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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, 6... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
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 oth... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
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, sor... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
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 pi... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
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 w... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
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 ... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
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 y... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
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 biase... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
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 wh... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
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 ge... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
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 lar... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
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, yo... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
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 t... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
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 vari... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
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 ap... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
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... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
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 m... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
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 perf... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 ac... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 hav... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 sam... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 buil... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 plu... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 d... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 sen... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 regre... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 o... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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,... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 o... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 ... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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.... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 i... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 sai... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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, whi... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 str... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 g... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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? | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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? | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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. | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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 mi... | Calculating the equation of a regression line AP Statistics Khan Academy.mp3 |
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,... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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 proporti... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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 inter... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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 w... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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%, t... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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 deviat... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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 va... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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 t... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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 ... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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 usin... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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 ... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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, the... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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 abo... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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. | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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 standar... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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 stand... | Critical value (z ) for a given confidence level AP Statistics Khan Academy.mp3 |
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 f... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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 sur... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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 ... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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 ... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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 great... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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 befo... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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 peopl... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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, ... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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 ... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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 t... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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 certa... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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. The... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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 ... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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 abou... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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 woul... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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 th... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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... | Examples identifying conditions for inference on two proportions AP Statistics Khan Academy.mp3 |
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 t... | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
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 abo... | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
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 ... | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
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... | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
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. ... | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
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 i... | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
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 delaye... | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
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. | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
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 nume... | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
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, ... | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
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 ... | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
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... | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
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 yo... | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
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... | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
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 numbe... | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
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. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
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... | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
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... | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
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... | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
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