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Well, actually, you want a negative number, because you want your z-score to be positive or negative. Negative would mean to the left of the mean, and positive would mean to the right of the mean. So we say 65 minus 81. So that's literally how far away we are, but we want that in terms of standard deviations. So we divide that by the length, or the magnitude, of our standard deviation. So 65 minus 81. Let's see, 81 minus 65 is what? | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
So that's literally how far away we are, but we want that in terms of standard deviations. So we divide that by the length, or the magnitude, of our standard deviation. So 65 minus 81. Let's see, 81 minus 65 is what? It is 5 plus 11, it's 16. So this is going to be minus 16 over 6.3. And take our calculator out, and let's see, if we have minus 16 divided by 6.3, you get minus 2.54. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
Let's see, 81 minus 65 is what? It is 5 plus 11, it's 16. So this is going to be minus 16 over 6.3. And take our calculator out, and let's see, if we have minus 16 divided by 6.3, you get minus 2.54. So approximately equal to minus 2.54. That's the z-score for a grade of 65. Pretty straightforward. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
And take our calculator out, and let's see, if we have minus 16 divided by 6.3, you get minus 2.54. So approximately equal to minus 2.54. That's the z-score for a grade of 65. Pretty straightforward. Let's do a couple more. Let's do all of them. 83. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
Pretty straightforward. Let's do a couple more. Let's do all of them. 83. So how far is away from the mean? Well, it's 83 minus 81. It's two grades above the mean. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
83. So how far is away from the mean? Well, it's 83 minus 81. It's two grades above the mean. But we want it in terms of standard deviations. How many standard deviations? So this was part a. a was right here. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
It's two grades above the mean. But we want it in terms of standard deviations. How many standard deviations? So this was part a. a was right here. We were at 2.5 standard deviations below the mean. So this was part a. 1, 2, and then 0.5, so this was a right there, 65. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
So this was part a. a was right here. We were at 2.5 standard deviations below the mean. So this was part a. 1, 2, and then 0.5, so this was a right there, 65. And then part b, 83. 83 is going to be right here, a little bit higher. We're right here. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
1, 2, and then 0.5, so this was a right there, 65. And then part b, 83. 83 is going to be right here, a little bit higher. We're right here. And the z-score here, 83 minus 81 divided by 6.3 will get us, let's see, clear the calculator. So we have, well, 83 minus 81 is 2. Divided by 6.3, 0.32, roughly. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
We're right here. And the z-score here, 83 minus 81 divided by 6.3 will get us, let's see, clear the calculator. So we have, well, 83 minus 81 is 2. Divided by 6.3, 0.32, roughly. So here we get 0.32. So 83 is 0.32 standard deviations above the mean. And so it would be roughly 1 third of the standard deviation along the way, right? | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
Divided by 6.3, 0.32, roughly. So here we get 0.32. So 83 is 0.32 standard deviations above the mean. And so it would be roughly 1 third of the standard deviation along the way, right? Because this was one whole standard deviation. So we're 0.3 of a standard deviation above the mean. Choice number c. Or not choice, part c, I guess I should call it. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
And so it would be roughly 1 third of the standard deviation along the way, right? Because this was one whole standard deviation. So we're 0.3 of a standard deviation above the mean. Choice number c. Or not choice, part c, I guess I should call it. 93. Well, we do the same exercise. 93 is how much above the mean? | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
Choice number c. Or not choice, part c, I guess I should call it. 93. Well, we do the same exercise. 93 is how much above the mean? Well, 93 minus 81 is 12. But we want it in terms of standard deviations. So 12 is how many standard deviations above the mean? | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
93 is how much above the mean? Well, 93 minus 81 is 12. But we want it in terms of standard deviations. So 12 is how many standard deviations above the mean? Well, it's going to be almost 2. Let's take the calculator out. So we get 12 divided by 6.3. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
So 12 is how many standard deviations above the mean? Well, it's going to be almost 2. Let's take the calculator out. So we get 12 divided by 6.3. It's 1.9 standard deviations. It's z-score. It's z-score is 1.9, which means it's 1.9 standard deviations above the mean. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
So we get 12 divided by 6.3. It's 1.9 standard deviations. It's z-score. It's z-score is 1.9, which means it's 1.9 standard deviations above the mean. So the mean is 81. We go one whole standard deviation, and then 0.9 standard deviations. And that's where a score of 93 would lie right there. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
It's z-score is 1.9, which means it's 1.9 standard deviations above the mean. So the mean is 81. We go one whole standard deviation, and then 0.9 standard deviations. And that's where a score of 93 would lie right there. It's z-score is 1.9. That all that means is 1.9 standard deviations above the mean. Let's do the last one. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
And that's where a score of 93 would lie right there. It's z-score is 1.9. That all that means is 1.9 standard deviations above the mean. Let's do the last one. I'll do it in magenta. D, part d. Score of 100. We don't even need the problem anymore. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
Let's do the last one. I'll do it in magenta. D, part d. Score of 100. We don't even need the problem anymore. A score of 100. Well, same thing. We figure out how far is 100 above the mean. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
We don't even need the problem anymore. A score of 100. Well, same thing. We figure out how far is 100 above the mean. Remember, the mean was 81. And we divide that by the length or the size or the magnitude of our standard deviation. So 100 minus 81 is equal to 19 over 6.3. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
We figure out how far is 100 above the mean. Remember, the mean was 81. And we divide that by the length or the size or the magnitude of our standard deviation. So 100 minus 81 is equal to 19 over 6.3. So it's going to be a little over 3 standard deviations. And in the next problem, we'll see what does that imply in terms of the probability of that actually occurring. But if we just want to figure out the z-score, 19 divided by 6.3 is equal to 3.01. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
So 100 minus 81 is equal to 19 over 6.3. So it's going to be a little over 3 standard deviations. And in the next problem, we'll see what does that imply in terms of the probability of that actually occurring. But if we just want to figure out the z-score, 19 divided by 6.3 is equal to 3.01. So it's very close. 3.02, really, if I were to round. So it's very close to 3.02. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
But if we just want to figure out the z-score, 19 divided by 6.3 is equal to 3.01. So it's very close. 3.02, really, if I were to round. So it's very close to 3.02. It's z-score is 3.02. Or a grade of 100 is 3.02 standard deviations above the mean. So remember, this was the mean right here. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
So it's very close to 3.02. It's z-score is 3.02. Or a grade of 100 is 3.02 standard deviations above the mean. So remember, this was the mean right here. Right here at 81. We go one standard deviation above the mean. Two standard deviations above the mean. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
So remember, this was the mean right here. Right here at 81. We go one standard deviation above the mean. Two standard deviations above the mean. The third standard deviation above the mean is right there. So we're sitting right there on our chart. A little bit above that. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
Two standard deviations above the mean. The third standard deviation above the mean is right there. So we're sitting right there on our chart. A little bit above that. 3.02 standard deviations above the mean, that's where a score of 100 would be. And you can see the probability, the height of this, that's what the chart tells us. It's actually a very low probability. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
A little bit above that. 3.02 standard deviations above the mean, that's where a score of 100 would be. And you can see the probability, the height of this, that's what the chart tells us. It's actually a very low probability. And actually, it's not just a very low probability of getting something higher than that. Because we've learned before in a probability density function, the probability, if this is a continuous, not a discrete, the probability of getting exactly that is 0 if this wasn't discrete. But since this is a score, it's not a test, we know that it's actually a discrete probability function. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
It's actually a very low probability. And actually, it's not just a very low probability of getting something higher than that. Because we've learned before in a probability density function, the probability, if this is a continuous, not a discrete, the probability of getting exactly that is 0 if this wasn't discrete. But since this is a score, it's not a test, we know that it's actually a discrete probability function. But the probability is low of getting higher than that. Because you can see where we sit in the bell curve. Well, anyway, hopefully this at least clarified how to solve for z-scores, which is pretty straightforward mathematically. | ck12.org normal distribution problems z-score Probability and Statistics Khan Academy.mp3 |
At one of its large factories, 2% of the chips produced are defective in some way. A quality check involves randomly selecting and testing 500 chips. What are the mean and standard deviation of the number of defective processing chips in these samples? So like always, try to pause this video and have a go at it on your own, and then we will work through it together. All right, so let me define a random variable that we're gonna find the mean and standard deviation of. And I'm gonna make that random variable the number of defective processing chips in a 500-chip sample. So let's let X be equal to the number of defective chips in 500-chip sample. | Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3 |
So like always, try to pause this video and have a go at it on your own, and then we will work through it together. All right, so let me define a random variable that we're gonna find the mean and standard deviation of. And I'm gonna make that random variable the number of defective processing chips in a 500-chip sample. So let's let X be equal to the number of defective chips in 500-chip sample. So the first thing to recognize is that this will be a binomial variable. This is binomial. How do we know it's binomial? | Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3 |
So let's let X be equal to the number of defective chips in 500-chip sample. So the first thing to recognize is that this will be a binomial variable. This is binomial. How do we know it's binomial? Well, it's made up of 500, it's a finite number of trials right over here. The probability of getting a defective chip, you could view this as a probability of success. It's a little bit counterintuitive that a defective chip would be a success, but we're summing up the defective chips. | Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3 |
How do we know it's binomial? Well, it's made up of 500, it's a finite number of trials right over here. The probability of getting a defective chip, you could view this as a probability of success. It's a little bit counterintuitive that a defective chip would be a success, but we're summing up the defective chips. So we would view the probability of a defect, or I should say a defective chip, it is constant across these 500 trials, and we will assume that they are independent of each other, 0.02. You might be saying, hey, well, are we replacing the chips before or after? But we're assuming it's from a functionally infinite population, or if you wanna make it feel better, you could say, well, maybe you are replacing the chips. | Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3 |
It's a little bit counterintuitive that a defective chip would be a success, but we're summing up the defective chips. So we would view the probability of a defect, or I should say a defective chip, it is constant across these 500 trials, and we will assume that they are independent of each other, 0.02. You might be saying, hey, well, are we replacing the chips before or after? But we're assuming it's from a functionally infinite population, or if you wanna make it feel better, you could say, well, maybe you are replacing the chips. They're not really telling us that right over here. So we'll assume that each of these trials are independent of each other, and that the probability of getting a defective chip stays constant here. And so this is a binomial random variable, or binomial variable, and we know the formulas for the mean and standard deviation of a binomial variable. | Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3 |
But we're assuming it's from a functionally infinite population, or if you wanna make it feel better, you could say, well, maybe you are replacing the chips. They're not really telling us that right over here. So we'll assume that each of these trials are independent of each other, and that the probability of getting a defective chip stays constant here. And so this is a binomial random variable, or binomial variable, and we know the formulas for the mean and standard deviation of a binomial variable. So the mean, the mean of x, which is the same thing as the expected value of x, is going to be equal to the number of trials, n, times the probability of success on each trial, times p. So what is this going to be? Well, this is going to be equal to, we have 500 trials, and then the probability of success on each of these trials is 0.02. So it's 500 times 0.02. | Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3 |
And so this is a binomial random variable, or binomial variable, and we know the formulas for the mean and standard deviation of a binomial variable. So the mean, the mean of x, which is the same thing as the expected value of x, is going to be equal to the number of trials, n, times the probability of success on each trial, times p. So what is this going to be? Well, this is going to be equal to, we have 500 trials, and then the probability of success on each of these trials is 0.02. So it's 500 times 0.02. And what is this going to be? 500 times 2 hundredths is going to be, it's going to be equal to 10. So that is your expected value of the number of defective processing chips, or the mean. | Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3 |
So it's 500 times 0.02. And what is this going to be? 500 times 2 hundredths is going to be, it's going to be equal to 10. So that is your expected value of the number of defective processing chips, or the mean. Now what about the standard deviation? So the standard deviation of our random variable, x, well that's just going to be equal to the square root of the variance of our random variable, x. So I could just write it, I'm just writing it all the different ways that you might see it, because sometimes the notation is the most confusing part in statistics. | Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3 |
So that is your expected value of the number of defective processing chips, or the mean. Now what about the standard deviation? So the standard deviation of our random variable, x, well that's just going to be equal to the square root of the variance of our random variable, x. So I could just write it, I'm just writing it all the different ways that you might see it, because sometimes the notation is the most confusing part in statistics. And so this is going to be the square root of what? Well, the variance of a binomial variable is going to be equal to the number of trials, times the probability of success in each trial, times one minus the probability of success in each trial. And so in this context, this is going to be equal to, you're gonna have the 500, 500, times 0.02, 0.02, times one minus 0.02 is.98. | Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3 |
So I could just write it, I'm just writing it all the different ways that you might see it, because sometimes the notation is the most confusing part in statistics. And so this is going to be the square root of what? Well, the variance of a binomial variable is going to be equal to the number of trials, times the probability of success in each trial, times one minus the probability of success in each trial. And so in this context, this is going to be equal to, you're gonna have the 500, 500, times 0.02, 0.02, times one minus 0.02 is.98. So times 0.98, and all of this is under the radical sign. I didn't make that radical sign long enough. And so what is this going to be? | Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3 |
And so in this context, this is going to be equal to, you're gonna have the 500, 500, times 0.02, 0.02, times one minus 0.02 is.98. So times 0.98, and all of this is under the radical sign. I didn't make that radical sign long enough. And so what is this going to be? Well, let's see, 500 times 0.02, we already said that this is going to be 10, 10 times 0.98, this is going to be equal to the square root of 9.8. So it's going to be, I don't know, three point something. If we want, we can get a calculator out to feel a little bit better about this value. | Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3 |
And so what is this going to be? Well, let's see, 500 times 0.02, we already said that this is going to be 10, 10 times 0.98, this is going to be equal to the square root of 9.8. So it's going to be, I don't know, three point something. If we want, we can get a calculator out to feel a little bit better about this value. So I'm gonna take 9.8, and then take the square root of it, and I get three point, if I round to the nearest hundredth, 3.13. So this is approximately 3.13 for the standard deviation. If I wanted the variance, it would be 9.8. | Finding the mean and standard deviation of a binomial random variable AP Statistics Khan Academy.mp3 |
What we're going to do in this video is talk about the idea of a sampling distribution. Now, just to make things a little bit concrete, let's imagine that we have a population of some kind. Let's say it's a bunch of balls, each of them have a number written on it. For that population, we could calculate parameters. So a parameter you could view as a truth about that population. We've covered this in other videos. So for example, you could have the population mean, the mean of the numbers written on top of that ball. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
For that population, we could calculate parameters. So a parameter you could view as a truth about that population. We've covered this in other videos. So for example, you could have the population mean, the mean of the numbers written on top of that ball. You could have the population standard deviation. You could have the proportion of balls that are even, whatever, these are all population parameters. Now, we know from many other videos that you might not know the population parameter or it might not even be easy to find. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
So for example, you could have the population mean, the mean of the numbers written on top of that ball. You could have the population standard deviation. You could have the proportion of balls that are even, whatever, these are all population parameters. Now, we know from many other videos that you might not know the population parameter or it might not even be easy to find. And so the way that we try to estimate a population parameter is by taking a sample. So this right over here is a sample of size n, sample of size n. And then we can calculate a statistic from that sample, based on that sample, maybe we picked n balls from there. And so from that, we can calculate a statistic that is used to estimate this parameter. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
Now, we know from many other videos that you might not know the population parameter or it might not even be easy to find. And so the way that we try to estimate a population parameter is by taking a sample. So this right over here is a sample of size n, sample of size n. And then we can calculate a statistic from that sample, based on that sample, maybe we picked n balls from there. And so from that, we can calculate a statistic that is used to estimate this parameter. But we know that this is a random sample right over here. So every time we take a sample, the statistic that we calculate for that sample is not necessarily going to be the same as the population parameter. In fact, if we were to take a random sample of size n again and then we were to calculate the statistic again, we could very well get a different value. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And so from that, we can calculate a statistic that is used to estimate this parameter. But we know that this is a random sample right over here. So every time we take a sample, the statistic that we calculate for that sample is not necessarily going to be the same as the population parameter. In fact, if we were to take a random sample of size n again and then we were to calculate the statistic again, we could very well get a different value. So these are all going to be estimates of this parameter. And so an interesting question is, is what is the distribution of the values that I could get for these statistics? What is the frequency with which I could get different values for the statistic that is trying to estimate this parameter? | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
In fact, if we were to take a random sample of size n again and then we were to calculate the statistic again, we could very well get a different value. So these are all going to be estimates of this parameter. And so an interesting question is, is what is the distribution of the values that I could get for these statistics? What is the frequency with which I could get different values for the statistic that is trying to estimate this parameter? And that distribution is what a sampling distribution is. So let's make this even a little bit more concrete. Let's imagine where our population, I'm gonna make this a very simple example. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
What is the frequency with which I could get different values for the statistic that is trying to estimate this parameter? And that distribution is what a sampling distribution is. So let's make this even a little bit more concrete. Let's imagine where our population, I'm gonna make this a very simple example. Let's say our population has three balls in it, one, two, three, and they're numbered one, two, and three. And it's very easy to calculate, let's say the parameter that we care about right over here is the population mean. And that of course is going to be one plus two plus three, all of that over three, which is six divided by three, which is two. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
Let's imagine where our population, I'm gonna make this a very simple example. Let's say our population has three balls in it, one, two, three, and they're numbered one, two, and three. And it's very easy to calculate, let's say the parameter that we care about right over here is the population mean. And that of course is going to be one plus two plus three, all of that over three, which is six divided by three, which is two. So that is our population parameter. But let's say that we wanted to take samples, let's say samples of two balls at a time, and every time we take a ball, we'll replace it. So each ball we take, it is an independent pick. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And that of course is going to be one plus two plus three, all of that over three, which is six divided by three, which is two. So that is our population parameter. But let's say that we wanted to take samples, let's say samples of two balls at a time, and every time we take a ball, we'll replace it. So each ball we take, it is an independent pick. And we're gonna use those samples of two balls at a time in order to estimate the population mean. So for example, this could be our first sample of size two. And let's say in that first sample, I pick a one and let's say I pick a two. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
So each ball we take, it is an independent pick. And we're gonna use those samples of two balls at a time in order to estimate the population mean. So for example, this could be our first sample of size two. And let's say in that first sample, I pick a one and let's say I pick a two. Well, then I can calculate the sample statistic here. In this case, it would be the sample mean, which is used to estimate the population mean. And in this, for this sample of two, it's going to be 1.5. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And let's say in that first sample, I pick a one and let's say I pick a two. Well, then I can calculate the sample statistic here. In this case, it would be the sample mean, which is used to estimate the population mean. And in this, for this sample of two, it's going to be 1.5. Then I can do it again. And let's say I get a one and I get a three. Well, now when I calculate the sample mean, the average of one and three or the mean of one and three is going to be equal to two. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And in this, for this sample of two, it's going to be 1.5. Then I can do it again. And let's say I get a one and I get a three. Well, now when I calculate the sample mean, the average of one and three or the mean of one and three is going to be equal to two. Let's think about all of the different scenarios of samples we can get and what the associated sample means are going to be. And then we can get see the frequency of getting those sample means. And so let me draw a little bit of a table here. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
Well, now when I calculate the sample mean, the average of one and three or the mean of one and three is going to be equal to two. Let's think about all of the different scenarios of samples we can get and what the associated sample means are going to be. And then we can get see the frequency of getting those sample means. And so let me draw a little bit of a table here. So make a table right over here. And let's see, these are the numbers that we pick. And remember, when we pick one ball, we'll record that number, then we'll put it back in and then we'll pick another ball. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And so let me draw a little bit of a table here. So make a table right over here. And let's see, these are the numbers that we pick. And remember, when we pick one ball, we'll record that number, then we'll put it back in and then we'll pick another ball. So these are going to be independent events and it's gonna be with replacement. And so let's say we could pick a one and then a one. We could pick a one then a two, a one and a three. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And remember, when we pick one ball, we'll record that number, then we'll put it back in and then we'll pick another ball. So these are going to be independent events and it's gonna be with replacement. And so let's say we could pick a one and then a one. We could pick a one then a two, a one and a three. We could pick a two and then a one. We could pick a two and a two, a two and a three. We could pick a three and a one, a three and a two, or three and a three. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
We could pick a one then a two, a one and a three. We could pick a two and then a one. We could pick a two and a two, a two and a three. We could pick a three and a one, a three and a two, or three and a three. There's three possible balls for the first pick and three possible balls for the second because we're doing it with replacement. And so what is the sample mean in each of these for all of these combinations? So for this one, the sample mean is one. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
We could pick a three and a one, a three and a two, or three and a three. There's three possible balls for the first pick and three possible balls for the second because we're doing it with replacement. And so what is the sample mean in each of these for all of these combinations? So for this one, the sample mean is one. Here it is 1.5. Here it is two. Here it is 1.5. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
So for this one, the sample mean is one. Here it is 1.5. Here it is two. Here it is 1.5. Here it is two. Here it is 2.5. Here it is two. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
Here it is 1.5. Here it is two. Here it is 2.5. Here it is two. Here it is 2.5. And then here it is three. And so we can now plot the frequencies of these possible sample means that we can get. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
Here it is two. Here it is 2.5. And then here it is three. And so we can now plot the frequencies of these possible sample means that we can get. And that plot will be a sampling distribution of the sample means. So let's do that. So let me make a little chart right over here, a little graph right over here. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And so we can now plot the frequencies of these possible sample means that we can get. And that plot will be a sampling distribution of the sample means. So let's do that. So let me make a little chart right over here, a little graph right over here. So these are the possible, possible sample means. We can get a one, we could get a 1.5. We could get a two, we could get a 2.5. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
So let me make a little chart right over here, a little graph right over here. So these are the possible, possible sample means. We can get a one, we could get a 1.5. We could get a two, we could get a 2.5. Or we can get a three. And now let's see the frequency of it. And I will put that over here. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
We could get a two, we could get a 2.5. Or we can get a three. And now let's see the frequency of it. And I will put that over here. And so let's see, how many ones out of our nine possibilities we have, how many were one? Well, only one of the sample means was one. And so the relative frequency, if we just said the number, we could make this line go up one, or we could just say, hey, this is going to be one out of the nine possibilities. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And I will put that over here. And so let's see, how many ones out of our nine possibilities we have, how many were one? Well, only one of the sample means was one. And so the relative frequency, if we just said the number, we could make this line go up one, or we could just say, hey, this is going to be one out of the nine possibilities. And so let me just make that, I'll call this right over here. This is 1 9th. Now what about 1.5? | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And so the relative frequency, if we just said the number, we could make this line go up one, or we could just say, hey, this is going to be one out of the nine possibilities. And so let me just make that, I'll call this right over here. This is 1 9th. Now what about 1.5? Well, let's see, there's one, two of these possibilities out of nine. So 1.5, it would look like this. This right over here is two over nine. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
Now what about 1.5? Well, let's see, there's one, two of these possibilities out of nine. So 1.5, it would look like this. This right over here is two over nine. And now what about two? Well, we can see there's one, two, three. So three out of the nine possibilities, we got a two. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
This right over here is two over nine. And now what about two? Well, we can see there's one, two, three. So three out of the nine possibilities, we got a two. So we could say this is two, or we could say this is 3 9th, which is the same thing, of course, as 1 3rd. So this right over here is three over nine. And then what about 2.5? | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
So three out of the nine possibilities, we got a two. So we could say this is two, or we could say this is 3 9th, which is the same thing, of course, as 1 3rd. So this right over here is three over nine. And then what about 2.5? Well, there's two 2.5s, so two out of the nine times. Another way you could interpret this is, when you take a random sample with replacement of two balls, you have a 2 9th chance of having a sample mean of 2.5. And then last but not least, right over here, there's one scenario out of the nine where you get 2 3, so 1 9th. | Introduction to sampling distributions Sampling distributions AP Statistics Khan Academy.mp3 |
And we're gonna talk about two things. The different conclusions you might make based on the different significance levels that you might set, and also why it's important to set your significance levels ahead of time, before you conduct an experiment and calculate the p-values, for frankly, ethical purposes. So to help us get this, let's look at a scenario right over here, which tells us, Rahim heard that spinning, rather than flipping a penny, raises the probability above 50% that the penny lands showing heads. That's actually quite fascinating, if that's true. He tested this by spinning 10 different pennies, 10 times each, so that would be a total of 100 spins. His hypotheses were, his null hypothesis is that, by spinning, your proportion doesn't change, rather, versus flipping, it's still 50%. And his alternative hypothesis is that, by spinning, your proportion of heads is greater than 50%, where p is the true proportion of spins that a penny would land showing heads. | Comparing P-values to different significance levels AP Statistics Khan Academy.mp3 |
That's actually quite fascinating, if that's true. He tested this by spinning 10 different pennies, 10 times each, so that would be a total of 100 spins. His hypotheses were, his null hypothesis is that, by spinning, your proportion doesn't change, rather, versus flipping, it's still 50%. And his alternative hypothesis is that, by spinning, your proportion of heads is greater than 50%, where p is the true proportion of spins that a penny would land showing heads. In his 100 spins, the penny landed showing heads in 59 spins. Rahim calculated that the statistic, so this is the sample proportion here, it's 59 out of 100 were heads, so that's 0.59, or 59 hundredths, and he calculated, had an associated p-value of approximately 0.036. So, based on this scenario, if ahead of time, Rahim had set his significance level at 0.05, what conclusions would he now make? | Comparing P-values to different significance levels AP Statistics Khan Academy.mp3 |
And his alternative hypothesis is that, by spinning, your proportion of heads is greater than 50%, where p is the true proportion of spins that a penny would land showing heads. In his 100 spins, the penny landed showing heads in 59 spins. Rahim calculated that the statistic, so this is the sample proportion here, it's 59 out of 100 were heads, so that's 0.59, or 59 hundredths, and he calculated, had an associated p-value of approximately 0.036. So, based on this scenario, if ahead of time, Rahim had set his significance level at 0.05, what conclusions would he now make? And, while you're pausing it, think about how that may or may not have been different if he set his significance levels ahead of time at 0.01. Pause the video and try to figure that out. So, let's, first of all, remind ourselves what a p-value even is. | Comparing P-values to different significance levels AP Statistics Khan Academy.mp3 |
So, based on this scenario, if ahead of time, Rahim had set his significance level at 0.05, what conclusions would he now make? And, while you're pausing it, think about how that may or may not have been different if he set his significance levels ahead of time at 0.01. Pause the video and try to figure that out. So, let's, first of all, remind ourselves what a p-value even is. You could view it as the probability of getting a sample proportion at least this large if you assume that the null hypothesis is true. And if that is low enough, if it is some, if it's below some threshold, which is our significance level, then we will reject the null hypothesis. And so, in this scenario, we do see that 0.036, our p-value, is indeed less than alpha. | Comparing P-values to different significance levels AP Statistics Khan Academy.mp3 |
So, let's, first of all, remind ourselves what a p-value even is. You could view it as the probability of getting a sample proportion at least this large if you assume that the null hypothesis is true. And if that is low enough, if it is some, if it's below some threshold, which is our significance level, then we will reject the null hypothesis. And so, in this scenario, we do see that 0.036, our p-value, is indeed less than alpha. It is indeed less than 0.05. And because of that, we would reject the null hypothesis. And in everyday language, rejecting the null hypothesis is rejecting the notion that the true proportion of spins that a penny would land showing heads is 50%. | Comparing P-values to different significance levels AP Statistics Khan Academy.mp3 |
And so, in this scenario, we do see that 0.036, our p-value, is indeed less than alpha. It is indeed less than 0.05. And because of that, we would reject the null hypothesis. And in everyday language, rejecting the null hypothesis is rejecting the notion that the true proportion of spins that a penny would land showing heads is 50%. And if you reject your null hypothesis, you could also say that suggests our alternative hypothesis that the true proportion of spins that a penny would land showing heads is greater than 50%. Now, what about the situation where our significance level was lower? Well, in this situation, our p-value, our probability of getting that sample statistic if we assumed our null hypothesis were true, in this situation, it's greater than or equal to, and it's greater than in this particular situation, than our threshold, than our significance level. | Comparing P-values to different significance levels AP Statistics Khan Academy.mp3 |
And in everyday language, rejecting the null hypothesis is rejecting the notion that the true proportion of spins that a penny would land showing heads is 50%. And if you reject your null hypothesis, you could also say that suggests our alternative hypothesis that the true proportion of spins that a penny would land showing heads is greater than 50%. Now, what about the situation where our significance level was lower? Well, in this situation, our p-value, our probability of getting that sample statistic if we assumed our null hypothesis were true, in this situation, it's greater than or equal to, and it's greater than in this particular situation, than our threshold, than our significance level. And so here, we would say that we fail, fail to reject our null hypothesis. So we're failing to reject this right over here, and it will not help us suggest our alternative hypothesis. And so, because of the difference between what you would conclude given this change in significance levels, that's why it's really important to set these levels ahead of time, because you could imagine it's human nature. | Comparing P-values to different significance levels AP Statistics Khan Academy.mp3 |
Well, in this situation, our p-value, our probability of getting that sample statistic if we assumed our null hypothesis were true, in this situation, it's greater than or equal to, and it's greater than in this particular situation, than our threshold, than our significance level. And so here, we would say that we fail, fail to reject our null hypothesis. So we're failing to reject this right over here, and it will not help us suggest our alternative hypothesis. And so, because of the difference between what you would conclude given this change in significance levels, that's why it's really important to set these levels ahead of time, because you could imagine it's human nature. If you're a researcher of some kind, you want to have an interesting result. You want to discover something. You wanna be able to tell your friends, hey, my alternative hypothesis, it actually is suggested. | Comparing P-values to different significance levels AP Statistics Khan Academy.mp3 |
And so, because of the difference between what you would conclude given this change in significance levels, that's why it's really important to set these levels ahead of time, because you could imagine it's human nature. If you're a researcher of some kind, you want to have an interesting result. You want to discover something. You wanna be able to tell your friends, hey, my alternative hypothesis, it actually is suggested. We can reject the assumption, the status quo. I found something that actually makes a difference. And so it's very tempting for a researcher to calculate your p-values and then say, oh, well, maybe no one will notice if I then set my significance values so that it's just high enough so that I can reject my null hypothesis. | Comparing P-values to different significance levels AP Statistics Khan Academy.mp3 |
You wanna be able to tell your friends, hey, my alternative hypothesis, it actually is suggested. We can reject the assumption, the status quo. I found something that actually makes a difference. And so it's very tempting for a researcher to calculate your p-values and then say, oh, well, maybe no one will notice if I then set my significance values so that it's just high enough so that I can reject my null hypothesis. If you did that, that would be very unethical. In future videos, we'll start thinking about the question of, okay, if I'm doing it ahead of time, if I'm setting my significance level ahead of time, how do I decide to set the threshold? When should it be 1 100th? | Comparing P-values to different significance levels AP Statistics Khan Academy.mp3 |
And so it's very tempting for a researcher to calculate your p-values and then say, oh, well, maybe no one will notice if I then set my significance values so that it's just high enough so that I can reject my null hypothesis. If you did that, that would be very unethical. In future videos, we'll start thinking about the question of, okay, if I'm doing it ahead of time, if I'm setting my significance level ahead of time, how do I decide to set the threshold? When should it be 1 100th? When should it be 5 100ths? When should it be 10 100ths? Or when should it be something else? | Comparing P-values to different significance levels AP Statistics Khan Academy.mp3 |
And so let's say we can set that, and let's make that 60% of the gumballs are green. But let's say someone else comes along and they don't actually know the proportion of gumballs that are green, but they can take samples. And so let's say they take samples of 50 at a time. And so they draw a sample. The sample proportion right over here actually just happened to be 0.6. But then they could draw another sample. This time the sample proportion is 0.52, or 52% of those 50 gumballs happened to be green. | Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3 |
And so they draw a sample. The sample proportion right over here actually just happened to be 0.6. But then they could draw another sample. This time the sample proportion is 0.52, or 52% of those 50 gumballs happened to be green. Now you could say, all right, well, these are all different estimates, but for any given estimate, how confident are we that a certain range around that estimate actually contains the true population proportion? And so if we look at this tab right over here, that's what confidence intervals are good for. And in a previous video, we talked about how you calculate the confidence interval. | Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3 |
This time the sample proportion is 0.52, or 52% of those 50 gumballs happened to be green. Now you could say, all right, well, these are all different estimates, but for any given estimate, how confident are we that a certain range around that estimate actually contains the true population proportion? And so if we look at this tab right over here, that's what confidence intervals are good for. And in a previous video, we talked about how you calculate the confidence interval. What we wanna do is say, well, there's a 95% chance, and we get that from this confidence level, and 95% is the confidence level people typically use. And so there's a 95% chance that whatever our sample proportion is, that it's within two standard deviations of the true proportion, or that the true proportion is going to be contained in an interval that are two standard deviations on either side of our sample proportion. Well, if you don't know the true proportion, the way that you estimate the standard deviation is with the standard error, which we've done in previous videos. | Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3 |
And in a previous video, we talked about how you calculate the confidence interval. What we wanna do is say, well, there's a 95% chance, and we get that from this confidence level, and 95% is the confidence level people typically use. And so there's a 95% chance that whatever our sample proportion is, that it's within two standard deviations of the true proportion, or that the true proportion is going to be contained in an interval that are two standard deviations on either side of our sample proportion. Well, if you don't know the true proportion, the way that you estimate the standard deviation is with the standard error, which we've done in previous videos. And so this is two standard errors to the right and two standard errors to the left of our sample proportion. And our confidence interval is this entire interval going from this left point to this right point. And as we draw more samples, you can see it's not obvious, but our intervals change depending on what our actual sample proportion is, because we use our sample proportion to calculate our confidence interval, because we're assuming whoever's doing the sampling does not actually know the true population proportion. | Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3 |
Well, if you don't know the true proportion, the way that you estimate the standard deviation is with the standard error, which we've done in previous videos. And so this is two standard errors to the right and two standard errors to the left of our sample proportion. And our confidence interval is this entire interval going from this left point to this right point. And as we draw more samples, you can see it's not obvious, but our intervals change depending on what our actual sample proportion is, because we use our sample proportion to calculate our confidence interval, because we're assuming whoever's doing the sampling does not actually know the true population proportion. Now, what's interesting here about this simulation is that we can see what percentage of the time does our confidence interval, does it actually contain the true parameter? So let me just draw 25 samples at a time. And so you can see here that right now, 93%, for 93% of our samples, did our confidence interval actually contain our population parameter? | Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3 |
And as we draw more samples, you can see it's not obvious, but our intervals change depending on what our actual sample proportion is, because we use our sample proportion to calculate our confidence interval, because we're assuming whoever's doing the sampling does not actually know the true population proportion. Now, what's interesting here about this simulation is that we can see what percentage of the time does our confidence interval, does it actually contain the true parameter? So let me just draw 25 samples at a time. And so you can see here that right now, 93%, for 93% of our samples, did our confidence interval actually contain our population parameter? And we can keep sampling over here. And we can see the more samples that we take, it really is approaching that close to 95% of the time, our confidence interval does indeed contain the true parameter. And so once again, we did all that math in the previous video, but here you can see that confidence intervals, calculated the way that we've calculated them, actually do a pretty good job of what they claim to do. | Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3 |
And so you can see here that right now, 93%, for 93% of our samples, did our confidence interval actually contain our population parameter? And we can keep sampling over here. And we can see the more samples that we take, it really is approaching that close to 95% of the time, our confidence interval does indeed contain the true parameter. And so once again, we did all that math in the previous video, but here you can see that confidence intervals, calculated the way that we've calculated them, actually do a pretty good job of what they claim to do. That if we calculate a confidence interval based on a confidence level of 95%, that it is indeed the case that roughly 95% of the time, the true parameter, the population proportion, will be contained in that interval. And I could just draw more and more and more samples. And we can actually see that happening. | Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3 |
And so once again, we did all that math in the previous video, but here you can see that confidence intervals, calculated the way that we've calculated them, actually do a pretty good job of what they claim to do. That if we calculate a confidence interval based on a confidence level of 95%, that it is indeed the case that roughly 95% of the time, the true parameter, the population proportion, will be contained in that interval. And I could just draw more and more and more samples. And we can actually see that happening. Every now and then for sure, you get a sample where even when you calculate your confidence interval, the true parameter, the true population proportion is not contained. But that is the exception that happens very infrequently. 95% of the time, your true population parameter is contained in that interval. | Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3 |
And we can actually see that happening. Every now and then for sure, you get a sample where even when you calculate your confidence interval, the true parameter, the true population proportion is not contained. But that is the exception that happens very infrequently. 95% of the time, your true population parameter is contained in that interval. Now another interesting thing to see is if we increase our sample size, our confidence interval is going to get narrower. So if we increase our sample size, we'll just make it 200. Now let's draw some samples. | Confidence interval simulation Confidence intervals AP Statistics Khan Academy.mp3 |
Aubrey wanted to see if there's a connection between the time a given exam takes place and the average score of this exam. She collected data about exams from the previous year. Plot the data in a scatter plot, and let's see, they give us a couple of rows here. This is the class, then they give us the period of the day that the class happened, and then they give us the average score on an exam. And one could, you know, we have to be a little careful with this study, maybe there's some correlation depending on what subject is taught during what period, but let's just use her data, at least just based on her data see if, see, well, definitely do what they're asking us, plot a scatter plot, and then see if there is any connection. So let's see, on the vertical, on the horizontal axis, we have period, and on this investigation, this exploration she's doing, she's trying to see, well, does the period of the day somehow drive average scores? So that's why period is on the horizontal axis, and the thing that's being, the thing that's driving is on the horizontal, the thing that's being driven is on the vertical. | Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3 |
This is the class, then they give us the period of the day that the class happened, and then they give us the average score on an exam. And one could, you know, we have to be a little careful with this study, maybe there's some correlation depending on what subject is taught during what period, but let's just use her data, at least just based on her data see if, see, well, definitely do what they're asking us, plot a scatter plot, and then see if there is any connection. So let's see, on the vertical, on the horizontal axis, we have period, and on this investigation, this exploration she's doing, she's trying to see, well, does the period of the day somehow drive average scores? So that's why period is on the horizontal axis, and the thing that's being, the thing that's driving is on the horizontal, the thing that's being driven is on the vertical. So let's plot each of these points. Period one, average score 93. Period one, average score 93, right over there. | Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3 |
So that's why period is on the horizontal axis, and the thing that's being, the thing that's driving is on the horizontal, the thing that's being driven is on the vertical. So let's plot each of these points. Period one, average score 93. Period one, average score 93, right over there. Period six, 87. Period six, 87. 80, oh, that's not the right place, and then we can move it if we want. | Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3 |
Period one, average score 93, right over there. Period six, 87. Period six, 87. 80, oh, that's not the right place, and then we can move it if we want. 87, right over there. Period two, 70. Two, period two, 70. | Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3 |
80, oh, that's not the right place, and then we can move it if we want. 87, right over there. Period two, 70. Two, period two, 70. Period four, 62. Four, four and 62, right over there. Period four and 86. | Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3 |
Two, period two, 70. Period four, 62. Four, four and 62, right over there. Period four and 86. Period four and again an 86, that's right over there. Period one, 73. Period one, 73. | Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3 |
Period four and 86. Period four and again an 86, that's right over there. Period one, 73. Period one, 73. Period three, average score of 73 as well. Period three, 73. Period one, 80, average score of 80. | Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3 |
Period one, 73. Period three, average score of 73 as well. Period three, 73. Period one, 80, average score of 80. So period one, average score of 80. And then period three, average score of 96. Period three, average score of 96. | Constructing a scatter plot Regression Probability and Statistics Khan Academy.mp3 |
The scatter plot below displays a set of bivariate data along with its least squares regression line. Consider removing the outlier at 95 comma one. So 95 comma one, we're talking about that outlier right over there, and calculating a new least squares regression line. What effects would removing the outlier have? Choose all answers that apply. Like always, pause this video and see if you could figure it out. Well let's see, even with this outlier here, we have an upward sloping regression line. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
What effects would removing the outlier have? Choose all answers that apply. Like always, pause this video and see if you could figure it out. Well let's see, even with this outlier here, we have an upward sloping regression line. And so it looks like our r already is going to be greater than zero, and of course it's going to be less than one. So our r is going to be greater than zero and less than one. We know it's not going to be equal one because then we would go perfectly through all of the dots. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
Well let's see, even with this outlier here, we have an upward sloping regression line. And so it looks like our r already is going to be greater than zero, and of course it's going to be less than one. So our r is going to be greater than zero and less than one. We know it's not going to be equal one because then we would go perfectly through all of the dots. And it's clear that this point right over here is indeed an outlier. The residual between this point and the line is quite high. We have a pretty big distance right over here. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
We know it's not going to be equal one because then we would go perfectly through all of the dots. And it's clear that this point right over here is indeed an outlier. The residual between this point and the line is quite high. We have a pretty big distance right over here. It would be a negative residual. And so this point is definitely bringing down the r, and it's definitely bringing down the slope of the regression line. If we were to remove this point, we're more likely to have a line that looks something like this, in which case it looks like we would get a much, much, much, much better fit. | Impact of removing outliers on regression lines AP Statistics Khan Academy.mp3 |
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