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So this, already, we've kind of come up with a neat way of writing the variance. You can essentially take the average of the squares of all of the numbers, in this case a population, and then subtract from that the mean squared of your population. So this could be, depending on how you're calculating things, maybe a slightly faster way of calculating the variance. So just playing with a little algebra we got from this thing, where you have to, each time, take each of your data points, subtract the mean from it, and then square it. And then, of course, before you had to do anything, you had to calculate the mean. And you take the square, then you sum it all up, then you take the average, essentially, when you divide it, when you sum and divide it by n. We've simplified it, just using a little bit of algebra, to this formula. And this is, we're getting to something called the raw score method. | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
So just playing with a little algebra we got from this thing, where you have to, each time, take each of your data points, subtract the mean from it, and then square it. And then, of course, before you had to do anything, you had to calculate the mean. And you take the square, then you sum it all up, then you take the average, essentially, when you divide it, when you sum and divide it by n. We've simplified it, just using a little bit of algebra, to this formula. And this is, we're getting to something called the raw score method. What we want to do is write this right here, just in terms of xi's. And then we really are at what you call the raw score method, which is oftentimes a faster way of calculating the variance. So let's see, what is mu equal to? | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
And this is, we're getting to something called the raw score method. What we want to do is write this right here, just in terms of xi's. And then we really are at what you call the raw score method, which is oftentimes a faster way of calculating the variance. So let's see, what is mu equal to? What is the mean? The mean is just equal to the sum from i is equal to 1 to n of each of the terms. You just take the sum of each of the terms, and you divide by the number of terms there are. | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
So let's see, what is mu equal to? What is the mean? The mean is just equal to the sum from i is equal to 1 to n of each of the terms. You just take the sum of each of the terms, and you divide by the number of terms there are. So that is equal to, so if we look at this thing, this thing can be written as, let me draw a line here, this thing can be written as the sum from i is equal to 1 to n of xi squared, all of that over n, minus mu squared. Well, mu is this. So this thing squared. | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
You just take the sum of each of the terms, and you divide by the number of terms there are. So that is equal to, so if we look at this thing, this thing can be written as, let me draw a line here, this thing can be written as the sum from i is equal to 1 to n of xi squared, all of that over n, minus mu squared. Well, mu is this. So this thing squared. So this thing squared is what? This is x sub i, take the sum to n, i is equal to 1. You're going to square this thing, and then you're going to divide it by, we squared, right? | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
So this thing squared. So this thing squared is what? This is x sub i, take the sum to n, i is equal to 1. You're going to square this thing, and then you're going to divide it by, we squared, right? You divide it by n squared. And this might seem like a more, out of all of them, this is actually seems like the simplest formula for me, where you essentially just take, if you know the mean of your population, you just say, OK, my mean is whatever, and I can just square that, and just put that aside for a second. But first, I can just take each of the numbers, square them, and then sum them up, and divide by the number of numbers I have. | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
You're going to square this thing, and then you're going to divide it by, we squared, right? You divide it by n squared. And this might seem like a more, out of all of them, this is actually seems like the simplest formula for me, where you essentially just take, if you know the mean of your population, you just say, OK, my mean is whatever, and I can just square that, and just put that aside for a second. But first, I can just take each of the numbers, square them, and then sum them up, and divide by the number of numbers I have. I don't know if I wrote, no, I've erased the last set of numbers, but we could show you that you'll get to the same variance. So to me, this is almost the simplest formula. But this one's even faster in a lot of ways, because you don't really have to even calculate the mean ahead of time. | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
But first, I can just take each of the numbers, square them, and then sum them up, and divide by the number of numbers I have. I don't know if I wrote, no, I've erased the last set of numbers, but we could show you that you'll get to the same variance. So to me, this is almost the simplest formula. But this one's even faster in a lot of ways, because you don't really have to even calculate the mean ahead of time. You can just say, OK, for each xi, I just perform this operation, and then I divide by n squared or n accordingly, and I'll also get to the variance. So you don't have to do this calculation before you figure out the whole variance. But anyway, I thought it would be instructive and hopefully give you a little bit more intuition behind the algebra dealing with sigma notation, if we kind of worked out these other ways to write variances. | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
But this one's even faster in a lot of ways, because you don't really have to even calculate the mean ahead of time. You can just say, OK, for each xi, I just perform this operation, and then I divide by n squared or n accordingly, and I'll also get to the variance. So you don't have to do this calculation before you figure out the whole variance. But anyway, I thought it would be instructive and hopefully give you a little bit more intuition behind the algebra dealing with sigma notation, if we kind of worked out these other ways to write variances. And frankly, some books will just kind of say, oh yeah, you know what, the variance could be written like this, or, and we're talking about the variance of a population, or it could be written like this, or maybe they'll even write it like this. And it's good to know that you can just do a little bit of simple algebraic manipulation and get from one to the other. Anyway, I've run out of time. | Statistics Alternate variance formulas Probability and Statistics Khan Academy.mp3 |
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. Her sample size is 120. | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
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. And then she calculates the test statistic for these results was z is approximately equal to 1.83. | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
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. But that's what this z statistic is, is, well, how many standard deviations above the assumed proportion is that? | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
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. That is just going to be the assumed proportion, so it would be just this. | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
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. So they did all of that for us. | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
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? 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. | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
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. The mean of the sampling distribution right over here would be the assumed population proportion. | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
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. So that would be 1.83 standard deviations. | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
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. So notice, this z table gives us the area to the left of a certain z value. | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
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. And so we could look at that on this z table right over here. | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
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. So there we have it. | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
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. 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. | Calculating a P-value given a z statistic AP Statistics Khan Academy.mp3 |
Let's say you're in the babysitting business and you like to keep a log of whom you are babysitting. So in the last month, you babysat six children and you wrote the ages of all six children in your log. 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? | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
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. So let's just think about how do we calculate, how would we calculate a mean if we knew what question mark is? | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
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. Mean times, and I'll just write times the number, times number of data points, or number of ages. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
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. 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. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
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. Well, we have six ages here. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
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. This is going to be equal to the mean. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
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. Let's see if we can simplify this. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
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. Five plus two is seven. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
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. We're left with 21 plus question mark over six is equal to four. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
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. We can multiply both sides times six. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
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. Six in the numerator, six in the denominator, those cancel. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
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. We get 21 plus question mark. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
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. 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. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
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. What's the question mark? | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
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. Which, of course, let me just write this down. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
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. It's exciting. | How to find a missing value given the mean Data and statistics 6th grade Khan Academy.mp3 |
Who knows what that's in some blood pressure units. Construct a 95% confidence interval for the true expected blood pressure increase for all patients in a population. So there's some population distribution here. It's a reasonable assumption to think that it is normal. It's a biological process. So if you gave this drug to every person who has ever lived, that will result in some mean increase in blood pressure. Or who knows, maybe it's actually a decrease. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
It's a reasonable assumption to think that it is normal. It's a biological process. So if you gave this drug to every person who has ever lived, that will result in some mean increase in blood pressure. Or who knows, maybe it's actually a decrease. And there's also going to be some standard deviation here. It is a normal distribution. And the reason why it's reasonable to assume that it's a normal distribution is because it's a biological process. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
Or who knows, maybe it's actually a decrease. And there's also going to be some standard deviation here. It is a normal distribution. And the reason why it's reasonable to assume that it's a normal distribution is because it's a biological process. It's going to be the sum of many thousands and millions of random events. And things that are sums of many millions and thousands of random events tend to be normal distribution. So this is a population distribution. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And the reason why it's reasonable to assume that it's a normal distribution is because it's a biological process. It's going to be the sum of many thousands and millions of random events. And things that are sums of many millions and thousands of random events tend to be normal distribution. So this is a population distribution. This is the population distribution. And we don't know anything really about it outside of the sample that we have here. Now, what we can do is, and this tends to be a good thing to do when you do have a sample, is just figure out everything that you can figure out about that sample from the get go. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So this is a population distribution. This is the population distribution. And we don't know anything really about it outside of the sample that we have here. Now, what we can do is, and this tends to be a good thing to do when you do have a sample, is just figure out everything that you can figure out about that sample from the get go. So we have our seven data points. And you can add them up and divide by 7 and get your sample mean. So our sample mean here is 2.34. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
Now, what we can do is, and this tends to be a good thing to do when you do have a sample, is just figure out everything that you can figure out about that sample from the get go. So we have our seven data points. And you can add them up and divide by 7 and get your sample mean. So our sample mean here is 2.34. And then you can also calculate your sample standard deviation. Find the square distance from each of these points to your sample mean, add them up, divide by n minus 1 because it's a sample, then take the square root, and you get your sample standard deviation. And I did this ahead of time just to save time. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So our sample mean here is 2.34. And then you can also calculate your sample standard deviation. Find the square distance from each of these points to your sample mean, add them up, divide by n minus 1 because it's a sample, then take the square root, and you get your sample standard deviation. And I did this ahead of time just to save time. Sample standard deviation is 1.04. And we don't know anything about the population distribution. The thing that we've been doing from the get go is estimating that character with our sample standard deviation. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And I did this ahead of time just to save time. Sample standard deviation is 1.04. And we don't know anything about the population distribution. The thing that we've been doing from the get go is estimating that character with our sample standard deviation. So we've been estimating the true standard deviation of the population with our sample standard deviation. Now, in this problem, this exact problem, we're going to run into a problem. We're estimating our standard deviation with an n of only 7. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
The thing that we've been doing from the get go is estimating that character with our sample standard deviation. So we've been estimating the true standard deviation of the population with our sample standard deviation. Now, in this problem, this exact problem, we're going to run into a problem. We're estimating our standard deviation with an n of only 7. So this is probably going to be a not so good estimate. Let me just write, because n is small. In general, this is considered a bad estimate if n is less than 30. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
We're estimating our standard deviation with an n of only 7. So this is probably going to be a not so good estimate. Let me just write, because n is small. In general, this is considered a bad estimate if n is less than 30. Above 30, you're dealing in the realm of pretty good estimates. And so the whole focus of this video is when we think about the sampling distribution, which is what we're going to use to generate our interval, instead of assuming that the sampling distribution is normal, like we did in many other videos using the central limit theorem and all of that, we're going to tweak the sampling distribution. We're not going to assume it's a normal distribution, because this is a bad estimate. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
In general, this is considered a bad estimate if n is less than 30. Above 30, you're dealing in the realm of pretty good estimates. And so the whole focus of this video is when we think about the sampling distribution, which is what we're going to use to generate our interval, instead of assuming that the sampling distribution is normal, like we did in many other videos using the central limit theorem and all of that, we're going to tweak the sampling distribution. We're not going to assume it's a normal distribution, because this is a bad estimate. We're going to assume that it's something called a t distribution. And the t distribution is essentially, the best way to think about it is it's almost engineered. It's almost engineered, so it gives a better estimate of your confidence intervals and all of that when you do have a small sample size. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
We're not going to assume it's a normal distribution, because this is a bad estimate. We're going to assume that it's something called a t distribution. And the t distribution is essentially, the best way to think about it is it's almost engineered. It's almost engineered, so it gives a better estimate of your confidence intervals and all of that when you do have a small sample size. And it looks very similar to a normal distribution. It has some mean. So this is your mean of your sampling distribution still. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
It's almost engineered, so it gives a better estimate of your confidence intervals and all of that when you do have a small sample size. And it looks very similar to a normal distribution. It has some mean. So this is your mean of your sampling distribution still. But it also has fatter tails. And the way I think about why it has fatter tails is when you make an assumption that this is the standard deviation for, well, let me take one more step. So normally what we do is we find the estimate of the true standard deviation. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So this is your mean of your sampling distribution still. But it also has fatter tails. And the way I think about why it has fatter tails is when you make an assumption that this is the standard deviation for, well, let me take one more step. So normally what we do is we find the estimate of the true standard deviation. And then we say that the standard deviation of the sampling distribution is equal to the true standard deviation of our population divided by the square root of n. In this case, n is equal to 7. And then we say, OK, we never know the true standard, or we seldom know. Sometimes you do know. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So normally what we do is we find the estimate of the true standard deviation. And then we say that the standard deviation of the sampling distribution is equal to the true standard deviation of our population divided by the square root of n. In this case, n is equal to 7. And then we say, OK, we never know the true standard, or we seldom know. Sometimes you do know. We seldom know the true standard deviation. So if we don't know that, the best thing we can put in there is our sample standard deviation. So the best thing we can put in there is our sample standard deviation. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
Sometimes you do know. We seldom know the true standard deviation. So if we don't know that, the best thing we can put in there is our sample standard deviation. So the best thing we can put in there is our sample standard deviation. And this right here, this is the whole reason why we don't say that this is just a 95 probability interval. This is the whole reason why we call it a confidence interval, because we're making some assumptions here. This thing is going to change from sample to sample. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So the best thing we can put in there is our sample standard deviation. And this right here, this is the whole reason why we don't say that this is just a 95 probability interval. This is the whole reason why we call it a confidence interval, because we're making some assumptions here. This thing is going to change from sample to sample. And in particular, this is going to be a particularly bad estimate when we have a small sample size, a size less than 30. So when you are estimating the standard deviation where you don't know it, you're estimating it with your sample standard deviation. And your sample size is small. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
This thing is going to change from sample to sample. And in particular, this is going to be a particularly bad estimate when we have a small sample size, a size less than 30. So when you are estimating the standard deviation where you don't know it, you're estimating it with your sample standard deviation. And your sample size is small. And you're going to use this to estimate the standard deviation of your sampling distribution. You don't assume your sampling distribution is a normal distribution. You assume it has fatter tails. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And your sample size is small. And you're going to use this to estimate the standard deviation of your sampling distribution. You don't assume your sampling distribution is a normal distribution. You assume it has fatter tails. And it has fatter tails, because you're essentially underestimating the standard deviation over here. Anyway, with all of that said, let's just actually go through this problem. So we need to think about a 95% confidence interval around this mean right over here. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
You assume it has fatter tails. And it has fatter tails, because you're essentially underestimating the standard deviation over here. Anyway, with all of that said, let's just actually go through this problem. So we need to think about a 95% confidence interval around this mean right over here. So a 95% confidence interval, if this was a normal distribution, you would just look it up in a z table. But it's not. This is a t distribution. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So we need to think about a 95% confidence interval around this mean right over here. So a 95% confidence interval, if this was a normal distribution, you would just look it up in a z table. But it's not. This is a t distribution. This is a t distribution. We're looking for a 95% confidence interval. So some interval around the mean that encapsulates 95% of the area. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
This is a t distribution. This is a t distribution. We're looking for a 95% confidence interval. So some interval around the mean that encapsulates 95% of the area. For t distribution, you use a t table. And I have a t table ahead of time right over here. And what you want to do is use the two-sided. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So some interval around the mean that encapsulates 95% of the area. For t distribution, you use a t table. And I have a t table ahead of time right over here. And what you want to do is use the two-sided. You want to use a two-sided row for what we're doing right over here. And the best way to think about it is that we're symmetric around the mean. And that's why they call it two-sided. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And what you want to do is use the two-sided. You want to use a two-sided row for what we're doing right over here. And the best way to think about it is that we're symmetric around the mean. And that's why they call it two-sided. One-sided if it was kind of a cumulative percentage up to some critical threshold. But in this case, it's two-sided. We're symmetric. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And that's why they call it two-sided. One-sided if it was kind of a cumulative percentage up to some critical threshold. But in this case, it's two-sided. We're symmetric. Or another way to think about it is we're excluding the two sides. So we want the 95% in the middle. And this is a sampling distribution of the sample mean for n is equal to 7. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
We're symmetric. Or another way to think about it is we're excluding the two sides. So we want the 95% in the middle. And this is a sampling distribution of the sample mean for n is equal to 7. And I won't go into the details here. But when n is equal to 7, you have 6 degrees of freedom, or n minus 1. And the way that t tables are set up, you go and find the degrees of freedom. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And this is a sampling distribution of the sample mean for n is equal to 7. And I won't go into the details here. But when n is equal to 7, you have 6 degrees of freedom, or n minus 1. And the way that t tables are set up, you go and find the degrees of freedom. So you don't go to the n. You go to the n minus 1. So you go to the 6 right here. So if you want to encapsulate 95% of this right over here, and you have an n of 6, you have to go 2.447 standard deviations in each direction. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And the way that t tables are set up, you go and find the degrees of freedom. So you don't go to the n. You go to the n minus 1. So you go to the 6 right here. So if you want to encapsulate 95% of this right over here, and you have an n of 6, you have to go 2.447 standard deviations in each direction. And this t table assumes that you are approximating that standard deviation using your sample standard deviation. So it's another way to think of it. You have to go 2.447 of these approximated standard deviations. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So if you want to encapsulate 95% of this right over here, and you have an n of 6, you have to go 2.447 standard deviations in each direction. And this t table assumes that you are approximating that standard deviation using your sample standard deviation. So it's another way to think of it. You have to go 2.447 of these approximated standard deviations. So let me go right here. So you have to go 2.447. This distance right here is 2.447 times this approximated standard deviation. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
You have to go 2.447 of these approximated standard deviations. So let me go right here. So you have to go 2.447. This distance right here is 2.447 times this approximated standard deviation. And sometimes, you'll see this in some statistics books, this thing right here, this exact number, is shown like this. They put a little hat on top of the standard deviation to show that it has been approximated using the sample standard deviation. So we'll put a little hat over here, because frankly, this is the only thing that we can calculate. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
This distance right here is 2.447 times this approximated standard deviation. And sometimes, you'll see this in some statistics books, this thing right here, this exact number, is shown like this. They put a little hat on top of the standard deviation to show that it has been approximated using the sample standard deviation. So we'll put a little hat over here, because frankly, this is the only thing that we can calculate. So this is how far you have to go in each direction. And we know what this value is. We know what the sample distribution is. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So we'll put a little hat over here, because frankly, this is the only thing that we can calculate. So this is how far you have to go in each direction. And we know what this value is. We know what the sample distribution is. So let's get our calculator out. So we know our sample standard deviation is 1.04. And we want to divide that by the square root of 7. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
We know what the sample distribution is. So let's get our calculator out. So we know our sample standard deviation is 1.04. And we want to divide that by the square root of 7. So we get 0.39. So this right here is 0.39. And so if we want to find the distance around this population mean that encapsulates 95% of the population, or of the sampling distribution, we have to multiply 0.39 times 2.447. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And we want to divide that by the square root of 7. So we get 0.39. So this right here is 0.39. And so if we want to find the distance around this population mean that encapsulates 95% of the population, or of the sampling distribution, we have to multiply 0.39 times 2.447. So let's do that. So times 2.447 is equal to 0.96. So this distance right here is 0.96. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And so if we want to find the distance around this population mean that encapsulates 95% of the population, or of the sampling distribution, we have to multiply 0.39 times 2.447. So let's do that. So times 2.447 is equal to 0.96. So this distance right here is 0.96. And then this distance right here is 0.96. So if you take a random sample, and that's exactly what we did when we found these seven samples. When we took these seven samples and took their mean, that mean can be viewed as a random sample from the sampling distribution. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So this distance right here is 0.96. And then this distance right here is 0.96. So if you take a random sample, and that's exactly what we did when we found these seven samples. When we took these seven samples and took their mean, that mean can be viewed as a random sample from the sampling distribution. And so the probability, so we can view it. We can say that there's a 95% chance. And we have to actually caveat everything with a confident, because we're doing all of these estimations here. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
When we took these seven samples and took their mean, that mean can be viewed as a random sample from the sampling distribution. And so the probability, so we can view it. We can say that there's a 95% chance. And we have to actually caveat everything with a confident, because we're doing all of these estimations here. So it's not a true, precise 95% chance. We're just confident that there's a 95% chance that our random sampling mean right here, so that 2.34, which we can kind of use. We just pick that 2.34 from this distribution right here. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
And we have to actually caveat everything with a confident, because we're doing all of these estimations here. So it's not a true, precise 95% chance. We're just confident that there's a 95% chance that our random sampling mean right here, so that 2.34, which we can kind of use. We just pick that 2.34 from this distribution right here. So there's a 95% chance that 2.34 is within 0.96 of the true sampling distribution mean, which we know is also the same thing as the population mean. So I'll just say of the population mean. Or we can just rearrange the sentence and say that there is a 95% chance that the mean, the true mean, which is the same thing as a sampling distribution mean, is within 0.96 of our sample mean of 2.34. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
We just pick that 2.34 from this distribution right here. So there's a 95% chance that 2.34 is within 0.96 of the true sampling distribution mean, which we know is also the same thing as the population mean. So I'll just say of the population mean. Or we can just rearrange the sentence and say that there is a 95% chance that the mean, the true mean, which is the same thing as a sampling distribution mean, is within 0.96 of our sample mean of 2.34. So at the low end, so if you go 2.34 minus 0.96, that's the low end of our confidence interval, 1.38. And the high end of our confidence interval, 2.34 plus 0.96 is equal to 3.3. So our 95% confidence interval is from 1.38 to 3.3. | Small sample size confidence intervals Probability and Statistics Khan Academy.mp3 |
So there's four suits, each of them have nine cards, so that gives us 36 unique cards. A hand is a collection of nine cards, which can be sorted however the player chooses. So they're essentially telling us that order does not matter. What is the probability of getting all four of the ones? So they want to know the probability of getting all four of the ones. So all four ones in my hand of nine. Now this is kind of daunting at first. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
What is the probability of getting all four of the ones? So they want to know the probability of getting all four of the ones. So all four ones in my hand of nine. Now this is kind of daunting at first. You're like, gee, I have nine cards and I'm kicking them out of 36. I have to figure out how do I get all of the ones. But if we think about it in very simple terms, all a probability is saying is the number of events, or I guess you could say the number of ways in which this action or this event happens. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Now this is kind of daunting at first. You're like, gee, I have nine cards and I'm kicking them out of 36. I have to figure out how do I get all of the ones. But if we think about it in very simple terms, all a probability is saying is the number of events, or I guess you could say the number of ways in which this action or this event happens. So this is what the definition of the probability is. It's going to be the number of ways in which event can happen, and when we talk about the event, we're talking about having all four ones in my hand. That's the event. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
But if we think about it in very simple terms, all a probability is saying is the number of events, or I guess you could say the number of ways in which this action or this event happens. So this is what the definition of the probability is. It's going to be the number of ways in which event can happen, and when we talk about the event, we're talking about having all four ones in my hand. That's the event. And all of these different ways, that's sometimes called the event space. But we actually want to count how many ways that if I get a hand of nine, picking from 36, that I can get the four ones in it, so this is the number of ways in which my event can happen. And we want to divide that into all of the possibilities. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
That's the event. And all of these different ways, that's sometimes called the event space. But we actually want to count how many ways that if I get a hand of nine, picking from 36, that I can get the four ones in it, so this is the number of ways in which my event can happen. And we want to divide that into all of the possibilities. Or maybe I should write it this way. The total number of hands that I can get. So the numerator in blue is the number of hands, or the number of different hands, where I have the four ones, and we're dividing it to divide the total number of hands. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And we want to divide that into all of the possibilities. Or maybe I should write it this way. The total number of hands that I can get. So the numerator in blue is the number of hands, or the number of different hands, where I have the four ones, and we're dividing it to divide the total number of hands. Now let's figure out the total number of hands first, because at some level this might be more intuitive, and we've actually done this before. Now, the total number of hands, we're picking nine cards, and we're picking them from a set of 36 unique cards. And we've done this many, many times. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So the numerator in blue is the number of hands, or the number of different hands, where I have the four ones, and we're dividing it to divide the total number of hands. Now let's figure out the total number of hands first, because at some level this might be more intuitive, and we've actually done this before. Now, the total number of hands, we're picking nine cards, and we're picking them from a set of 36 unique cards. And we've done this many, many times. Let me write this. Total number of hands, or total number of possible hands. That's equal to, you can imagine you have nine cards to pick from. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And we've done this many, many times. Let me write this. Total number of hands, or total number of possible hands. That's equal to, you can imagine you have nine cards to pick from. The first card you pick is going to be one of 36 cards, then the next one's going to be one of 35, then the next one's going to be one of 34, 33, 32, 31. We're going to do this nine times, 1, 2, 3, 4, 5, 6, 7, 8, and 9. So that would be the total number of hands if order mattered. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
That's equal to, you can imagine you have nine cards to pick from. The first card you pick is going to be one of 36 cards, then the next one's going to be one of 35, then the next one's going to be one of 34, 33, 32, 31. We're going to do this nine times, 1, 2, 3, 4, 5, 6, 7, 8, and 9. So that would be the total number of hands if order mattered. But we know, and we've gone over this before, that we don't care about the order. All we care about the actual cards that are in there. So we're over counting here. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So that would be the total number of hands if order mattered. But we know, and we've gone over this before, that we don't care about the order. All we care about the actual cards that are in there. So we're over counting here. We're over counting for all of the different rearrangements that these cards would have. It doesn't matter whether the ace of diamonds is the first card I pick or the last card I pick. The way I've counted them right now, we are counting those as two separate hands. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So we're over counting here. We're over counting for all of the different rearrangements that these cards would have. It doesn't matter whether the ace of diamonds is the first card I pick or the last card I pick. The way I've counted them right now, we are counting those as two separate hands. But they aren't two separate hands, so order doesn't matter. So what we have to do is, we have to divide this by the number of ways you can arrange nine things. So you could put nine of the things in the first position, then eight in the second, seven in the third, so forth and so on. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
The way I've counted them right now, we are counting those as two separate hands. But they aren't two separate hands, so order doesn't matter. So what we have to do is, we have to divide this by the number of ways you can arrange nine things. So you could put nine of the things in the first position, then eight in the second, seven in the third, so forth and so on. It essentially becomes 9 factorial, times 2, times 1. And we've seen this multiple times. This is essentially 36 choose 9. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So you could put nine of the things in the first position, then eight in the second, seven in the third, so forth and so on. It essentially becomes 9 factorial, times 2, times 1. And we've seen this multiple times. This is essentially 36 choose 9. This expression right here is the same thing, just so you can relate it to the, I guess, combinatorics formulas that you might be familiar with. This is the same thing as 36 factorial over 36 minus 9 factorial, that's what this orange part is over here, divided by 9 factorial, or over 9 factorial. What's green is what's green, and what is orange is what's orange there. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
This is essentially 36 choose 9. This expression right here is the same thing, just so you can relate it to the, I guess, combinatorics formulas that you might be familiar with. This is the same thing as 36 factorial over 36 minus 9 factorial, that's what this orange part is over here, divided by 9 factorial, or over 9 factorial. What's green is what's green, and what is orange is what's orange there. So that's the total number of hands. Now, a little bit more of a nuanced thought process is, how do we figure out the number of ways in which the event can happen, in which we can have all four 1's? So let's figure that out. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
What's green is what's green, and what is orange is what's orange there. So that's the total number of hands. Now, a little bit more of a nuanced thought process is, how do we figure out the number of ways in which the event can happen, in which we can have all four 1's? So let's figure that out. So number of ways, or maybe we should say this, number of hands with four 1's. And just as a little bit of thought experiment, imagine if we were only taking four cards, if a hand only had four cards in it. Well, if a card only had four hands, if a hand only had four cards in it, then the number of ways to get a hand with four 1's, there would only be one way, one combination. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So let's figure that out. So number of ways, or maybe we should say this, number of hands with four 1's. And just as a little bit of thought experiment, imagine if we were only taking four cards, if a hand only had four cards in it. Well, if a card only had four hands, if a hand only had four cards in it, then the number of ways to get a hand with four 1's, there would only be one way, one combination. You just have four 1's. That's the only combination with four 1's if we were only picking four cards. But here, we're not only picking four cards. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
Well, if a card only had four hands, if a hand only had four cards in it, then the number of ways to get a hand with four 1's, there would only be one way, one combination. You just have four 1's. That's the only combination with four 1's if we were only picking four cards. But here, we're not only picking four cards. Four of the cards are going to be 1's. So four of the cards are going to be 1's. I mean, 1, 2, 3, 4, but the other five cards are going to be different. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
But here, we're not only picking four cards. Four of the cards are going to be 1's. So four of the cards are going to be 1's. I mean, 1, 2, 3, 4, but the other five cards are going to be different. So 1, 2, 3, 4, 5. So for the other five cards, if you imagine this slot, considering that of the 36, we would have to pick four of them already in order for us to have four 1's. So there's another, well, we've used up four of them, so there's 32 possible cards over in that position of the hand. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
I mean, 1, 2, 3, 4, but the other five cards are going to be different. So 1, 2, 3, 4, 5. So for the other five cards, if you imagine this slot, considering that of the 36, we would have to pick four of them already in order for us to have four 1's. So there's another, well, we've used up four of them, so there's 32 possible cards over in that position of the hand. And then there'd be 31 in that position of the hand. And then there'd be 30, because every time we're picking a card, we're using it up. And now we only have 30 to pick from. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So there's another, well, we've used up four of them, so there's 32 possible cards over in that position of the hand. And then there'd be 31 in that position of the hand. And then there'd be 30, because every time we're picking a card, we're using it up. And now we only have 30 to pick from. Then we only have 29 to pick from. And then we have 28 to pick from. And just like we did before, we don't care about order. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And now we only have 30 to pick from. Then we only have 29 to pick from. And then we have 28 to pick from. And just like we did before, we don't care about order. We don't care if we pick the five of clubs first or whether we pick the five of clubs last. So we shouldn't double count it. So we have to divide by the different number of ways that five cards can be arranged. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
And just like we did before, we don't care about order. We don't care if we pick the five of clubs first or whether we pick the five of clubs last. So we shouldn't double count it. So we have to divide by the different number of ways that five cards can be arranged. So we have to divide this by the different ways that five cards can be arranged. So the first card or the first position could be any one of five cards, then four cards, then three cards, then two cards, then one cards. So the number of hands with four 1's is actually just this number. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So we have to divide by the different number of ways that five cards can be arranged. So we have to divide this by the different ways that five cards can be arranged. So the first card or the first position could be any one of five cards, then four cards, then three cards, then two cards, then one cards. So the number of hands with four 1's is actually just this number. You're actually looking at all of the different ways you can fill up the remaining cards. These four 1's are just going to be four 1's. There's only one way to get that. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
So the number of hands with four 1's is actually just this number. You're actually looking at all of the different ways you can fill up the remaining cards. These four 1's are just going to be four 1's. There's only one way to get that. It's the remaining cards that's going to give all of the different combinations of having four 1's. So this will be a count of all of the different combinations. Because all of the different extra stuff that you have will be all of the different hands. | Example Combinatorics and probability Probability and combinatorics Precalculus Khan Academy.mp3 |
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