Sentence stringlengths 102 4.09k | video_title stringlengths 27 104 |
|---|---|
And I encourage you to pause the video and try to think of what that number would actually be. Well, a big clue was, when we essentially wrote all of the permutations where we've picked a group of three people, we see that there's six ways of arranging the three people. And you pick a certain group of three people that turned into six permutations. And so if all you want to do is care about, well, how many different ways are there to choose three from the six, you would take your whole permutations, you would take your number of permutations, and then you would divide it by the number of ways to arrange three people. Number of ways to arrange three people. And we see that you can arrange three people, or even three letters, you can arrange it in six different ways. So this would be equal to 120 divided by six, or this would be equal to 20. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
And so if all you want to do is care about, well, how many different ways are there to choose three from the six, you would take your whole permutations, you would take your number of permutations, and then you would divide it by the number of ways to arrange three people. Number of ways to arrange three people. And we see that you can arrange three people, or even three letters, you can arrange it in six different ways. So this would be equal to 120 divided by six, or this would be equal to 20. So there were 120 permutations here. If you said, how many different arrangements are there of taking six people and putting them into three chairs, that's 120. But now we're asking another thing. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
So this would be equal to 120 divided by six, or this would be equal to 20. So there were 120 permutations here. If you said, how many different arrangements are there of taking six people and putting them into three chairs, that's 120. But now we're asking another thing. We're saying, if we start with 120 people, and we want to choose, and we want to, sorry, if we're starting, if we're starting with six people, and we want to figure out how many ways, how many combinations, how many ways are there for us to choose three of them, then we end up with 20 combinations. Combinations of people. This right over here, once again, this right over here is just one combination. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
But now we're asking another thing. We're saying, if we start with 120 people, and we want to choose, and we want to, sorry, if we're starting, if we're starting with six people, and we want to figure out how many ways, how many combinations, how many ways are there for us to choose three of them, then we end up with 20 combinations. Combinations of people. This right over here, once again, this right over here is just one combination. It's the combination A, B, C. I don't care what order they sit in. I've chosen them. I've chosen these three of the six. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
This right over here, once again, this right over here is just one combination. It's the combination A, B, C. I don't care what order they sit in. I've chosen them. I've chosen these three of the six. This is a combination of people. I don't care about the order. This right over here is another combination. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
I've chosen these three of the six. This is a combination of people. I don't care about the order. This right over here is another combination. It is F, C, and B. Once again, I don't care about the order. I just care that I've chosen these three people. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
This right over here is another combination. It is F, C, and B. Once again, I don't care about the order. I just care that I've chosen these three people. So how many ways are there to choose three people out of six? It is 20. It's the total number of permutations, it's 120, divided by the number of ways you can arrange three people. | Introduction to combinations Probability and Statistics Khan Academy.mp3 |
So the probability of rolling even numbers. So even roll on six-sided die. So let's think about that probability. Well, how many total outcomes are there? How many possible rolls could we get? Well, you could get 1, 2, 3, 4, 5, 6. And how many of them satisfy these conditions, that it's an even number? | Die rolling probability with independent events Precalculus Khan Academy.mp3 |
Well, how many total outcomes are there? How many possible rolls could we get? Well, you could get 1, 2, 3, 4, 5, 6. And how many of them satisfy these conditions, that it's an even number? Well, it could be a 2, it could be a 4, or it could be a 6. So the probability is the events that match what you need, your condition for right here. So three of the possible events are an even roll. | Die rolling probability with independent events Precalculus Khan Academy.mp3 |
And how many of them satisfy these conditions, that it's an even number? Well, it could be a 2, it could be a 4, or it could be a 6. So the probability is the events that match what you need, your condition for right here. So three of the possible events are an even roll. And it's out of a total of six possible events. So there is a 3 over 6, the same thing as 1 half probability of rolling even on each roll. Now, they want to roll even three times. | Die rolling probability with independent events Precalculus Khan Academy.mp3 |
So three of the possible events are an even roll. And it's out of a total of six possible events. So there is a 3 over 6, the same thing as 1 half probability of rolling even on each roll. Now, they want to roll even three times. And these are all going to be independent events. Every time you roll, it's not going to affect what happens in the next roll, despite what some gamblers might think. It has no impact on what happens on the next roll. | Die rolling probability with independent events Precalculus Khan Academy.mp3 |
Now, they want to roll even three times. And these are all going to be independent events. Every time you roll, it's not going to affect what happens in the next roll, despite what some gamblers might think. It has no impact on what happens on the next roll. So the probability of rolling even three times is equal to the probability of an even roll one time, or even roll on six-sided die, this thing over here, is equal to that thing times that thing again. That's our first roll. Let me copy and paste it. | Die rolling probability with independent events Precalculus Khan Academy.mp3 |
It has no impact on what happens on the next roll. So the probability of rolling even three times is equal to the probability of an even roll one time, or even roll on six-sided die, this thing over here, is equal to that thing times that thing again. That's our first roll. Let me copy and paste it. Times that thing, and then times that thing again. That's our first roll, which is that. That's our second roll. | Die rolling probability with independent events Precalculus Khan Academy.mp3 |
Let me copy and paste it. Times that thing, and then times that thing again. That's our first roll, which is that. That's our second roll. That's our third roll. They're independent events. So this is going to be equal to 1 half, that's the same 1 half right there, times 1 half, times 1 half, which is equal to 1 over 8. | Die rolling probability with independent events Precalculus Khan Academy.mp3 |
Administrators at Riverview High School surveyed a random sample of 100 of their seniors to see how they felt about the lunch offerings at the school's cafeteria. So you have all of the seniors, I'm assuming there's more than 100 of them, and then they sampled 100 of them. So this is the sample. So the population is all of the seniors at the school. That's the population, all of the seniors. And they sampled 100 of them. So the 100 seniors that they talked to, that is the sample. | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
So the population is all of the seniors at the school. That's the population, all of the seniors. And they sampled 100 of them. So the 100 seniors that they talked to, that is the sample. That is the sample. So they tell us, identify the population and the sample in the setting. So let's just see which of these choices actually match up to what I just said. | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
So the 100 seniors that they talked to, that is the sample. That is the sample. So they tell us, identify the population and the sample in the setting. So let's just see which of these choices actually match up to what I just said. And like always, I encourage you to pause the video and see if you can work through it on your own. So the population is all high school seniors in the world. The sample is all of the seniors at Riverview High. | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
So let's just see which of these choices actually match up to what I just said. And like always, I encourage you to pause the video and see if you can work through it on your own. So the population is all high school seniors in the world. The sample is all of the seniors at Riverview High. No, this is not right. We're not trying to figure out, we're not trying to get an indication of how all of the high school seniors in the world feel about the food at Riverview High School. We're trying to get an indication of how the seniors at Riverview High School feel about the lunch at the school's cafeteria. | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
The sample is all of the seniors at Riverview High. No, this is not right. We're not trying to figure out, we're not trying to get an indication of how all of the high school seniors in the world feel about the food at Riverview High School. We're trying to get an indication of how the seniors at Riverview High School feel about the lunch at the school's cafeteria. So they did a sample of 100 of them. So this is definitely not going to be, let me cross this one out. The population is all students at Riverview High. | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
We're trying to get an indication of how the seniors at Riverview High School feel about the lunch at the school's cafeteria. So they did a sample of 100 of them. So this is definitely not going to be, let me cross this one out. The population is all students at Riverview High. The sample is all of the seniors at Riverview High. Well, they clearly didn't sample all of the seniors. They sample 100 of the seniors. | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
The population is all students at Riverview High. The sample is all of the seniors at Riverview High. Well, they clearly didn't sample all of the seniors. They sample 100 of the seniors. So this isn't gonna be right either. Let's hope that the third choice is working out. The population is all seniors at Riverview High. | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
They sample 100 of the seniors. So this isn't gonna be right either. Let's hope that the third choice is working out. The population is all seniors at Riverview High. The sample is the 100 seniors surveyed. Yep, that's exactly what we talked here. We're trying to get an indication about how all of the seniors at Riverview High feel about the food, the lunch offerings. | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
The population is all seniors at Riverview High. The sample is the 100 seniors surveyed. Yep, that's exactly what we talked here. We're trying to get an indication about how all of the seniors at Riverview High feel about the food, the lunch offerings. We probably think it's impractical, or the administrators feel it's impractical to talk to everyone so they get exactly what the population thinks. So instead, they're gonna do a random sample of 100 of them. So the sample is 100 seniors who are actually surveyed. | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
So they decide to ask a simple random sample of 160 students if they have experienced extreme levels of stress during the past month. Subsequently, they find that 10% of the sample replied yes to the question. Assuming the true proportion is 15%, which they tell us up here, they say 15% of the population of the 1,750 students actually have experienced extreme levels of stress during the past month. So that is the true proportion. So let me just write that. The true proportion for our population is 0.15. What is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month? | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
So that is the true proportion. So let me just write that. The true proportion for our population is 0.15. What is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month? So pause this video and see if you can answer it on your own, and there are four choices here. I'll scroll down a little bit and see if you can answer this on your own. So the way that we're going to tackle this is we're gonna think about the sampling distribution of our sample proportions. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
What is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month? So pause this video and see if you can answer it on your own, and there are four choices here. I'll scroll down a little bit and see if you can answer this on your own. So the way that we're going to tackle this is we're gonna think about the sampling distribution of our sample proportions. And first, we're gonna say, well, is this sampling distribution approximately normal? Is it approximately normal? And if it is, then we can use its mean and its standard deviation and create a normal distribution that has that same mean and standard deviation in order to approximate the probability that they're asking for. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
So the way that we're going to tackle this is we're gonna think about the sampling distribution of our sample proportions. And first, we're gonna say, well, is this sampling distribution approximately normal? Is it approximately normal? And if it is, then we can use its mean and its standard deviation and create a normal distribution that has that same mean and standard deviation in order to approximate the probability that they're asking for. So first, this first part, how do we decide this? Well, the rule of thumb we have here, and it is a rule of thumb, is that if we take our sample size times our population proportion, and that is greater than or equal to 10, and our sample size times one minus our population proportion is greater than or equal to 10, then if both of these are true, then our sampling distribution of our sample proportions is going to be approximately normal. So in this case, the newspaper is asking 160 students. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
And if it is, then we can use its mean and its standard deviation and create a normal distribution that has that same mean and standard deviation in order to approximate the probability that they're asking for. So first, this first part, how do we decide this? Well, the rule of thumb we have here, and it is a rule of thumb, is that if we take our sample size times our population proportion, and that is greater than or equal to 10, and our sample size times one minus our population proportion is greater than or equal to 10, then if both of these are true, then our sampling distribution of our sample proportions is going to be approximately normal. So in this case, the newspaper is asking 160 students. That's the sample size. So 160, the true population proportion is 0.15, and that needs to be greater than or equal to 10. And so let's see, this is going to be 16 plus eight, which is 24, and 24 is indeed greater than or equal to 10, so that checks out. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
So in this case, the newspaper is asking 160 students. That's the sample size. So 160, the true population proportion is 0.15, and that needs to be greater than or equal to 10. And so let's see, this is going to be 16 plus eight, which is 24, and 24 is indeed greater than or equal to 10, so that checks out. And then if I take our sample size times one minus P, well, one minus 15 hundredths is going to be 85 hundredths. And this is definitely going to be greater than or equal to 10. Let's see, this is going to be 24 less than 160, so this is going to be 136, which is way larger than 10, so that checks out. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
And so let's see, this is going to be 16 plus eight, which is 24, and 24 is indeed greater than or equal to 10, so that checks out. And then if I take our sample size times one minus P, well, one minus 15 hundredths is going to be 85 hundredths. And this is definitely going to be greater than or equal to 10. Let's see, this is going to be 24 less than 160, so this is going to be 136, which is way larger than 10, so that checks out. And so the sampling distribution of our sample proportions is approximately going to be normal. And so what is the mean and standard deviation of our sampling distribution? So the mean of our sampling distribution is just going to be our population proportion. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
Let's see, this is going to be 24 less than 160, so this is going to be 136, which is way larger than 10, so that checks out. And so the sampling distribution of our sample proportions is approximately going to be normal. And so what is the mean and standard deviation of our sampling distribution? So the mean of our sampling distribution is just going to be our population proportion. We've seen that in other videos, which is equal to 0.15. And our standard deviation of our sampling distribution, of our sample proportions, is going to be equal to the square root of P times one minus P over N, which is equal to the square root of 0.15 times 0.85, all of that over our sample size, 160. So now let's get our calculator out. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
So the mean of our sampling distribution is just going to be our population proportion. We've seen that in other videos, which is equal to 0.15. And our standard deviation of our sampling distribution, of our sample proportions, is going to be equal to the square root of P times one minus P over N, which is equal to the square root of 0.15 times 0.85, all of that over our sample size, 160. So now let's get our calculator out. So I'm gonna take the square root of 0.15 times 0.85 divided by 160, and let me close those parentheses. And so what is this going to give me? So it's going to give me approximately 0.028, and I'll go to the thousands place here. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
So now let's get our calculator out. So I'm gonna take the square root of 0.15 times 0.85 divided by 160, and let me close those parentheses. And so what is this going to give me? So it's going to give me approximately 0.028, and I'll go to the thousands place here. So this is approximately 0.028. This is going to be approximately a normal distribution. So you could draw your classic bell curve for a normal distribution, so something like this. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
So it's going to give me approximately 0.028, and I'll go to the thousands place here. So this is approximately 0.028. This is going to be approximately a normal distribution. So you could draw your classic bell curve for a normal distribution, so something like this. And our normal distribution is going to have a mean. It's going to have a mean right over here of, so this is the mean of our sampling distribution. So this is going to be equal to the same thing as our population proportion, 0.15. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
So you could draw your classic bell curve for a normal distribution, so something like this. And our normal distribution is going to have a mean. It's going to have a mean right over here of, so this is the mean of our sampling distribution. So this is going to be equal to the same thing as our population proportion, 0.15. And we also know that our standard deviation here is going to be approximately equal to 0.028. And what we wanna know is what is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month? So we could say that 10% would be right over here. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
So this is going to be equal to the same thing as our population proportion, 0.15. And we also know that our standard deviation here is going to be approximately equal to 0.028. And what we wanna know is what is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month? So we could say that 10% would be right over here. I'll say 0.10. And so the probability that in a sample of 160, you get a proportion for that sample, a sample proportion that is larger than 10% would be this area right over here. So this right over here would be the probability that your sample proportion is greater than, they say is more than 10%, is more than 0.1. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
So we could say that 10% would be right over here. I'll say 0.10. And so the probability that in a sample of 160, you get a proportion for that sample, a sample proportion that is larger than 10% would be this area right over here. So this right over here would be the probability that your sample proportion is greater than, they say is more than 10%, is more than 0.1. I could write one zero just like that. And then to calculate it, I can get out our calculator again. So here I'm gonna go to my distribution menu right over there. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
So this right over here would be the probability that your sample proportion is greater than, they say is more than 10%, is more than 0.1. I could write one zero just like that. And then to calculate it, I can get out our calculator again. So here I'm gonna go to my distribution menu right over there. And then I'm gonna do a normal cumulative distribution function. So let me click Enter there. And so what is my lower bound? | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
So here I'm gonna go to my distribution menu right over there. And then I'm gonna do a normal cumulative distribution function. So let me click Enter there. And so what is my lower bound? Well, my lower bound is 10%, 0.1. What is my upper bound? Well, we'll just make this one because that is the highest proportion you could have for a sampling distribution of sample proportions. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
And so what is my lower bound? Well, my lower bound is 10%, 0.1. What is my upper bound? Well, we'll just make this one because that is the highest proportion you could have for a sampling distribution of sample proportions. Now what is our mean? Well, we already know that's 0.15. What is the standard deviation of our sampling distribution? | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
Well, we'll just make this one because that is the highest proportion you could have for a sampling distribution of sample proportions. Now what is our mean? Well, we already know that's 0.15. What is the standard deviation of our sampling distribution? Well, it's approximately 0.028. And then I can click Enter. And if you're taking an AP exam, you actually should write this. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
What is the standard deviation of our sampling distribution? Well, it's approximately 0.028. And then I can click Enter. And if you're taking an AP exam, you actually should write this. You should say, you should tell the graders what you're actually typing in in your normal CDF function. But if we click Enter right over here, and then Enter, there we have it. It's approximately 96%. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
And if you're taking an AP exam, you actually should write this. You should say, you should tell the graders what you're actually typing in in your normal CDF function. But if we click Enter right over here, and then Enter, there we have it. It's approximately 96%. So this is approximately 0.96. And then out of our choices, it would be this one right over here. If you're taking this on the AP exam, you would say that called, called normal, normal CDF, where you have your lower bound, lower bound, and you would put in your 0.10. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
It's approximately 96%. So this is approximately 0.96. And then out of our choices, it would be this one right over here. If you're taking this on the AP exam, you would say that called, called normal, normal CDF, where you have your lower bound, lower bound, and you would put in your 0.10. You would say that you use an upper bound, upper bound of one. You would say that you gave a mean of 0.15, and then you gave a standard deviation of 0.028, just so people know that you knew what you were doing. But hopefully this is helpful. | Probability of sample proportions example Sampling distributions AP Statistics Khan Academy.mp3 |
Here is computer output on the sample data. So we have some statistics calculated on the reaction time, on the memory time, and then he had his computer do a regression for the data that he collected, and then we're told assume that all conditions for inference have been met, calculate the test statistic that should be used for testing a null hypothesis that the population slope is actually zero. So pause this video and have a go at it. All right, so let's just make sure we understand what is going on. So let's first think about the population. So I'll do that right over here. So in the population, there might be some true linear relationship. | Calculating t statistic for slope of regression line AP Statistics Khan Academy.mp3 |
All right, so let's just make sure we understand what is going on. So let's first think about the population. So I'll do that right over here. So in the population, there might be some true linear relationship. So in theory, on our x-axis, we would have our reaction time, and on our y-axis, you have your memory time. If you were able to plot every single possible data point, it might even be an infinite or near infinite, so it would be very hard to do it, but if there was just some truth in the universe that says, yes, there actually is a positive linear relationship, and it looks like this, and you could describe that regression line as y hat, it's a regression line, is equal to some true population parameter, which would be this y-intercept, so we could call that alpha, plus some true population parameter that would be the slope of this regression line, we could call that beta, times x. Now, we don't know what this truth of the universe is, of the linear relationship between reaction time and memory time, but we can try to estimate it, and that's what Jian is trying to do. | Calculating t statistic for slope of regression line AP Statistics Khan Academy.mp3 |
So in the population, there might be some true linear relationship. So in theory, on our x-axis, we would have our reaction time, and on our y-axis, you have your memory time. If you were able to plot every single possible data point, it might even be an infinite or near infinite, so it would be very hard to do it, but if there was just some truth in the universe that says, yes, there actually is a positive linear relationship, and it looks like this, and you could describe that regression line as y hat, it's a regression line, is equal to some true population parameter, which would be this y-intercept, so we could call that alpha, plus some true population parameter that would be the slope of this regression line, we could call that beta, times x. Now, we don't know what this truth of the universe is, of the linear relationship between reaction time and memory time, but we can try to estimate it, and that's what Jian is trying to do. So he's taking a sample of 24, so samples, samples, 24 data, data points, and that's much easier to then, you could even visualize it on a scatter plot like this, so you'd have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24. You input those data points into a computer, and it does a regression line, and it's trying to minimize the squared distance to all of these points, and so let's say it gets a regression line that looks something like this, where this regression line can be described as some estimate of the true y-intercept, so this would actually be a statistic right over here that's estimating this parameter, plus some estimate of the true slope of the regression line, so this is just a statistic. This b is just a statistic that is trying to estimate the true parameter beta. | Calculating t statistic for slope of regression line AP Statistics Khan Academy.mp3 |
Now, we don't know what this truth of the universe is, of the linear relationship between reaction time and memory time, but we can try to estimate it, and that's what Jian is trying to do. So he's taking a sample of 24, so samples, samples, 24 data, data points, and that's much easier to then, you could even visualize it on a scatter plot like this, so you'd have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24. You input those data points into a computer, and it does a regression line, and it's trying to minimize the squared distance to all of these points, and so let's say it gets a regression line that looks something like this, where this regression line can be described as some estimate of the true y-intercept, so this would actually be a statistic right over here that's estimating this parameter, plus some estimate of the true slope of the regression line, so this is just a statistic. This b is just a statistic that is trying to estimate the true parameter beta. Now, when we went and inputted these data points into a computer, we got values for a and b right over here. A is equal to this, the constant coefficient, and then the reaction coefficient, this is just telling us, hey, for every incremental change in the reaction, how much would we expect the memory time to change, or for every change in x, how much would we expect for a change in y, so this is actually our estimate of the slope of the regression line. Now, you could imagine, every time you take a different sample, you might get a different estimate of these things, and when we're doing inferential statistics, we set up hypotheses, you set up a null and an alternative hypothesis, and the null hypothesis is always the no news here, and no news, when you're dealing with regressions, is that even though you might suspect there's a positive linear relationship, even though you might see it in the data you got, it's, for your null hypothesis, you wanna assume that there is no positive linear relationship, so our null hypothesis here would be that the true slope of the true regression line, this, the parameter right over here, is equal to zero, so beta is equal to zero, so our null hypothesis is actually, might be that our true regression line might look something like this, that what y is is somewhat independent of what x is, and that if you suspect that there is a positive linear relationship, you could say something like, well, my alternative hypothesis is that my beta is greater than zero, or if you suspect that there's just some linear relationship, you don't know if it's positive or negative, then you might say that the beta is not equal to zero, but here it says he noticed, or he suspects, a positive linear relationship, so this would be his alternative hypothesis, but what you then want to do to test your null hypothesis, which we've done multiple, multiple times, is find a test statistic that is associated with the statistic for b that you actually got. | Calculating t statistic for slope of regression line AP Statistics Khan Academy.mp3 |
This b is just a statistic that is trying to estimate the true parameter beta. Now, when we went and inputted these data points into a computer, we got values for a and b right over here. A is equal to this, the constant coefficient, and then the reaction coefficient, this is just telling us, hey, for every incremental change in the reaction, how much would we expect the memory time to change, or for every change in x, how much would we expect for a change in y, so this is actually our estimate of the slope of the regression line. Now, you could imagine, every time you take a different sample, you might get a different estimate of these things, and when we're doing inferential statistics, we set up hypotheses, you set up a null and an alternative hypothesis, and the null hypothesis is always the no news here, and no news, when you're dealing with regressions, is that even though you might suspect there's a positive linear relationship, even though you might see it in the data you got, it's, for your null hypothesis, you wanna assume that there is no positive linear relationship, so our null hypothesis here would be that the true slope of the true regression line, this, the parameter right over here, is equal to zero, so beta is equal to zero, so our null hypothesis is actually, might be that our true regression line might look something like this, that what y is is somewhat independent of what x is, and that if you suspect that there is a positive linear relationship, you could say something like, well, my alternative hypothesis is that my beta is greater than zero, or if you suspect that there's just some linear relationship, you don't know if it's positive or negative, then you might say that the beta is not equal to zero, but here it says he noticed, or he suspects, a positive linear relationship, so this would be his alternative hypothesis, but what you then want to do to test your null hypothesis, which we've done multiple, multiple times, is find a test statistic that is associated with the statistic for b that you actually got. Now, ideally, you would take your b, you would take your b, and from that, subtract the slope assumed in the null hypothesis, so the slope of the regression line you get, minus the slope that's assumed from the null hypothesis, and then divide by the standard deviation of the sampling distribution of the slope of the regression line, and if you did this, you would get a, it would be appropriate to use a z statistic over here. Now, the problem is is that we don't know exactly what the standard deviation of the sampling distribution is, but we can estimate it. We can calculate the slope that we got for our sample regression line, minus the slope we're assuming in our null hypothesis, which is going to be equal to zero, so we know what we're assuming, and we can calculate the standard error of the sampling distribution. | Calculating t statistic for slope of regression line AP Statistics Khan Academy.mp3 |
Now, you could imagine, every time you take a different sample, you might get a different estimate of these things, and when we're doing inferential statistics, we set up hypotheses, you set up a null and an alternative hypothesis, and the null hypothesis is always the no news here, and no news, when you're dealing with regressions, is that even though you might suspect there's a positive linear relationship, even though you might see it in the data you got, it's, for your null hypothesis, you wanna assume that there is no positive linear relationship, so our null hypothesis here would be that the true slope of the true regression line, this, the parameter right over here, is equal to zero, so beta is equal to zero, so our null hypothesis is actually, might be that our true regression line might look something like this, that what y is is somewhat independent of what x is, and that if you suspect that there is a positive linear relationship, you could say something like, well, my alternative hypothesis is that my beta is greater than zero, or if you suspect that there's just some linear relationship, you don't know if it's positive or negative, then you might say that the beta is not equal to zero, but here it says he noticed, or he suspects, a positive linear relationship, so this would be his alternative hypothesis, but what you then want to do to test your null hypothesis, which we've done multiple, multiple times, is find a test statistic that is associated with the statistic for b that you actually got. Now, ideally, you would take your b, you would take your b, and from that, subtract the slope assumed in the null hypothesis, so the slope of the regression line you get, minus the slope that's assumed from the null hypothesis, and then divide by the standard deviation of the sampling distribution of the slope of the regression line, and if you did this, you would get a, it would be appropriate to use a z statistic over here. Now, the problem is is that we don't know exactly what the standard deviation of the sampling distribution is, but we can estimate it. We can calculate the slope that we got for our sample regression line, minus the slope we're assuming in our null hypothesis, which is going to be equal to zero, so we know what we're assuming, and we can calculate the standard error of the sampling distribution. In fact, our computer has already done it for us, and this is an estimate of this, and we know what number that is, so we know what all of these numbers are, but if you're using an estimate of the standard deviation of the sampling distribution, and we've seen this before when we've done inferential statistics using for means, it is appropriate to use a t statistic, but with that said, pause the video. What is this going to be equal to? Well, this is going to be equal to the slope for our sample regression line. | Calculating t statistic for slope of regression line AP Statistics Khan Academy.mp3 |
We can calculate the slope that we got for our sample regression line, minus the slope we're assuming in our null hypothesis, which is going to be equal to zero, so we know what we're assuming, and we can calculate the standard error of the sampling distribution. In fact, our computer has already done it for us, and this is an estimate of this, and we know what number that is, so we know what all of these numbers are, but if you're using an estimate of the standard deviation of the sampling distribution, and we've seen this before when we've done inferential statistics using for means, it is appropriate to use a t statistic, but with that said, pause the video. What is this going to be equal to? Well, this is going to be equal to the slope for our sample regression line. We know it's 14.686, minus our assumed true population parameter, the slope of the true regression line. Well, we're assuming that is zero, so minus zero, and then we divide that by the standard error, which is going to be, we could view this as a standard error for B, and so this is divided by 13.329, so it's just gonna be 14.686 divided by 13.329, and if we assume, if we're doing a one-sided test here, what we would then do is take this t statistic and think about the degrees of freedom, and then say, and then calculate a p value. What is the probability of getting a result at least this far above t is equal to zero, or what is the probability of getting a t statistic this high or higher, and that would be our p value, and if that's below some threshold, let's say, hey, that's pretty unlikely, then we would reject the null, and that which would suggest the alternative, but they're not asking us to do all of that. | Calculating t statistic for slope of regression line AP Statistics Khan Academy.mp3 |
He decides to test his null hypothesis is that the mean number of years of experience is five years and his alternative hypothesis is that the true mean years of experience is less than five years using a sample of 25 teachers. His sample mean was four years and his sample standard deviation was two years. Rory wants to use these sample data to conduct a t-test on the mean. Assume that all conditions for inference have been met. Calculate the test statistic for Rory's test. So I always just like to remind ourselves what's going on. So you have your null hypothesis here that the mean number of years of experience for teachers in the district is five and then the alternative hypothesis is that the mean years of experience is less than five for teachers in the district. | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
Assume that all conditions for inference have been met. Calculate the test statistic for Rory's test. So I always just like to remind ourselves what's going on. So you have your null hypothesis here that the mean number of years of experience for teachers in the district is five and then the alternative hypothesis is that the mean years of experience is less than five for teachers in the district. So if this represents all the teachers in the district, the population, then what he did is he took a sample and said he used a sample of 25 teachers. So n here is equal to 25. And then from that sample, he was able to calculate some statistics. | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
So you have your null hypothesis here that the mean number of years of experience for teachers in the district is five and then the alternative hypothesis is that the mean years of experience is less than five for teachers in the district. So if this represents all the teachers in the district, the population, then what he did is he took a sample and said he used a sample of 25 teachers. So n here is equal to 25. And then from that sample, he was able to calculate some statistics. He was able to calculate the sample mean. So that sample mean was four years. The sample mean was four years. | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
And then from that sample, he was able to calculate some statistics. He was able to calculate the sample mean. So that sample mean was four years. The sample mean was four years. And then he was also able to calculate the sample standard deviation. The sample standard deviation was equal to two years. Now, the whole point that we do or the main thing we do when we do significance tests is we say, all right, if we assume the null hypothesis is true, what's the probability of getting a sample mean this low or lower? | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
The sample mean was four years. And then he was also able to calculate the sample standard deviation. The sample standard deviation was equal to two years. Now, the whole point that we do or the main thing we do when we do significance tests is we say, all right, if we assume the null hypothesis is true, what's the probability of getting a sample mean this low or lower? And if that probability is below a preset significance level, then we reject the null hypothesis and it suggests the alternative. But in order to figure out that probability, we need to figure out a test statistic. Sometimes we use a z-test. | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
Now, the whole point that we do or the main thing we do when we do significance tests is we say, all right, if we assume the null hypothesis is true, what's the probability of getting a sample mean this low or lower? And if that probability is below a preset significance level, then we reject the null hypothesis and it suggests the alternative. But in order to figure out that probability, we need to figure out a test statistic. Sometimes we use a z-test. If we're dealing with proportions. But when we deal with means, we tend to use a t-test. And the reason why is if you wanted to figure out a z-statistic, what you would do is you would take your sample mean, subtract from that the assumed mean from the null hypothesis. | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
Sometimes we use a z-test. If we're dealing with proportions. But when we deal with means, we tend to use a t-test. And the reason why is if you wanted to figure out a z-statistic, what you would do is you would take your sample mean, subtract from that the assumed mean from the null hypothesis. So mu, and I'll just put a little zero, sub zero there. So this is the assumed mean from the null hypothesis. And then you would want to divide by the standard deviation of the sampling distribution of the sample mean. | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
And the reason why is if you wanted to figure out a z-statistic, what you would do is you would take your sample mean, subtract from that the assumed mean from the null hypothesis. So mu, and I'll just put a little zero, sub zero there. So this is the assumed mean from the null hypothesis. And then you would want to divide by the standard deviation of the sampling distribution of the sample mean. So you'd wanna divide by that. But this, we don't know. And so that's why instead we do a t-statistic, in which case we take the difference between our sample mean and our assumed population mean, the population parameter, and we try to estimate this. | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
And then you would want to divide by the standard deviation of the sampling distribution of the sample mean. So you'd wanna divide by that. But this, we don't know. And so that's why instead we do a t-statistic, in which case we take the difference between our sample mean and our assumed population mean, the population parameter, and we try to estimate this. And we estimate that with our sample standard deviation divided by the square root of our sample size. And so if you're inspired, I encourage you, even if you're not inspired, I encourage you to pause this video and try to calculate this t-statistic. Well, this is going to be equal to, let's see, our sample mean is four minus our assumed mean is five, our assumed population mean is five. | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
And so that's why instead we do a t-statistic, in which case we take the difference between our sample mean and our assumed population mean, the population parameter, and we try to estimate this. And we estimate that with our sample standard deviation divided by the square root of our sample size. And so if you're inspired, I encourage you, even if you're not inspired, I encourage you to pause this video and try to calculate this t-statistic. Well, this is going to be equal to, let's see, our sample mean is four minus our assumed mean is five, our assumed population mean is five. Our sample standard deviation is two. All of that over the square root of the sample size, all of that over the square root of 25. So this is going to be equal, our numerator is negative one. | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
Well, this is going to be equal to, let's see, our sample mean is four minus our assumed mean is five, our assumed population mean is five. Our sample standard deviation is two. All of that over the square root of the sample size, all of that over the square root of 25. So this is going to be equal, our numerator is negative one. So it's negative one divided by two over five, which is equal to negative one times five over two. And so this is going to be equal to, equal to negative five over two, or negative 2.5. And then what we would do in this, what Rory would do, is then look this t-value up on a t-table and say, so if you look at a distribution of a t-statistic, something like that, and say, okay, we are negative 2.5 below the mean. | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
So this is going to be equal, our numerator is negative one. So it's negative one divided by two over five, which is equal to negative one times five over two. And so this is going to be equal to, equal to negative five over two, or negative 2.5. And then what we would do in this, what Rory would do, is then look this t-value up on a t-table and say, so if you look at a distribution of a t-statistic, something like that, and say, okay, we are negative 2.5 below the mean. So negative, negative 2.5. And so what he would wanna do is figure out this area here, because this would be the probability of being that far below the mean or even further below the mean. And so that would give us our p-value. | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
And then what we would do in this, what Rory would do, is then look this t-value up on a t-table and say, so if you look at a distribution of a t-statistic, something like that, and say, okay, we are negative 2.5 below the mean. So negative, negative 2.5. And so what he would wanna do is figure out this area here, because this would be the probability of being that far below the mean or even further below the mean. And so that would give us our p-value. And then if that p-value is below some preset significance level that Rory should have set, maybe 5% or 1%, then he'll reject the null hypothesis, which would suggest his suspicion that the true mean of years of experience for the teachers in his district is less than five. Now another really important thing to keep in mind is, they told us that assume all conditions for inference have been met. And so that's the, assuming that this was truly a random sample, that each, the individual observations are either truly independent or roughly independent, that maybe he observed either with replacement or it's less than 10% of the population, and he feels good that the sampling distribution is going to be roughly normal. | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
You are planning a full day nature trip for 50 men, and will bring 110 liters of water. What is the probability that you will run out of water? So let's think about what's happening here. So there's some distribution of how many liters the average man needs when they're active outdoors. And let me just draw an example. It might look something like this. So they're all going to need at least more than 0 liters. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So there's some distribution of how many liters the average man needs when they're active outdoors. And let me just draw an example. It might look something like this. So they're all going to need at least more than 0 liters. So this would be 0 liters over here. The average male, so the mean of the amount of water a man needs when active outdoors, is 2 liters. So 2 liters would be right over here. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So they're all going to need at least more than 0 liters. So this would be 0 liters over here. The average male, so the mean of the amount of water a man needs when active outdoors, is 2 liters. So 2 liters would be right over here. So the mean is equal to 2 liters. It has a standard deviation of 0.7 liters, or 0.7 liters. So the standard deviation, maybe I'll draw it this way. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So 2 liters would be right over here. So the mean is equal to 2 liters. It has a standard deviation of 0.7 liters, or 0.7 liters. So the standard deviation, maybe I'll draw it this way. So this distribution, once again, we don't know whether it's a normal distribution or not. It could just be some type of crazy distribution. So maybe some people need almost close to, well, everyone needs a little bit of water, but maybe some people need very, very little water. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So the standard deviation, maybe I'll draw it this way. So this distribution, once again, we don't know whether it's a normal distribution or not. It could just be some type of crazy distribution. So maybe some people need almost close to, well, everyone needs a little bit of water, but maybe some people need very, very little water. Then you have a lot of people who need that, maybe some people who need more, and maybe no one can drink more than maybe this is like 4 liters of water. So maybe this is the actual distribution. And then one standard deviation is going to be 0.7 liters away. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So maybe some people need almost close to, well, everyone needs a little bit of water, but maybe some people need very, very little water. Then you have a lot of people who need that, maybe some people who need more, and maybe no one can drink more than maybe this is like 4 liters of water. So maybe this is the actual distribution. And then one standard deviation is going to be 0.7 liters away. So this is 1, 0.7 liters is, so this would be 1 liter, 2 liters, 3 liters. So one standard deviation is going to be about that far away from the mean. If you go above it, it'll be about that far if you go below it. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
And then one standard deviation is going to be 0.7 liters away. So this is 1, 0.7 liters is, so this would be 1 liter, 2 liters, 3 liters. So one standard deviation is going to be about that far away from the mean. If you go above it, it'll be about that far if you go below it. So let me draw. This is the standard deviation. That right there is the standard deviation to the right. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
If you go above it, it'll be about that far if you go below it. So let me draw. This is the standard deviation. That right there is the standard deviation to the right. That's the standard deviation to the left. And we know that the standard deviation is equal to, I'll write the 0 in front, 0.7 liters. So that's the actual distribution of how much water the average man needs when active. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
That right there is the standard deviation to the right. That's the standard deviation to the left. And we know that the standard deviation is equal to, I'll write the 0 in front, 0.7 liters. So that's the actual distribution of how much water the average man needs when active. Now what's interesting about this problem, we are planning a full day nature trip for 50 men and we'll bring 110 liters of water. What is the probability that you will run out? So the probability that you will run out, let me write this down. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So that's the actual distribution of how much water the average man needs when active. Now what's interesting about this problem, we are planning a full day nature trip for 50 men and we'll bring 110 liters of water. What is the probability that you will run out? So the probability that you will run out, let me write this down. The probability that I will or that you will run out is equal or is the same thing as the probability that we use more than 110 liters on our outdoor nature day, whatever we're doing, which is the same thing as the probability. If we use more than 110 liters, that means that on average, because we have 50 men, so 110 divided by 50 is what? That's 2. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So the probability that you will run out, let me write this down. The probability that I will or that you will run out is equal or is the same thing as the probability that we use more than 110 liters on our outdoor nature day, whatever we're doing, which is the same thing as the probability. If we use more than 110 liters, that means that on average, because we have 50 men, so 110 divided by 50 is what? That's 2. Let me get the calculator out just so we don't make any mistakes here. So this is going to be the calculator out. So on average, if we have 110 liters and it's going to be drunk by 50 men, including ourselves, I guess, that means that it's the problem. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
That's 2. Let me get the calculator out just so we don't make any mistakes here. So this is going to be the calculator out. So on average, if we have 110 liters and it's going to be drunk by 50 men, including ourselves, I guess, that means that it's the problem. So we would run out if on average more than 2.2 liters is used per man. So this is the same thing as the probability of the average, or maybe we should say the sample mean, or let me write it this way, that the average water use per man of our 50 men is greater than, or we could say greater than or equal to, let me say greater than, well, I'll say greater than, because if we write on the money, then we won't run out of water, is greater than 2.2 liters per man. So let's think about this. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So on average, if we have 110 liters and it's going to be drunk by 50 men, including ourselves, I guess, that means that it's the problem. So we would run out if on average more than 2.2 liters is used per man. So this is the same thing as the probability of the average, or maybe we should say the sample mean, or let me write it this way, that the average water use per man of our 50 men is greater than, or we could say greater than or equal to, let me say greater than, well, I'll say greater than, because if we write on the money, then we won't run out of water, is greater than 2.2 liters per man. So let's think about this. We are essentially taking 50 men out of a universal sample. And we got this data. Who knows where we got this data from, that the average man drinks 2 liters, and that the standard deviation is this. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So let's think about this. We are essentially taking 50 men out of a universal sample. And we got this data. Who knows where we got this data from, that the average man drinks 2 liters, and that the standard deviation is this. Maybe there's some huge study, and this was the best estimate of what the population parameters are, that this is the mean and this is the standard deviation. Now we're sampling 50 men. And what we need to do is figure out, essentially, what is the probability that the mean of this sample, that the sample mean is going to be greater than 2.2 liters. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
Who knows where we got this data from, that the average man drinks 2 liters, and that the standard deviation is this. Maybe there's some huge study, and this was the best estimate of what the population parameters are, that this is the mean and this is the standard deviation. Now we're sampling 50 men. And what we need to do is figure out, essentially, what is the probability that the mean of this sample, that the sample mean is going to be greater than 2.2 liters. And to do that, we have to figure out the distribution of the sampling mean. And we know what that's called. It's the sampling distribution of the sample means. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
And what we need to do is figure out, essentially, what is the probability that the mean of this sample, that the sample mean is going to be greater than 2.2 liters. And to do that, we have to figure out the distribution of the sampling mean. And we know what that's called. It's the sampling distribution of the sample means. And we know that that is going to be a normal distribution. We know a few of the properties of that normal distribution. So this is the distribution of just all men. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
It's the sampling distribution of the sample means. And we know that that is going to be a normal distribution. We know a few of the properties of that normal distribution. So this is the distribution of just all men. And then if you take samples of, say, 50 men. So this will be, let me write this down. So down here, I'm going to draw the sampling distribution of the sample mean when n. So when our sample size is equal to 50. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So this is the distribution of just all men. And then if you take samples of, say, 50 men. So this will be, let me write this down. So down here, I'm going to draw the sampling distribution of the sample mean when n. So when our sample size is equal to 50. So this is essentially telling us the likelihood of the different means when we are sampling 50 men from this population and taking their average water use. So let me draw that. So let's say this is the frequency, and then here are the different values. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So down here, I'm going to draw the sampling distribution of the sample mean when n. So when our sample size is equal to 50. So this is essentially telling us the likelihood of the different means when we are sampling 50 men from this population and taking their average water use. So let me draw that. So let's say this is the frequency, and then here are the different values. Now, the mean value of this, the mean of the sampling distribution of the sample mean, this x bar, that's really just the sample mean right over there, is equal to, it's going to be equal to, if we were to do this millions and millions of times, if we were to plot all of the means when we keep taking samples of 50 and then we were to plot them all out, we would show that this mean of distribution is actually going to be the mean of our actual population. So it's going to be the same value. I want to do it in that same blue. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So let's say this is the frequency, and then here are the different values. Now, the mean value of this, the mean of the sampling distribution of the sample mean, this x bar, that's really just the sample mean right over there, is equal to, it's going to be equal to, if we were to do this millions and millions of times, if we were to plot all of the means when we keep taking samples of 50 and then we were to plot them all out, we would show that this mean of distribution is actually going to be the mean of our actual population. So it's going to be the same value. I want to do it in that same blue. It's going to be the same value as this population over here. So that is going to be 2 liters. So we're still centered at 2 liters. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
I want to do it in that same blue. It's going to be the same value as this population over here. So that is going to be 2 liters. So we're still centered at 2 liters. But what's neat about this is that the sampling distribution of the sample mean, so you take 50 people, find their mean, plot the frequency. 50 people, find the mean. This is actually going to be a normal distribution regardless of, this one just has a well-defined standard deviation. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So we're still centered at 2 liters. But what's neat about this is that the sampling distribution of the sample mean, so you take 50 people, find their mean, plot the frequency. 50 people, find the mean. This is actually going to be a normal distribution regardless of, this one just has a well-defined standard deviation. It's not normal. Even though this one isn't normal, this one over here will be. And we've seen it in multiple videos already. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
This is actually going to be a normal distribution regardless of, this one just has a well-defined standard deviation. It's not normal. Even though this one isn't normal, this one over here will be. And we've seen it in multiple videos already. So this is going to be a normal distribution. And the standard deviation, and we saw this in the last video, and hopefully we've got a little bit of intuition for why this is true. The standard deviation, actually maybe put it a better way. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
And we've seen it in multiple videos already. So this is going to be a normal distribution. And the standard deviation, and we saw this in the last video, and hopefully we've got a little bit of intuition for why this is true. The standard deviation, actually maybe put it a better way. The variance of the sample mean is going to be the variance. So remember, this is standard deviation. So it's going to be the variance of the population divided by n. And if you wanted the standard deviation of this distribution right here, you just take the square root of both sides. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
The standard deviation, actually maybe put it a better way. The variance of the sample mean is going to be the variance. So remember, this is standard deviation. So it's going to be the variance of the population divided by n. And if you wanted the standard deviation of this distribution right here, you just take the square root of both sides. You take the square root of both sides of that. We have the standard deviation of the sample mean is going to be equal to, the square root of this side over here, is going to be equal to the standard deviation of the population divided by the square root of n. And what's this going to be in our case? We know what the standard deviation of the population is. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So it's going to be the variance of the population divided by n. And if you wanted the standard deviation of this distribution right here, you just take the square root of both sides. You take the square root of both sides of that. We have the standard deviation of the sample mean is going to be equal to, the square root of this side over here, is going to be equal to the standard deviation of the population divided by the square root of n. And what's this going to be in our case? We know what the standard deviation of the population is. It is 0.7. And what is n? We have 50 men. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
We know what the standard deviation of the population is. It is 0.7. And what is n? We have 50 men. So 0.7 over the square root of 50. Now let's figure out what that is with the calculator. So we have 0.7 divided by the square root of 50. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
We have 50 men. So 0.7 over the square root of 50. Now let's figure out what that is with the calculator. So we have 0.7 divided by the square root of 50. And we have 0.098. It was pretty close to 0.99. So I'll just write that down. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So we have 0.7 divided by the square root of 50. And we have 0.098. It was pretty close to 0.99. So I'll just write that down. So this is equal to 0.099. That's going to be the standard deviation of this. So it's going to have a lower standard deviation. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So I'll just write that down. So this is equal to 0.099. That's going to be the standard deviation of this. So it's going to have a lower standard deviation. So it's going to look, the distribution is going to be normal. It's going to look something like this. So this is 3 liters over here. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So it's going to have a lower standard deviation. So it's going to look, the distribution is going to be normal. It's going to look something like this. So this is 3 liters over here. This is 1 liter. The standard deviation is almost a tenth. So it's going to be a much narrower distribution. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So this is 3 liters over here. This is 1 liter. The standard deviation is almost a tenth. So it's going to be a much narrower distribution. It's going to look something. I'm trying my best to draw it. It's going to look something like this. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So it's going to be a much narrower distribution. It's going to look something. I'm trying my best to draw it. It's going to look something like this. You get the idea. Where the standard deviation right now is almost 0.1. So it's 0.09, almost a tenth. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
It's going to look something like this. You get the idea. Where the standard deviation right now is almost 0.1. So it's 0.09, almost a tenth. So it's going to be one standard deviation away. It's going to look something like that. So we have our distribution. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So it's 0.09, almost a tenth. So it's going to be one standard deviation away. It's going to look something like that. So we have our distribution. It's a normal distribution. And now let's go back to our question that we're asking. We want to know the probability that our sample will have an average greater than 2.2. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So we have our distribution. It's a normal distribution. And now let's go back to our question that we're asking. We want to know the probability that our sample will have an average greater than 2.2. So this is the distribution of all of the possible samples, the means of all of the possible samples. Now to be greater than 2.2, 2.2 is going to be right around here. 2.2 is going to be right around here. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.