 plus 0.2 is one, or 100%. So just like that, we know that this is 0.8. If for kicks, we wanted to figure out this question mark right over here, we could just say that, look, have to add up to one. So we could say the probability of exactly four is going to be equal to one minus 0.2 minus 0.16 minus 0.128. I get one minus 0.2 minus 0.16 minus 0.128 is equal to 0.512, is equal to 0.512. 0.512."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "If for kicks, we wanted to figure out this question mark right over here, we could just say that, look, have to add up to one. So we could say the probability of exactly four is going to be equal to one minus 0.2 minus 0.16 minus 0.128. I get one minus 0.2 minus 0.16 minus 0.128 is equal to 0.512, is equal to 0.512. 0.512. You might immediately say, wait, wait, this seems like a very high probability. There's more than a 50% chance that he buys four packs. And you have to remember, he has to stop at four."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "0.512. You might immediately say, wait, wait, this seems like a very high probability. There's more than a 50% chance that he buys four packs. And you have to remember, he has to stop at four. Even if on the fourth, he doesn't get the card he wants, he still has to stop there. So there's a high probability that that's where we end up. There is a little less than 50% chance that he gets the card he's looking for before that point."}, {"video_title": "Matched pairs experiment design Study design AP Statistics Khan Academy.mp3", "Sentence": "We have a whole video on it on Khan Academy, but it is an average measure of your blood sugar over roughly a three-month period. So that's the explanatory variable, whether or not you're taking the pill, and the response variable is, well, what does it do to your hemoglobin A1c? We constructed a somewhat classic experiment where we had a control group and a treatment group, and we randomly assigned folks into either the control or the treatment group. And to ensure that one group or the other, or I guess both of them, don't end up with an imbalance of, in the case of the last video, an imbalance of men or women, we did what we call block design, where we took our 100 people, and we just happened to have 60 women and 40 men, and we said, okay, well, let's split the 60 women randomly between the two groups, and let's split the 40 men between these two groups so that we have at least an even distribution with respect to sex. And so we would measure folks' A1cs before they get the treatment or the placebo. Then we would wait three months of getting either the treatment or the placebo, and then we'll see if there's a statistically significant improvement. Now, this was a pretty good, and it's a bit of a classic experimental design."}, {"video_title": "Matched pairs experiment design Study design AP Statistics Khan Academy.mp3", "Sentence": "And to ensure that one group or the other, or I guess both of them, don't end up with an imbalance of, in the case of the last video, an imbalance of men or women, we did what we call block design, where we took our 100 people, and we just happened to have 60 women and 40 men, and we said, okay, well, let's split the 60 women randomly between the two groups, and let's split the 40 men between these two groups so that we have at least an even distribution with respect to sex. And so we would measure folks' A1cs before they get the treatment or the placebo. Then we would wait three months of getting either the treatment or the placebo, and then we'll see if there's a statistically significant improvement. Now, this was a pretty good, and it's a bit of a classic experimental design. We would also do it so that the patients don't know which one they're getting, placebo or the actual treatment, so it's a blind experiment. And it would probably be good if even the nurses or the doctors who are administering the pills, who are giving the pills, also don't know which one they're giving, so it would be a double blind experiment. But this doesn't mean that it's a perfect experiment, and there seldom is a perfect experiment, and that's why it should be able to be replicated."}, {"video_title": "Matched pairs experiment design Study design AP Statistics Khan Academy.mp3", "Sentence": "Now, this was a pretty good, and it's a bit of a classic experimental design. We would also do it so that the patients don't know which one they're getting, placebo or the actual treatment, so it's a blind experiment. And it would probably be good if even the nurses or the doctors who are administering the pills, who are giving the pills, also don't know which one they're giving, so it would be a double blind experiment. But this doesn't mean that it's a perfect experiment, and there seldom is a perfect experiment, and that's why it should be able to be replicated. Other people should try to prove the same thing, maybe in different ways. But even the way that we designed it, there's still a possibility that there are some lurking variables in here. Maybe, you know, we took care to make sure that our distribution of men and women was roughly even across both of these groups, but maybe by, through that random sampling, we got a disproportionate number of young people in the treatment group, and maybe young people responded better to taking a pill."}, {"video_title": "Matched pairs experiment design Study design AP Statistics Khan Academy.mp3", "Sentence": "But this doesn't mean that it's a perfect experiment, and there seldom is a perfect experiment, and that's why it should be able to be replicated. Other people should try to prove the same thing, maybe in different ways. But even the way that we designed it, there's still a possibility that there are some lurking variables in here. Maybe, you know, we took care to make sure that our distribution of men and women was roughly even across both of these groups, but maybe by, through that random sampling, we got a disproportionate number of young people in the treatment group, and maybe young people responded better to taking a pill. Maybe it changes their behaviors in other ways, or maybe older people, when they take a pill, they decide to eat worse because they say, oh, this pill's gonna solve all my problems. And so you could have these other lurking variables, like age, or where in the country they live, or other types of things, that just by the random process, you might have things get uneven in one way or another. Now, one technique to help control for this a little bit, and I shouldn't use the word control too much, another technique to help mitigate this is something called matched pairs design."}, {"video_title": "Matched pairs experiment design Study design AP Statistics Khan Academy.mp3", "Sentence": "Maybe, you know, we took care to make sure that our distribution of men and women was roughly even across both of these groups, but maybe by, through that random sampling, we got a disproportionate number of young people in the treatment group, and maybe young people responded better to taking a pill. Maybe it changes their behaviors in other ways, or maybe older people, when they take a pill, they decide to eat worse because they say, oh, this pill's gonna solve all my problems. And so you could have these other lurking variables, like age, or where in the country they live, or other types of things, that just by the random process, you might have things get uneven in one way or another. Now, one technique to help control for this a little bit, and I shouldn't use the word control too much, another technique to help mitigate this is something called matched pairs design. Matched, matched pairs, pairs design of an experiment, and it's essentially, instead of going through all of this trouble saying, oh boy, maybe we do block design, all this random sampling, instead, you randomly put people first into either the control or the treatment group, and then we do another round, you measure, and then you do another round where you switch, where the people who are in the treatment go into the control, and the people who are in the control go into the treatment. So we could even extend from what we have here, we could imagine a world where the first three months, we have the 50 people in this treatment group, we have another 50 people in this control group that are taking the placebo, we see what happens to the A1Cs, and then we switch, where this group over here, then, and they don't know, they don't know, first of all, ideally, it's a blind experiment, so they don't even know they were in the treatment groups, and hopefully the pills look identical, so now, that same group, for the next three months, is now going to be the control group, and so they got the medicine for the first three months, and we saw what happens to their A1C, and now they're gonna get the placebo, they're going to get the placebo for the second three months, and then we are going to see what happens to their A1C, and likewise, the other group is going to be switched around. The thing that, the folks that used to be getting the placebo could now get, could now get the treatment."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "Vera rents bicycles to tourists. She recorded the height in centimeters of each customer and the frame size in centimeters of the bicycle that customer rented. After plotting her results, Vera noticed that the relationship between the two variables was fairly linear, so she used the data to calculate the following least squares regression equation for predicting bicycle frame size from the height of the customer, and this is the equation. So before I even look at this question, let's just think about what she did. So she had a bunch of customers, and she recorded, given the height of the customer, what size frame that person rented, and so she might have had something like this, where in the horizontal axis, you have height measured in centimeters, and in the vertical axis, you have frame size that's also measured in centimeters, and so there might have been someone who measures 100 centimeters in height who gets a 25-centimeter frame. I don't know if that's reasonable or not for you bicycle experts, but let's just go with it, and so she would have plotted it there. Maybe there was another person of 100 centimeters in height who got a frame that was slightly larger, and she plotted it there, and then she did a least squares regression, and a least squares regression is trying to fit a line to this data."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "So before I even look at this question, let's just think about what she did. So she had a bunch of customers, and she recorded, given the height of the customer, what size frame that person rented, and so she might have had something like this, where in the horizontal axis, you have height measured in centimeters, and in the vertical axis, you have frame size that's also measured in centimeters, and so there might have been someone who measures 100 centimeters in height who gets a 25-centimeter frame. I don't know if that's reasonable or not for you bicycle experts, but let's just go with it, and so she would have plotted it there. Maybe there was another person of 100 centimeters in height who got a frame that was slightly larger, and she plotted it there, and then she did a least squares regression, and a least squares regression is trying to fit a line to this data. Oftentimes, you would use a spreadsheet or you use a computer, and that line is trying to minimize the square of the distance between these points, and so the least squares regression, maybe it would look something like this, and this is just a rough estimate of it. It might look something, actually, let me get my ruler tool. It might look something like, it might look something like this."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "Maybe there was another person of 100 centimeters in height who got a frame that was slightly larger, and she plotted it there, and then she did a least squares regression, and a least squares regression is trying to fit a line to this data. Oftentimes, you would use a spreadsheet or you use a computer, and that line is trying to minimize the square of the distance between these points, and so the least squares regression, maybe it would look something like this, and this is just a rough estimate of it. It might look something, actually, let me get my ruler tool. It might look something like, it might look something like this. So let me plot it. So this, that would be the line, so our regression line, y-hat, is equal to 1 3rd plus 1 3rd x, and so you could view this as a way of predicting or either modeling the relationship or predicting that, hey, if I get a new person, I could take their height and put it as an x and figure out what frame size they're likely to rent, but they ask us, what is the residual of a customer with a frame, with a height of 155 centimeters who rents a bike with a 51-centimeter frame? So how do we think about this?"}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "It might look something like, it might look something like this. So let me plot it. So this, that would be the line, so our regression line, y-hat, is equal to 1 3rd plus 1 3rd x, and so you could view this as a way of predicting or either modeling the relationship or predicting that, hey, if I get a new person, I could take their height and put it as an x and figure out what frame size they're likely to rent, but they ask us, what is the residual of a customer with a frame, with a height of 155 centimeters who rents a bike with a 51-centimeter frame? So how do we think about this? Well, the residual is going to be the difference between what they actually produce and what the line, what our regression line would have predicted, so we could say residual, let me write it this way, residual is going to be actual, actual minus predicted. So if predicted is larger than actual, this is actually going to be a negative number. If predicted is smaller than actual, this is going to be a positive number."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "So how do we think about this? Well, the residual is going to be the difference between what they actually produce and what the line, what our regression line would have predicted, so we could say residual, let me write it this way, residual is going to be actual, actual minus predicted. So if predicted is larger than actual, this is actually going to be a negative number. If predicted is smaller than actual, this is going to be a positive number. Well, we know the actual, they tell us that, they tell us that they rent, it's a, the 155-centimeter person rents a bike with a 51-centimeter frame, so this is 51 centimeters, but what is the predicted? Well, that's where we can use our regression equation that Vera came up with. The predicted, I'll do that in orange, the predicted is going to be equal to 1 3rd plus 1 3rd times a person's height."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "If predicted is smaller than actual, this is going to be a positive number. Well, we know the actual, they tell us that, they tell us that they rent, it's a, the 155-centimeter person rents a bike with a 51-centimeter frame, so this is 51 centimeters, but what is the predicted? Well, that's where we can use our regression equation that Vera came up with. The predicted, I'll do that in orange, the predicted is going to be equal to 1 3rd plus 1 3rd times a person's height. Their height is 155. That's the predicted. Y hat is what our linear regression predicts, our line predicts, so what is this going to be?"}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "The predicted, I'll do that in orange, the predicted is going to be equal to 1 3rd plus 1 3rd times a person's height. Their height is 155. That's the predicted. Y hat is what our linear regression predicts, our line predicts, so what is this going to be? This is going to be equal to 1 3rd plus 155 over three, which is equal to 156 over three, which comes out nicely to 52. So the predicted on our line is 52, and so here, so this person is 155, we can plot them right over here, 155. They're coming in slightly below the line."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "Y hat is what our linear regression predicts, our line predicts, so what is this going to be? This is going to be equal to 1 3rd plus 155 over three, which is equal to 156 over three, which comes out nicely to 52. So the predicted on our line is 52, and so here, so this person is 155, we can plot them right over here, 155. They're coming in slightly below the line. So they're coming in slightly below the line right there, and that distance, which is, and we can see that they are below the line, so that distance is going to be, or in this case, the residual is going to be negative, so this is going to be negative one. And so if we were to zoom in right over here, you can't see it that well, but let me draw, so if we zoom in, let's say we were to zoom in the line, and it looks like this, and our data point is right, our data point is right over here. We know we're below the line, and this is going to be a negative residual, and the magnitude of that residual is how far we are below the line."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "They're coming in slightly below the line. So they're coming in slightly below the line right there, and that distance, which is, and we can see that they are below the line, so that distance is going to be, or in this case, the residual is going to be negative, so this is going to be negative one. And so if we were to zoom in right over here, you can't see it that well, but let me draw, so if we zoom in, let's say we were to zoom in the line, and it looks like this, and our data point is right, our data point is right over here. We know we're below the line, and this is going to be a negative residual, and the magnitude of that residual is how far we are below the line. And in this case, it is negative one. And so that is our residual. This is what the actual data minus what was predicted by our regression line."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So what's the probability of the different possible outcomes or the different possible values for this random variable? And we'll plot them to see how that distribution is spread out amongst those possible outcomes. So let's think about all of the different values that you could get when you flip a fair coin three times. So you could get all heads. Heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So you could get all heads. Heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails. You could have tails, heads, head. You could have tails, head, tails."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "You could get heads, tails, heads. You could get heads, tails, tails. You could have tails, heads, head. You could have tails, head, tails. You could have tails, tails, heads. And then you could have all tails. So when you do the actual experiment, there's eight equally likely outcomes here."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "You could have tails, head, tails. You could have tails, tails, heads. And then you could have all tails. So when you do the actual experiment, there's eight equally likely outcomes here. But which of them, how would these relate to the value of this random variable? So let's think about what's the probability. There is a situation where you have zero heads."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So when you do the actual experiment, there's eight equally likely outcomes here. But which of them, how would these relate to the value of this random variable? So let's think about what's the probability. There is a situation where you have zero heads. So we could say, what's the probability that our random variable X is equal to 0? Well, that's this situation right over here where you have zero heads. It is one out of the eight equally likely outcomes."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "There is a situation where you have zero heads. So we could say, what's the probability that our random variable X is equal to 0? Well, that's this situation right over here where you have zero heads. It is one out of the eight equally likely outcomes. So that's going to be 1 over 8. What's the probability that our random variable capital X is equal to 1? Well, let's see."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "It is one out of the eight equally likely outcomes. So that's going to be 1 over 8. What's the probability that our random variable capital X is equal to 1? Well, let's see. Which of these outcomes gets us exactly one head? We have this one right over here. We have that one right over there."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "Well, let's see. Which of these outcomes gets us exactly one head? We have this one right over here. We have that one right over there. We have this one right over there. And I think that's all of them. So three out of the eight equally likely outcomes get us to one head, which is the same thing as saying that our random variable equals 1."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "We have that one right over there. We have this one right over there. And I think that's all of them. So three out of the eight equally likely outcomes get us to one head, which is the same thing as saying that our random variable equals 1. So this has a 3 8's probability. Now, what's the probability? I think you're maybe getting the hang for it at this point."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So three out of the eight equally likely outcomes get us to one head, which is the same thing as saying that our random variable equals 1. So this has a 3 8's probability. Now, what's the probability? I think you're maybe getting the hang for it at this point. What's the probability that our random variable X is going to be equal to 2? Well, for X to be equal to 2, that means we got two heads when we flipped the coin three times. So this outcome meets that constraint."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "I think you're maybe getting the hang for it at this point. What's the probability that our random variable X is going to be equal to 2? Well, for X to be equal to 2, that means we got two heads when we flipped the coin three times. So this outcome meets that constraint. This outcome would get our random variable to be equal to 2. And this outcome would make our random variable equal to 2. And this is three out of the eight equally likely outcomes."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So this outcome meets that constraint. This outcome would get our random variable to be equal to 2. And this outcome would make our random variable equal to 2. And this is three out of the eight equally likely outcomes. So this has a 3 8's probability. And then finally, we could say, what is the probability that our random variable X is equal to 3? Well, how does our random variable X equal 3?"}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "And this is three out of the eight equally likely outcomes. So this has a 3 8's probability. And then finally, we could say, what is the probability that our random variable X is equal to 3? Well, how does our random variable X equal 3? Well, we would have to get three heads when we flip the coin. So there's only one out of the eight equally likely outcomes that meets that constraint. So it's a 1 8 probability."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "Well, how does our random variable X equal 3? Well, we would have to get three heads when we flip the coin. So there's only one out of the eight equally likely outcomes that meets that constraint. So it's a 1 8 probability. So now we just have to think about how we plot this to really see how it's distributed. So let me draw over here on the vertical axis. I'll draw this will be the probability."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So it's a 1 8 probability. So now we just have to think about how we plot this to really see how it's distributed. So let me draw over here on the vertical axis. I'll draw this will be the probability. And it's going to be between 0 and 1. You can have a probability larger than 1. So just like this."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "I'll draw this will be the probability. And it's going to be between 0 and 1. You can have a probability larger than 1. So just like this. So let's see. If this is 1 right over here. And let's see."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So just like this. So let's see. If this is 1 right over here. And let's see. Everything here, it looks like it's an eighth. So let's put everything in terms of eighths. So that's half."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "And let's see. Everything here, it looks like it's an eighth. So let's put everything in terms of eighths. So that's half. This is a fourth. That's a fourth. That's not quite a fourth."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So that's half. This is a fourth. That's a fourth. That's not quite a fourth. This is a fourth right over here. And then we can do it in terms of eighths. So that's a pretty good rough approximation."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "That's not quite a fourth. This is a fourth right over here. And then we can do it in terms of eighths. So that's a pretty good rough approximation. And then over here, we could have the outcomes. And so outcomes, I'll say outcomes for, or let's write this so value. So value for X."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So that's a pretty good rough approximation. And then over here, we could have the outcomes. And so outcomes, I'll say outcomes for, or let's write this so value. So value for X. So X could be 0, 1. Actually, let me do those same colors. X could be 0."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So value for X. So X could be 0, 1. Actually, let me do those same colors. X could be 0. X could be 1. X could be 2. X could be equal to 2."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "X could be 0. X could be 1. X could be 2. X could be equal to 2. And X could be equal to 3. These are the possible values for X. And now we're just going to plot the probability."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "X could be equal to 2. And X could be equal to 3. These are the possible values for X. And now we're just going to plot the probability. The probability that X has a value of 0 is 1 eighth. So I'll make a little bar right over here that goes up to 1 eighth. So actually, let me draw it like this."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "And now we're just going to plot the probability. The probability that X has a value of 0 is 1 eighth. So I'll make a little bar right over here that goes up to 1 eighth. So actually, let me draw it like this. So this is 1 eighth right over here. The probability that X equals 1 is 3 eighths. So that's 2 eighths, 3 eighths."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So actually, let me draw it like this. So this is 1 eighth right over here. The probability that X equals 1 is 3 eighths. So that's 2 eighths, 3 eighths. Gets us right over. Let me do that in that purple color. So probability of 1, that's 3 eighths."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So that's 2 eighths, 3 eighths. Gets us right over. Let me do that in that purple color. So probability of 1, that's 3 eighths. That's right over there. That's 3 eighths. So let me draw that bar."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So probability of 1, that's 3 eighths. That's right over there. That's 3 eighths. So let me draw that bar. Just like that. The probability that X equals 2 is also 3 eighths. So that's going to be that same level, just like that."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So let me draw that bar. Just like that. The probability that X equals 2 is also 3 eighths. So that's going to be that same level, just like that. And then the probability that X equals 3, well, that's 1 eighth. So it's going to be the same height as this thing right over here. So actually, I'm using the wrong color."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So that's going to be that same level, just like that. And then the probability that X equals 3, well, that's 1 eighth. So it's going to be the same height as this thing right over here. So actually, I'm using the wrong color. So it's going to look like this. It's going to look like this. And actually, let me just write this a little bit neater."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So actually, I'm using the wrong color. So it's going to look like this. It's going to look like this. And actually, let me just write this a little bit neater. I can move that 3. So cut and paste. Let me move that 3 a little bit closer in, just so it looks a little bit neater."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "And actually, let me just write this a little bit neater. I can move that 3. So cut and paste. Let me move that 3 a little bit closer in, just so it looks a little bit neater. And I can move that 2 in, actually, as well. So cut and paste. So I can move that 2."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "Let me move that 3 a little bit closer in, just so it looks a little bit neater. And I can move that 2 in, actually, as well. So cut and paste. So I can move that 2. And there you have it. We have made a probability distribution for the random variable X. And the random variable X can only take on these discrete values."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So I can move that 2. And there you have it. We have made a probability distribution for the random variable X. And the random variable X can only take on these discrete values. It can't take on the value half or the value pi or anything like that. And so what we've just done here is we've just constructed a discrete probability distribution. Let me write that down."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "And the random variable X can only take on these discrete values. It can't take on the value half or the value pi or anything like that. And so what we've just done here is we've just constructed a discrete probability distribution. Let me write that down. So this right over here is a discrete. The random variable only takes on discrete values. It can't take on any value in between these things."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Perhaps you want to do this because over time, you're trying to breed watermelons that have fewer seeds, and you should see whether you are actually making progress. And you don't want to cut open every watermelon in your watermelon farm or patch or whatever it might be called, because you want to sell most of them. You just want to sample a few watermelons and then take samples of those watermelons to figure out how dense the seeds are, and hope that you can calculate statistics on those samples that are decent estimates of the parameters for the population. So let's start doing that. So let's say that you take these little cubic inch chunks out of a random sample of your watermelons, and then you count the number of seeds in them. And you have eight samples like this. So in one of them, you found four seeds."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So let's start doing that. So let's say that you take these little cubic inch chunks out of a random sample of your watermelons, and then you count the number of seeds in them. And you have eight samples like this. So in one of them, you found four seeds. In the next, you found three, five, seven, two, nine, 11, and seven. So this is a sample just to make sure we're visualizing it right. If this is a population of all of the chunks, these little cubic, I guess if we view this as a cubic inch, the cubic inch chunks in my entire watermelon farm, I'm sampling a very small sample of them."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So in one of them, you found four seeds. In the next, you found three, five, seven, two, nine, 11, and seven. So this is a sample just to make sure we're visualizing it right. If this is a population of all of the chunks, these little cubic, I guess if we view this as a cubic inch, the cubic inch chunks in my entire watermelon farm, I'm sampling a very small sample of them. So I'm sampling a very small sample. Maybe I could have had a million over here. A million chunks of watermelon could have been produced from my farm, but I'm only sampling."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "If this is a population of all of the chunks, these little cubic, I guess if we view this as a cubic inch, the cubic inch chunks in my entire watermelon farm, I'm sampling a very small sample of them. So I'm sampling a very small sample. Maybe I could have had a million over here. A million chunks of watermelon could have been produced from my farm, but I'm only sampling. So capital N would be 1 million, lowercase n is equal to 8. And once again, you might want to have more samples, but this will make our math easy. Now, let's think about what statistics we can measure."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "A million chunks of watermelon could have been produced from my farm, but I'm only sampling. So capital N would be 1 million, lowercase n is equal to 8. And once again, you might want to have more samples, but this will make our math easy. Now, let's think about what statistics we can measure. Well, the first one that we often do is a measure of central tendency, and that's the arithmetic mean. But here, we were trying to estimate the population mean by coming up with the sample mean. So what is the sample mean going to be?"}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Now, let's think about what statistics we can measure. Well, the first one that we often do is a measure of central tendency, and that's the arithmetic mean. But here, we were trying to estimate the population mean by coming up with the sample mean. So what is the sample mean going to be? Well, all we have to do is add up these points, add up these measurements, and then divide by the number of measurements we have. So let's get our calculator out for that. Actually, maybe I don't need my calculator."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So what is the sample mean going to be? Well, all we have to do is add up these points, add up these measurements, and then divide by the number of measurements we have. So let's get our calculator out for that. Actually, maybe I don't need my calculator. Let's see. So 4 plus 3 is 7. 7 plus 5 is 12."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Actually, maybe I don't need my calculator. Let's see. So 4 plus 3 is 7. 7 plus 5 is 12. 12 plus 7 is 19. 19 plus 2 is 21. Plus 9 is 30."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "7 plus 5 is 12. 12 plus 7 is 19. 19 plus 2 is 21. Plus 9 is 30. Plus 11 is 41. Plus 7 is 48. So I'm going to get 48 over 8 data points."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Plus 9 is 30. Plus 11 is 41. Plus 7 is 48. So I'm going to get 48 over 8 data points. So this worked out quite well. 48 divided by 8 is equal to 6. So our sample mean is 6."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So I'm going to get 48 over 8 data points. So this worked out quite well. 48 divided by 8 is equal to 6. So our sample mean is 6. It's our estimate of what the population mean might be. But we also want to think about how much in our population, we want to estimate how much in our population, how much spread is there? How much do our measurements vary from this mean?"}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So our sample mean is 6. It's our estimate of what the population mean might be. But we also want to think about how much in our population, we want to estimate how much in our population, how much spread is there? How much do our measurements vary from this mean? So there we say, well, we can try to estimate the population variance by calculating the sample variance. And we're going to calculate the unbiased sample variance. Hopefully we're fairly convinced at this point why we divide by n minus 1."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "How much do our measurements vary from this mean? So there we say, well, we can try to estimate the population variance by calculating the sample variance. And we're going to calculate the unbiased sample variance. Hopefully we're fairly convinced at this point why we divide by n minus 1. So we're going to calculate the unbiased sample variance. And if we do that, what do we get? Well, it's just going to be, I'll do this in a different color, it's going to be 4 minus 6 squared plus 3 minus 6 squared plus 5 minus 6 squared plus 7 minus 6 squared plus 2 minus 6 squared plus 9 minus 6 squared plus 11 minus 6 squared plus 7 minus 6 squared."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Hopefully we're fairly convinced at this point why we divide by n minus 1. So we're going to calculate the unbiased sample variance. And if we do that, what do we get? Well, it's just going to be, I'll do this in a different color, it's going to be 4 minus 6 squared plus 3 minus 6 squared plus 5 minus 6 squared plus 7 minus 6 squared plus 2 minus 6 squared plus 9 minus 6 squared plus 11 minus 6 squared plus 7 minus 6 squared. All of that divided by, not by 8. Remember, we want the unbiased sample variance. We're going to divide it by 8 minus 1."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Well, it's just going to be, I'll do this in a different color, it's going to be 4 minus 6 squared plus 3 minus 6 squared plus 5 minus 6 squared plus 7 minus 6 squared plus 2 minus 6 squared plus 9 minus 6 squared plus 11 minus 6 squared plus 7 minus 6 squared. All of that divided by, not by 8. Remember, we want the unbiased sample variance. We're going to divide it by 8 minus 1. So we're going to divide by 7. And so this is going to be equal to, let me give myself a little bit more real estate, the unbiased sample variance, and I could even denote it by this to make it clear that we're dividing by lowercase n minus 1, is going to be equal to, let's see, 4 minus 6 is negative 2. That squared is positive 4."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "We're going to divide it by 8 minus 1. So we're going to divide by 7. And so this is going to be equal to, let me give myself a little bit more real estate, the unbiased sample variance, and I could even denote it by this to make it clear that we're dividing by lowercase n minus 1, is going to be equal to, let's see, 4 minus 6 is negative 2. That squared is positive 4. So I did that one. 3 minus 6 is negative 3. That squared is going to be 9."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "That squared is positive 4. So I did that one. 3 minus 6 is negative 3. That squared is going to be 9. 5 minus 6 squared is 1 squared, which is 1. 7 minus 6 is, once again, 1 squared, which is 1. 2 minus 6, negative 4 squared, negative 4 squared is 16."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "That squared is going to be 9. 5 minus 6 squared is 1 squared, which is 1. 7 minus 6 is, once again, 1 squared, which is 1. 2 minus 6, negative 4 squared, negative 4 squared is 16. 9 minus 6 squared, well, that's going to be 9. 11 minus 6 squared, that is 25. And then finally, 7 minus 6 squared, that's another 1."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "2 minus 6, negative 4 squared, negative 4 squared is 16. 9 minus 6 squared, well, that's going to be 9. 11 minus 6 squared, that is 25. And then finally, 7 minus 6 squared, that's another 1. And we're going to divide it by 7. Now let's see if we can add this up in our heads. 4 plus 9 is 13."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "And then finally, 7 minus 6 squared, that's another 1. And we're going to divide it by 7. Now let's see if we can add this up in our heads. 4 plus 9 is 13. Plus 1 is 14, 15, 31, 40, 65, 66. So this is going to be equal to 66 over 7. And we could either divide."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "4 plus 9 is 13. Plus 1 is 14, 15, 31, 40, 65, 66. So this is going to be equal to 66 over 7. And we could either divide. That's 9 and 3 sevenths. We could write that as 9 and 3 sevenths. Or if we want to write that as a decimal, I can just take 66."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "And we could either divide. That's 9 and 3 sevenths. We could write that as 9 and 3 sevenths. Or if we want to write that as a decimal, I can just take 66. 66 divided by 7 gives us 9 point, I'll just round it. So it's approximately 9.43. So this is approximately 9.43."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Or if we want to write that as a decimal, I can just take 66. 66 divided by 7 gives us 9 point, I'll just round it. So it's approximately 9.43. So this is approximately 9.43. Now that gave us our unbiased sample variance. Well, how could we calculate a sample standard deviation? We want to somehow get at an estimate of what the population standard deviation might be."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So this is approximately 9.43. Now that gave us our unbiased sample variance. Well, how could we calculate a sample standard deviation? We want to somehow get at an estimate of what the population standard deviation might be. Well, the logic should, I guess, is reasonable to say, well, this is our unbiased sample variance. It's our best estimate of what the true population variance is. When we think about population parameters to get the population standard deviation, we just take the square root of the population variance."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "We want to somehow get at an estimate of what the population standard deviation might be. Well, the logic should, I guess, is reasonable to say, well, this is our unbiased sample variance. It's our best estimate of what the true population variance is. When we think about population parameters to get the population standard deviation, we just take the square root of the population variance. So if we want to get an estimate of the sample standard deviation, why don't we just take the square root of the unbiased sample variance? So that's what we'll do. So we'll define it that way."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "When we think about population parameters to get the population standard deviation, we just take the square root of the population variance. So if we want to get an estimate of the sample standard deviation, why don't we just take the square root of the unbiased sample variance? So that's what we'll do. So we'll define it that way. We'll call the sample standard deviation, we're going to define it to be equal to the square root of the unbiased sample variance. So it's going to be the square root of this quantity. And we could take our calculator out."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So we'll define it that way. We'll call the sample standard deviation, we're going to define it to be equal to the square root of the unbiased sample variance. So it's going to be the square root of this quantity. And we could take our calculator out. It's going to be the square root of what I just typed in. I could do second answer. It'll be the last entry here."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "And we could take our calculator out. It's going to be the square root of what I just typed in. I could do second answer. It'll be the last entry here. So the square root of that is, and I'll just round, it's approximately equal to 3.07. Now I'm going to tell you something very counterintuitive. Or at least initially it's counterintuitive, but hopefully you'll appreciate this over time."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "It'll be the last entry here. So the square root of that is, and I'll just round, it's approximately equal to 3.07. Now I'm going to tell you something very counterintuitive. Or at least initially it's counterintuitive, but hopefully you'll appreciate this over time. This we've already talked about in some depth. People have even created simulations to show that this is an unbiased estimate of population variance when we divide it by n minus 1. And that's a good starting point if we're going to take the square root of anything."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Or at least initially it's counterintuitive, but hopefully you'll appreciate this over time. This we've already talked about in some depth. People have even created simulations to show that this is an unbiased estimate of population variance when we divide it by n minus 1. And that's a good starting point if we're going to take the square root of anything. But it actually turns out that because the square root function is nonlinear, that this sample standard deviation, and this is how it tends to be defined, sample standard deviation, that this sample standard deviation, which is the square root of our sample variance, so from i equals 1 to n of our unbiased sample variance, so we divide it by n minus 1. This is how we literally divide our sample standard deviation. Because the square root function is nonlinear, it turns out that this is not an unbiased estimate of the true population standard deviation."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "And that's a good starting point if we're going to take the square root of anything. But it actually turns out that because the square root function is nonlinear, that this sample standard deviation, and this is how it tends to be defined, sample standard deviation, that this sample standard deviation, which is the square root of our sample variance, so from i equals 1 to n of our unbiased sample variance, so we divide it by n minus 1. This is how we literally divide our sample standard deviation. Because the square root function is nonlinear, it turns out that this is not an unbiased estimate of the true population standard deviation. And I encourage people to make simulations of that if they're interested. But then you might say, OK, well, we went through great pains to divide by n minus 1 here in order to get an unbiased estimate of the population variance. Why don't we go through similar pains and somehow figure out a formula for an unbiased estimate of the population standard deviation?"}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Because the square root function is nonlinear, it turns out that this is not an unbiased estimate of the true population standard deviation. And I encourage people to make simulations of that if they're interested. But then you might say, OK, well, we went through great pains to divide by n minus 1 here in order to get an unbiased estimate of the population variance. Why don't we go through similar pains and somehow figure out a formula for an unbiased estimate of the population standard deviation? And the reason why that's difficult is to unbiased the sample variance, we just have to divide by n minus 1 instead of n. And that worked for any probability distribution for our population. It turns out to do the same thing for the standard deviation. It's not that easy."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Why don't we go through similar pains and somehow figure out a formula for an unbiased estimate of the population standard deviation? And the reason why that's difficult is to unbiased the sample variance, we just have to divide by n minus 1 instead of n. And that worked for any probability distribution for our population. It turns out to do the same thing for the standard deviation. It's not that easy. It's actually dependent on how that population is actually distributed. So in statistics, we just define the sample standard deviation. And the one that we typically use is based on the square root of the unbiased sample variance."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Let's do a little bit of probability with playing cards. And for the sake of this video, we're going to assume that our deck has no jokers in it. You could do the same problems with the joker. You'll just get slightly different numbers. So with that out of the way, let's first just think about how many cards we have in a standard playing deck. So you have four suits. So you have four suits."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "You'll just get slightly different numbers. So with that out of the way, let's first just think about how many cards we have in a standard playing deck. So you have four suits. So you have four suits. And the suits are the spades, the diamonds, the clubs, and the hearts. You have four suits. And then in each of those suits, you have 13 different types of cards."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So you have four suits. And the suits are the spades, the diamonds, the clubs, and the hearts. You have four suits. And then in each of those suits, you have 13 different types of cards. Or sometimes it's called the rank. So each suit has 13 types of cards. You have the ace, then you have the two, the three, the four, the five, the six, seven, eight, nine, 10."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "And then in each of those suits, you have 13 different types of cards. Or sometimes it's called the rank. So each suit has 13 types of cards. You have the ace, then you have the two, the three, the four, the five, the six, seven, eight, nine, 10. And then you have the jack, the king, and the queen. And that is 13 cards. So for each suit, you can have any of these."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "You have the ace, then you have the two, the three, the four, the five, the six, seven, eight, nine, 10. And then you have the jack, the king, and the queen. And that is 13 cards. So for each suit, you can have any of these. For any of these, you can have any of the suits. So you could have a jack of diamonds, a jack of clubs, a jack of spades, or a jack of hearts. So if you just multiply these two things, you could take a deck of playing cards and actually count them, take out the jokers and count them."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So for each suit, you can have any of these. For any of these, you can have any of the suits. So you could have a jack of diamonds, a jack of clubs, a jack of spades, or a jack of hearts. So if you just multiply these two things, you could take a deck of playing cards and actually count them, take out the jokers and count them. But if you just multiply this, you have four suits. Each of those suits have 13 types. So you're going to have 4 times 13 cards."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So if you just multiply these two things, you could take a deck of playing cards and actually count them, take out the jokers and count them. But if you just multiply this, you have four suits. Each of those suits have 13 types. So you're going to have 4 times 13 cards. Or you're going to have 52 cards in a standard playing deck. Another way you could say it, you're like, look, I'm going to have these ranks or types. And each of those come in four different suits, 13 times 4."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So you're going to have 4 times 13 cards. Or you're going to have 52 cards in a standard playing deck. Another way you could say it, you're like, look, I'm going to have these ranks or types. And each of those come in four different suits, 13 times 4. Once again, you would have gotten 52 cards. Now with that out of the way, let's think about the probabilities of different events. So let's say I shuffle that deck."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "And each of those come in four different suits, 13 times 4. Once again, you would have gotten 52 cards. Now with that out of the way, let's think about the probabilities of different events. So let's say I shuffle that deck. I shuffle it really, really well. And then I randomly pick a card from that deck. And I want to think about what is the probability that I pick a jack."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So let's say I shuffle that deck. I shuffle it really, really well. And then I randomly pick a card from that deck. And I want to think about what is the probability that I pick a jack. Well, how many equally likely events are there? Well, I could pick any one of those 52 cards. So there's 52 possibilities for when I pick that card."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "And I want to think about what is the probability that I pick a jack. Well, how many equally likely events are there? Well, I could pick any one of those 52 cards. So there's 52 possibilities for when I pick that card. And how many of those 52 possibilities are jacks? Well, you have the jack of spades, the jack of diamonds, the jack of clubs, and the jack of hearts. There's four jacks in that deck."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So there's 52 possibilities for when I pick that card. And how many of those 52 possibilities are jacks? Well, you have the jack of spades, the jack of diamonds, the jack of clubs, and the jack of hearts. There's four jacks in that deck. So it is 4 over 52. These are both divisible by 4. 4 divided by 4 is 1."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "There's four jacks in that deck. So it is 4 over 52. These are both divisible by 4. 4 divided by 4 is 1. 52 divided by 4 is 13. Now let's think about the probability. So we're going to start over."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "4 divided by 4 is 1. 52 divided by 4 is 13. Now let's think about the probability. So we're going to start over. I'm going to put that jack back in. I'm going to reshuffle the deck. So once again, I still have 52 cards."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So we're going to start over. I'm going to put that jack back in. I'm going to reshuffle the deck. So once again, I still have 52 cards. So what's the probability that I get a hearts? What's the probability that I just randomly pick a card from a shuffled deck and it is a hearts? Its suit is a heart."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So once again, I still have 52 cards. So what's the probability that I get a hearts? What's the probability that I just randomly pick a card from a shuffled deck and it is a hearts? Its suit is a heart. Well, once again, there's 52 possible cards I could pick from, 52 possible equally likely events that we're dealing with. And how many of those have our hearts? Well, essentially 13 of them are hearts."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Its suit is a heart. Well, once again, there's 52 possible cards I could pick from, 52 possible equally likely events that we're dealing with. And how many of those have our hearts? Well, essentially 13 of them are hearts. For each of those suits, you have 13 types. So there are 13 hearts in that deck. There are 13 diamonds in that deck."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Well, essentially 13 of them are hearts. For each of those suits, you have 13 types. So there are 13 hearts in that deck. There are 13 diamonds in that deck. There are 13 spades in that deck. There are 13 clubs in that deck. So 13 of the 52 would result in hearts."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "There are 13 diamonds in that deck. There are 13 spades in that deck. There are 13 clubs in that deck. So 13 of the 52 would result in hearts. And both of these are divisible by 13. This is the same thing as 1 fourth. 1 and 4 times, I will pick it out or I have a 1 and 4 probability of getting a hearts when I go to that, when I randomly pick a card from that shuffle deck."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So 13 of the 52 would result in hearts. And both of these are divisible by 13. This is the same thing as 1 fourth. 1 and 4 times, I will pick it out or I have a 1 and 4 probability of getting a hearts when I go to that, when I randomly pick a card from that shuffle deck. Now let's do something that's a little bit more interesting. Or maybe it's a little obvious. What's the probability that I pick something that is a jack?"}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "1 and 4 times, I will pick it out or I have a 1 and 4 probability of getting a hearts when I go to that, when I randomly pick a card from that shuffle deck. Now let's do something that's a little bit more interesting. Or maybe it's a little obvious. What's the probability that I pick something that is a jack? I'll just write J. It's a jack and it is a hearts. Well, if you're reasonably familiar with cards, you'll know that there's actually only one card that is both a jack and a heart."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "What's the probability that I pick something that is a jack? I'll just write J. It's a jack and it is a hearts. Well, if you're reasonably familiar with cards, you'll know that there's actually only one card that is both a jack and a heart. It is literally the jack of hearts. So we're saying, what is the probability that we pick the exact card, the jack of hearts? Well, there's only one event, one card, that meets this criteria right over here."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Well, if you're reasonably familiar with cards, you'll know that there's actually only one card that is both a jack and a heart. It is literally the jack of hearts. So we're saying, what is the probability that we pick the exact card, the jack of hearts? Well, there's only one event, one card, that meets this criteria right over here. And there's 52 possible cards. So there's a 1 in 52 chance that I pick the jack of hearts, something that is both a jack and it's a heart. Now let's do something a little bit more interesting."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Well, there's only one event, one card, that meets this criteria right over here. And there's 52 possible cards. So there's a 1 in 52 chance that I pick the jack of hearts, something that is both a jack and it's a heart. Now let's do something a little bit more interesting. What is the probability? You might want to pause this and think about this a little bit before I give you the answer. What is the probability of, so I once again, I have a deck of 52 cards."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Now let's do something a little bit more interesting. What is the probability? You might want to pause this and think about this a little bit before I give you the answer. What is the probability of, so I once again, I have a deck of 52 cards. I shuffle it, randomly pick a card from that deck. What is the probability that that card that I pick from that deck is a jack or a heart? So it could be the jack of hearts or it could be the jack of diamonds or it could be the jack of spades or it could be the queen of hearts or it could be the two of hearts."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "What is the probability of, so I once again, I have a deck of 52 cards. I shuffle it, randomly pick a card from that deck. What is the probability that that card that I pick from that deck is a jack or a heart? So it could be the jack of hearts or it could be the jack of diamonds or it could be the jack of spades or it could be the queen of hearts or it could be the two of hearts. So what is the probability of this? And this is a little bit more of an interesting thing because we know, first of all, that there are 52 possibilities. There are 52 possibilities."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So it could be the jack of hearts or it could be the jack of diamonds or it could be the jack of spades or it could be the queen of hearts or it could be the two of hearts. So what is the probability of this? And this is a little bit more of an interesting thing because we know, first of all, that there are 52 possibilities. There are 52 possibilities. But how many of those possibilities meet the criteria, meet these conditions that it is a jack or a heart? And to understand that, I'll draw a Venn diagram. Sounds kind of fancy, but nothing fancy here."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "There are 52 possibilities. But how many of those possibilities meet the criteria, meet these conditions that it is a jack or a heart? And to understand that, I'll draw a Venn diagram. Sounds kind of fancy, but nothing fancy here. So imagine that this rectangle I'm drawing here represents all of the outcomes. So if you want, you can imagine it has an area of 52. So this is 52 possible outcomes."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Sounds kind of fancy, but nothing fancy here. So imagine that this rectangle I'm drawing here represents all of the outcomes. So if you want, you can imagine it has an area of 52. So this is 52 possible outcomes. Now, how many of those outcomes result in a jack? So we already learned, it's one out of 13 of those outcomes result in a jack. So I could draw a little circle here where that area, and I'm approximating, that represents the probability of a jack."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So this is 52 possible outcomes. Now, how many of those outcomes result in a jack? So we already learned, it's one out of 13 of those outcomes result in a jack. So I could draw a little circle here where that area, and I'm approximating, that represents the probability of a jack. So it should be roughly 1 13th or 4 52nds of this area right over here. So I'll just draw it like this. So this right over here is the probability of a jack."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So I could draw a little circle here where that area, and I'm approximating, that represents the probability of a jack. So it should be roughly 1 13th or 4 52nds of this area right over here. So I'll just draw it like this. So this right over here is the probability of a jack. The probability of the jack. It is four, there's four possible cards out of the 52. So that is four 52nds or one out of 13."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So this right over here is the probability of a jack. The probability of the jack. It is four, there's four possible cards out of the 52. So that is four 52nds or one out of 13. 1 13. Now, what's the probability of getting a hearts? Well, I'll draw another little circle here that represents that."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So that is four 52nds or one out of 13. 1 13. Now, what's the probability of getting a hearts? Well, I'll draw another little circle here that represents that. 13 out of 52. 13 out of these 52 cards represent a heart. And actually, one of them represents both a heart and a jack."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Well, I'll draw another little circle here that represents that. 13 out of 52. 13 out of these 52 cards represent a heart. And actually, one of them represents both a heart and a jack. So I'm actually going to overlap them, and hopefully this will make sense in a second. So there's actually 13 cards that are a heart. So this is the number of hearts."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "And actually, one of them represents both a heart and a jack. So I'm actually going to overlap them, and hopefully this will make sense in a second. So there's actually 13 cards that are a heart. So this is the number of hearts. Number of hearts. And actually, let me write this top thing that way as well. That makes it a little bit clearer that we're actually looking at."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So this is the number of hearts. Number of hearts. And actually, let me write this top thing that way as well. That makes it a little bit clearer that we're actually looking at. So let me clear that. So the number of jacks. Number of jacks."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "That makes it a little bit clearer that we're actually looking at. So let me clear that. So the number of jacks. Number of jacks. And of course, this overlap right here is the number of jacks and hearts. The number of items out of this 52 that are both a jack and a heart. It is in both sets here."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Number of jacks. And of course, this overlap right here is the number of jacks and hearts. The number of items out of this 52 that are both a jack and a heart. It is in both sets here. It is in this green circle, and it is in this orange circle. So this right over here, let me do that in yellow since I did that problem in yellow. This right over here is the number of jacks and hearts."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "It is in both sets here. It is in this green circle, and it is in this orange circle. So this right over here, let me do that in yellow since I did that problem in yellow. This right over here is the number of jacks and hearts. So let me draw a little arrow there. It's getting a little cluttered. Maybe I should have drawn a little bit bigger."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "This right over here is the number of jacks and hearts. So let me draw a little arrow there. It's getting a little cluttered. Maybe I should have drawn a little bit bigger. Number of jacks and hearts. Number of jacks and hearts. And that's an overlap over there."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Maybe I should have drawn a little bit bigger. Number of jacks and hearts. Number of jacks and hearts. And that's an overlap over there. So what is the probability of getting a jack or a heart? So if you think about it, the probability is going to be the number of events that meet these conditions over the total number of events. We already know the total number of events are 52, but how many meet these conditions?"}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "And that's an overlap over there. So what is the probability of getting a jack or a heart? So if you think about it, the probability is going to be the number of events that meet these conditions over the total number of events. We already know the total number of events are 52, but how many meet these conditions? So it's going to be the number, it's going to be, you could say, well, look, the green circle right there says the number that gives us a jack, and the orange circle tells us that the number that gives us a heart. So you might want to say, well, why don't we add up the green and the orange? But if you did that, you would be double counting."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "We already know the total number of events are 52, but how many meet these conditions? So it's going to be the number, it's going to be, you could say, well, look, the green circle right there says the number that gives us a jack, and the orange circle tells us that the number that gives us a heart. So you might want to say, well, why don't we add up the green and the orange? But if you did that, you would be double counting. Because if you added up, if you just did four, if you did four plus 13, what are we saying? We're saying that there are four jacks, and we're saying that there are 13 hearts. But in both of these, when we do it this way, in both cases, we are counting the jack of hearts."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "But if you did that, you would be double counting. Because if you added up, if you just did four, if you did four plus 13, what are we saying? We're saying that there are four jacks, and we're saying that there are 13 hearts. But in both of these, when we do it this way, in both cases, we are counting the jack of hearts. We're putting the jack of hearts here, and we're putting the jack of hearts here. So we're counting the jack of hearts twice, even though there's only one card there. So you would have to subtract out where they're common."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "But in both of these, when we do it this way, in both cases, we are counting the jack of hearts. We're putting the jack of hearts here, and we're putting the jack of hearts here. So we're counting the jack of hearts twice, even though there's only one card there. So you would have to subtract out where they're common. You would have to subtract out the item that is both a jack and a heart. So you would subtract out a one. Another way to think about it is, you really want to figure out the total area here."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So you would have to subtract out where they're common. You would have to subtract out the item that is both a jack and a heart. So you would subtract out a one. Another way to think about it is, you really want to figure out the total area here. You want to figure out the total area here. You want to figure out this total area. And let me zoom in, and I'll generalize it a little bit."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Another way to think about it is, you really want to figure out the total area here. You want to figure out the total area here. You want to figure out this total area. And let me zoom in, and I'll generalize it a little bit. So if you have one circle like that, and then you have another overlapping circle like that, and you wanted to figure out the total area of the circles combined, you would look at the area of this circle, you would look at the area of this circle, and then you could add it to the area of this circle. But when you do that, you'll see that when you add the two areas, you're counting this area twice. So in order to only count that area once, you have to subtract that area from the sum."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "And let me zoom in, and I'll generalize it a little bit. So if you have one circle like that, and then you have another overlapping circle like that, and you wanted to figure out the total area of the circles combined, you would look at the area of this circle, you would look at the area of this circle, and then you could add it to the area of this circle. But when you do that, you'll see that when you add the two areas, you're counting this area twice. So in order to only count that area once, you have to subtract that area from the sum. So if this area has A, this area is B, and the intersection where they overlap is C, the combined area is going to be A plus B minus where they overlap, minus C. So that's the same thing over here. We're counting all the jacks, and that includes the jack of hearts. We're counting all the hearts, and that includes the jack of hearts."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So in order to only count that area once, you have to subtract that area from the sum. So if this area has A, this area is B, and the intersection where they overlap is C, the combined area is going to be A plus B minus where they overlap, minus C. So that's the same thing over here. We're counting all the jacks, and that includes the jack of hearts. We're counting all the hearts, and that includes the jack of hearts. So we counted the jack of hearts twice, so we have to subtract one out of that. So it's gonna be four plus 13 minus one, or this is going to be 1650 seconds, and both of these things are divisible by four, so this is going to be the same thing as, divide 16 by four, you get four. 52 divided by four is 13."}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "He randomly selects 30 of the 150 total seniors and finds that seven of those sampled would order the vegetarian option. Which conditions for constructing this confidence interval did Ali's sample meet? So pause this video, and you can select more than one of these. All right, now let's work through this together. So one thing that you might be wondering is, well, what is a one-sample z-interval? Well, you could really interpret that as he's gonna take one sample and then construct a confidence interval based on that. The reason why it might be called a z-interval is the whole idea behind a confidence interval is you're going to pick a number of standard deviations above and below the true parameter that you are actually trying to estimate, and then use that to make your inferences."}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "All right, now let's work through this together. So one thing that you might be wondering is, well, what is a one-sample z-interval? Well, you could really interpret that as he's gonna take one sample and then construct a confidence interval based on that. The reason why it might be called a z-interval is the whole idea behind a confidence interval is you're going to pick a number of standard deviations above and below the true parameter that you are actually trying to estimate, and then use that to make your inferences. And one way of thinking about the number of standard deviations, people will often call that a z-score, or z is often used as a variable for the number of standard deviations above or below something. So really, he's just trying to construct a confidence interval. But remember, in order to construct a confidence interval, we have to make some assumptions."}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "The reason why it might be called a z-interval is the whole idea behind a confidence interval is you're going to pick a number of standard deviations above and below the true parameter that you are actually trying to estimate, and then use that to make your inferences. And one way of thinking about the number of standard deviations, people will often call that a z-score, or z is often used as a variable for the number of standard deviations above or below something. So really, he's just trying to construct a confidence interval. But remember, in order to construct a confidence interval, we have to make some assumptions. He's taking, there's 150 students right over here. He's finding it impractical to survey all 150 to figure out the true population proportion. So instead, he samples 30 of the seniors."}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "But remember, in order to construct a confidence interval, we have to make some assumptions. He's taking, there's 150 students right over here. He's finding it impractical to survey all 150 to figure out the true population proportion. So instead, he samples 30 of the seniors. So n is equal to 30. And from that, he calculates a sample proportion. It looks like seven out of the 30 are they want the vegetarian option."}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "So instead, he samples 30 of the seniors. So n is equal to 30. And from that, he calculates a sample proportion. It looks like seven out of the 30 are they want the vegetarian option. And he's going to determine some confidence level and then construct a confidence interval. But remember the conditions that we've talked about in previous videos. The first thing is we have to be confident that is this a random sample?"}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "It looks like seven out of the 30 are they want the vegetarian option. And he's going to determine some confidence level and then construct a confidence interval. But remember the conditions that we've talked about in previous videos. The first thing is we have to be confident that is this a random sample? So that would be the random condition. And that's what choice A is telling us. The data is a random sample from the population of interest."}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "The first thing is we have to be confident that is this a random sample? So that would be the random condition. And that's what choice A is telling us. The data is a random sample from the population of interest. Do we know that? Well, it tells us in the passage here, he randomly selects 30 of the total seniors. So I guess we'll take their word for it."}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "The data is a random sample from the population of interest. Do we know that? Well, it tells us in the passage here, he randomly selects 30 of the total seniors. So I guess we'll take their word for it. We don't know his methodology of what he considers random, but we'll take their word for it that yes, this has been met. The data is a random, random sample. If it said he sampled the football team, well, that would not have been a random sample."}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "So I guess we'll take their word for it. We don't know his methodology of what he considers random, but we'll take their word for it that yes, this has been met. The data is a random, random sample. If it said he sampled the football team, well, that would not have been a random sample. The next condition here, it looks all mathematical, but this is really the normal condition. And the idea behind the normal condition is that in order to construct these confidence intervals, we're assuming that the sampling distribution of the sample proportions is roughly normal. And it is not skewed to the right or skewed to the left like this."}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "If it said he sampled the football team, well, that would not have been a random sample. The next condition here, it looks all mathematical, but this is really the normal condition. And the idea behind the normal condition is that in order to construct these confidence intervals, we're assuming that the sampling distribution of the sample proportions is roughly normal. And it is not skewed to the right or skewed to the left like this. And so right here it says, look, the sample size times our sample proportion has to be greater than or equal to 10. Or our sample size times one minus our sample proportion has to be greater than or equal to 10. Well, another way to think about this is our successes, our successes in our sample need to be greater than or equal to 10."}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "And it is not skewed to the right or skewed to the left like this. And so right here it says, look, the sample size times our sample proportion has to be greater than or equal to 10. Or our sample size times one minus our sample proportion has to be greater than or equal to 10. Well, another way to think about this is our successes, our successes in our sample need to be greater than or equal to 10. And our failures need to be greater than or equal to 10. Well, how many successes were there? There were seven, seven."}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "Well, another way to think about this is our successes, our successes in our sample need to be greater than or equal to 10. And our failures need to be greater than or equal to 10. Well, how many successes were there? There were seven, seven. And you could even say, look, our n is 30 times our sample proportion is seven over 30, which is going to be seven. So our successes is less than 10. So actually we violate the normal condition."}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "There were seven, seven. And you could even say, look, our n is 30 times our sample proportion is seven over 30, which is going to be seven. So our successes is less than 10. So actually we violate the normal condition. And once again, this is a rule of thumb, but this is telling us that our actual sampling distribution might be skewed. Remember, this is just based on one sample, what we're able to figure out. This is one sample z interval."}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "So actually we violate the normal condition. And once again, this is a rule of thumb, but this is telling us that our actual sampling distribution might be skewed. Remember, this is just based on one sample, what we're able to figure out. This is one sample z interval. We might be wrong, but we wouldn't feel good that we're meeting the normal condition here. So I would rule this one out. Individual observations can be considered independent."}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "This is one sample z interval. We might be wrong, but we wouldn't feel good that we're meeting the normal condition here. So I would rule this one out. Individual observations can be considered independent. Well, if he randomly selected people with replacement, then they could be independent. Or if the people he is selecting, if his sample size is less than 10% of the total population, then it could be considered independent, even though it wouldn't be perfectly independent. But we see here that he sampled 30 people out of 150."}, {"video_title": "Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3", "Sentence": "Individual observations can be considered independent. Well, if he randomly selected people with replacement, then they could be independent. Or if the people he is selecting, if his sample size is less than 10% of the total population, then it could be considered independent, even though it wouldn't be perfectly independent. But we see here that he sampled 30 people out of 150. So his sample size was 30 out of 150, which is the same thing as 1 5th of the population, which is the same thing as 20%. And since this is greater than 10%, we are violating the independence condition. We could have met the independence condition if he was sampling with replacement, which it doesn't seem like he is, or if this thing right over here was less than 10%."}, {"video_title": "Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3", "Sentence": "Adriana gathered data on different schools' winning percentages and the average yearly salary of their head coaches in millions of dollars in the years 2000 to 2011. She then created the following scatter plot and trend line. So this is salary in millions of dollars and the winning percentage. And so here we have a coach who made over $4 million and it looks like they won over 80% of their games. Then you have this coach over here who has a salary of a little over a million and a half dollars and they are winning over 85%. And so each of one of these data points is a coach and it's plotting their salary or their winning percentage against their salary. Assuming the line correctly shows the trend in the data and it's a bit of an assumption."}, {"video_title": "Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3", "Sentence": "And so here we have a coach who made over $4 million and it looks like they won over 80% of their games. Then you have this coach over here who has a salary of a little over a million and a half dollars and they are winning over 85%. And so each of one of these data points is a coach and it's plotting their salary or their winning percentage against their salary. Assuming the line correctly shows the trend in the data and it's a bit of an assumption. There are some outliers here that are well away from the model and this isn't a, it looks like there's a linear, a positive linear correlation here, but it's not super tight and there's a bunch of coaches right over here in the lower salary area going all the way from 20 something percent to over 60%. Assuming the line correctly shows the trend in the data, what does it mean that the lie's y-intercept is 39? Well if you believe the model, then a y-intercept of being 39 would be, the model is saying that if someone makes no money, that they could, zero dollars, that they could win, that the model would expect them to win 39% of their gains, which seems a little unrealistic because you would expect most coaches to get paid something."}, {"video_title": "Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3", "Sentence": "Assuming the line correctly shows the trend in the data and it's a bit of an assumption. There are some outliers here that are well away from the model and this isn't a, it looks like there's a linear, a positive linear correlation here, but it's not super tight and there's a bunch of coaches right over here in the lower salary area going all the way from 20 something percent to over 60%. Assuming the line correctly shows the trend in the data, what does it mean that the lie's y-intercept is 39? Well if you believe the model, then a y-intercept of being 39 would be, the model is saying that if someone makes no money, that they could, zero dollars, that they could win, that the model would expect them to win 39% of their gains, which seems a little unrealistic because you would expect most coaches to get paid something. But anyway, let's see which of these choices actually describe that. So let me look at the choices. The average salary was $39 million."}, {"video_title": "Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3", "Sentence": "Well if you believe the model, then a y-intercept of being 39 would be, the model is saying that if someone makes no money, that they could, zero dollars, that they could win, that the model would expect them to win 39% of their gains, which seems a little unrealistic because you would expect most coaches to get paid something. But anyway, let's see which of these choices actually describe that. So let me look at the choices. The average salary was $39 million. No, no one on our chart made 39 million. On average, each million dollar increase in salary was associated with a 39% increase in winning percentage. So that would be something related to the slope, and the slope was definitely not 39."}, {"video_title": "Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3", "Sentence": "The average salary was $39 million. No, no one on our chart made 39 million. On average, each million dollar increase in salary was associated with a 39% increase in winning percentage. So that would be something related to the slope, and the slope was definitely not 39. The average winning percentage was 39%. No, that wasn't the case either. The model indicates that teams with coaches who had a salary of zero million dollars will average a winning percentage of approximately 39%."}, {"video_title": "Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3", "Sentence": "So that would be something related to the slope, and the slope was definitely not 39. The average winning percentage was 39%. No, that wasn't the case either. The model indicates that teams with coaches who had a salary of zero million dollars will average a winning percentage of approximately 39%. Yeah, this is the closest statement to what we just said, that if you believe that model, and that's a big if, if you believe this model, then this model says someone making zero dollars will get 39%, and this is frankly why you have to be skeptical of models. They're not going to be perfect, especially at extreme cases oftentimes, but who knows? Anyway, hopefully you found that useful."}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So it says here I have a 0.35 probability of making a free throw. What is the probability of making four out of seven free throws? Well, this is a classic binomial random variable question. If we said the binomial random variable X is equal to number of made free throws from seven, I could say seven trials or seven shots, seven trials with the probability of success is equal to 0.35 for each free throw. So really this question amounts to what is the probability that my binomial random variable X is equal to four? Now what we're going to see is we can use a function on our TI-84 named binomec, or binomepdf I should say, binomepdf, which is short for binomial probability distribution function. And what you're going to want to do here is use three arguments."}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "If we said the binomial random variable X is equal to number of made free throws from seven, I could say seven trials or seven shots, seven trials with the probability of success is equal to 0.35 for each free throw. So really this question amounts to what is the probability that my binomial random variable X is equal to four? Now what we're going to see is we can use a function on our TI-84 named binomec, or binomepdf I should say, binomepdf, which is short for binomial probability distribution function. And what you're going to want to do here is use three arguments. So the first one is the number of trials. So in this case it is seven. And if you're doing it on the free response section of the AP test, you should make it very clear that that right over there is your N. The graders will actually look for that to make sure that you're not just guessing what goes where."}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "And what you're going to want to do here is use three arguments. So the first one is the number of trials. So in this case it is seven. And if you're doing it on the free response section of the AP test, you should make it very clear that that right over there is your N. The graders will actually look for that to make sure that you're not just guessing what goes where. So you would say that is my N and then you would say your probability, 0.35. And once again, if you're taking the test, you should mark that. That is your P. And then last but not least, what is the probability that a binomial variable, when you're taking seven trials with a probability of success of each of them being 0.35, that you have exactly four successes?"}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "And if you're doing it on the free response section of the AP test, you should make it very clear that that right over there is your N. The graders will actually look for that to make sure that you're not just guessing what goes where. So you would say that is my N and then you would say your probability, 0.35. And once again, if you're taking the test, you should mark that. That is your P. And then last but not least, what is the probability that a binomial variable, when you're taking seven trials with a probability of success of each of them being 0.35, that you have exactly four successes? So now let's get our calculator out and actually do that. All right, so now we have our graphing calculator out. So there's a couple of ways to input this."}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "That is your P. And then last but not least, what is the probability that a binomial variable, when you're taking seven trials with a probability of success of each of them being 0.35, that you have exactly four successes? So now let's get our calculator out and actually do that. All right, so now we have our graphing calculator out. So there's a couple of ways to input this. You could just type it in directly. That could take time. You could do second and this little blue distribution here."}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So there's a couple of ways to input this. You could just type it in directly. That could take time. You could do second and this little blue distribution here. So there you have it. In order to get to the function, you could either scroll down or you could scroll up to get to the bottom of the list and you see it right over here, binomPDF. You could do alpha A to go there really fast or you could just scroll up here, click Enter."}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "You could do second and this little blue distribution here. So there you have it. In order to get to the function, you could either scroll down or you could scroll up to get to the bottom of the list and you see it right over here, binomPDF. You could do alpha A to go there really fast or you could just scroll up here, click Enter. And then you have the number of trials that you wanna deal with. Well, we're gonna take seven trials. The probability of success in each trial is 0.35."}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "You could do alpha A to go there really fast or you could just scroll up here, click Enter. And then you have the number of trials that you wanna deal with. Well, we're gonna take seven trials. The probability of success in each trial is 0.35. And then my X value, well, I wanna find the probability that my binomial random variable is equal to four, four successes out of the trials. And now let me go to Paste and this is actually going to type in exactly what we had before. Notice this is the exact same thing."}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "The probability of success in each trial is 0.35. And then my X value, well, I wanna find the probability that my binomial random variable is equal to four, four successes out of the trials. And now let me go to Paste and this is actually going to type in exactly what we had before. Notice this is the exact same thing. So I have seven trials, P is equal to 0.35 and I wanna know the probability of having exactly four successes. And then I just click Enter and I get, there you go, 0.14. So this is equal to approximately 0.14."}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "Notice this is the exact same thing. So I have seven trials, P is equal to 0.35 and I wanna know the probability of having exactly four successes. And then I just click Enter and I get, there you go, 0.14. So this is equal to approximately 0.14. Now based on the same binomial random variable, if we're then asked what is the probability of making less than five free throws? So we could say this is the probability that X is less than five or we could say this is the probability that X is less than or equal to four. And the reason why I write it this way is because using it this way, you can now use the binomial cumulative distribution function on my calculator."}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So this is equal to approximately 0.14. Now based on the same binomial random variable, if we're then asked what is the probability of making less than five free throws? So we could say this is the probability that X is less than five or we could say this is the probability that X is less than or equal to four. And the reason why I write it this way is because using it this way, you can now use the binomial cumulative distribution function on my calculator. So if I just type in binom, and once again, I'm gonna take seven, or binom CDF, I should say, cumulative distribution function, and I'm gonna take seven trials and the probability of success in each trial is 0.35. And now when I type in four here, it doesn't mean what is the probability that I make exactly four free throws. It is the probability that I make zero, one, two, three, or four free throws."}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "And the reason why I write it this way is because using it this way, you can now use the binomial cumulative distribution function on my calculator. So if I just type in binom, and once again, I'm gonna take seven, or binom CDF, I should say, cumulative distribution function, and I'm gonna take seven trials and the probability of success in each trial is 0.35. And now when I type in four here, it doesn't mean what is the probability that I make exactly four free throws. It is the probability that I make zero, one, two, three, or four free throws. So all of the possible outcomes of my binomial random variable up to and including this value right over here. So let me get that, let me get my calculator back. So once again, I can go to second, distribution."}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "It is the probability that I make zero, one, two, three, or four free throws. So all of the possible outcomes of my binomial random variable up to and including this value right over here. So let me get that, let me get my calculator back. So once again, I can go to second, distribution. I'll scroll up to go to the bottom of the list and here you see it, binomial cumulative distribution function. So let me go there, click enter. And once again, seven trials."}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "So once again, I can go to second, distribution. I'll scroll up to go to the bottom of the list and here you see it, binomial cumulative distribution function. So let me go there, click enter. And once again, seven trials. My P is 0.35. And my X value is four, but now this is gonna give me the probability that my binomial random variable equals four. This is going to give me the probability that I get any value up to and including four."}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "And once again, seven trials. My P is 0.35. And my X value is four, but now this is gonna give me the probability that my binomial random variable equals four. This is going to give me the probability that I get any value up to and including four. So this should be a higher probability. And there you have it. It is approximately 0.94."}, {"video_title": "Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3", "Sentence": "This is going to give me the probability that I get any value up to and including four. So this should be a higher probability. And there you have it. It is approximately 0.94. So this is approximately 0.94. So hopefully you found that helpful. These calculators can be very useful, especially on something like an AP Stats exam."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "And homogeneity or homogeneity, in everyday language, this means how similar things are. And that's what we're essentially going to test here. We're gonna look at two different groups and see whether the distributions of those groups for a certain variable are similar or not. And so the question I'm going to think about or we're going to think about together in this video is, let's say we were thinking about left-handed versus right-handed people. And we're wondering, do they have the same preferences for subject domains? Are they equally inclined to science, technology, engineering, math, humanities, or neither? And so we can set up our null and alternative hypotheses."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "And so the question I'm going to think about or we're going to think about together in this video is, let's say we were thinking about left-handed versus right-handed people. And we're wondering, do they have the same preferences for subject domains? Are they equally inclined to science, technology, engineering, math, humanities, or neither? And so we can set up our null and alternative hypotheses. Our null hypothesis is that there is no difference in the distribution between left-handed and right-handed people in terms of their preference for subject domains. So no difference in subject, subject preference for left and right, for left and right-handed folks. And then the alternative hypothesis, well, no, there is a difference."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "And so we can set up our null and alternative hypotheses. Our null hypothesis is that there is no difference in the distribution between left-handed and right-handed people in terms of their preference for subject domains. So no difference in subject, subject preference for left and right, for left and right-handed folks. And then the alternative hypothesis, well, no, there is a difference. So there is a difference. So how would we go about testing this? Well, we've done hypothesis testing many times in many videos already."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "And then the alternative hypothesis, well, no, there is a difference. So there is a difference. So how would we go about testing this? Well, we've done hypothesis testing many times in many videos already. But here, we're going to sample from two different groups. So let's say that this is the population of right-handed folks, and this is the population of left-handed folks. Let's say from that sample of right-handed folks, I take a sample of 60, and then I do the same thing for the left-handed folks."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "Well, we've done hypothesis testing many times in many videos already. But here, we're going to sample from two different groups. So let's say that this is the population of right-handed folks, and this is the population of left-handed folks. Let's say from that sample of right-handed folks, I take a sample of 60, and then I do the same thing for the left-handed folks. And these don't even have to be the same sample sizes. So the left-handed folks, let's say I sample 40 folks. And here is the data that I actually collect."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "Let's say from that sample of right-handed folks, I take a sample of 60, and then I do the same thing for the left-handed folks. And these don't even have to be the same sample sizes. So the left-handed folks, let's say I sample 40 folks. And here is the data that I actually collect. So for those 60 right-handed folks, 30 of them preferred the STEM subjects, science, technology, engineering, math. 15 preferred humanities. And 15 were indifferent."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "And here is the data that I actually collect. So for those 60 right-handed folks, 30 of them preferred the STEM subjects, science, technology, engineering, math. 15 preferred humanities. And 15 were indifferent. They liked them equally. And then for the 40 left-handed folks, I got 10 preferring STEM, 25 preferring humanities, and five viewed them equally. And then you see the total number of right-handed folks, total number of left-handed folks."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "And 15 were indifferent. They liked them equally. And then for the 40 left-handed folks, I got 10 preferring STEM, 25 preferring humanities, and five viewed them equally. And then you see the total number of right-handed folks, total number of left-handed folks. And then you have the total number from both groups that preferred STEM, total number from both groups that preferred humanities, total number from both groups that had no preference. So let's just start thinking about what the expected data would be if we are assuming that the null hypothesis is true, that there's no difference in preference between right and left-handed folks. This is the right-handed column."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "And then you see the total number of right-handed folks, total number of left-handed folks. And then you have the total number from both groups that preferred STEM, total number from both groups that preferred humanities, total number from both groups that had no preference. So let's just start thinking about what the expected data would be if we are assuming that the null hypothesis is true, that there's no difference in preference between right and left-handed folks. This is the right-handed column. This is the left-handed column. Well, assuming that the null hypothesis is true, that there's no difference between right and left-handed people in terms of their preference, our best estimate of what the distribution of preference would be in the population generally would come from this total column. Since we're assuming no difference, we would assume that in either group, 40 out of every 100 would prefer STEM, or 40%."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "This is the right-handed column. This is the left-handed column. Well, assuming that the null hypothesis is true, that there's no difference between right and left-handed people in terms of their preference, our best estimate of what the distribution of preference would be in the population generally would come from this total column. Since we're assuming no difference, we would assume that in either group, 40 out of every 100 would prefer STEM, or 40%. 40% would prefer humanities. And 20% would have no preference. And so our expected would be that 40% of the 60 right-handed folks would prefer STEM."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "Since we're assuming no difference, we would assume that in either group, 40 out of every 100 would prefer STEM, or 40%. 40% would prefer humanities. And 20% would have no preference. And so our expected would be that 40% of the 60 right-handed folks would prefer STEM. So what's 40% of 60? .4 times 60 is 24. And similarly, we would expect 40% preferring humanities."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "And so our expected would be that 40% of the 60 right-handed folks would prefer STEM. So what's 40% of 60? .4 times 60 is 24. And similarly, we would expect 40% preferring humanities. 40% times 60 is 24 again. And then we would expect 20% of the right-handed group to have no preference. So 20% of 60 is 12."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "And similarly, we would expect 40% preferring humanities. 40% times 60 is 24 again. And then we would expect 20% of the right-handed group to have no preference. So 20% of 60 is 12. And these, once again, they add up to 60. And then for the left-handed folks, we would go through the same process. We would expect that 40% of them prefer STEM."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "So 20% of 60 is 12. And these, once again, they add up to 60. And then for the left-handed folks, we would go through the same process. We would expect that 40% of them prefer STEM. 40% of 40, that is 16. On the humanities, again, 40% of 40 is 16. And equal, 20% of 40 is eight."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "We would expect that 40% of them prefer STEM. 40% of 40, that is 16. On the humanities, again, 40% of 40 is 16. And equal, 20% of 40 is eight. And then all of these add up to 40. Once you calculate these expected values, it's a good time to make sure you're meeting your conditions for conducting a chi-squared test. The first is the random condition."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "And equal, 20% of 40 is eight. And then all of these add up to 40. Once you calculate these expected values, it's a good time to make sure you're meeting your conditions for conducting a chi-squared test. The first is the random condition. And so these need to be truly random samples. So hopefully we met that condition. The second is that the expected value for any of these data points have to be at least equal to five."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "The first is the random condition. And so these need to be truly random samples. So hopefully we met that condition. The second is that the expected value for any of these data points have to be at least equal to five. And so we have met that condition. These are all at least equal to five. And then the last condition is the independence condition that we are either sampling with replacement, or if we're not sampling with replacement, we have to feel good that our samples are no more than 10% of the population."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "The second is that the expected value for any of these data points have to be at least equal to five. And so we have met that condition. These are all at least equal to five. And then the last condition is the independence condition that we are either sampling with replacement, or if we're not sampling with replacement, we have to feel good that our samples are no more than 10% of the population. So let's assume that that is the case as well. And now we're ready to calculate our chi-squared statistic. We would get our chi-squared statistic is going to be equal to the difference between what we got and the expected squared, so 30 minus 24 squared, divided by the expected, divided by 24."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "And then the last condition is the independence condition that we are either sampling with replacement, or if we're not sampling with replacement, we have to feel good that our samples are no more than 10% of the population. So let's assume that that is the case as well. And now we're ready to calculate our chi-squared statistic. We would get our chi-squared statistic is going to be equal to the difference between what we got and the expected squared, so 30 minus 24 squared, divided by the expected, divided by 24. And we'll do it for all six of these data points. So then I will go to the next one. So then this is going to be, so plus, and if I look at this and this here, I'm going to have 10 minus 16 squared over expected, 16."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "We would get our chi-squared statistic is going to be equal to the difference between what we got and the expected squared, so 30 minus 24 squared, divided by the expected, divided by 24. And we'll do it for all six of these data points. So then I will go to the next one. So then this is going to be, so plus, and if I look at this and this here, I'm going to have 10 minus 16 squared over expected, 16. And then I'm going to have, I'll look at that data point and that expected, and I would get 15 minus 24 squared over expected, over 24. I'm running out of colors. And then we would look at that, those two numbers, and we would say plus 25 minus 16 squared divided by expected."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "So then this is going to be, so plus, and if I look at this and this here, I'm going to have 10 minus 16 squared over expected, 16. And then I'm going to have, I'll look at that data point and that expected, and I would get 15 minus 24 squared over expected, over 24. I'm running out of colors. And then we would look at that, those two numbers, and we would say plus 25 minus 16 squared divided by expected. And then we would get, we would look at these two, plus 15 minus 12 squared over expected, over 12. And then last but not least, let me find a color I haven't used, we would look at that and that, and we would say plus five minus eight squared over expected, over eight. Now once you get that value for the chi-square statistic, the next question is what are the degrees of freedom?"}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "And then we would look at that, those two numbers, and we would say plus 25 minus 16 squared divided by expected. And then we would get, we would look at these two, plus 15 minus 12 squared over expected, over 12. And then last but not least, let me find a color I haven't used, we would look at that and that, and we would say plus five minus eight squared over expected, over eight. Now once you get that value for the chi-square statistic, the next question is what are the degrees of freedom? Now a simple rule of thumb is to just look at your data and think about the number of rows and the number of columns. And we have three rows and two columns. And so your degrees of freedom are going to be the number of rows minus one, three minus one, times the number of columns minus one, two minus one."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "Now once you get that value for the chi-square statistic, the next question is what are the degrees of freedom? Now a simple rule of thumb is to just look at your data and think about the number of rows and the number of columns. And we have three rows and two columns. And so your degrees of freedom are going to be the number of rows minus one, three minus one, times the number of columns minus one, two minus one. And so this is going to be equal to two times one, which is equal to two. Now the reason why that makes intuitive sense is, think about it, if you knew two of these data points, and if you knew all of the totals, then you could figure out the other data points. If you knew these two data points, you could figure out that."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "And so your degrees of freedom are going to be the number of rows minus one, three minus one, times the number of columns minus one, two minus one. And so this is going to be equal to two times one, which is equal to two. Now the reason why that makes intuitive sense is, think about it, if you knew two of these data points, and if you knew all of the totals, then you could figure out the other data points. If you knew these two data points, you could figure out that. If you knew this data point and you knew the total, you could figure out that. If you knew this data point and you knew the total, you could figure out that. And if you figured out that and that, then you could figure out this right over here."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "If you knew these two data points, you could figure out that. If you knew this data point and you knew the total, you could figure out that. If you knew this data point and you knew the total, you could figure out that. And if you figured out that and that, then you could figure out this right over here. And so that's why this rule of thumb works. The number of rows minus one times the number of columns minus one gives you your degrees of freedom. Now, given this chi-squared statistic that I haven't calculated, but you could type this into a calculator and figure it out, and this degrees of freedom, we could then figure out the p-value."}, {"video_title": "Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3", "Sentence": "And if you figured out that and that, then you could figure out this right over here. And so that's why this rule of thumb works. The number of rows minus one times the number of columns minus one gives you your degrees of freedom. Now, given this chi-squared statistic that I haven't calculated, but you could type this into a calculator and figure it out, and this degrees of freedom, we could then figure out the p-value. We could figure out the probability of getting a chi-squared statistic this extreme or more extreme. And if this is less than our significance level, which we should have set ahead of time, then we would reject the null hypothesis and it would suggest the alternative. If this is not less than our significance level, then it does not allow us to reject the null hypothesis."}, {"video_title": "Probability for a geometric random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "Alright, this is the telltale signs of a geometric random variable. How many trials do I have to take until I get a success? Let m be the number of shots it takes Jeremiah to successfully make his first three-point shot. Okay, so they're defining the random variable here, the number of shots it takes, the number of trials it takes until we get a successful three-point shot. Assume that the results of each shot are independent. Alright, the probability that he makes a given shot is not dependent on whether he made or missed the previous shots. Find the probability that Jeremiah's first successful shot occurs on his third attempt."}, {"video_title": "Probability for a geometric random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "Okay, so they're defining the random variable here, the number of shots it takes, the number of trials it takes until we get a successful three-point shot. Assume that the results of each shot are independent. Alright, the probability that he makes a given shot is not dependent on whether he made or missed the previous shots. Find the probability that Jeremiah's first successful shot occurs on his third attempt. So like always, pause this video and see if you can have a go at it. Alright, now let's work through this together. So we wanna find the probability that, so m is the number of shots it takes until Jeremiah makes his first successful one."}, {"video_title": "Probability for a geometric random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "Find the probability that Jeremiah's first successful shot occurs on his third attempt. So like always, pause this video and see if you can have a go at it. Alright, now let's work through this together. So we wanna find the probability that, so m is the number of shots it takes until Jeremiah makes his first successful one. And so what they're really asking is find the probability that m is equal to three, that his first successful shot occurs on his third attempt. So m is equal to three. So that the number of shots it takes Jeremiah to make his first successful shot is three."}, {"video_title": "Probability for a geometric random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So we wanna find the probability that, so m is the number of shots it takes until Jeremiah makes his first successful one. And so what they're really asking is find the probability that m is equal to three, that his first successful shot occurs on his third attempt. So m is equal to three. So that the number of shots it takes Jeremiah to make his first successful shot is three. So how do we do this? Well, what's just the probability of that happening? Well, that means he has to miss his first two shots and then make his third shot."}, {"video_title": "Probability for a geometric random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So that the number of shots it takes Jeremiah to make his first successful shot is three. So how do we do this? Well, what's just the probability of that happening? Well, that means he has to miss his first two shots and then make his third shot. So what's the probability of him missing his first shot? Well, if he has a 1 4th chance of making his shots, he has a 3 4th probability of missing his shots. So this will be 3 4ths, so he misses the first shot, times he has to miss the second shot, and then he has to make his third shot."}, {"video_title": "Probability for a geometric random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, that means he has to miss his first two shots and then make his third shot. So what's the probability of him missing his first shot? Well, if he has a 1 4th chance of making his shots, he has a 3 4th probability of missing his shots. So this will be 3 4ths, so he misses the first shot, times he has to miss the second shot, and then he has to make his third shot. So there you have it, that's the probability. Miss, miss, make. And so what is this going to be?"}, {"video_title": "Probability for a geometric random variable Random variables AP Statistics Khan Academy.mp3", "Sentence": "So this will be 3 4ths, so he misses the first shot, times he has to miss the second shot, and then he has to make his third shot. So there you have it, that's the probability. Miss, miss, make. And so what is this going to be? This is equal to nine over 60 4ths. So there you have it. If you wanted to have this as a decimal, we could get a calculator out real fast."}, {"video_title": "Creating a bar chart Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "20 teachers were asked about their favorite course. Seven teachers said language, three teachers said history, nine teachers said geometry, one teacher said chemistry, zero teachers said physics. Create a bar chart showing everyone's favorite courses. So we've got the bar chart right over here, and let's see what we need to plot. So it said zero teachers said physics, which is surprising to me, because since physics is arguably my favorite course. But let's plot what the data has. So physics, so right now it looks like it's halfway between zero and one, so I actually have to bring the physics down to zero."}, {"video_title": "Creating a bar chart Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "So we've got the bar chart right over here, and let's see what we need to plot. So it said zero teachers said physics, which is surprising to me, because since physics is arguably my favorite course. But let's plot what the data has. So physics, so right now it looks like it's halfway between zero and one, so I actually have to bring the physics down to zero. See chemistry, they said, let's see, one teacher said chemistry, so we gotta bring chemistry up to one. Now, nine teachers said geometry. So geometry, let's bring that up to nine."}, {"video_title": "Creating a bar chart Applying mathematical reasoning Pre-Algebra Khan Academy.mp3", "Sentence": "So physics, so right now it looks like it's halfway between zero and one, so I actually have to bring the physics down to zero. See chemistry, they said, let's see, one teacher said chemistry, so we gotta bring chemistry up to one. Now, nine teachers said geometry. So geometry, let's bring that up to nine. One teacher said chemistry, oh, I already read that. History, history, three teachers said history, so let's bring history up to three. And then language, seven teachers said language."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "One of the most commonly used tools in all of statistics is the notion of a z-score. And one way to think about a z-score is it's just the number of standard deviations away from the mean that a certain data point is. So let me write that down. Number of standard deviations, I'll write it like this. Number of standard deviations from our population mean for a particular, particular data point. Now let's make that a little bit concrete. Let's say that you're some type of marine biologist and you've discovered a new species of winged turtles."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Number of standard deviations, I'll write it like this. Number of standard deviations from our population mean for a particular, particular data point. Now let's make that a little bit concrete. Let's say that you're some type of marine biologist and you've discovered a new species of winged turtles. And there's a total of seven winged turtles. The entire population of these winged turtles is seven. And so you go and you're actually able to measure all the winged turtles."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Let's say that you're some type of marine biologist and you've discovered a new species of winged turtles. And there's a total of seven winged turtles. The entire population of these winged turtles is seven. And so you go and you're actually able to measure all the winged turtles. So, and you care about their length. And you also wanna care about how are those lengths distributed? Lengths of winged turtles."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And so you go and you're actually able to measure all the winged turtles. So, and you care about their length. And you also wanna care about how are those lengths distributed? Lengths of winged turtles. All right. And let's say, and this is all in centimeters. These are very small turtles."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Lengths of winged turtles. All right. And let's say, and this is all in centimeters. These are very small turtles. So you discover, and these are all adults, so there's a two-centimeter one, there's another two-centimeter one, there's a three-centimeter one, there's another two-centimeter one, there's a five-centimeter one, a one-centimeter one, and a six-centimeter one. So we have seven data points. And from this, and you, I encourage you at any point, if you want, pause this video and see if you wanna calculate, what does the population mean here?"}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "These are very small turtles. So you discover, and these are all adults, so there's a two-centimeter one, there's another two-centimeter one, there's a three-centimeter one, there's another two-centimeter one, there's a five-centimeter one, a one-centimeter one, and a six-centimeter one. So we have seven data points. And from this, and you, I encourage you at any point, if you want, pause this video and see if you wanna calculate, what does the population mean here? We're assuming that this is the population of all the winged turtles. Well, the mean in this situation is going to be equal to, you could add up all of these numbers and divide by seven, and you would then get three. And then using these data points and the mean, you can calculate the population standard deviation."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And from this, and you, I encourage you at any point, if you want, pause this video and see if you wanna calculate, what does the population mean here? We're assuming that this is the population of all the winged turtles. Well, the mean in this situation is going to be equal to, you could add up all of these numbers and divide by seven, and you would then get three. And then using these data points and the mean, you can calculate the population standard deviation. And once again, as review, I always encourage you to pause this video and see if you can do it on your own, but I've calculated that ahead of time. The population standard deviation in this situation is approximately, I'll round to the hundredths place, 1.69. So with this information, you should be able to calculate the z-score for each of these data points."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And then using these data points and the mean, you can calculate the population standard deviation. And once again, as review, I always encourage you to pause this video and see if you can do it on your own, but I've calculated that ahead of time. The population standard deviation in this situation is approximately, I'll round to the hundredths place, 1.69. So with this information, you should be able to calculate the z-score for each of these data points. Pause this video and see if you can do that. So let me make a new column here. So here I'm gonna put our z-score."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So with this information, you should be able to calculate the z-score for each of these data points. Pause this video and see if you can do that. So let me make a new column here. So here I'm gonna put our z-score. And if you just look at the definition, what you're going to do for each of these data points, let's say each data point is x, you're going to subtract from that the mean, and then you're going to divide that by the standard deviation. The numerator ID over here is gonna tell you how far you are above or below the mean, but you wanna know how many standard deviations you are from the mean. So then you'll divide by the population standard deviation."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So here I'm gonna put our z-score. And if you just look at the definition, what you're going to do for each of these data points, let's say each data point is x, you're going to subtract from that the mean, and then you're going to divide that by the standard deviation. The numerator ID over here is gonna tell you how far you are above or below the mean, but you wanna know how many standard deviations you are from the mean. So then you'll divide by the population standard deviation. So for example, this first data point right over here, if I wanna calculate its z-score, I will take two. From that, I will subtract three, and then I will divide by 1.69. I will divide by 1.69."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So then you'll divide by the population standard deviation. So for example, this first data point right over here, if I wanna calculate its z-score, I will take two. From that, I will subtract three, and then I will divide by 1.69. I will divide by 1.69. And if you got a calculator out, this is going to be negative one divided by 1.69. And if you use a calculator, you would get this is going to be approximately negative 0.59. And the z-score for this data point is going to be the same."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "I will divide by 1.69. And if you got a calculator out, this is going to be negative one divided by 1.69. And if you use a calculator, you would get this is going to be approximately negative 0.59. And the z-score for this data point is going to be the same. That is also going to be negative 0.59. One way to interpret this is, this is a little bit more than half a standard deviation below the mean. And we could do a similar calculation for data points that are above the mean."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And the z-score for this data point is going to be the same. That is also going to be negative 0.59. One way to interpret this is, this is a little bit more than half a standard deviation below the mean. And we could do a similar calculation for data points that are above the mean. Let's say this data point right over here what is its z-score? Pause this video and see if you can figure that out. Well, it's going to be six minus our mean, so minus three."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And we could do a similar calculation for data points that are above the mean. Let's say this data point right over here what is its z-score? Pause this video and see if you can figure that out. Well, it's going to be six minus our mean, so minus three. All of that over the standard deviation. All of that over 1.69. And this, if you have a calculator, and I calculated it ahead of time, this is going to be approximately 1.77."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Well, it's going to be six minus our mean, so minus three. All of that over the standard deviation. All of that over 1.69. And this, if you have a calculator, and I calculated it ahead of time, this is going to be approximately 1.77. So more than one, but less than two standard deviations above the mean. I encourage you to pause this video and now try to figure out the z-scores for these other data points. Now an obvious question that some of you might be asking is why?"}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And this, if you have a calculator, and I calculated it ahead of time, this is going to be approximately 1.77. So more than one, but less than two standard deviations above the mean. I encourage you to pause this video and now try to figure out the z-scores for these other data points. Now an obvious question that some of you might be asking is why? Why do we care how many standard deviations above or below the mean a data point is? In your future statistical life, z-scores are going to be a really useful way to think about how usual or how unusual a certain data point is. And that's going to be really valuable once we start making inferences based on our data."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Now an obvious question that some of you might be asking is why? Why do we care how many standard deviations above or below the mean a data point is? In your future statistical life, z-scores are going to be a really useful way to think about how usual or how unusual a certain data point is. And that's going to be really valuable once we start making inferences based on our data. So I will leave you there. Just keep in mind, it's a very useful idea, but at the heart of it, a fairly simple one. If you know the mean, you know the standard deviation."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "What we're going to do in this video is start to compare distributions. So for example here we have two distributions that show the various temperatures different cities get during the month of January. This is the distribution for Portland, for example they get eight days between one and four degrees Celsius, they get 12 days between four and seven degrees Celsius, so forth and so on, and then this is the distribution for Minneapolis. Now when we make these comparisons, what we're going to focus on is the center of the distributions to compare that, and also the spread. Sometimes people will talk about the variability of the distributions, and so these are the things that we're going to compare. And in making the comparison, we're actually just going to try to eyeball it. We're not gonna try to pick a measure of central tendency, say the mean or the median, and then calculate precisely what those numbers are for these."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "Now when we make these comparisons, what we're going to focus on is the center of the distributions to compare that, and also the spread. Sometimes people will talk about the variability of the distributions, and so these are the things that we're going to compare. And in making the comparison, we're actually just going to try to eyeball it. We're not gonna try to pick a measure of central tendency, say the mean or the median, and then calculate precisely what those numbers are for these. We might wanna do those if they're close, but if we can eyeball it, that would be even better. Similar for the spread and variability. In either of these cases, there are multiple measures in our statistical toolkit."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "We're not gonna try to pick a measure of central tendency, say the mean or the median, and then calculate precisely what those numbers are for these. We might wanna do those if they're close, but if we can eyeball it, that would be even better. Similar for the spread and variability. In either of these cases, there are multiple measures in our statistical toolkit. Center, mean, median is, mean, median is valuable for the center. For spread variability, the range, the interquartile range, the mean absolute deviation, the standard deviation, these are all measures. But sometimes you can just kind of gauge it by looking."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "In either of these cases, there are multiple measures in our statistical toolkit. Center, mean, median is, mean, median is valuable for the center. For spread variability, the range, the interquartile range, the mean absolute deviation, the standard deviation, these are all measures. But sometimes you can just kind of gauge it by looking. So in this first comparison, which distribution has a higher center, or are they comparable? Well, if you look at the distribution for Portland, the center of this distribution, let's say if we were to just think about the mean, although I think the mean and the median would be reasonably close right over here, it seems like it would be around, it would be around seven, or maybe a little bit lower than seven, so it would be kind of in that range, maybe between five and seven, would be our central tendency, would be either our mean or our median. While for Minneapolis, it looks like our center is much closer to maybe negative two or negative three degrees Celsius."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "But sometimes you can just kind of gauge it by looking. So in this first comparison, which distribution has a higher center, or are they comparable? Well, if you look at the distribution for Portland, the center of this distribution, let's say if we were to just think about the mean, although I think the mean and the median would be reasonably close right over here, it seems like it would be around, it would be around seven, or maybe a little bit lower than seven, so it would be kind of in that range, maybe between five and seven, would be our central tendency, would be either our mean or our median. While for Minneapolis, it looks like our center is much closer to maybe negative two or negative three degrees Celsius. So here, even though we don't know precisely what the mean or the median is of each of these distributions, you can say that Portland, Portland distribution has a higher center, has higher center, however you wanna measure it, either mean or median. Now what about the spread or variability? Well, if you just superficially thought about range, you see here that there's nothing below one degree Celsius and nothing above 13, so you have about a 13-degree range at most right over here."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "While for Minneapolis, it looks like our center is much closer to maybe negative two or negative three degrees Celsius. So here, even though we don't know precisely what the mean or the median is of each of these distributions, you can say that Portland, Portland distribution has a higher center, has higher center, however you wanna measure it, either mean or median. Now what about the spread or variability? Well, if you just superficially thought about range, you see here that there's nothing below one degree Celsius and nothing above 13, so you have about a 13-degree range at most right over here. In fact, what might be contributing to this first column might be a bunch of things at three degrees or even 3.9 degrees, and similarly, what's contributing to this last column might be a bunch of things at 10.1 degrees, but at most, you have a 12-degree range right over here, while over here, it looks like you have, well, it looks like it's approaching a 27-degree range. So based on that, and even if you just eyeball it, this is just, we're using the same scales for our horizontal axes here, the temperature axes, and this is just a much wider distribution than what you see over here, and so you would say that the Minneapolis distribution has more spread or higher spread or more variability, so higher spread right over here. Let's do another example, and we'll use a different representation for the data here."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "Well, if you just superficially thought about range, you see here that there's nothing below one degree Celsius and nothing above 13, so you have about a 13-degree range at most right over here. In fact, what might be contributing to this first column might be a bunch of things at three degrees or even 3.9 degrees, and similarly, what's contributing to this last column might be a bunch of things at 10.1 degrees, but at most, you have a 12-degree range right over here, while over here, it looks like you have, well, it looks like it's approaching a 27-degree range. So based on that, and even if you just eyeball it, this is just, we're using the same scales for our horizontal axes here, the temperature axes, and this is just a much wider distribution than what you see over here, and so you would say that the Minneapolis distribution has more spread or higher spread or more variability, so higher spread right over here. Let's do another example, and we'll use a different representation for the data here. So we're told at the Olympic Games, many events have several rounds of competition. One of these events is the men's 100-meter backstroke. The upper dot plot shows the times in seconds of the top eight finishers in the final round of the 2012 Olympics, so that's in green right over here, the final round."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "Let's do another example, and we'll use a different representation for the data here. So we're told at the Olympic Games, many events have several rounds of competition. One of these events is the men's 100-meter backstroke. The upper dot plot shows the times in seconds of the top eight finishers in the final round of the 2012 Olympics, so that's in green right over here, the final round. The lower dot plot shows the times of the same eight swimmers but in the semifinal round. So given these distributions, which one has a higher center? Well, once again, I mean, and here, you can actually, it's a little bit easier to eyeball even what the median might be."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "The upper dot plot shows the times in seconds of the top eight finishers in the final round of the 2012 Olympics, so that's in green right over here, the final round. The lower dot plot shows the times of the same eight swimmers but in the semifinal round. So given these distributions, which one has a higher center? Well, once again, I mean, and here, you can actually, it's a little bit easier to eyeball even what the median might be. The mean, I would probably have to do a little bit more mathematics, but let's say the median, let's say there's one, two, three, four, five, six, seven, eight data points, so the median is gonna sit between the lower four and the upper four, so the central tendency right over here is for the final round, is looks like it's around 57.1 seconds, while the, especially if we think about the median, while the central tendency for the semifinal round, let's see, one, two, three, four, five, six, seven, eight, looks like it is right about there, so this is about 57, more than 57.3 seconds, so the semifinal round seems to have a higher central tendency, which is a little bit counterintuitive. You would expect the finalists to be running faster on average than the semifinalists, but that's not what this data is showing, so the semifinal round has higher center, higher, higher center, and I just eyeballed the median, and I suspect that the mean would also be higher in this second distribution, and now what about variability? Well, once again, if you just looked at range, and these are both at the same scale, if you just visually look, the variability here, the range for the final round, is larger than the range for the semifinal round, so you would say that the final round has higher variability, variability, it has a higher range."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "Well, once again, I mean, and here, you can actually, it's a little bit easier to eyeball even what the median might be. The mean, I would probably have to do a little bit more mathematics, but let's say the median, let's say there's one, two, three, four, five, six, seven, eight data points, so the median is gonna sit between the lower four and the upper four, so the central tendency right over here is for the final round, is looks like it's around 57.1 seconds, while the, especially if we think about the median, while the central tendency for the semifinal round, let's see, one, two, three, four, five, six, seven, eight, looks like it is right about there, so this is about 57, more than 57.3 seconds, so the semifinal round seems to have a higher central tendency, which is a little bit counterintuitive. You would expect the finalists to be running faster on average than the semifinalists, but that's not what this data is showing, so the semifinal round has higher center, higher, higher center, and I just eyeballed the median, and I suspect that the mean would also be higher in this second distribution, and now what about variability? Well, once again, if you just looked at range, and these are both at the same scale, if you just visually look, the variability here, the range for the final round, is larger than the range for the semifinal round, so you would say that the final round has higher variability, variability, it has a higher range. Eyeballing it, it looks like it has a higher spread, and there's, of course, times where one distribution could have a higher range, but then it might have a lower standard deviation. For example, you could have data that's like, you know, two data points that are really far apart, but then all of the other data just sits right, it's really, really closely packed, so for example, a distribution like this, and I'll draw the horizontal axis here, just so you can imagine it as a distribution. A distribution like this might have a higher range, but lower standard deviation than a distribution like this."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "Well, once again, if you just looked at range, and these are both at the same scale, if you just visually look, the variability here, the range for the final round, is larger than the range for the semifinal round, so you would say that the final round has higher variability, variability, it has a higher range. Eyeballing it, it looks like it has a higher spread, and there's, of course, times where one distribution could have a higher range, but then it might have a lower standard deviation. For example, you could have data that's like, you know, two data points that are really far apart, but then all of the other data just sits right, it's really, really closely packed, so for example, a distribution like this, and I'll draw the horizontal axis here, just so you can imagine it as a distribution. A distribution like this might have a higher range, but lower standard deviation than a distribution like this. Let me just, I'm just drawing a very rough example. A distribution like this has a lower range, but actually might have a higher standard deviation, might have a higher standard deviation than the one above it. In fact, I can make that even better."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "A distribution like this might have a higher range, but lower standard deviation than a distribution like this. Let me just, I'm just drawing a very rough example. A distribution like this has a lower range, but actually might have a higher standard deviation, might have a higher standard deviation than the one above it. In fact, I can make that even better. A distribution like this would have a lower range, but it would also have a higher standard deviation. So you can't just look at, it's not always the case that just by looking at one of these measures, the range or the standard deviation, you'll know for sure, but in cases like this, it's safe to say when you're looking at it by inspection that look, this green, the final round data does seem to have a higher range, higher variability, and so I'd feel pretty good at this. This is very high-level comparison."}, {"video_title": "Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Researchers used these results to test the null hypothesis is that the proportion is 0.5, the alternative hypothesis is that it's greater than 0.5, where P is the true proportion of adults that support the tax increase. They calculated a test statistic of z is approximately equal to 1.84 and a corresponding P value of approximately 0.033. Assuming the conditions for inference were met, which of these is an appropriate conclusion? And we have our four conclusions here. At any point, I encourage you to pause this video and see if you can answer it for yourself. But now we will do it together and just make sure we understand what's going on. Before we even cut to the chase and get to the answer."}, {"video_title": "Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "And we have our four conclusions here. At any point, I encourage you to pause this video and see if you can answer it for yourself. But now we will do it together and just make sure we understand what's going on. Before we even cut to the chase and get to the answer. So what we do is we have this population and we are going to sample it. So n is equal to 200. From that sample, we calculate a sample proportion of adults that support the tax increase."}, {"video_title": "Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "Before we even cut to the chase and get to the answer. So what we do is we have this population and we are going to sample it. So n is equal to 200. From that sample, we calculate a sample proportion of adults that support the tax increase. We see 113 out of 200 support it, which is going to be equal to, let's see, that is the same thing as 56.5%. So 56.5%. And so the key is is to figure out the P value."}, {"video_title": "Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "From that sample, we calculate a sample proportion of adults that support the tax increase. We see 113 out of 200 support it, which is going to be equal to, let's see, that is the same thing as 56.5%. So 56.5%. And so the key is is to figure out the P value. What is the probability of getting a result this much above the assumed proportion or greater, at least this much above the assumed proportion if we assume that the null hypothesis is true? And if that probability, if that P value is below a preset threshold, if it's below our significance level, they haven't told it to us yet, it looks like they're gonna give some in the choices, well, then we would reject the null hypothesis, which would suggest the alternative. If the P value is not lower than this, then we will fail to reject the null hypothesis."}, {"video_title": "Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "And so the key is is to figure out the P value. What is the probability of getting a result this much above the assumed proportion or greater, at least this much above the assumed proportion if we assume that the null hypothesis is true? And if that probability, if that P value is below a preset threshold, if it's below our significance level, they haven't told it to us yet, it looks like they're gonna give some in the choices, well, then we would reject the null hypothesis, which would suggest the alternative. If the P value is not lower than this, then we will fail to reject the null hypothesis. Now, to calculate that P value, to calculate that probability, what we figure out is, well, how many, in our sampling distribution, how many standard deviations above the mean of the sampling distribution, and the mean of the sampling distribution would be our assumed population proportion, how many standard deviations above that mean is this right over here? And that is what this test statistic is. And then we can use this to look at a z-table and say, all right, well, in a normal distribution, what percentage or what is the area under the normal curve that is further than 1.84 standard deviations or at least 1.84 or more standard deviations above the mean?"}, {"video_title": "Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "If the P value is not lower than this, then we will fail to reject the null hypothesis. Now, to calculate that P value, to calculate that probability, what we figure out is, well, how many, in our sampling distribution, how many standard deviations above the mean of the sampling distribution, and the mean of the sampling distribution would be our assumed population proportion, how many standard deviations above that mean is this right over here? And that is what this test statistic is. And then we can use this to look at a z-table and say, all right, well, in a normal distribution, what percentage or what is the area under the normal curve that is further than 1.84 standard deviations or at least 1.84 or more standard deviations above the mean? And they give that for us as well. So really, what we just need to do is compare this P value right over here to the significance level. If the P value is less than our significance level, then we reject, reject our null hypothesis, and that would suggest the alternative."}, {"video_title": "Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "And then we can use this to look at a z-table and say, all right, well, in a normal distribution, what percentage or what is the area under the normal curve that is further than 1.84 standard deviations or at least 1.84 or more standard deviations above the mean? And they give that for us as well. So really, what we just need to do is compare this P value right over here to the significance level. If the P value is less than our significance level, then we reject, reject our null hypothesis, and that would suggest the alternative. If this is not true, then we would fail to reject the null hypothesis. So let's look at these choices. And if you didn't answer it the first time, I encourage you to pause the video again."}, {"video_title": "Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "If the P value is less than our significance level, then we reject, reject our null hypothesis, and that would suggest the alternative. If this is not true, then we would fail to reject the null hypothesis. So let's look at these choices. And if you didn't answer it the first time, I encourage you to pause the video again. So at the alpha is equal to 0.01 significance level, they should conclude that more than 50% of adults support the tax increase. So if the alpha is 1 hundredth, the P value right over here is over 3 hundredths. It's roughly 3.3%."}, {"video_title": "Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "And if you didn't answer it the first time, I encourage you to pause the video again. So at the alpha is equal to 0.01 significance level, they should conclude that more than 50% of adults support the tax increase. So if the alpha is 1 hundredth, the P value right over here is over 3 hundredths. It's roughly 3.3%. So this is a situation where our P value, our P value is greater than or equal to alpha. In fact, it's definitely greater than alpha here. And so here we would fail to reject, we would fail to reject our null hypothesis."}, {"video_title": "Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "It's roughly 3.3%. So this is a situation where our P value, our P value is greater than or equal to alpha. In fact, it's definitely greater than alpha here. And so here we would fail to reject, we would fail to reject our null hypothesis. And so we wouldn't conclude that more than 50% of adults support the tax increase. Because remember, our null hypothesis is that 50% do, and we're failing to reject this. So that's not gonna be true."}, {"video_title": "Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "And so here we would fail to reject, we would fail to reject our null hypothesis. And so we wouldn't conclude that more than 50% of adults support the tax increase. Because remember, our null hypothesis is that 50% do, and we're failing to reject this. So that's not gonna be true. At that same significance level, they should conclude that less than 50% of adults support the tax increase. No, we can't say that either. We just failed to reject this null hypothesis, that the true proportion is 50%."}, {"video_title": "Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "So that's not gonna be true. At that same significance level, they should conclude that less than 50% of adults support the tax increase. No, we can't say that either. We just failed to reject this null hypothesis, that the true proportion is 50%. At the alpha is equal to, so at the alpha equals to 5 hundredth significance level, they should conclude that more than 50% of adults support the tax increase. Well yeah, in this situation, we have our P value, which is 0.033. It is indeed less than our significance level, in which case we reject, reject the null hypothesis."}, {"video_title": "Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3", "Sentence": "We just failed to reject this null hypothesis, that the true proportion is 50%. At the alpha is equal to, so at the alpha equals to 5 hundredth significance level, they should conclude that more than 50% of adults support the tax increase. Well yeah, in this situation, we have our P value, which is 0.033. It is indeed less than our significance level, in which case we reject, reject the null hypothesis. And if we reject the null hypothesis, that would suggest the alternative, that the true proportion is greater than 50%. And so I would pick this choice right over here. And then choice D, at that same significance level, they should conclude that less than 50% of adults support the tax increase."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "What I have here is the list of ages of the students in a class. And what I want to explore in this video is different ways of representing this data and then see if we can answer questions about the data. So the first way we can think about it is as a frequency table. Frequency table. Frequency table. And what we're going to do is we're going to look at each, for each age, for each possible age that we've measured here to see how many students in the class are of that age. So we could say the age is one column."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Frequency table. Frequency table. And what we're going to do is we're going to look at each, for each age, for each possible age that we've measured here to see how many students in the class are of that age. So we could say the age is one column. And then the number. The number of students of that age. Or we could even say the frequency."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So we could say the age is one column. And then the number. The number of students of that age. Or we could even say the frequency. Frequency. When people say, how frequent do you do something, they're saying, how often does it happen? How often do you do that thing?"}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Or we could even say the frequency. Frequency. When people say, how frequent do you do something, they're saying, how often does it happen? How often do you do that thing? Frequency. Or we could also say, actually, let me just write number. I'm always a fan of the simpler number at age, which we could also consider the frequency at that age."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "How often do you do that thing? Frequency. Or we could also say, actually, let me just write number. I'm always a fan of the simpler number at age, which we could also consider the frequency at that age. Frequency of students. All right. So what's the lowest age that we have here?"}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "I'm always a fan of the simpler number at age, which we could also consider the frequency at that age. Frequency of students. All right. So what's the lowest age that we have here? Well, the lowest age is five. So I'll start with five. And how many students in the class are age five?"}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So what's the lowest age that we have here? Well, the lowest age is five. So I'll start with five. And how many students in the class are age five? How frequent is the number five? Let's see. There is one, two."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And how many students in the class are age five? How frequent is the number five? Let's see. There is one, two. Let me keep scanning. Looks like there's only two fives. So I could write a two here."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "There is one, two. Let me keep scanning. Looks like there's only two fives. So I could write a two here. There are two fives. And now let's go to six. How many sixes are there?"}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So I could write a two here. There are two fives. And now let's go to six. How many sixes are there? Let's see. There is one sixth. There's only one six-year-old in the class."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "How many sixes are there? Let's see. There is one sixth. There's only one six-year-old in the class. All right. Seven-year-olds. See, there's one, two, three, four seven-year-olds."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "There's only one six-year-old in the class. All right. Seven-year-olds. See, there's one, two, three, four seven-year-olds. Now what about eight-year-olds? I'm going to do this in a color that I have not used yet. Eight-year-olds, we have no eight-year-olds."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "See, there's one, two, three, four seven-year-olds. Now what about eight-year-olds? I'm going to do this in a color that I have not used yet. Eight-year-olds, we have no eight-year-olds. Zero eight-year-olds. And then we have nine-year-olds. Let's see."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Eight-year-olds, we have no eight-year-olds. Zero eight-year-olds. And then we have nine-year-olds. Let's see. Nine-year-olds, we have one, two, three, four nine-year-olds. 10-year-olds, what do we have? We have one 10-year-old right over there."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let's see. Nine-year-olds, we have one, two, three, four nine-year-olds. 10-year-olds, what do we have? We have one 10-year-old right over there. And then 11-year-olds. There are no 11-year-olds. And then let me scroll up a little bit."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We have one 10-year-old right over there. And then 11-year-olds. There are no 11-year-olds. And then let me scroll up a little bit. And then finally 12-year-olds. 12-year-olds, there are one, two 12-year-olds. So what we have just constructed is a frequency table."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then let me scroll up a little bit. And then finally 12-year-olds. 12-year-olds, there are one, two 12-year-olds. So what we have just constructed is a frequency table. It's a frequency table. You can see for each age how many students are at that age. So it's giving you the same information as we have up here."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So what we have just constructed is a frequency table. It's a frequency table. You can see for each age how many students are at that age. So it's giving you the same information as we have up here. You could take this table and construct what we have up here. You would just write down two fives, one sixth, four sevens, no eights, four nines, one 10, no 11s, and two 12s. And then you would just have this list of numbers."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So it's giving you the same information as we have up here. You could take this table and construct what we have up here. You would just write down two fives, one sixth, four sevens, no eights, four nines, one 10, no 11s, and two 12s. And then you would just have this list of numbers. Now a way to visually look at a frequency table is a dot plot. So let me draw a dot plot right over here. So a dot plot."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then you would just have this list of numbers. Now a way to visually look at a frequency table is a dot plot. So let me draw a dot plot right over here. So a dot plot. And a dot plot, we essentially just take the same information and even think about it the same way, but we just show it visually. So in a dot plot, what we would have of, actually let me just not draw an even arrow there. We have the different age groups."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So a dot plot. And a dot plot, we essentially just take the same information and even think about it the same way, but we just show it visually. So in a dot plot, what we would have of, actually let me just not draw an even arrow there. We have the different age groups. So five, six, seven, eight, nine, 10, 11, and 12. And we have a dot to represent, or we use a dot for each student at that age. So there's two five-year-olds, so I'll do two dots."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We have the different age groups. So five, six, seven, eight, nine, 10, 11, and 12. And we have a dot to represent, or we use a dot for each student at that age. So there's two five-year-olds, so I'll do two dots. One and two. There's one six-year-old, so that's going to be one dot right over here. There's four seven-year-olds, so one, two, three, four dots."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So there's two five-year-olds, so I'll do two dots. One and two. There's one six-year-old, so that's going to be one dot right over here. There's four seven-year-olds, so one, two, three, four dots. There's no eight-year-olds. There's four nine-year-olds, so one, two, three, and four. There's one 10-year-old, so let's put a dot, one dot, right over there for that one 10-year-old."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "There's four seven-year-olds, so one, two, three, four dots. There's no eight-year-olds. There's four nine-year-olds, so one, two, three, and four. There's one 10-year-old, so let's put a dot, one dot, right over there for that one 10-year-old. There's no 11-year-olds, so I'm not going to put any dots there. And then there's two 12-year-olds. So one 12-year-old and another 12-year-old."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "There's one 10-year-old, so let's put a dot, one dot, right over there for that one 10-year-old. There's no 11-year-olds, so I'm not going to put any dots there. And then there's two 12-year-olds. So one 12-year-old and another 12-year-old. So there you go. We have frequency table, dot plot, list of numbers. These are all showing the same data, just in different ways."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So one 12-year-old and another 12-year-old. So there you go. We have frequency table, dot plot, list of numbers. These are all showing the same data, just in different ways. And once you have it represented in any of these ways, we can start to ask questions about it. So we could say, what is the most frequent age? Well, the most frequent age, when you look at it visually, or the easiest thing might be just look at the dot plot, because you see it visually, the most frequent age are the two highest stacks."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "These are all showing the same data, just in different ways. And once you have it represented in any of these ways, we can start to ask questions about it. So we could say, what is the most frequent age? Well, the most frequent age, when you look at it visually, or the easiest thing might be just look at the dot plot, because you see it visually, the most frequent age are the two highest stacks. So there's actually seven and nine are tied for the most frequent age. You would have also seen it here, where seven and nine are tied at four. And if you just had this data, you'd have to count all of them to kind of come up with this again and say, OK, there's four sevens, four nines."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, the most frequent age, when you look at it visually, or the easiest thing might be just look at the dot plot, because you see it visually, the most frequent age are the two highest stacks. So there's actually seven and nine are tied for the most frequent age. You would have also seen it here, where seven and nine are tied at four. And if you just had this data, you'd have to count all of them to kind of come up with this again and say, OK, there's four sevens, four nines. That's the largest number. So if you're looking for what's the most frequent age, when you just visually inspect here, it probably pops out at you the fastest. But there's other questions we can ask ourselves."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And if you just had this data, you'd have to count all of them to kind of come up with this again and say, OK, there's four sevens, four nines. That's the largest number. So if you're looking for what's the most frequent age, when you just visually inspect here, it probably pops out at you the fastest. But there's other questions we can ask ourselves. We can ask ourselves, what is the range of ages in the classroom? And this is, once again, where maybe the dot plot is the most, it jumps out at you the most, because the range is just the maximum age in your, or the maximum data point minus the minimum data point. So what's the maximum age here?"}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "But there's other questions we can ask ourselves. We can ask ourselves, what is the range of ages in the classroom? And this is, once again, where maybe the dot plot is the most, it jumps out at you the most, because the range is just the maximum age in your, or the maximum data point minus the minimum data point. So what's the maximum age here? Well, the maximum age here, we see it from the dot plot, is 12. And the minimum age here, you see, is five. So there's a range of seven."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So what's the maximum age here? Well, the maximum age here, we see it from the dot plot, is 12. And the minimum age here, you see, is five. So there's a range of seven. The difference between the maximum and the minimum is seven. But you could have also done that over here. You could say, hey, the maximum age here is 12."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So there's a range of seven. The difference between the maximum and the minimum is seven. But you could have also done that over here. You could say, hey, the maximum age here is 12. Minimum age here is five. And so you find the difference between 12 and five, which is seven. Here, you still could have done it."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "You could say, hey, the maximum age here is 12. Minimum age here is five. And so you find the difference between 12 and five, which is seven. Here, you still could have done it. You could say, OK, what's the lowest? Let's look at five. Are there any fours here?"}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Here, you still could have done it. You could say, OK, what's the lowest? Let's look at five. Are there any fours here? Nope, there's no fours. So five's the minimum age. And what's the largest?"}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Are there any fours here? Nope, there's no fours. So five's the minimum age. And what's the largest? Is it seven? No, is it nine? Nine, not even 10?"}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And what's the largest? Is it seven? No, is it nine? Nine, not even 10? Oh, 12. 12. Are there any 13s?"}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Nine, not even 10? Oh, 12. 12. Are there any 13s? No, 12 is the maximum. So you say 12 minus five is seven to get the range. But then we could ask ourselves other questions."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Are there any 13s? No, 12 is the maximum. So you say 12 minus five is seven to get the range. But then we could ask ourselves other questions. We could say, how many older than nine is a question we could ask ourselves. And then if we were to look at the dot plot, we say, OK, this is nine. And we care about how many are older than nine."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "But then we could ask ourselves other questions. We could say, how many older than nine is a question we could ask ourselves. And then if we were to look at the dot plot, we say, OK, this is nine. And we care about how many are older than nine. So that would be this one, two, and three. Or you could look over here. How many are older than nine?"}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And we care about how many are older than nine. So that would be this one, two, and three. Or you could look over here. How many are older than nine? Well, it's the one person who's 10, and then the two who are 12. So there are three. And over here, if you said how many are older than nine, well, then you would just have to go through the list and say, OK, no, no, no, no, no, no, no, no."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "How many are older than nine? Well, it's the one person who's 10, and then the two who are 12. So there are three. And over here, if you said how many are older than nine, well, then you would just have to go through the list and say, OK, no, no, no, no, no, no, no, no. OK, here, one, two, three. And then not that person right over there. So hopefully, this is just an appreciation for yet another two ways of looking at data, frequency tables and dot plots."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "Choose all answers that apply. Pause this video, have a go at that. All right, now let's go through each of these choices. So choice A says the probability that one of these tickets wins a prize is 0.95 on average. Well, I see where they're getting that 0.95. They're getting it from right over here, but that's not the probability that you're winning. That's the expected return."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "So choice A says the probability that one of these tickets wins a prize is 0.95 on average. Well, I see where they're getting that 0.95. They're getting it from right over here, but that's not the probability that you're winning. That's the expected return. The probability that you win is much lower. If the odds are one to 50, that means that the probability of winning is one to 51. So it's a much lower probability than this right over here."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "That's the expected return. The probability that you win is much lower. If the odds are one to 50, that means that the probability of winning is one to 51. So it's a much lower probability than this right over here. So definitely rule that out. Someone who buys this ticket is most likely to win 95 cents. That is not necessarily the case either."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "So it's a much lower probability than this right over here. So definitely rule that out. Someone who buys this ticket is most likely to win 95 cents. That is not necessarily the case either. We don't know what the different outcomes are for the prize. It's very likely that there's no outcome for that prize where you win exactly 95 cents. Instead, there's likely to be outcomes that are much larger than that with very low probabilities, and then when you take the weighted average of all of the outcomes, then you get an expected return of 95 cents."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "That is not necessarily the case either. We don't know what the different outcomes are for the prize. It's very likely that there's no outcome for that prize where you win exactly 95 cents. Instead, there's likely to be outcomes that are much larger than that with very low probabilities, and then when you take the weighted average of all of the outcomes, then you get an expected return of 95 cents. So it's actually maybe even impossible to win exactly 95 cents, so I would rule that out. If we looked at many of these tickets, the average return would be about 95 cents per ticket. That one feels pretty interesting because we're looking at many of these tickets, and so across many of them, you would expect to, on average, get the expected return as your return."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "Instead, there's likely to be outcomes that are much larger than that with very low probabilities, and then when you take the weighted average of all of the outcomes, then you get an expected return of 95 cents. So it's actually maybe even impossible to win exactly 95 cents, so I would rule that out. If we looked at many of these tickets, the average return would be about 95 cents per ticket. That one feels pretty interesting because we're looking at many of these tickets, and so across many of them, you would expect to, on average, get the expected return as your return. And so this is what we are saying here. The average return would be about that, would be approximately that. So I like that choice."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "That one feels pretty interesting because we're looking at many of these tickets, and so across many of them, you would expect to, on average, get the expected return as your return. And so this is what we are saying here. The average return would be about that, would be approximately that. So I like that choice. That is a good interpretation of expected value. And then choice D, if 1,000 people each bought one of these tickets, they'd expect a net gain of about $950 in total. This one is tempting."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "So I like that choice. That is a good interpretation of expected value. And then choice D, if 1,000 people each bought one of these tickets, they'd expect a net gain of about $950 in total. This one is tempting. Instead of net gain, if it just said return, this would make a lot of sense. In fact, it would be completely consistent with choice C. If you have 1,000 people, that would be many tickets, and if on average, if their average return is about 95 cents per ticket, then their total return would be about $950. But they didn't write return here."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "This one is tempting. Instead of net gain, if it just said return, this would make a lot of sense. In fact, it would be completely consistent with choice C. If you have 1,000 people, that would be many tickets, and if on average, if their average return is about 95 cents per ticket, then their total return would be about $950. But they didn't write return here. They wrote net gain. Net gain would be how much you get minus how much you paid. And 1,000 people would have to pay, if they each got a ticket, would pay $2,000."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "But they didn't write return here. They wrote net gain. Net gain would be how much you get minus how much you paid. And 1,000 people would have to pay, if they each got a ticket, would pay $2,000. So they would pay 2,000. They would expect a return of $950. Their net gain would actually be negative $1,050."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "After randomization, each child was asked to watch a cartoon in a private room, containing a large bowl of Goldfish crackers. The cartoon included two commercial breaks. The first group watched food commercials, mostly snacks, while the second group watched non-food commercials, games and entertainment products. Once the child finished watching the cartoon, the conductors of the experiment weighed the cracker bowls to measure how many grams of crackers the child ate. They found that the mean amount of crackers eaten by the children who watched food commercials is 10 grams greater than the mean amount of crackers eaten by the children who watched non-food commercials. So let's just think about what happens up to this point. They took 500 children and then they randomly assigned them to two different groups."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "Once the child finished watching the cartoon, the conductors of the experiment weighed the cracker bowls to measure how many grams of crackers the child ate. They found that the mean amount of crackers eaten by the children who watched food commercials is 10 grams greater than the mean amount of crackers eaten by the children who watched non-food commercials. So let's just think about what happens up to this point. They took 500 children and then they randomly assigned them to two different groups. So you have group 1 over here and you have group 2. So let's say that this right over here is the first group. The first group watched food commercials."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "They took 500 children and then they randomly assigned them to two different groups. So you have group 1 over here and you have group 2. So let's say that this right over here is the first group. The first group watched food commercials. So this is group number 1. So they watched food commercials. We could call this the treatment group."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "The first group watched food commercials. So this is group number 1. So they watched food commercials. We could call this the treatment group. We're trying to see what's the effect of watching food commercials. And then they tell us the second group watched non-food commercials. So this is the control group."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "We could call this the treatment group. We're trying to see what's the effect of watching food commercials. And then they tell us the second group watched non-food commercials. So this is the control group. So number 2, this is non-food commercials. So this is the control right over here. Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "So this is the control group. So number 2, this is non-food commercials. So this is the control right over here. Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here. Which, just looking at that data, makes you believe that something maybe happened over here. That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers. But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance?"}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here. Which, just looking at that data, makes you believe that something maybe happened over here. That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers. But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance? That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one. It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more. So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance?"}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance? That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one. It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more. So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance? Well, the way you do it is what they do right over here. Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups. They repeated the simulation 150 times and plotted the resulting differences as given below."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance? Well, the way you do it is what they do right over here. Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups. They repeated the simulation 150 times and plotted the resulting differences as given below. So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500. And for each child, they measured how much was the weight of the crackers that they ate. So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "They repeated the simulation 150 times and plotted the resulting differences as given below. So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500. And for each child, they measured how much was the weight of the crackers that they ate. So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams. And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group. But what they're doing now is they're taking these same results and they're re-randomizing it. So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams. And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group. But what they're doing now is they're taking these same results and they're re-randomizing it. So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1. So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means? So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean. So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1. So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means? So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean. So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups. So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely. So this is, let's see, is this the difference, the difference between the means of the new groups. So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups. So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely. So this is, let's see, is this the difference, the difference between the means of the new groups. So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times. So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times. There's 150 dots here. So that is on the order of 2%, or actually it's less than 2%."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times. So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times. There's 150 dots here. So that is on the order of 2%, or actually it's less than 2%. It's between 1 and 2%. And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times. It happened less frequently than 1 in 100 times."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "So that is on the order of 2%, or actually it's less than 2%. It's between 1 and 2%. And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times. It happened less frequently than 1 in 100 times. It happened only 1 in 150 times. So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%. So to me, and then to most statisticians, that tells us that our experiment was significant, that the probability of getting the results that you got, so the children who watch food commercials being 10 grams greater than the mean amount of crackers eaten by the children who watch non-food commercials, if you just randomly put 500 kids into 2 different buckets based on the simulation results, it looks like there's only, if you run the simulation 150 times, that only happened 1 out of 150 times."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "It happened less frequently than 1 in 100 times. It happened only 1 in 150 times. So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%. So to me, and then to most statisticians, that tells us that our experiment was significant, that the probability of getting the results that you got, so the children who watch food commercials being 10 grams greater than the mean amount of crackers eaten by the children who watch non-food commercials, if you just randomly put 500 kids into 2 different buckets based on the simulation results, it looks like there's only, if you run the simulation 150 times, that only happened 1 out of 150 times. So it seems like this was very, it's very unlikely that this was purely due to chance. If this was just a chance event, this would only happen roughly 1 in 150 times. But the fact that this happened in your experiment makes you feel pretty confident that your experiment is significant."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "So I have some data here in a spreadsheet. You could use Microsoft Excel or you could use Google Spreadsheets. And we're gonna use the spreadsheet to quickly calculate some parameters. Let's say this is the population. Let's say this is, we're looking at a population of students and we wanna calculate some parameters, and this is their ages, and we wanna calculate some parameters on that. And so, first I'm gonna calculate it using the spreadsheet. And then we're gonna think about how those parameters change as we do things to the data."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "Let's say this is the population. Let's say this is, we're looking at a population of students and we wanna calculate some parameters, and this is their ages, and we wanna calculate some parameters on that. And so, first I'm gonna calculate it using the spreadsheet. And then we're gonna think about how those parameters change as we do things to the data. If we were to shift the data up or down, or if we were to multiply all the points by some value, what does that do to the actual parameters? So the first parameter I'm gonna calculate is the mean. Then I'm gonna calculate the standard deviation."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "And then we're gonna think about how those parameters change as we do things to the data. If we were to shift the data up or down, or if we were to multiply all the points by some value, what does that do to the actual parameters? So the first parameter I'm gonna calculate is the mean. Then I'm gonna calculate the standard deviation. Then I wanna calculate the median. And then I wanna calculate, let's say the interquartile range. Inter, I'll call it IQR."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "Then I'm gonna calculate the standard deviation. Then I wanna calculate the median. And then I wanna calculate, let's say the interquartile range. Inter, I'll call it IQR. So let's do this. Let's first look at the measures of central tendency. So the mean, the function on most spreadsheets is the average function."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "Inter, I'll call it IQR. So let's do this. Let's first look at the measures of central tendency. So the mean, the function on most spreadsheets is the average function. And then I could use my mouse and select all of these, or I could press shift with my arrow button and select all of those. Okay, that's the mean of that data. Now let's think about what happens if I take all of that data and if I were to add a fixed amount to it."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "So the mean, the function on most spreadsheets is the average function. And then I could use my mouse and select all of these, or I could press shift with my arrow button and select all of those. Okay, that's the mean of that data. Now let's think about what happens if I take all of that data and if I were to add a fixed amount to it. So if I took all the data and if I were to add five to it. So an easy way to do that in a spreadsheet is you select that, you add five, and then I can scroll down. And notice, for every data point I had before, I now have five more than that."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "Now let's think about what happens if I take all of that data and if I were to add a fixed amount to it. So if I took all the data and if I were to add five to it. So an easy way to do that in a spreadsheet is you select that, you add five, and then I can scroll down. And notice, for every data point I had before, I now have five more than that. So this is my new data set, which I'm calling data plus five. And let's see what the mean of that is. So the mean of that, notice, is exactly five more."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "And notice, for every data point I had before, I now have five more than that. So this is my new data set, which I'm calling data plus five. And let's see what the mean of that is. So the mean of that, notice, is exactly five more. And the same would have been true if I added or subtracted any number. The mean would change by the amount that I add or subtract. And so, and that shouldn't surprise you, because when you're calculating the mean, you're adding all the numbers up and you're dividing by the numbers you have."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "So the mean of that, notice, is exactly five more. And the same would have been true if I added or subtracted any number. The mean would change by the amount that I add or subtract. And so, and that shouldn't surprise you, because when you're calculating the mean, you're adding all the numbers up and you're dividing by the numbers you have. And so, if all the numbers are five more, you're gonna add five, in this case, how many numbers are there? One, two, three, four, five, six, seven, eight, nine, 10, 11, 12. You're gonna add 12 more fives and then you're gonna divide by 12, and so it makes sense that your mean goes up by five."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "And so, and that shouldn't surprise you, because when you're calculating the mean, you're adding all the numbers up and you're dividing by the numbers you have. And so, if all the numbers are five more, you're gonna add five, in this case, how many numbers are there? One, two, three, four, five, six, seven, eight, nine, 10, 11, 12. You're gonna add 12 more fives and then you're gonna divide by 12, and so it makes sense that your mean goes up by five. Let's think about how the mean changes if you multiply. So if you take your data and if I were to multiply it times five, what happens? So this equals this times five."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "You're gonna add 12 more fives and then you're gonna divide by 12, and so it makes sense that your mean goes up by five. Let's think about how the mean changes if you multiply. So if you take your data and if I were to multiply it times five, what happens? So this equals this times five. So now, all the data points are five times more. Now what happens to my mean? Notice, my mean is now five times as much."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "So this equals this times five. So now, all the data points are five times more. Now what happens to my mean? Notice, my mean is now five times as much. So the measures of central tendency, if I add or subtract, well, I'm gonna add or subtract the mean by that amount. And if I scale it up by five or if I scaled it down by five, well, my mean would scale up or down by that same amount. And if you numerically looked at how you calculate a mean, it would make sense that this is happening mathematically."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "Notice, my mean is now five times as much. So the measures of central tendency, if I add or subtract, well, I'm gonna add or subtract the mean by that amount. And if I scale it up by five or if I scaled it down by five, well, my mean would scale up or down by that same amount. And if you numerically looked at how you calculate a mean, it would make sense that this is happening mathematically. Now let's look at the other measure, the other typical measure of central tendency, and that is the median, to see if that has the same properties. So let's calculate the median here. So once again, you wanna order these numbers and just find the middle number, which isn't too hard, but a computer can do it awfully fast."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "And if you numerically looked at how you calculate a mean, it would make sense that this is happening mathematically. Now let's look at the other measure, the other typical measure of central tendency, and that is the median, to see if that has the same properties. So let's calculate the median here. So once again, you wanna order these numbers and just find the middle number, which isn't too hard, but a computer can do it awfully fast. So that's the median for that data set. What do you think the median's gonna be if you take all of the data plus five? Well, the middle number, if you ordered all of these numbers and made them all five more, the orders, you could think of it as being the same order, but now the one in the middle is gonna be five more."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "So once again, you wanna order these numbers and just find the middle number, which isn't too hard, but a computer can do it awfully fast. So that's the median for that data set. What do you think the median's gonna be if you take all of the data plus five? Well, the middle number, if you ordered all of these numbers and made them all five more, the orders, you could think of it as being the same order, but now the one in the middle is gonna be five more. So this should be 10.5. And yes, it is indeed 10.5. And what would happen if you multiply everything by five?"}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "Well, the middle number, if you ordered all of these numbers and made them all five more, the orders, you could think of it as being the same order, but now the one in the middle is gonna be five more. So this should be 10.5. And yes, it is indeed 10.5. And what would happen if you multiply everything by five? Well, once again, you still have the same ordering, and so it should just multiply that by five. Yep, the middle number is now gonna be five times larger. So both of these measures of central tendency, if you shift all the data points or if you scale them up, you're going to similarly shift or scale up these measures of central tendency."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "And what would happen if you multiply everything by five? Well, once again, you still have the same ordering, and so it should just multiply that by five. Yep, the middle number is now gonna be five times larger. So both of these measures of central tendency, if you shift all the data points or if you scale them up, you're going to similarly shift or scale up these measures of central tendency. Now let's think about these measures of spread. See if that's the same with these measures of spread. So standard deviation."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "So both of these measures of central tendency, if you shift all the data points or if you scale them up, you're going to similarly shift or scale up these measures of central tendency. Now let's think about these measures of spread. See if that's the same with these measures of spread. So standard deviation. So S-T-D-E-V, I'm gonna take the population standard deviation. I'm assuming that this is my entire population. So let me, why is it, so let me make sure I'm doing, so standard deviation of all of this is going to be 2.99."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "So standard deviation. So S-T-D-E-V, I'm gonna take the population standard deviation. I'm assuming that this is my entire population. So let me, why is it, so let me make sure I'm doing, so standard deviation of all of this is going to be 2.99. Let's see what happens when I shift everything by five. Actually, pause the video. What do you think is going to happen?"}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "So let me, why is it, so let me make sure I'm doing, so standard deviation of all of this is going to be 2.99. Let's see what happens when I shift everything by five. Actually, pause the video. What do you think is going to happen? This is a measure of spread. So if you shift, I'll tell you what I think. If I shift everything by the same amount, the mean shifts, but the distance of everything from the mean should not change."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "What do you think is going to happen? This is a measure of spread. So if you shift, I'll tell you what I think. If I shift everything by the same amount, the mean shifts, but the distance of everything from the mean should not change. So the standard deviation should not change, I don't think, in this example. And indeed, it does not change. So if we shift the data sets, in this case we shifted it up by five, or if we shifted it down by one, your measure of spread, in this case standard deviation, should not change, or at least the standard deviation measure of spread does not change."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "If I shift everything by the same amount, the mean shifts, but the distance of everything from the mean should not change. So the standard deviation should not change, I don't think, in this example. And indeed, it does not change. So if we shift the data sets, in this case we shifted it up by five, or if we shifted it down by one, your measure of spread, in this case standard deviation, should not change, or at least the standard deviation measure of spread does not change. But if we scale it, well, I think it should change, because you can imagine a very simple data set, the things that were a certain amount of distance from the mean are now going to be five times further from the mean. So I think this actually should, we should multiply by five here, and it does look like that is the case. If I multiply this by five."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "So if we shift the data sets, in this case we shifted it up by five, or if we shifted it down by one, your measure of spread, in this case standard deviation, should not change, or at least the standard deviation measure of spread does not change. But if we scale it, well, I think it should change, because you can imagine a very simple data set, the things that were a certain amount of distance from the mean are now going to be five times further from the mean. So I think this actually should, we should multiply by five here, and it does look like that is the case. If I multiply this by five. So scaling the data set will scale the standard deviation in a similar way. What about interquartile range? Where it's essentially we're taking the third quartile and subtracting from that the first quartile to figure out kind of the range of the middle 50%."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "If I multiply this by five. So scaling the data set will scale the standard deviation in a similar way. What about interquartile range? Where it's essentially we're taking the third quartile and subtracting from that the first quartile to figure out kind of the range of the middle 50%. And so let's do that. We can have the quartile function equals quartile, and then we wanna look at our data, and we want the third quartile. So that's gonna calculate the third quartile minus quartile, same data set."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "Where it's essentially we're taking the third quartile and subtracting from that the first quartile to figure out kind of the range of the middle 50%. And so let's do that. We can have the quartile function equals quartile, and then we wanna look at our data, and we want the third quartile. So that's gonna calculate the third quartile minus quartile, same data set. So now we wanna select it again. So same data set, but this is now gonna be the first quartile. So this is gonna give us our interquartile range."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "So that's gonna calculate the third quartile minus quartile, same data set. So now we wanna select it again. So same data set, but this is now gonna be the first quartile. So this is gonna give us our interquartile range. This is the, calculates the third quartile on that data set, and this calculates the first quartile on that data set. And we get 2.75. Now let's think about what the inter, whether the interquartile range should change."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "So this is gonna give us our interquartile range. This is the, calculates the third quartile on that data set, and this calculates the first quartile on that data set. And we get 2.75. Now let's think about what the inter, whether the interquartile range should change. And I don't think it will, because remember, everything shifts. And even though the first quartile is gonna be five more, but the third quartile is gonna be five more as well. So the difference shouldn't change."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "Now let's think about what the inter, whether the interquartile range should change. And I don't think it will, because remember, everything shifts. And even though the first quartile is gonna be five more, but the third quartile is gonna be five more as well. So the difference shouldn't change. And indeed, look, the distance does not change, or the difference does not change. But similarly, if we scale everything up, if we were to scale up the first quartile and the third quartile by five, well then their difference should scale up by five. And we see that right over there."}, {"video_title": "How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3", "Sentence": "So the difference shouldn't change. And indeed, look, the distance does not change, or the difference does not change. But similarly, if we scale everything up, if we were to scale up the first quartile and the third quartile by five, well then their difference should scale up by five. And we see that right over there. So the big takeaway here, and I just use the example of shifting up by five and scaling up by five, but you could subtract by any number, and you could divide by a number as well. The typical measures of central tendency, mean and median, they both shift and scale as you shift and scale the data. But your typical measures of spread, standard deviation and interquartile range, they don't change if you shift the data, but they do change and they scale as you scale the data."}, {"video_title": "Conditions for inference for difference of means AP Statistics Khan Academy.mp3", "Sentence": "Which of the following are conditions for this type of interval? So before I even look at these choices, and they say choose all answers that apply, so it might be more than one, let's just think about what the conditions for inference for this type of interval actually are. So we've done this many times in many different contexts, and so we first of all have the random condition. And that's the idea that each of our samples are random, or we are conducting some type of an experiment where we randomly assign folks to one, or eggs in this case, to one of two groups. In this case, we are taking two samples, and we would hope that they are truly random samples. The second is the normal condition. And the normal condition is a little bit different depending on whether we're talking about means or whether we're talking about proportions."}, {"video_title": "Conditions for inference for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And that's the idea that each of our samples are random, or we are conducting some type of an experiment where we randomly assign folks to one, or eggs in this case, to one of two groups. In this case, we are taking two samples, and we would hope that they are truly random samples. The second is the normal condition. And the normal condition is a little bit different depending on whether we're talking about means or whether we're talking about proportions. The random condition is essentially the same. The normal condition when we're talking about means, remember, they're looking at the difference between mean weights of eggs, is you would want your, there's actually several ways to meet the normal condition. One is if the underlying distribution is normal."}, {"video_title": "Conditions for inference for difference of means AP Statistics Khan Academy.mp3", "Sentence": "And the normal condition is a little bit different depending on whether we're talking about means or whether we're talking about proportions. The random condition is essentially the same. The normal condition when we're talking about means, remember, they're looking at the difference between mean weights of eggs, is you would want your, there's actually several ways to meet the normal condition. One is if the underlying distribution is normal. The second way is if your sample sizes for each of your samples are greater than or equal to 30. So if your first sample size is greater than or equal to 30 and your second sample size is greater than or equal to 30, or even if the underlying data, you don't know whether it's normal or if it isn't normal, and even if you aren't able to meet these, as long as your sample data is roughly symmetric and not skewed heavily in one way or the other, then that also roughly meets the normal condition. When we're dealing with means."}, {"video_title": "Conditions for inference for difference of means AP Statistics Khan Academy.mp3", "Sentence": "One is if the underlying distribution is normal. The second way is if your sample sizes for each of your samples are greater than or equal to 30. So if your first sample size is greater than or equal to 30 and your second sample size is greater than or equal to 30, or even if the underlying data, you don't know whether it's normal or if it isn't normal, and even if you aren't able to meet these, as long as your sample data is roughly symmetric and not skewed heavily in one way or the other, then that also roughly meets the normal condition. When we're dealing with means. And then the third condition, and we see this whether we're dealing with means or proportions or differences of means or differences of proportions, is the independence condition. And this is the idea that either your individual observations are done with replacement in both of your samples or that the sample size for both of your samples is no more than 10% of the population, then you have met this condition. So with that little bit of a review, let's see which of these apply."}, {"video_title": "Conditions for inference for difference of means AP Statistics Khan Academy.mp3", "Sentence": "When we're dealing with means. And then the third condition, and we see this whether we're dealing with means or proportions or differences of means or differences of proportions, is the independence condition. And this is the idea that either your individual observations are done with replacement in both of your samples or that the sample size for both of your samples is no more than 10% of the population, then you have met this condition. So with that little bit of a review, let's see which of these apply. They observe at least 10 heavy eggs and 10 light eggs in each sample. So this actually is the normal condition when we are dealing with proportions, not for means. So I would rule this out."}, {"video_title": "Conditions for inference for difference of means AP Statistics Khan Academy.mp3", "Sentence": "So with that little bit of a review, let's see which of these apply. They observe at least 10 heavy eggs and 10 light eggs in each sample. So this actually is the normal condition when we are dealing with proportions, not for means. So I would rule this out. It's a good distractor choice. The eggs in each sample are randomly selected from their population. Yep, that's the random condition right over there, so I would select that."}, {"video_title": "Conditions for inference for difference of means AP Statistics Khan Academy.mp3", "Sentence": "So I would rule this out. It's a good distractor choice. The eggs in each sample are randomly selected from their population. Yep, that's the random condition right over there, so I would select that. They sample an equal number of each type of egg. So this is a common misconception that whether we're dealing with means or proportions, when we're thinking about the difference between, say, means or the difference between proportions, that somehow your sample sizes have to be the exact same. That is not the case."}, {"video_title": "Conditions for inference for difference of means AP Statistics Khan Academy.mp3", "Sentence": "Yep, that's the random condition right over there, so I would select that. They sample an equal number of each type of egg. So this is a common misconception that whether we're dealing with means or proportions, when we're thinking about the difference between, say, means or the difference between proportions, that somehow your sample sizes have to be the exact same. That is not the case. Your sample sizes do not have to be the exact case or do not have to be the exact same. So we would rule this out as well. So right over here, they have listed the random condition."}, {"video_title": "Writing hypotheses for a significance test about a mean AP Statistics Khan Academy.mp3", "Sentence": "The amount in the sample had a mean of 503 milliliters and a standard deviation of five milliliters. They want to test if this is convincing evidence that the mean amount for bottles in this batch is different than the target value of 500 milliliters. Let mu be the mean amount of liquid in each bottle in the batch. Write an appropriate set of hypotheses for their significance test, for the significance test that the quality-control expert is running. So pause this video and see if you can do that. All right, now let's do this together. So first, you're going to have two hypotheses."}, {"video_title": "Writing hypotheses for a significance test about a mean AP Statistics Khan Academy.mp3", "Sentence": "Write an appropriate set of hypotheses for their significance test, for the significance test that the quality-control expert is running. So pause this video and see if you can do that. All right, now let's do this together. So first, you're going to have two hypotheses. You're gonna have your null hypothesis and your alternative hypothesis. Your null hypothesis is going to be a hypothesis about the population parameter that you care about, and it's going to assume kind of the status quo, no news here. And so the parameter that we care about is the mean amount of liquid in the bottles in the batch."}, {"video_title": "Writing hypotheses for a significance test about a mean AP Statistics Khan Academy.mp3", "Sentence": "So first, you're going to have two hypotheses. You're gonna have your null hypothesis and your alternative hypothesis. Your null hypothesis is going to be a hypothesis about the population parameter that you care about, and it's going to assume kind of the status quo, no news here. And so the parameter that we care about is the mean amount of liquid in the bottles in the batch. So that's mu right over there. And what would be the assumption that that would be, the no news here? Well, it would be 500 milliliters."}, {"video_title": "Writing hypotheses for a significance test about a mean AP Statistics Khan Academy.mp3", "Sentence": "And so the parameter that we care about is the mean amount of liquid in the bottles in the batch. So that's mu right over there. And what would be the assumption that that would be, the no news here? Well, it would be 500 milliliters. That's the target value. So it's reasonable to say, well, you know, the null is it's doing what it's supposed to, that where the actual mean for the batch is actually what the target needs to be, is actually 500 milliliters. Some of you might have said, hey, wait, didn't they say the amounts in the sample had a mean of 503 milliliters?"}, {"video_title": "Writing hypotheses for a significance test about a mean AP Statistics Khan Academy.mp3", "Sentence": "Well, it would be 500 milliliters. That's the target value. So it's reasonable to say, well, you know, the null is it's doing what it's supposed to, that where the actual mean for the batch is actually what the target needs to be, is actually 500 milliliters. Some of you might have said, hey, wait, didn't they say the amounts in the sample had a mean of 503 milliliters? Why isn't this 503? Remember, your hypothesis is going to be about the population parameter, your assumption about the population parameter. This 503 milliliters right over here, this is a sample statistic."}, {"video_title": "Writing hypotheses for a significance test about a mean AP Statistics Khan Academy.mp3", "Sentence": "Some of you might have said, hey, wait, didn't they say the amounts in the sample had a mean of 503 milliliters? Why isn't this 503? Remember, your hypothesis is going to be about the population parameter, your assumption about the population parameter. This 503 milliliters right over here, this is a sample statistic. This is a sample mean that's trying to estimate this thing right over here. When we do our significance test, we're going to incorporate this 533 milliliters. We're going to think about, well, what's the probability of getting a sample statistic, a sample mean, this far or further away from the assumed mean if we assume that the null hypothesis is true, and if that probability is below a threshold, our significance level, then we reject the null hypothesis, and it would suggest the alternative."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "What we're going to do in this video is get some practice classifying whether a random variable is a binomial variable, and we're gonna do it by looking at a few exercises from Khan Academy. So this says a manager oversees 11 female employees and nine male employees. They need to pick three of these employees to go on a business trip, so the manager places all 20 names in a hat and chooses at random. Let x equal the number of female employees chosen. So they're going to do three trials, and on each of those trials, you could say success is if they pick a female employee, and then the random variable x is the number of females out of those three. Is x a binomial variable? Why or why not?"}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Let x equal the number of female employees chosen. So they're going to do three trials, and on each of those trials, you could say success is if they pick a female employee, and then the random variable x is the number of females out of those three. Is x a binomial variable? Why or why not? So pause this video and see if you can work through this on your own. All right, now let's go through each choice. Choice A says each trial isn't being classified as a success or failure, so x is not a binomial variable."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Why or why not? So pause this video and see if you can work through this on your own. All right, now let's go through each choice. Choice A says each trial isn't being classified as a success or failure, so x is not a binomial variable. I disagree with this. Each trial is being classified as a success or failure. It's either going to be female or not, and since we're counting the number of female employees, if in a trial we pick a female, that would be a success, so each trial is being classified as a success or failure."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Choice A says each trial isn't being classified as a success or failure, so x is not a binomial variable. I disagree with this. Each trial is being classified as a success or failure. It's either going to be female or not, and since we're counting the number of female employees, if in a trial we pick a female, that would be a success, so each trial is being classified as a success or failure. So this one over here isn't true. There is no fixed number of trials, so x is not a binomial variable. There is a fixed number of trials."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "It's either going to be female or not, and since we're counting the number of female employees, if in a trial we pick a female, that would be a success, so each trial is being classified as a success or failure. So this one over here isn't true. There is no fixed number of trials, so x is not a binomial variable. There is a fixed number of trials. They're doing three trials. They're picking three hats, three names out of a hat. The trials are not independent, so x is not a binomial variable."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "There is a fixed number of trials. They're doing three trials. They're picking three hats, three names out of a hat. The trials are not independent, so x is not a binomial variable. So this is interesting. So for example, trial one, what's the probability of success? Well, there are 20 employees, 20 names in the hat, and 11 of the outcomes would be success, so you have 11 20th probability of success, but in trial two, what's the probability of success given success in, I'll say, trial one, T1?"}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "The trials are not independent, so x is not a binomial variable. So this is interesting. So for example, trial one, what's the probability of success? Well, there are 20 employees, 20 names in the hat, and 11 of the outcomes would be success, so you have 11 20th probability of success, but in trial two, what's the probability of success given success in, I'll say, trial one, T1? Well, if you succeeded in trial one, that means that there's now only 10 female names in the hat out of 19, and if you don't have success in trial one, then you will have, then it'll be 11 out of 19, so your probability does change based on previous outcomes, and so the trials are not independent, and so x is not a binomial variable. So this one is true. The trials are not independent, so that violates that condition for being a binomial variable."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, there are 20 employees, 20 names in the hat, and 11 of the outcomes would be success, so you have 11 20th probability of success, but in trial two, what's the probability of success given success in, I'll say, trial one, T1? Well, if you succeeded in trial one, that means that there's now only 10 female names in the hat out of 19, and if you don't have success in trial one, then you will have, then it'll be 11 out of 19, so your probability does change based on previous outcomes, and so the trials are not independent, and so x is not a binomial variable. So this one is true. The trials are not independent, so that violates that condition for being a binomial variable. In order to be a binomial variable, all your trials have to be independent of each other, and so we'd rule this last one out because this last one says that x has a binomial distribution, or it is or does meet all the conditions for being a binomial variable. Let's do another example. So here we have different scenarios, and I have the conditions for binomial variable written right over here, and so once again, pause the video and look at each of these scenarios for random variables, and look at these conditions, and think about whether these random variables are binomial or not."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "The trials are not independent, so that violates that condition for being a binomial variable. In order to be a binomial variable, all your trials have to be independent of each other, and so we'd rule this last one out because this last one says that x has a binomial distribution, or it is or does meet all the conditions for being a binomial variable. Let's do another example. So here we have different scenarios, and I have the conditions for binomial variable written right over here, and so once again, pause the video and look at each of these scenarios for random variables, and look at these conditions, and think about whether these random variables are binomial or not. So let's look at the first one. In a game involving a standard deck of 52 playing cards, an individual randomly draws seven cards without replacement. Let y be equal to the number of aces drawn."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "So here we have different scenarios, and I have the conditions for binomial variable written right over here, and so once again, pause the video and look at each of these scenarios for random variables, and look at these conditions, and think about whether these random variables are binomial or not. So let's look at the first one. In a game involving a standard deck of 52 playing cards, an individual randomly draws seven cards without replacement. Let y be equal to the number of aces drawn. Well, in our introductory video to binomial variables, we talked about if we're doing without replacement, your probability of getting an ace on a given trial, where trial is you're taking a card out of the deck, it's going to be dependent on whether you got aces in previous trials, because if you got an ace in a previous trial, well, that ace, then you're gonna have fewer aces in the decks. So the trials in this case are not independent. Not independent, not independent trials."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Let y be equal to the number of aces drawn. Well, in our introductory video to binomial variables, we talked about if we're doing without replacement, your probability of getting an ace on a given trial, where trial is you're taking a card out of the deck, it's going to be dependent on whether you got aces in previous trials, because if you got an ace in a previous trial, well, that ace, then you're gonna have fewer aces in the decks. So the trials in this case are not independent. Not independent, not independent trials. Now, on the other hand, if on every trial you looked at whatever card you got and put it back in the deck, then they would be independent trials. The probability of getting an ace on each trial would be the same, but not when you have without replacement. So this is not binomial right over here, because you don't have independent trials."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Not independent, not independent trials. Now, on the other hand, if on every trial you looked at whatever card you got and put it back in the deck, then they would be independent trials. The probability of getting an ace on each trial would be the same, but not when you have without replacement. So this is not binomial right over here, because you don't have independent trials. The second scenario. 60% of a certain species of tomato live after transplanting from pot to garden. Eli transplants 16 of these tomato plants."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "So this is not binomial right over here, because you don't have independent trials. The second scenario. 60% of a certain species of tomato live after transplanting from pot to garden. Eli transplants 16 of these tomato plants. Assume that the plants live independently of each other. So whether one plant lives isn't dependent on whether another plant lives. Let T equal the number of tomato plants that live."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Eli transplants 16 of these tomato plants. Assume that the plants live independently of each other. So whether one plant lives isn't dependent on whether another plant lives. Let T equal the number of tomato plants that live. All right, so let's look at the conditions. The outcome of each trial can be classified as either success or failure. So each trial over here is one of the tomato plants, and we have 16 of those trials, and success is if the tomato plant lives, and failure is if it dies."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Let T equal the number of tomato plants that live. All right, so let's look at the conditions. The outcome of each trial can be classified as either success or failure. So each trial over here is one of the tomato plants, and we have 16 of those trials, and success is if the tomato plant lives, and failure is if it dies. So we have either success or failure. Each trial is independent of the others. They tell us the plants live independently of each other."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "So each trial over here is one of the tomato plants, and we have 16 of those trials, and success is if the tomato plant lives, and failure is if it dies. So we have either success or failure. Each trial is independent of the others. They tell us the plants live independently of each other. So whether or not a neighboring plant lives or dies doesn't affect whether the plant next to it lives or dies. So each trial is independent of the others. There is a fixed number of trials."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "They tell us the plants live independently of each other. So whether or not a neighboring plant lives or dies doesn't affect whether the plant next to it lives or dies. So each trial is independent of the others. There is a fixed number of trials. Yes, we have 16 right over there. The probability P of success on each trial remains constant. Well, yeah, according to at least the scenario, they're saying that we have a 60% chance for each tomato plant, which is each trial."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "There is a fixed number of trials. Yes, we have 16 right over there. The probability P of success on each trial remains constant. Well, yeah, according to at least the scenario, they're saying that we have a 60% chance for each tomato plant, which is each trial. So it meets all of the conditions right over here. So this one is binomial. Now let's look at this third scenario."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, yeah, according to at least the scenario, they're saying that we have a 60% chance for each tomato plant, which is each trial. So it meets all of the conditions right over here. So this one is binomial. Now let's look at this third scenario. In a game of luck, a turn consists of a player continuing to roll a pair of six-sided die until they roll a double, two of the same face values. Let x equal the number of rolls in one turn. So you're gonna keep rolling until they roll a double."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Now let's look at this third scenario. In a game of luck, a turn consists of a player continuing to roll a pair of six-sided die until they roll a double, two of the same face values. Let x equal the number of rolls in one turn. So you're gonna keep rolling until they roll a double. Well, the thing that jumps out at me is that you don't have a fixed number of trials, not fixed number of trials. You could say each trial, each roll is a trial. Success is getting a double, which has a fixed probability."}, {"video_title": "Recognizing binomial variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "So you're gonna keep rolling until they roll a double. Well, the thing that jumps out at me is that you don't have a fixed number of trials, not fixed number of trials. You could say each trial, each roll is a trial. Success is getting a double, which has a fixed probability. Whether or not you get a double on each trial is gonna be independent of the previous roll. So it meets all the other constraints, but it does not meet that there's a fixed number of trials. You're gonna keep someone, there's some chance you might have to roll 20 times, or 200 times, or who knows however many times, until they roll a double."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "Let's say that your school has a population of 80 students in it. Maybe it's not your whole school, maybe it's just your grade. So there's 80 students in your population, and you want to get an estimate of the average height in your population. And you think it's too hard for you to go and measure the height of all 80 students, so you decide to find a simple, or take a simple random sample. You think it's reasonable for you to measure the heights of 30 of these students. And so what you want to do is randomly sample 30 of the 80 students, and take their average height, and say, well, that's probably a pretty good estimate for the population parameter, for the average height of the entire population. So once you decide to do this, you say, well, how do I select those 30 students, and how do I select it so that I feel good that it is actually random?"}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "And you think it's too hard for you to go and measure the height of all 80 students, so you decide to find a simple, or take a simple random sample. You think it's reasonable for you to measure the heights of 30 of these students. And so what you want to do is randomly sample 30 of the 80 students, and take their average height, and say, well, that's probably a pretty good estimate for the population parameter, for the average height of the entire population. So once you decide to do this, you say, well, how do I select those 30 students, and how do I select it so that I feel good that it is actually random? And there's several ways that you could approach this. One way to do it is associate every person in your school with a piece of paper, and put them all in a bowl, and then pick them out. So let's do that."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "So once you decide to do this, you say, well, how do I select those 30 students, and how do I select it so that I feel good that it is actually random? And there's several ways that you could approach this. One way to do it is associate every person in your school with a piece of paper, and put them all in a bowl, and then pick them out. So let's do that. So let's say this is alphabetically the first person in the school, they're on a slip of paper, then the next slip of paper gets the next person, and you're gonna go all the way down, so you're gonna have 80 pieces of paper. They all should be the same size. And then you throw them all, you throw them all into a bowl of some kind."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "So let's do that. So let's say this is alphabetically the first person in the school, they're on a slip of paper, then the next slip of paper gets the next person, and you're gonna go all the way down, so you're gonna have 80 pieces of paper. They all should be the same size. And then you throw them all, you throw them all into a bowl of some kind. And this seems like a very basic way of doing it, but it's actually a pretty effective way of getting a simple, of getting a simple random sample. So I'll try to draw a little, I don't know, it looks like a little fishbowl or something. All right, so that's our bowl."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "And then you throw them all, you throw them all into a bowl of some kind. And this seems like a very basic way of doing it, but it's actually a pretty effective way of getting a simple, of getting a simple random sample. So I'll try to draw a little, I don't know, it looks like a little fishbowl or something. All right, so that's our bowl. And so all the pieces of paper go in there. And then you put a blindfold on someone, and they can't feel what names are there, and so they should pick out the first 30 without replacing them, because you obviously don't wanna pick the same, you don't wanna pick out the same name twice. And those 30 names that you pick, that would be your simple random sample, and then you could measure their heights to estimate the average height for the population."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "All right, so that's our bowl. And so all the pieces of paper go in there. And then you put a blindfold on someone, and they can't feel what names are there, and so they should pick out the first 30 without replacing them, because you obviously don't wanna pick the same, you don't wanna pick out the same name twice. And those 30 names that you pick, that would be your simple random sample, and then you could measure their heights to estimate the average height for the population. This would be a completely legitimate way of doing it. Other ways that you could do it, if you have a computer or a calculator, you could use a random number generator. And the random functions on computer programming languages or on your calculator, they tend to be something, you know, some place you'll see something like a math.rand, short for random."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "And those 30 names that you pick, that would be your simple random sample, and then you could measure their heights to estimate the average height for the population. This would be a completely legitimate way of doing it. Other ways that you could do it, if you have a computer or a calculator, you could use a random number generator. And the random functions on computer programming languages or on your calculator, they tend to be something, you know, some place you'll see something like a math.rand, short for random. You might see something like random. You might see, you might see something like random. Without anything passed into it, it might give you a number between zero and one, or zero and 100."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "And the random functions on computer programming languages or on your calculator, they tend to be something, you know, some place you'll see something like a math.rand, short for random. You might see something like random. You might see, you might see something like random. Without anything passed into it, it might give you a number between zero and one, or zero and 100. And you have to be very careful on how you use this to make sure that you have an even chance of picking certain numbers. But what you would do in this situation, if you had access to some random number generator, and it could even pick out a random number between one and 80, including one and 80, is you would maybe line up all the students' names alphabetically. And so the first student alphabetically, assign the number zero, one."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "Without anything passed into it, it might give you a number between zero and one, or zero and 100. And you have to be very careful on how you use this to make sure that you have an even chance of picking certain numbers. But what you would do in this situation, if you had access to some random number generator, and it could even pick out a random number between one and 80, including one and 80, is you would maybe line up all the students' names alphabetically. And so the first student alphabetically, assign the number zero, one. And you could just say one if you're using a random number generator, but I'll use two digits for it just because it'll be useful and consistent. And in a little bit, we'll use another technique where it's gonna be nice to be consistent with our number of digits. And so the next one, zero, two, and you go all the way to 79, and all the way to 80."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "And so the first student alphabetically, assign the number zero, one. And you could just say one if you're using a random number generator, but I'll use two digits for it just because it'll be useful and consistent. And in a little bit, we'll use another technique where it's gonna be nice to be consistent with our number of digits. And so the next one, zero, two, and you go all the way to 79, and all the way to 80. And then you use your random number generator to keep generating numbers from one to 80. And as long as you don't get repeats, you pick the first 30 to be your actual random sample. Another related technique, which is a little bit more old school, but is definitely the way that it has been done in the past and even done now sometimes, is to use a random digit table."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "And so the next one, zero, two, and you go all the way to 79, and all the way to 80. And then you use your random number generator to keep generating numbers from one to 80. And as long as you don't get repeats, you pick the first 30 to be your actual random sample. Another related technique, which is a little bit more old school, but is definitely the way that it has been done in the past and even done now sometimes, is to use a random digit table. You still start with these number associations with each student in the class, and then you use a randomly generated list of numbers. And so let's say that's our randomly generated list of numbers, and it keeps going well beyond this. And you start at the beginning."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "Another related technique, which is a little bit more old school, but is definitely the way that it has been done in the past and even done now sometimes, is to use a random digit table. You still start with these number associations with each student in the class, and then you use a randomly generated list of numbers. And so let's say that's our randomly generated list of numbers, and it keeps going well beyond this. And you start at the beginning. And you say, okay, we're interested in getting 30 two-digit numbers from one to 80, including one in 80. So one technique that you could use is you start it right at the beginning, and you could say, all right, this is a randomly generated list of numbers. So the first number here is 59."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "And you start at the beginning. And you say, okay, we're interested in getting 30 two-digit numbers from one to 80, including one in 80. So one technique that you could use is you start it right at the beginning, and you could say, all right, this is a randomly generated list of numbers. So the first number here is 59. Is 59 between one and 80? Sure is. As long as we, you know, if this was a zero one, that would have worked."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "So the first number here is 59. Is 59 between one and 80? Sure is. As long as we, you know, if this was a zero one, that would have worked. If this was an eight zero, that would have worked. If this was a zero zero, it wouldn't have worked. If this was an eight one, it wouldn't have worked."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "As long as we, you know, if this was a zero one, that would have worked. If this was an eight zero, that would have worked. If this was a zero zero, it wouldn't have worked. If this was an eight one, it wouldn't have worked. But this would be our, this right over here, that would be our first name that we, you could imagine the same as picking that first name out of the hat, whoever's associated with number 59. Now, you would move on. You get the next two digits."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "If this was an eight one, it wouldn't have worked. But this would be our, this right over here, that would be our first name that we, you could imagine the same as picking that first name out of the hat, whoever's associated with number 59. Now, you would move on. You get the next two digits. The next two digits are 83. They don't fall into our range from one to 80, so we're not going to use it. Then you look at the next two digits."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "You get the next two digits. The next two digits are 83. They don't fall into our range from one to 80, so we're not going to use it. Then you look at the next two digits. So we get a five and a nine. Well, that fits in our range, but we already picked 59. We already picked person 59, so we're not gonna pick 59 again."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "Then you look at the next two digits. So we get a five and a nine. Well, that fits in our range, but we already picked 59. We already picked person 59, so we're not gonna pick 59 again. So we keep moving on. Then we get a 37. Well, that's in our range."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "We already picked person 59, so we're not gonna pick 59 again. So we keep moving on. Then we get a 37. Well, that's in our range. We haven't picked that yet. We do that. Then we get a zero zero."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "Well, that's in our range. We haven't picked that yet. We do that. Then we get a zero zero. Once again, not in our range. I think you see where this is going. 91, not in our range."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "Then we get a zero zero. Once again, not in our range. I think you see where this is going. 91, not in our range. 23, it's in our range, and we haven't picked it yet. So we're gonna pick the 23. I think you see where this is going."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "91, not in our range. 23, it's in our range, and we haven't picked it yet. So we're gonna pick the 23. I think you see where this is going. We're gonna keep going down this list in the way that I've just described until we get 30 of these. We've just gotten three. We just have to keep on going."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "I think you see where this is going. We're gonna keep going down this list in the way that I've just described until we get 30 of these. We've just gotten three. We just have to keep on going. And this isn't an exhaustive list of all of the different ways that you can get random numbers, but it starts to give you some techniques in your toolkit. And you might say, oh, well, why don't I just randomly come up with some numbers in my head? And I would really suggest that you don't do that because humans are famously bad at being truly random."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "What we're going to do in this video is think about under which conditions does the sampling distribution of the sample proportions? In which situations does it look roughly normal? And under which situations does it look skewed right? So does it look something like this? And under which situations does it look skewed left? Maybe something like that. And the conditions that we're going to talk about, and this is a rough rule of thumb, that if we take our sample size and we multiply it by the population proportion that we care about, and that is greater than or equal to 10, and if we take the sample size and we multiply it times one minus the population proportion, and that also is greater than or equal to 10, if both of these are true, the rule of thumb tells us that this is going to be approximately normal in shape, the sampling distribution of the sample proportions."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "So does it look something like this? And under which situations does it look skewed left? Maybe something like that. And the conditions that we're going to talk about, and this is a rough rule of thumb, that if we take our sample size and we multiply it by the population proportion that we care about, and that is greater than or equal to 10, and if we take the sample size and we multiply it times one minus the population proportion, and that also is greater than or equal to 10, if both of these are true, the rule of thumb tells us that this is going to be approximately normal in shape, the sampling distribution of the sample proportions. So with that in our minds, let's do some examples here. So this first example says, Emiliana runs a restaurant that receives a shipment of 50 tangerines every day. According to the supplier, approximately 12% of the population of these tangerines is overripe."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "And the conditions that we're going to talk about, and this is a rough rule of thumb, that if we take our sample size and we multiply it by the population proportion that we care about, and that is greater than or equal to 10, and if we take the sample size and we multiply it times one minus the population proportion, and that also is greater than or equal to 10, if both of these are true, the rule of thumb tells us that this is going to be approximately normal in shape, the sampling distribution of the sample proportions. So with that in our minds, let's do some examples here. So this first example says, Emiliana runs a restaurant that receives a shipment of 50 tangerines every day. According to the supplier, approximately 12% of the population of these tangerines is overripe. Suppose that Emiliana calculates the daily proportion of overripe tangerines in her sample of 50. We can assume the supplier's claim is true and that the tangerines each day represent a random sample. What will be the shape of the sampling distribution, what will be the shape of the sampling distribution of the daily proportions of overripe tangerines?"}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "According to the supplier, approximately 12% of the population of these tangerines is overripe. Suppose that Emiliana calculates the daily proportion of overripe tangerines in her sample of 50. We can assume the supplier's claim is true and that the tangerines each day represent a random sample. What will be the shape of the sampling distribution, what will be the shape of the sampling distribution of the daily proportions of overripe tangerines? Pause this video, think about what we just talked about, and see if you can answer this. All right, so right over here, we're getting daily samples of 50 tangerines. So for this particular example, our n is equal to 50, and our population proportion, the proportion that is overripe is 12%, so p is 0.12."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "What will be the shape of the sampling distribution, what will be the shape of the sampling distribution of the daily proportions of overripe tangerines? Pause this video, think about what we just talked about, and see if you can answer this. All right, so right over here, we're getting daily samples of 50 tangerines. So for this particular example, our n is equal to 50, and our population proportion, the proportion that is overripe is 12%, so p is 0.12. So if we take n times p, what do we get? np is equal to 50 times 0.12. Well, 100 times this would be 12, so 50 times this is going to be equal to six."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "So for this particular example, our n is equal to 50, and our population proportion, the proportion that is overripe is 12%, so p is 0.12. So if we take n times p, what do we get? np is equal to 50 times 0.12. Well, 100 times this would be 12, so 50 times this is going to be equal to six. And this is less than or equal to 10. So this immediately violates this first condition, and so we know that we're not going to be dealing with a normal distribution. And so the question is, how is it going to be skewed?"}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "Well, 100 times this would be 12, so 50 times this is going to be equal to six. And this is less than or equal to 10. So this immediately violates this first condition, and so we know that we're not going to be dealing with a normal distribution. And so the question is, how is it going to be skewed? And the key realization is, remember, the mean of the sample proportions, of the sampling distribution of the sample proportions, or the mean of the sampling distribution of the daily proportions, that that's going to be the same thing as our population proportion. So the mean is going to be 12%. So if I were to draw it, let me see if I were to draw it right over here, where this is 50%, and this is 100%, our mean is gonna be right over here at 12%."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "And so the question is, how is it going to be skewed? And the key realization is, remember, the mean of the sample proportions, of the sampling distribution of the sample proportions, or the mean of the sampling distribution of the daily proportions, that that's going to be the same thing as our population proportion. So the mean is going to be 12%. So if I were to draw it, let me see if I were to draw it right over here, where this is 50%, and this is 100%, our mean is gonna be right over here at 12%. And so you're gonna have it really high over there, and then it's gonna be skewed to the right. You're gonna have a big, long tail. So this is going to be skewed to the right."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "So if I were to draw it, let me see if I were to draw it right over here, where this is 50%, and this is 100%, our mean is gonna be right over here at 12%. And so you're gonna have it really high over there, and then it's gonna be skewed to the right. You're gonna have a big, long tail. So this is going to be skewed to the right. Let's do another example. So here we're told, according to a Nielsen survey, radio reaches 88% of children each week. Suppose we took weekly random samples of N equals 125 children from this population, and computed the proportion of children in each sample whom radio reaches."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "So this is going to be skewed to the right. Let's do another example. So here we're told, according to a Nielsen survey, radio reaches 88% of children each week. Suppose we took weekly random samples of N equals 125 children from this population, and computed the proportion of children in each sample whom radio reaches. What will be the shape of the sampling distribution of the proportions of children the radio reaches? Once again, pause this video and see if you can figure it out. All right, well, let's just figure out what N and P are."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "Suppose we took weekly random samples of N equals 125 children from this population, and computed the proportion of children in each sample whom radio reaches. What will be the shape of the sampling distribution of the proportions of children the radio reaches? Once again, pause this video and see if you can figure it out. All right, well, let's just figure out what N and P are. Our sample size here, N, is equal to 125, and our population proportion of the proportion of children that are reached each week by radio is 88%. So P is 0.88. So now let's calculate NP."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "All right, well, let's just figure out what N and P are. Our sample size here, N, is equal to 125, and our population proportion of the proportion of children that are reached each week by radio is 88%. So P is 0.88. So now let's calculate NP. So N is 125 times P is 0.88. And is this going to be greater than or equal to 10? Well, we don't even have to calculate this exactly."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "So now let's calculate NP. So N is 125 times P is 0.88. And is this going to be greater than or equal to 10? Well, we don't even have to calculate this exactly. This is almost 90% of 125. This is actually going to be over 100. So it for sure is going to be greater than 10."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "Well, we don't even have to calculate this exactly. This is almost 90% of 125. This is actually going to be over 100. So it for sure is going to be greater than 10. So we meet this first condition. But what about the second condition? We could take N, 125, times one minus P. So this is times 0.12."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "So it for sure is going to be greater than 10. So we meet this first condition. But what about the second condition? We could take N, 125, times one minus P. So this is times 0.12. So this is 12% of 125. Well, even 10% of 125 would be 12.5. So 12% is for sure going to be greater than that."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "We could take N, 125, times one minus P. So this is times 0.12. So this is 12% of 125. Well, even 10% of 125 would be 12.5. So 12% is for sure going to be greater than that. So this too is going to be greater than 10. I didn't even have to calculate it. I could just estimate it."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "So 12% is for sure going to be greater than that. So this too is going to be greater than 10. I didn't even have to calculate it. I could just estimate it. And so we meet that second condition. So even though our population proportion is quite high, it's quite close to one here, because our sample size is so large, it still will be roughly normal. And one way to get the intuition for that is, so this is a proportion of zero."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "I could just estimate it. And so we meet that second condition. So even though our population proportion is quite high, it's quite close to one here, because our sample size is so large, it still will be roughly normal. And one way to get the intuition for that is, so this is a proportion of zero. Let's say this is 50%, and this is 100%. So our mean right over here is going to be 0.88 for our sampling distribution of the sample proportions. If we had a low sample size, then our standard deviation would be quite large."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "And one way to get the intuition for that is, so this is a proportion of zero. Let's say this is 50%, and this is 100%. So our mean right over here is going to be 0.88 for our sampling distribution of the sample proportions. If we had a low sample size, then our standard deviation would be quite large. And so then you would end up with a left skewed distribution. But we saw before, the higher your sample size, the smaller your standard deviation for the sampling distribution. And so what that does is it tightens up, it tightens up the standard deviation."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "If we had a low sample size, then our standard deviation would be quite large. And so then you would end up with a left skewed distribution. But we saw before, the higher your sample size, the smaller your standard deviation for the sampling distribution. And so what that does is it tightens up, it tightens up the standard deviation. And so it's going to look more normal. It's gonna look closer, it's gonna look closer to being normal. So we'll say approximately normal, because it met our conditions for this rule of thumb."}, {"video_title": "Normal conditions for sampling distributions of sample proportions AP Statistics Khan Academy.mp3", "Sentence": "And so what that does is it tightens up, it tightens up the standard deviation. And so it's going to look more normal. It's gonna look closer, it's gonna look closer to being normal. So we'll say approximately normal, because it met our conditions for this rule of thumb. Is it gonna be perfectly normal? No. In fact, if we didn't have this rule of thumb to kind of draw the line, some might even argue that, well, we still have a longer tail to the left than we do to the right."}, {"video_title": "P-values and significance tests AP Statistics Khan Academy.mp3", "Sentence": "Let's say that I run a website that currently has this off-white color for its background, and I know the mean amount of time that people spend on my website. Let's say it is 20 minutes, and I'm interested in making a change that will make people spend more time on my website. My idea is to make the background color of my website yellow. But after making that change, how do I feel good about this actually having the intended consequence? Well, that's where significance tests come into play. What I would do is first set up some hypotheses, a null hypothesis and an alternative hypothesis. The null hypothesis tends to be a statement that, hey, your change actually had no effect."}, {"video_title": "P-values and significance tests AP Statistics Khan Academy.mp3", "Sentence": "But after making that change, how do I feel good about this actually having the intended consequence? Well, that's where significance tests come into play. What I would do is first set up some hypotheses, a null hypothesis and an alternative hypothesis. The null hypothesis tends to be a statement that, hey, your change actually had no effect. There's no news here. And so this would be that your mean is still equal to 20 minutes, is still equal to 20 minutes after, after the change to yellow, in this case, for our background. And we would also have an alternative hypothesis."}, {"video_title": "P-values and significance tests AP Statistics Khan Academy.mp3", "Sentence": "The null hypothesis tends to be a statement that, hey, your change actually had no effect. There's no news here. And so this would be that your mean is still equal to 20 minutes, is still equal to 20 minutes after, after the change to yellow, in this case, for our background. And we would also have an alternative hypothesis. Our alternative hypothesis is actually that our mean is now greater because of the change, that people are spending more time on my site. So our mean is greater than 20 minutes after, after the change. Now, the next thing we do is we set up a threshold known as the significance level, and you will see how this comes into play in a second."}, {"video_title": "P-values and significance tests AP Statistics Khan Academy.mp3", "Sentence": "And we would also have an alternative hypothesis. Our alternative hypothesis is actually that our mean is now greater because of the change, that people are spending more time on my site. So our mean is greater than 20 minutes after, after the change. Now, the next thing we do is we set up a threshold known as the significance level, and you will see how this comes into play in a second. So your significance level, significance level, is usually denoted by the Greek letter alpha, and you tend to see significant levels like 1 1 100th or 5 1 100th or 1 10th or 1%, 5% or 10%. You might see other ones, but we're gonna set a significance level for this particular case. Let's just say it's going to be 0.05."}, {"video_title": "P-values and significance tests AP Statistics Khan Academy.mp3", "Sentence": "Now, the next thing we do is we set up a threshold known as the significance level, and you will see how this comes into play in a second. So your significance level, significance level, is usually denoted by the Greek letter alpha, and you tend to see significant levels like 1 1 100th or 5 1 100th or 1 10th or 1%, 5% or 10%. You might see other ones, but we're gonna set a significance level for this particular case. Let's just say it's going to be 0.05. And what we're going to now do is we're going to take a sample of people visiting this new yellow background website, and we're gonna calculate statistics, the sample mean, the sample standard deviation, and we're gonna say, hey, if we assume that the null hypothesis is true, what is the probability of getting a sample with the statistics that we get? And if that probability is lower than our significance level, if that probability is less than 5 100th, if it's less than 5%, then we reject the null hypothesis and say that we have evidence for the alternative. However, if the probability of getting the statistics for that sample are at the significance level or higher, then we say, hey, we can't reject the null hypothesis, and we aren't able to have evidence for the alternative."}, {"video_title": "P-values and significance tests AP Statistics Khan Academy.mp3", "Sentence": "Let's just say it's going to be 0.05. And what we're going to now do is we're going to take a sample of people visiting this new yellow background website, and we're gonna calculate statistics, the sample mean, the sample standard deviation, and we're gonna say, hey, if we assume that the null hypothesis is true, what is the probability of getting a sample with the statistics that we get? And if that probability is lower than our significance level, if that probability is less than 5 100th, if it's less than 5%, then we reject the null hypothesis and say that we have evidence for the alternative. However, if the probability of getting the statistics for that sample are at the significance level or higher, then we say, hey, we can't reject the null hypothesis, and we aren't able to have evidence for the alternative. So what we would then do, I will call this step three. In step three, we would take a sample, take sample, so let's say we take a sample size, let's say we take 100 folks who visit the new website, the yellow background website, and we measure sample statistics. We measure the sample mean here."}, {"video_title": "P-values and significance tests AP Statistics Khan Academy.mp3", "Sentence": "However, if the probability of getting the statistics for that sample are at the significance level or higher, then we say, hey, we can't reject the null hypothesis, and we aren't able to have evidence for the alternative. So what we would then do, I will call this step three. In step three, we would take a sample, take sample, so let's say we take a sample size, let's say we take 100 folks who visit the new website, the yellow background website, and we measure sample statistics. We measure the sample mean here. Let's say that for that sample, the mean is 25, 25 minutes. We are also likely to, if we don't know what the actual population standard deviation is, which we typically don't know, we would also calculate the sample standard deviation. Then the next step is we calculate a p-value, and the p-value, which stands for probability value, is the probability of getting a statistic at least this far away from the mean if we were to assume that the null hypothesis is true."}, {"video_title": "P-values and significance tests AP Statistics Khan Academy.mp3", "Sentence": "We measure the sample mean here. Let's say that for that sample, the mean is 25, 25 minutes. We are also likely to, if we don't know what the actual population standard deviation is, which we typically don't know, we would also calculate the sample standard deviation. Then the next step is we calculate a p-value, and the p-value, which stands for probability value, is the probability of getting a statistic at least this far away from the mean if we were to assume that the null hypothesis is true. So one way to think about it, it is a conditional probability. It is the probability that our sample mean, our sample mean, when we take a sample of size n equals 100, is greater than or equal to 25, 25 minutes given, given our null hypothesis is true. And in other videos, we have talked about how to do this."}, {"video_title": "P-values and significance tests AP Statistics Khan Academy.mp3", "Sentence": "Then the next step is we calculate a p-value, and the p-value, which stands for probability value, is the probability of getting a statistic at least this far away from the mean if we were to assume that the null hypothesis is true. So one way to think about it, it is a conditional probability. It is the probability that our sample mean, our sample mean, when we take a sample of size n equals 100, is greater than or equal to 25, 25 minutes given, given our null hypothesis is true. And in other videos, we have talked about how to do this. If we assume that the sampling distribution of the sample means is roughly normal, we can use the sample mean, we can use our sample size, we can use our sample standard deviation, perhaps we use a t-statistic to figure out roughly what this probability is going to be. And then we decide whether we can reject the null hypothesis. So let me call that step five."}, {"video_title": "P-values and significance tests AP Statistics Khan Academy.mp3", "Sentence": "And in other videos, we have talked about how to do this. If we assume that the sampling distribution of the sample means is roughly normal, we can use the sample mean, we can use our sample size, we can use our sample standard deviation, perhaps we use a t-statistic to figure out roughly what this probability is going to be. And then we decide whether we can reject the null hypothesis. So let me call that step five. So step five, there are two situations. If my p-value, if my p-value, if it is less than alpha, then I reject my null hypothesis, reject, reject my null hypothesis, and say that I have evidence for my alternative hypothesis. Now, if we have the other situation, if my p-value is greater than or equal to, in this case, 0.05, so if it's greater than or equal to my significance level, then I cannot reject the null hypothesis."}, {"video_title": "P-values and significance tests AP Statistics Khan Academy.mp3", "Sentence": "So let me call that step five. So step five, there are two situations. If my p-value, if my p-value, if it is less than alpha, then I reject my null hypothesis, reject, reject my null hypothesis, and say that I have evidence for my alternative hypothesis. Now, if we have the other situation, if my p-value is greater than or equal to, in this case, 0.05, so if it's greater than or equal to my significance level, then I cannot reject the null hypothesis. I wouldn't say that I accept the null hypothesis. I would just say that we do not, do not reject, reject the null hypothesis. And so let's say when I do all of these calculations, I get a p-value, which would put me in this scenario right over here, let's say that I get a p-value of 0.03."}, {"video_title": "P-values and significance tests AP Statistics Khan Academy.mp3", "Sentence": "Now, if we have the other situation, if my p-value is greater than or equal to, in this case, 0.05, so if it's greater than or equal to my significance level, then I cannot reject the null hypothesis. I wouldn't say that I accept the null hypothesis. I would just say that we do not, do not reject, reject the null hypothesis. And so let's say when I do all of these calculations, I get a p-value, which would put me in this scenario right over here, let's say that I get a p-value of 0.03. 0.03 is indeed less than 0.05, so I would reject the null hypothesis and say that I have evidence for the alternative. And this should hopefully make logical sense because what we're saying is, hey, look, we took a sample, and if we assume the null hypothesis, the probability of getting that sample is 3%. It's 3 100ths, and so since that probability is less than our probability threshold here, we'll reject it and say we have evidence for the alternative."}, {"video_title": "P-values and significance tests AP Statistics Khan Academy.mp3", "Sentence": "And so let's say when I do all of these calculations, I get a p-value, which would put me in this scenario right over here, let's say that I get a p-value of 0.03. 0.03 is indeed less than 0.05, so I would reject the null hypothesis and say that I have evidence for the alternative. And this should hopefully make logical sense because what we're saying is, hey, look, we took a sample, and if we assume the null hypothesis, the probability of getting that sample is 3%. It's 3 100ths, and so since that probability is less than our probability threshold here, we'll reject it and say we have evidence for the alternative. On the other hand, there might have been a scenario where we do all of the calculations here and we figure out a p-value, a p-value that we get is equal to 0.5, which you can interpret as saying that, hey, if we assume the null hypothesis is true, that there's no change due to making the background yellow, I would have a 50% chance of getting this result. And in that situation, since it's higher than my significance level, I wouldn't reject the null hypothesis. A world where the null hypothesis is true and I get this result, well, you know, it seems reasonably likely."}, {"video_title": "P-values and significance tests AP Statistics Khan Academy.mp3", "Sentence": "It's 3 100ths, and so since that probability is less than our probability threshold here, we'll reject it and say we have evidence for the alternative. On the other hand, there might have been a scenario where we do all of the calculations here and we figure out a p-value, a p-value that we get is equal to 0.5, which you can interpret as saying that, hey, if we assume the null hypothesis is true, that there's no change due to making the background yellow, I would have a 50% chance of getting this result. And in that situation, since it's higher than my significance level, I wouldn't reject the null hypothesis. A world where the null hypothesis is true and I get this result, well, you know, it seems reasonably likely. And so this is the basis for significant tests generally, and as you will see, it is applicable in almost every field you'll find yourself in. Now, there's one last point of clarification that I wanna make very, very, very clear. Our p-value, the thing that we're using to decide whether or not we reject the null hypothesis, this is the probability of getting your sample statistics given that the null hypothesis is true."}, {"video_title": "P-values and significance tests AP Statistics Khan Academy.mp3", "Sentence": "A world where the null hypothesis is true and I get this result, well, you know, it seems reasonably likely. And so this is the basis for significant tests generally, and as you will see, it is applicable in almost every field you'll find yourself in. Now, there's one last point of clarification that I wanna make very, very, very clear. Our p-value, the thing that we're using to decide whether or not we reject the null hypothesis, this is the probability of getting your sample statistics given that the null hypothesis is true. Sometimes people confuse this and they say, hey, is this the probability that the null hypothesis is true given the sample, given the sample statistics that we got? And I would say clearly, no, that is not the case. We are not trying to gauge the probability that the null hypothesis is true or not."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "What we're going to do in this video is calculate by hand the correlation coefficient for a set of bivariate data. And when I say bivariate, it's just a fancy way of saying for each x data point, there is a corresponding y data point. Now before I calculate the correlation coefficient, let's just make sure we understand some of these other statistics that they've given us. So we assume that these are samples of the x and the corresponding y from a broader population. And so we have the sample mean for x and the sample standard deviation for x. The sample mean for x is quite straightforward to calculate. It would just be one plus two plus two plus three over four, and this is eight over four, which is indeed equal to two."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "So we assume that these are samples of the x and the corresponding y from a broader population. And so we have the sample mean for x and the sample standard deviation for x. The sample mean for x is quite straightforward to calculate. It would just be one plus two plus two plus three over four, and this is eight over four, which is indeed equal to two. The sample standard deviation for x, we've also seen this before. This should be a little bit of review. It's gonna be the square root of the distance from each of these points to the sample mean squared."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "It would just be one plus two plus two plus three over four, and this is eight over four, which is indeed equal to two. The sample standard deviation for x, we've also seen this before. This should be a little bit of review. It's gonna be the square root of the distance from each of these points to the sample mean squared. So one minus two squared plus two minus two squared plus two minus two squared plus three minus two squared. All of that over, since we're talking about sample standard deviation, we have four data points, so one less than four is all of that over three. Now this actually simplifies quite nicely because this is zero, this is zero, this is one, this is one, and so you essentially get the square root of 2 3rds, which is, if you approximate, 0.816."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "It's gonna be the square root of the distance from each of these points to the sample mean squared. So one minus two squared plus two minus two squared plus two minus two squared plus three minus two squared. All of that over, since we're talking about sample standard deviation, we have four data points, so one less than four is all of that over three. Now this actually simplifies quite nicely because this is zero, this is zero, this is one, this is one, and so you essentially get the square root of 2 3rds, which is, if you approximate, 0.816. So that's that, and the same thing is true for y. The sample mean for y, if you just add up one plus two plus three plus six over four, four data points, this is 12 over four, which is indeed equal to three, and then the sample standard deviation for y, you would calculate the exact same way we did it for x, and you get 2.160. Now with all of that out of the way, let's think about how we calculate the correlation coefficient."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "Now this actually simplifies quite nicely because this is zero, this is zero, this is one, this is one, and so you essentially get the square root of 2 3rds, which is, if you approximate, 0.816. So that's that, and the same thing is true for y. The sample mean for y, if you just add up one plus two plus three plus six over four, four data points, this is 12 over four, which is indeed equal to three, and then the sample standard deviation for y, you would calculate the exact same way we did it for x, and you get 2.160. Now with all of that out of the way, let's think about how we calculate the correlation coefficient. Now right over here is a representation for the formula for the correlation coefficient, and at first it might seem a little intimidating. Until you realize a few things. All this is saying is, for each corresponding x and y, find the z-score for x."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "Now with all of that out of the way, let's think about how we calculate the correlation coefficient. Now right over here is a representation for the formula for the correlation coefficient, and at first it might seem a little intimidating. Until you realize a few things. All this is saying is, for each corresponding x and y, find the z-score for x. So we could call this z sub x for that particular x. So z sub x sub i, and we could say this is the z-score for that particular y. Z sub y sub i is one way that you could think about it."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "All this is saying is, for each corresponding x and y, find the z-score for x. So we could call this z sub x for that particular x. So z sub x sub i, and we could say this is the z-score for that particular y. Z sub y sub i is one way that you could think about it. Look, this is just saying for each data point, find the difference between it and its mean, and then divide by the sample standard deviation. And so that's how many sample standard deviations is it away from its mean. And so that's the z-score for that x data point, and this is the z-score for the corresponding y data point."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "Z sub y sub i is one way that you could think about it. Look, this is just saying for each data point, find the difference between it and its mean, and then divide by the sample standard deviation. And so that's how many sample standard deviations is it away from its mean. And so that's the z-score for that x data point, and this is the z-score for the corresponding y data point. How many sample standard deviations is it away from the sample mean? In the real world, you won't have only four pairs, and it will be very hard to do it by hand, and we typically use software, computer tools to do it, but it's really valuable to do it by hand to get an intuitive understanding of what's going on here. So in this particular situation, r is going to be equal to one over n minus one."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "And so that's the z-score for that x data point, and this is the z-score for the corresponding y data point. How many sample standard deviations is it away from the sample mean? In the real world, you won't have only four pairs, and it will be very hard to do it by hand, and we typically use software, computer tools to do it, but it's really valuable to do it by hand to get an intuitive understanding of what's going on here. So in this particular situation, r is going to be equal to one over n minus one. We have four pairs, so it's gonna be one over three, and it's gonna be times a sum of the products of the z-scores. So this first pair right over here, so the z-score for this one is going to be one minus how far it is away from the x sample mean divided by the x sample standard deviation, 0.816, that times one, now we're looking at the y variable, the y z-score, so it's one minus three, one minus three over the y sample standard deviation, 2.160, and we're just going to keep doing that. I'll do it like this."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "So in this particular situation, r is going to be equal to one over n minus one. We have four pairs, so it's gonna be one over three, and it's gonna be times a sum of the products of the z-scores. So this first pair right over here, so the z-score for this one is going to be one minus how far it is away from the x sample mean divided by the x sample standard deviation, 0.816, that times one, now we're looking at the y variable, the y z-score, so it's one minus three, one minus three over the y sample standard deviation, 2.160, and we're just going to keep doing that. I'll do it like this. So the next one, it's going to be two minus two over 0.816, this is where I got the two from, and I'm subtracting from that the sample mean right over here, times, now we're looking at this two, two minus three over 2.160 plus, I'm happy there's only four pairs here, two minus two again, two minus two over 0.816, times, now we're gonna have three minus three, three minus three over 2.160, and then the last pair, you're going to have three minus two over 0.816 times six minus three, six minus three over 2.160. So before I get a calculator out, let's see if there's some simplifications I can do. Two minus two, that's gonna be zero, zero times anything is zero, so this whole thing is zero."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "I'll do it like this. So the next one, it's going to be two minus two over 0.816, this is where I got the two from, and I'm subtracting from that the sample mean right over here, times, now we're looking at this two, two minus three over 2.160 plus, I'm happy there's only four pairs here, two minus two again, two minus two over 0.816, times, now we're gonna have three minus three, three minus three over 2.160, and then the last pair, you're going to have three minus two over 0.816 times six minus three, six minus three over 2.160. So before I get a calculator out, let's see if there's some simplifications I can do. Two minus two, that's gonna be zero, zero times anything is zero, so this whole thing is zero. Two minus two is zero, three minus three is zero, is gonna actually be zero times zero, so that whole thing is zero. Let's see, this is going to be one minus two, which is negative one, one minus three is negative two, so this is going to be R is equal to 1 3rd times, negative times negative is positive, and so this is going to be two over 0.816 times 2.160, and then plus three minus two is one, six minus three is three, so plus three over 0.816 times 2.160, well these are the same denominator, so actually I could rewrite, if I have two over this thing plus three over this thing, that's going to be five over this thing. So I could rewrite this whole thing, five over 0.816 times 2.160, and now I can just get a calculator out to actually calculate this."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "Two minus two, that's gonna be zero, zero times anything is zero, so this whole thing is zero. Two minus two is zero, three minus three is zero, is gonna actually be zero times zero, so that whole thing is zero. Let's see, this is going to be one minus two, which is negative one, one minus three is negative two, so this is going to be R is equal to 1 3rd times, negative times negative is positive, and so this is going to be two over 0.816 times 2.160, and then plus three minus two is one, six minus three is three, so plus three over 0.816 times 2.160, well these are the same denominator, so actually I could rewrite, if I have two over this thing plus three over this thing, that's going to be five over this thing. So I could rewrite this whole thing, five over 0.816 times 2.160, and now I can just get a calculator out to actually calculate this. So we have one divided by three times five divided by 0.816 times 2.160, the zero won't make a difference, but I'll just write it down, and then I will close that parentheses, and let's see what we get. We get an R of, and since everything else goes to the thousandths place, I'll just round to the thousandths place, an R of 0.946, so R is approximately 0.946. So what does this tell us?"}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "So I could rewrite this whole thing, five over 0.816 times 2.160, and now I can just get a calculator out to actually calculate this. So we have one divided by three times five divided by 0.816 times 2.160, the zero won't make a difference, but I'll just write it down, and then I will close that parentheses, and let's see what we get. We get an R of, and since everything else goes to the thousandths place, I'll just round to the thousandths place, an R of 0.946, so R is approximately 0.946. So what does this tell us? The correlation coefficient is a measure of how well a line can describe the relationship between X and Y. R is always going to be greater than or equal to negative one and less than or equal to one. If R is positive one, it means that an upward-sloping line can completely describe the relationship. If R is negative one, it means a downward-sloping line can completely describe the relationship."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "So what does this tell us? The correlation coefficient is a measure of how well a line can describe the relationship between X and Y. R is always going to be greater than or equal to negative one and less than or equal to one. If R is positive one, it means that an upward-sloping line can completely describe the relationship. If R is negative one, it means a downward-sloping line can completely describe the relationship. R anywhere in between says, well, it won't be as good. If R is zero, that means that a line isn't describing the relationships well at all. Now in our situation here, not to use a pun, in our situation here, our R is pretty close to one, which means that a line can get pretty close to describing the relationship between our Xs and our Ys."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "If R is negative one, it means a downward-sloping line can completely describe the relationship. R anywhere in between says, well, it won't be as good. If R is zero, that means that a line isn't describing the relationships well at all. Now in our situation here, not to use a pun, in our situation here, our R is pretty close to one, which means that a line can get pretty close to describing the relationship between our Xs and our Ys. So for example, I'm just going to try and hand-draw a line here, and it does turn out that our least squares line will always go through the mean of the X and the Y. So the mean of the X is two, mean of the Y is three. We'll study that in more depth in future videos."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "Now in our situation here, not to use a pun, in our situation here, our R is pretty close to one, which means that a line can get pretty close to describing the relationship between our Xs and our Ys. So for example, I'm just going to try and hand-draw a line here, and it does turn out that our least squares line will always go through the mean of the X and the Y. So the mean of the X is two, mean of the Y is three. We'll study that in more depth in future videos. But let's see, this actually does look like a pretty good line. So let me just draw it right over there. You see that I actually can draw a line that gets pretty close to describing."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "We'll study that in more depth in future videos. But let's see, this actually does look like a pretty good line. So let me just draw it right over there. You see that I actually can draw a line that gets pretty close to describing. It isn't perfect. If it went through every point, then I would have an R of one, but it gets pretty close to describing what is going on. Now the next thing I wanna do is focus on the intuition."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "You see that I actually can draw a line that gets pretty close to describing. It isn't perfect. If it went through every point, then I would have an R of one, but it gets pretty close to describing what is going on. Now the next thing I wanna do is focus on the intuition. What was actually going on here with these Z-scores, and how does taking products of corresponding Z-scores get us this property that I just talked about, where an R of one will be strong positive correlation, R of negative one would be strong negative correlation? Well, let's draw the sample means here. So the X sample mean is two."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "Now the next thing I wanna do is focus on the intuition. What was actually going on here with these Z-scores, and how does taking products of corresponding Z-scores get us this property that I just talked about, where an R of one will be strong positive correlation, R of negative one would be strong negative correlation? Well, let's draw the sample means here. So the X sample mean is two. This is our X axis here. This is X equals two. And our Y sample mean is three."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "So the X sample mean is two. This is our X axis here. This is X equals two. And our Y sample mean is three. This is the line Y is equal to three. Now we can also draw the standard deviations. This is, let's see, the standard deviation for X is 0.816, so I'll be approximating it."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "And our Y sample mean is three. This is the line Y is equal to three. Now we can also draw the standard deviations. This is, let's see, the standard deviation for X is 0.816, so I'll be approximating it. So if I go 0.816 less than our mean, it'll get us someplace around there. So that's one standard deviation below the mean. One standard deviation above the mean would put us someplace right over here."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "This is, let's see, the standard deviation for X is 0.816, so I'll be approximating it. So if I go 0.816 less than our mean, it'll get us someplace around there. So that's one standard deviation below the mean. One standard deviation above the mean would put us someplace right over here. And if I do the same thing in Y, one standard deviation above the mean, 2.160, so that would be 5.160, so it would put us someplace around there. And one standard deviation below the mean, so let's see, we're gonna go, if we took away two, we would go to one, and then we're gonna go take another 0.160, so it's gonna be someplace right around here. So for example, for this first pair, one comma one, what were we doing?"}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "One standard deviation above the mean would put us someplace right over here. And if I do the same thing in Y, one standard deviation above the mean, 2.160, so that would be 5.160, so it would put us someplace around there. And one standard deviation below the mean, so let's see, we're gonna go, if we took away two, we would go to one, and then we're gonna go take another 0.160, so it's gonna be someplace right around here. So for example, for this first pair, one comma one, what were we doing? Well, we said, all right, how many standard deviations is this below the mean? And that turns out to be negative one over 0.816. That's what we have right over here."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "So for example, for this first pair, one comma one, what were we doing? Well, we said, all right, how many standard deviations is this below the mean? And that turns out to be negative one over 0.816. That's what we have right over here. That's what this would have calculated. And then how many standard deviations for in the Y direction? And that is our negative two over 2.160."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "That's what we have right over here. That's what this would have calculated. And then how many standard deviations for in the Y direction? And that is our negative two over 2.160. But notice, since both of them were negative, it contributed to the R. This would become a positive value. And so one way to think about it, it might be helping us get closer to the one. If both of them have a negative Z score, that means that there is a positive correlation between the variables."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "And that is our negative two over 2.160. But notice, since both of them were negative, it contributed to the R. This would become a positive value. And so one way to think about it, it might be helping us get closer to the one. If both of them have a negative Z score, that means that there is a positive correlation between the variables. When one is below the mean, the other is, you could say, similarly below the mean. Now, if you go to the next data point, two comma two, right over here, what happened? Well, the X variable was right on the mean."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "If both of them have a negative Z score, that means that there is a positive correlation between the variables. When one is below the mean, the other is, you could say, similarly below the mean. Now, if you go to the next data point, two comma two, right over here, what happened? Well, the X variable was right on the mean. And because of that, that entire term became zero. The X Z score was zero. And so that would have taken away a little bit from our correlation coefficient."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "Well, the X variable was right on the mean. And because of that, that entire term became zero. The X Z score was zero. And so that would have taken away a little bit from our correlation coefficient. The reason why it would take away, even though it's not negative, you're not contributing to the sum, but you're going to be dividing by a slightly higher value by including that extra pair. If you had a data point where, let's say X was below the mean and Y was above the mean, something like this, if this was one of the points, this term would have been negative because the Y Z score would have been positive and the X Z score would have been negative. And so when you put it in the sum, it would have actually taken away from the sum."}, {"video_title": "Calculating correlation coefficient r AP Statistics Khan Academy.mp3", "Sentence": "And so that would have taken away a little bit from our correlation coefficient. The reason why it would take away, even though it's not negative, you're not contributing to the sum, but you're going to be dividing by a slightly higher value by including that extra pair. If you had a data point where, let's say X was below the mean and Y was above the mean, something like this, if this was one of the points, this term would have been negative because the Y Z score would have been positive and the X Z score would have been negative. And so when you put it in the sum, it would have actually taken away from the sum. And so it would have made the R score even lower. Similarly, something like this would have done, would have made the R score even lower because you would have a positive Z score for X and a negative Z score for Y. And so a product of a positive and a negative would be a negative."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "And we know a few other things about it. We know what the expected value of X is. It is equal to 16 ounces. In fact, they tell it to us on a box. They say, you know, net weight, 16 ounces. Now when you see that on a cereal box, it doesn't mean that every box is going to be exactly 16 ounces. Remember, you have a discrete number of these flakes in here."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "In fact, they tell it to us on a box. They say, you know, net weight, 16 ounces. Now when you see that on a cereal box, it doesn't mean that every box is going to be exactly 16 ounces. Remember, you have a discrete number of these flakes in here. They might have slightly different densities, slightly different shapes, depending on how they get packed into this volume. So there is some variation, which we can measure with standard deviation. So the standard deviation, let's just say for the sake of argument, for the random variable X, is 0.8 ounces."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Remember, you have a discrete number of these flakes in here. They might have slightly different densities, slightly different shapes, depending on how they get packed into this volume. So there is some variation, which we can measure with standard deviation. So the standard deviation, let's just say for the sake of argument, for the random variable X, is 0.8 ounces. And just to build our intuition a little bit later in this video, let's say that this, the random variable X, is always stays constrained within a range. That if it goes above a certain weight or below a certain weight, then the company that produces it just throws out that box. And so let's say that our random variable X is always greater than or equal to 15 ounces, and it is always less than or equal to 17 ounces, just for argument."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "So the standard deviation, let's just say for the sake of argument, for the random variable X, is 0.8 ounces. And just to build our intuition a little bit later in this video, let's say that this, the random variable X, is always stays constrained within a range. That if it goes above a certain weight or below a certain weight, then the company that produces it just throws out that box. And so let's say that our random variable X is always greater than or equal to 15 ounces, and it is always less than or equal to 17 ounces, just for argument. This will help us build our intuition later on. Now separately, let's consider a bowl. And we're always gonna consider the same size bowl."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so let's say that our random variable X is always greater than or equal to 15 ounces, and it is always less than or equal to 17 ounces, just for argument. This will help us build our intuition later on. Now separately, let's consider a bowl. And we're always gonna consider the same size bowl. Let's consider this a four-ounce bowl, because the expected value of Y, if you took a random one of these bowls, always the same bowl, and if, or if you took the same bowl and you, someone filled it with mathies, the expected weight of the mathies in that bowl is going to be four ounces. But once again, there's going to be some variation. Depends who filled it in, how it packed in, did they shake it before, while they were filling it?"}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "And we're always gonna consider the same size bowl. Let's consider this a four-ounce bowl, because the expected value of Y, if you took a random one of these bowls, always the same bowl, and if, or if you took the same bowl and you, someone filled it with mathies, the expected weight of the mathies in that bowl is going to be four ounces. But once again, there's going to be some variation. Depends who filled it in, how it packed in, did they shake it before, while they were filling it? There could be all sorts of things that could make some variation here. And so for the sake of argument, let's say that variation can be measured by standard deviation, and it's 0.6 ounces. And let's say whoever the bowl fillers are, they are also, they don't like bowls that are too heavy or too light, and so they'll also throw out bowls."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Depends who filled it in, how it packed in, did they shake it before, while they were filling it? There could be all sorts of things that could make some variation here. And so for the sake of argument, let's say that variation can be measured by standard deviation, and it's 0.6 ounces. And let's say whoever the bowl fillers are, they are also, they don't like bowls that are too heavy or too light, and so they'll also throw out bowls. So we can say that Y can, its maximum value that it'll ever take on is five ounces, and the minimum value that it could ever take on, let's say it is three ounces. So given all of this information, what I wanna do is, let's just say I take a random box of mathies, and I take a random filled bowl, and I wanna think about the combined weight in the closed box and the filled bowl. So what I wanna think about is really X plus Y. I wanna think about the sum of the random variables."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "And let's say whoever the bowl fillers are, they are also, they don't like bowls that are too heavy or too light, and so they'll also throw out bowls. So we can say that Y can, its maximum value that it'll ever take on is five ounces, and the minimum value that it could ever take on, let's say it is three ounces. So given all of this information, what I wanna do is, let's just say I take a random box of mathies, and I take a random filled bowl, and I wanna think about the combined weight in the closed box and the filled bowl. So what I wanna think about is really X plus Y. I wanna think about the sum of the random variables. So in previous videos, we already know that the expected value of this is just going to be the sum of the expected values of each of the random variables. So it would be the expected value of X plus the expected value of Y, and so it would be 16 plus four ounces. In this case, this would be equal to 20 ounces."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "So what I wanna think about is really X plus Y. I wanna think about the sum of the random variables. So in previous videos, we already know that the expected value of this is just going to be the sum of the expected values of each of the random variables. So it would be the expected value of X plus the expected value of Y, and so it would be 16 plus four ounces. In this case, this would be equal to 20 ounces. But what about the variation? Can we just add up the standard deviations? If I wanna figure out the standard deviation of X plus Y, how can I do this?"}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "In this case, this would be equal to 20 ounces. But what about the variation? Can we just add up the standard deviations? If I wanna figure out the standard deviation of X plus Y, how can I do this? Well, it turns out that you can't just add up the standard deviations, but you can add up the variances. So it is the case that the variance of X plus Y is equal to the variance of X plus the variance of Y. And so this is gonna have an X right over here, X, and then we have plus Y and our Y."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "If I wanna figure out the standard deviation of X plus Y, how can I do this? Well, it turns out that you can't just add up the standard deviations, but you can add up the variances. So it is the case that the variance of X plus Y is equal to the variance of X plus the variance of Y. And so this is gonna have an X right over here, X, and then we have plus Y and our Y. And actually, both of these assume independent random variables. So it assumes X and Y are independent. I'm gonna write it in caps."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so this is gonna have an X right over here, X, and then we have plus Y and our Y. And actually, both of these assume independent random variables. So it assumes X and Y are independent. I'm gonna write it in caps. In a future video, I'm going to give you, hopefully, a better intuition for why this must be true, that they're independent, in order to make this claim right over here. I'm not gonna prove it in this video, but we could build a little bit of intuition. Here, for each of these random variables, we have a range of two ounces over which this random variable can take, and that's true for both of them."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "I'm gonna write it in caps. In a future video, I'm going to give you, hopefully, a better intuition for why this must be true, that they're independent, in order to make this claim right over here. I'm not gonna prove it in this video, but we could build a little bit of intuition. Here, for each of these random variables, we have a range of two ounces over which this random variable can take, and that's true for both of them. But what about this sum? Well, this sum here could get as high as, so let me write it this way. So X plus Y, X plus Y, what's the maximum value that it could take on?"}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Here, for each of these random variables, we have a range of two ounces over which this random variable can take, and that's true for both of them. But what about this sum? Well, this sum here could get as high as, so let me write it this way. So X plus Y, X plus Y, what's the maximum value that it could take on? Well, if you get a heavy version of each of these, then it's going to be 17 plus five, so this has to be less than 22 ounces, and it's going to be greater than or equal to, well, what's the lightest possible scenario? Well, you could get a 15-ouncer here and a three-ouncer here, and it is 18 ounces. And so notice, now the variation for the sum is larger."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "So X plus Y, X plus Y, what's the maximum value that it could take on? Well, if you get a heavy version of each of these, then it's going to be 17 plus five, so this has to be less than 22 ounces, and it's going to be greater than or equal to, well, what's the lightest possible scenario? Well, you could get a 15-ouncer here and a three-ouncer here, and it is 18 ounces. And so notice, now the variation for the sum is larger. We have a range that this thing can take on now of four, while the range for each of these was just two. Or another way you could think about it is these upper and lower ends of the range are further from the mean than these upper and lower ends of the range were from their respective means. So hopefully this gives you an intuition for why this makes sense."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so notice, now the variation for the sum is larger. We have a range that this thing can take on now of four, while the range for each of these was just two. Or another way you could think about it is these upper and lower ends of the range are further from the mean than these upper and lower ends of the range were from their respective means. So hopefully this gives you an intuition for why this makes sense. But let me ask you another question. What if I were to say, what about the variance, what about the variance of X minus Y? What would this be?"}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "So hopefully this gives you an intuition for why this makes sense. But let me ask you another question. What if I were to say, what about the variance, what about the variance of X minus Y? What would this be? Would you subtract the variances of each of the random variables here? Well, let's just do the exact same exercise. Let's take X minus Y, X minus Y, and think about it."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "What would this be? Would you subtract the variances of each of the random variables here? Well, let's just do the exact same exercise. Let's take X minus Y, X minus Y, and think about it. What would be the lowest value that X minus Y could take on? Well, the lowest value is if you have a low X and you have a high Y, so it'd be 15 minus five. So this would be 10 right over here."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Let's take X minus Y, X minus Y, and think about it. What would be the lowest value that X minus Y could take on? Well, the lowest value is if you have a low X and you have a high Y, so it'd be 15 minus five. So this would be 10 right over here. That would be the lowest value that you could take on. And what would be the highest value? Well, the highest value is if you have a high X and a low Y, so 17 minus three is 14."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "So this would be 10 right over here. That would be the lowest value that you could take on. And what would be the highest value? Well, the highest value is if you have a high X and a low Y, so 17 minus three is 14. So notice, just as we saw in this case of the sum, even in the difference, your variability seems to have increased. This is still going to be, the extremes are still further than the mean of the difference. The mean of the difference would be 16 minus four is 12."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, the highest value is if you have a high X and a low Y, so 17 minus three is 14. So notice, just as we saw in this case of the sum, even in the difference, your variability seems to have increased. This is still going to be, the extremes are still further than the mean of the difference. The mean of the difference would be 16 minus four is 12. These extreme values are two away from the 12. And this is just to give us an intuition. Once again, it's not a rigorous proof."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "The mean of the difference would be 16 minus four is 12. These extreme values are two away from the 12. And this is just to give us an intuition. Once again, it's not a rigorous proof. So it actually turns out that in either case, when you're taking the variance of X plus Y or X minus Y, you would sum the variances, assuming X and Y are independent variables. Now with that out of the way, let's just calculate the standard deviation of X plus Y. Well, we know this."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Once again, it's not a rigorous proof. So it actually turns out that in either case, when you're taking the variance of X plus Y or X minus Y, you would sum the variances, assuming X and Y are independent variables. Now with that out of the way, let's just calculate the standard deviation of X plus Y. Well, we know this. Let me just write it using this sigma notation. So another way of writing the variance of X plus Y is to write the standard deviation of X plus Y squared. And that's going to be equal to the variance of X plus the variance of Y."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, we know this. Let me just write it using this sigma notation. So another way of writing the variance of X plus Y is to write the standard deviation of X plus Y squared. And that's going to be equal to the variance of X plus the variance of Y. Now what is the variance of X? Well, it's the standard deviation of X squared, 0.8 squared. This is 0.64, 0.64."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "And that's going to be equal to the variance of X plus the variance of Y. Now what is the variance of X? Well, it's the standard deviation of X squared, 0.8 squared. This is 0.64, 0.64. The standard deviation of Y is 0.6. You square it to get the variance. That's 0.36."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "This is 0.64, 0.64. The standard deviation of Y is 0.6. You square it to get the variance. That's 0.36. You add these two up, and you are going to get one. So the variance of the sum is one. And then if you take the square root of both of these, you get the standard deviation of the sum is also going to be one."}, {"video_title": "Variance of sum and difference of random variables Random variables AP Statistics Khan Academy.mp3", "Sentence": "That's 0.36. You add these two up, and you are going to get one. So the variance of the sum is one. And then if you take the square root of both of these, you get the standard deviation of the sum is also going to be one. And that just happened to work out because we're dealing with the scenario where the variance, where the square root of one is, well, one. So this hopefully builds your intuition, whether we are adding or subtracting two independent random variables, the variance of that sum or the difference, the variability will increase. In the next video, we'll go into some depth talking about getting an intuition for why independence is an important condition for making this statement, this claim."}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "This random variable can take on values from one to five and has an equal probability of taking on any of these values from one to five. Find the probability that x is less than four. So x can go from one to four, there's no probability that it'll be less than one. So we know the entire area under the density curve is going to be one. So if we can find the fraction of the area that meets our criteria, then we know the answer to the question. So what we're going to look at is we want to go from one to four. The reason why I know we can start at one is there's no probability, there's zero chances that I'll get a value less than one."}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "So we know the entire area under the density curve is going to be one. So if we can find the fraction of the area that meets our criteria, then we know the answer to the question. So what we're going to look at is we want to go from one to four. The reason why I know we can start at one is there's no probability, there's zero chances that I'll get a value less than one. We see that from the density curve. And so we just need to think about what is the area here? What is this area right over here?"}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "The reason why I know we can start at one is there's no probability, there's zero chances that I'll get a value less than one. We see that from the density curve. And so we just need to think about what is the area here? What is this area right over here? Well, this is just a rectangle where the height is 0.25 and the width is one, two, three. So our area is going to be 0.25 times three, which is equal to 0.75. So the probability that x is less than four is 0.75 or you could say it's a 75% probability."}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "What is this area right over here? Well, this is just a rectangle where the height is 0.25 and the width is one, two, three. So our area is going to be 0.25 times three, which is equal to 0.75. So the probability that x is less than four is 0.75 or you could say it's a 75% probability. Let's do another one of these with a slightly more involved density curve. A set of middle school students' heights are normally distributed with a mean of 150 centimeters and a standard deviation of 20 centimeters. Let h be the height of a randomly selected student from this set."}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "So the probability that x is less than four is 0.75 or you could say it's a 75% probability. Let's do another one of these with a slightly more involved density curve. A set of middle school students' heights are normally distributed with a mean of 150 centimeters and a standard deviation of 20 centimeters. Let h be the height of a randomly selected student from this set. Find and interpret the probability that h, that the height of a randomly selected student from the set is greater than 170 centimeters. So let's first visualize the density curve. It is a normal distribution."}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "Let h be the height of a randomly selected student from this set. Find and interpret the probability that h, that the height of a randomly selected student from the set is greater than 170 centimeters. So let's first visualize the density curve. It is a normal distribution. They tell us that the mean is 150 centimeters. So let me draw that. So the mean, that is 150."}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "It is a normal distribution. They tell us that the mean is 150 centimeters. So let me draw that. So the mean, that is 150. And they also say that we have a standard deviation of 20 centimeters. So 20 centimeters above the mean, one standard deviation above the mean is 170. One standard deviation below the mean is 130."}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "So the mean, that is 150. And they also say that we have a standard deviation of 20 centimeters. So 20 centimeters above the mean, one standard deviation above the mean is 170. One standard deviation below the mean is 130. And we want the probability of, if we randomly select from these middle school students, what's the probability that the height is greater than 170? So that's going to be this area under this normal distribution curve. It's going to be that area."}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "One standard deviation below the mean is 130. And we want the probability of, if we randomly select from these middle school students, what's the probability that the height is greater than 170? So that's going to be this area under this normal distribution curve. It's going to be that area. So how can we figure that out? Well, there's several ways to do it. We know that this is the area above one standard deviation above the mean."}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "It's going to be that area. So how can we figure that out? Well, there's several ways to do it. We know that this is the area above one standard deviation above the mean. You could use a z-table, or you could use some generally useful knowledge about normal distributions. And that's that the area between one standard deviation below the mean and one standard deviation above the mean, this area right over here, is roughly 68%. It's closer to 68.2%."}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "We know that this is the area above one standard deviation above the mean. You could use a z-table, or you could use some generally useful knowledge about normal distributions. And that's that the area between one standard deviation below the mean and one standard deviation above the mean, this area right over here, is roughly 68%. It's closer to 68.2%. For our purposes, 68 will work fine. And so if we're looking at just from the mean to one standard deviation above the mean, it would be half of that. So this is going to be approximately 34%."}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "It's closer to 68.2%. For our purposes, 68 will work fine. And so if we're looking at just from the mean to one standard deviation above the mean, it would be half of that. So this is going to be approximately 34%. Now, we also know that for a normal distribution, the area below the mean is going to be 50%. So we know all of that is 50%. And so the combined area below 170, below one standard deviation above the mean, is going to be 84%, or approximately 84%."}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "So this is going to be approximately 34%. Now, we also know that for a normal distribution, the area below the mean is going to be 50%. So we know all of that is 50%. And so the combined area below 170, below one standard deviation above the mean, is going to be 84%, or approximately 84%. And so that helps us figure out what is the area above one standard deviation above the mean, which will answer our question. The entire area under this density curve, under any density curve, is going to be equal to one. And so if the entire area is one, this green area is 84%, or 0.84, well, then we just subtract that from one to get this blue area."}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so the combined area below 170, below one standard deviation above the mean, is going to be 84%, or approximately 84%. And so that helps us figure out what is the area above one standard deviation above the mean, which will answer our question. The entire area under this density curve, under any density curve, is going to be equal to one. And so if the entire area is one, this green area is 84%, or 0.84, well, then we just subtract that from one to get this blue area. So this is going to be one minus 0.84, or I'll say approximately. And so that's going to be approximately 0.16. If you want a slightly more precise value, you could use a z-table."}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so if the entire area is one, this green area is 84%, or 0.84, well, then we just subtract that from one to get this blue area. So this is going to be one minus 0.84, or I'll say approximately. And so that's going to be approximately 0.16. If you want a slightly more precise value, you could use a z-table. The area below one standard deviation above the mean will be closer to about 84.1%, in which case this would be about 15.9%, or 0.159. But you can see that we got pretty close by knowing the general rule that it's roughly 68% between one standard deviation below the mean and one standard deviation above the mean for a normal distribution."}, {"video_title": "Probabilities from density curves Random variables AP Statistics Khan Academy.mp3", "Sentence": "If you want a slightly more precise value, you could use a z-table. The area below one standard deviation above the mean will be closer to about 84.1%, in which case this would be about 15.9%, or 0.159. But you can see that we got pretty close by knowing the general rule that it's roughly 68% between one standard deviation below the mean and one standard deviation above the mean for a normal distribution."}]