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Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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 unifor... |
Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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... |
Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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 s... |
Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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 t... |
Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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 varianc... |
Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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 re... |
Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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 horizont... |
Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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 t... |
Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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 tha... |
Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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 vertica... |
Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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 ... |
Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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 ... |
Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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... |
Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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 ... |
Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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 t... |
Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance.mp3 | 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 ... |
Types of studies AP Statistics Khan Academy.mp3 | 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 you... |
Types of studies AP Statistics Khan Academy.mp3 | 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... |
Types of studies AP Statistics Khan Academy.mp3 | 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,... |
Types of studies AP Statistics Khan Academy.mp3 | 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 observati... |
Types of studies AP Statistics Khan Academy.mp3 | 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,... |
Types of studies AP Statistics Khan Academy.mp3 | 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 th... |
Types of studies AP Statistics Khan Academy.mp3 | 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 ... |
Types of studies AP Statistics Khan Academy.mp3 | 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 tryi... |
Types of studies AP Statistics Khan Academy.mp3 | 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,... |
Types of studies AP Statistics Khan Academy.mp3 | 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... |
Types of studies AP Statistics Khan Academy.mp3 | 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 o... |
Types of studies AP Statistics Khan Academy.mp3 | 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 forwa... |
Types of studies AP Statistics Khan Academy.mp3 | 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? Wel... |
Types of studies AP Statistics Khan Academy.mp3 | 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 ... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 c... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 ... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 tak... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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, ... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 ... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 ... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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, ... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 level... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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, whic... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 th... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 ... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 ... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 ... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 on... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 ... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 getti... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 lo... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 he... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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? S... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 catego... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 ... |
Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 | 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 proba... |
Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 | 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... |
Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 | 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 tal... |
Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 | 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 s... |
Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 | 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 ... |
Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 | 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 si... |
Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 | 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 ... |
Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 | 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 kn... |
Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 | 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 ... |
Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 | 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... |
Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 | 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... |
Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 | 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 ca... |
Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 | 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 ... |
Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3 | 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 pe... |
Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3 | 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... |
Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3 | 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 a... |
Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3 | 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 ... |
Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3 | 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 als... |
Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3 | 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... |
Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3 | 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 t... |
Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3 | 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... |
Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3 | 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 ... |
Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3 | 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 th... |
Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3 | 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... |
Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3 | 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 ti... |
Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3 | 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 y... |
Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3 | 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 he... |
Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 | 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 ... |
Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 | 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 tha... |
Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 | 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 ca... |
Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 | 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 distributio... |
Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 | 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-dis... |
Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 | 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 2.7... |
Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 | 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, a... |
Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 | 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... |
Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 | 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 l... |
Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 | 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, th... |
Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 | 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 on... |
Using a table to estimate P-value from t statistic AP Statistics Khan Academy.mp3 | 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... |
Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 | 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... |
Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 | 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 ... |
Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 | 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 variab... |
Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 | 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. |
Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 | 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 kee... |
Cumulative geometric probability (greater than a value) AP Statistics Khan Academy.mp3 | 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 ... |
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