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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 t... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 lite... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 |
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 a... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 w... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 ... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 d... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 ... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 me... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 al... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 sa... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 thi... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 ... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 goin... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 ... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 ... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 ... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 s... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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 s... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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. And we have 0.098. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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. So this is equal to 0.099. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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. So it's going to look, the distribution is going to be normal. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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. This is 1 liter. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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. It's going to look something. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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. You get the idea. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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. So it's going to be one standard deviation away. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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. It's a normal distribution. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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. So this is the distribution of... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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. So we are essentially asking, we will run out if our sample mean falls into this bucket over here. So we essential... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So we are essentially asking, we will run out if our sample mean falls into this bucket over here. So we essentially need to figure out what is, you can even view it as, what's this area under this curve there. And to figure that out, we just have to figure out how many standard deviations above the mean we are, which ... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
And then we could use a z-table to figure out what this area right over here is. So we want to know, when we are above 2.2 liters, so 2.2 liters, we could even do it in our head, 2.2 liters is what we care about. That's right over here. That is, our mean is 2, so we are 0.2 above the mean. And if we want that in terms ... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
That is, our mean is 2, so we are 0.2 above the mean. And if we want that in terms of standard deviations, we just divide this by the standard deviation of this distribution over here. And we figured out what that is. The standard deviation of this distribution is 0.099. So if we take, and you'll see a formula where yo... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
The standard deviation of this distribution is 0.099. So if we take, and you'll see a formula where you take this value minus the mean and divide it by the standard deviation, that's all we're doing. We're just figuring out how many standard deviations above the mean we are. So you just take this number right over here... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So you just take this number right over here, divided by the standard deviation, so 0.099, or 0.099. And then we get, let's get our calculator. And actually, we had the exact number over here. So we could just take 0.2, we could just take this 0.2, 0.2, divided by this value over here. And on this calculator, when I pr... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So we could just take 0.2, we could just take this 0.2, 0.2, divided by this value over here. And on this calculator, when I press second answer, it just means the last answer. So I'm taking 0.2 divided by this value over there. And I get 2.020. So this means, so that means that this value, or I should write this proba... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
And I get 2.020. So this means, so that means that this value, or I should write this probability is the same probability of being 2.02 standard deviations. Maybe I should write it this way. More than, let me write it down here where I have more space. So this all boils down to the probability of running out of water i... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
More than, let me write it down here where I have more space. So this all boils down to the probability of running out of water is the probability that the sample mean will be more than just the 50 that we happen to select. Remember, if we take a bunch of samples of 50 and plot all of them, we'll get this whole distrib... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
But the 150, the group of 50 that we happen to select, the probability of running out of water is the same thing as the probability of the mean of those people will be more than 2.020 standard deviations above the mean. Above the mean of this distribution, which is actually the same distribution. So what is that going ... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
And here we just have to look up our z table. Remember, this 2.02 is just this value right here. 0.2 divided by 0.09. I just had to pause the video because there's some type of fighter jet outside or something. Anyway, hopefully they won't come back. But anyway, so we need to figure out the probability that the sample ... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
I just had to pause the video because there's some type of fighter jet outside or something. Anyway, hopefully they won't come back. But anyway, so we need to figure out the probability that the sample mean will be more than 2.02 standard deviations above the mean. And to figure that out, we go to a z table. And you ca... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
And to figure that out, we go to a z table. And you can find this pretty much anywhere. Usually it's in any stat book or on the internet, wherever. And so essentially, we want to know the probability. The z table will tell you how much area is below this value. So if you go to z of 2.02, that was the value that we were... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
And so essentially, we want to know the probability. The z table will tell you how much area is below this value. So if you go to z of 2.02, that was the value that we were dealing with, right? Yeah, 2.02. It was, so you go for the first digit. We go to 2.0, and it was 2.02. 2.02 is right over there, right? | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
Yeah, 2.02. It was, so you go for the first digit. We go to 2.0, and it was 2.02. 2.02 is right over there, right? So we had 2.0, and then the next digit you go up here. So it's 2.02 is right over there. So this 0.9783, let me write it down over here. | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
2.02 is right over there, right? So we had 2.0, and then the next digit you go up here. So it's 2.02 is right over there. So this 0.9783, let me write it down over here. This 0.9783, I want to be very careful, 0.9783. That z table, that's not this value over here. This 0.9783 on the z table, that is giving us this whol... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So this 0.9783, let me write it down over here. This 0.9783, I want to be very careful, 0.9783. That z table, that's not this value over here. This 0.9783 on the z table, that is giving us this whole area over here. It's giving us the probability that we are below that value, that we are less than 2.02 standard deviati... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
This 0.9783 on the z table, that is giving us this whole area over here. It's giving us the probability that we are below that value, that we are less than 2.02 standard deviations above the mean. So it's giving us that value over here. So to answer our question, to answer this probability, we just have to subtract thi... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So to answer our question, to answer this probability, we just have to subtract this from 1, because these will all add up to 1. So we just take our calculator back out, and we just take 1 minus 0.9783 is equal to 0.0217. So this right here is 0.0217. Or another way you could say it, it is a 2.17% probability that we w... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
You have a 99 out of 100 chance that any time you apply this test, that it is going to be accurate. Now let's say that you, and you just magically know that, we're just assuming that. Now let's just say that you get 100 folks into this room, and you apply this test to all 100 of them. So apply, apply test 100 times. So... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
So apply, apply test 100 times. So what are some of the possible outcomes here? Is it for sure that 99, exactly 99 out of 100 are going to be accurate, and that one out of 100 is going to be inaccurate? Well that's definitely a likely possibility, but it's also possible you get a little lucky and all 100 are accurate, ... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
Well that's definitely a likely possibility, but it's also possible you get a little lucky and all 100 are accurate, or you get a little unlucky, and that 98 are accurate, and that two are inaccurate. And actually I calculated the probabilities ahead of time, and the goal of this video isn't to go into the probability ... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
I rounded to the nearest tenth of a percent. So it's a little better than a third chance that you'll actually get all of, all of the people are going to get an accurate result, even though for any one of them, there's a 99% chance that it's accurate. Now we could keep going. The probability that it is accurate, I'm jus... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
The probability that it is accurate, I'm just going to put these quotes here so I don't have to rewrite accurate over and over again. The probability that it is accurate 99 out of 100 times, I calculated it ahead of time, it is approximately 37.0%. So this is what you would expect. Getting 100 out of 100 doesn't seem t... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
Getting 100 out of 100 doesn't seem that unlikely if each of the times you apply it has a 99% chance of being accurate. But it makes sense that you, that you would expect 99 out of 100 to be even more likely, slightly more likely. And we can of course keep going. The probability that it is accurate 98 out of 100 times ... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
The probability that it is accurate 98 out of 100 times is approximately 18.5%. And I'm just going to do a few more. The probability that it is accurate 97 out of 100 times, and once again I calculated all of these ahead of time, is 6%. So it's definitely in the realm of possibility, but it's, the probability is much l... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
So it's definitely in the realm of possibility, but it's, the probability is much lower than getting, having 99 out of 100 or 100 out of 100 being accurate. And then the probability, let me put the double quotes here, the probability that it's accurate 96 out of 100 times is approximately 1.5%. And then the probability... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
But I'll just, and you see, you see what's happening to the probabilities as we have fewer and fewer of them being accurate. It becomes less and less probable. So the probability that 95 out of the 100 are accurate is approximately 0.3%. So this was just kind of a, I guess you could say a thought experiment. If we had ... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
So this was just kind of a, I guess you could say a thought experiment. If we had a test that we know for sure that every time you administer it the probability that it's accurate is 99%, then these are the probabilities that if you administer it 100 times that you get 100 out of 100 accurate, the probability that you ... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
So let's put all that in the back of our minds. And let's say that you have devised a new test. You have a new test and you don't know how accurate it is. You have a new cholesterol test. You don't know how accurate it is. You know that in order for it to be approved by whatever governing body, it has to be accurate 99... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
You have a new cholesterol test. You don't know how accurate it is. You know that in order for it to be approved by whatever governing body, it has to be accurate 99, the probability of it being accurate has to be 99%. So needs to have probability of accurate equal to 99%. You don't know if this is true. You just know ... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
So needs to have probability of accurate equal to 99%. You don't know if this is true. You just know that that's what it needs to be. And so you have your test and let's say you set up a hypothesis and your hypothesis could be a lot of things and once you get deeper into statistics, there's null hypothesis and alternat... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
And so you have your test and let's say you set up a hypothesis and your hypothesis could be a lot of things and once you get deeper into statistics, there's null hypothesis and alternate hypotheses, but let's just start with just a simple hypothesis. You're hopeful your hypothesis is that the probability that your new... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
So then you go off and you apply it 100 times. So you apply your new test. You don't know the actual probability of it being accurate. You apply the test 100 times. And let's say out of those 100 times, you get that they are accurate, you get that it is accurate and you're able to use some other test that you know, som... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
You apply the test 100 times. And let's say out of those 100 times, you get that they are accurate, you get that it is accurate and you're able to use some other test that you know, some for sure test, some super accurate test to verify your own test results and you see that it is accurate 95 out of the 100 times. So t... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
Will you accept this hypothesis? Well, what you say is, well, if my hypothesis was true, if my test were accurate 99, if the probability of my test being accurate is 99%, what's the probability of me getting this outcome? Well, we figured that out. If it really was accurate 99% of the time, then the probability of gett... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
If it really was accurate 99% of the time, then the probability of getting this outcome is only 0.3%. So if you assume true, if you assume hypothesis, I'll just write hype. If you assume the hypothesis is true, the probability of the outcome you got, probability of observed outcome is approximately 0.3%. And so you say... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
And so you say, look, you know, maybe it's definitely possible that I just got very, very, very, very unlikely, but based on this, I probably should reject my hypothesis because the probability of me getting this outcome, if the hypothesis was true, is very, very, very, very low. And as we go deeper into statistics, yo... | Idea behind hypothesis testing Probability and Statistics Khan Academy.mp3 |
The manager looks at the difference between the proportions of cars with the defect in each sample. So they're looking at the difference of sample proportions every month. Describe the distribution of the difference of sample proportions in terms of its mean, standard deviation, and shape. So let's take these step by s... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
So let's take these step by step. So first, let's think about the mean of the difference of our sample proportions. Pause this video and try to figure out what that's going to be. Well, we have seen this in previous videos, that if we have the mean of the difference of two random variables, that's the same as the diffe... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
Well, we have seen this in previous videos, that if we have the mean of the difference of two random variables, that's the same as the difference of the means. Or another way to think about it is, if we wanna figure out the mean of this, so sample proportion from plant A minus sample proportion from plant B, this is ju... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
Well, what's the mean of the sample proportion of plant A? It's just going to be the true population proportion for plant A, and they tell us that. They tell us that 8% of all cars produced at plant A have a certain defect. So this could be 8%, or we could write it as 0.08. And then from that, we are going to subtract ... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
So this could be 8%, or we could write it as 0.08. And then from that, we are going to subtract the mean of the sample proportion from plant B. And we know what that mean's going to be. The mean of a sample proportion is going to be the population proportion. The parameter of the population, which we know for plant B i... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
The mean of a sample proportion is going to be the population proportion. The parameter of the population, which we know for plant B is 6%, 0.06. And then that gets us a mean of the difference of 0.02, or 2% having a, or 2% difference in defect rate would be the mean. Now let's think about the standard deviation. So in... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
Now let's think about the standard deviation. So instead of thinking in terms of standard deviation, let's think about the square of the standard deviation, which is variance. And from there, we can go back to standard deviation by taking a square root. So if we're looking at the variance, and let me write it this way.... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
So if we're looking at the variance, and let me write it this way. If we're looking at the variance of the difference of the sample proportions, so the sample proportion from plant A minus the sample proportion from plant B, but just as a review, if you assume that we're sampling independently from each of the plants, ... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
Some of you might be saying, wait, aren't we taking the difference of sample proportions here? Why are we adding? And the reminder is, remember, variance is a measure of spread. And whether you're now taking the difference of random variables or you're taking the sum of them, when you have more variables, you're going ... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
And whether you're now taking the difference of random variables or you're taking the sum of them, when you have more variables, you're going to have more spread. So regardless of whether this is a negative or a positive over here, this is going to be a positive. So what is this going to be equal to? Well, we could tak... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
Well, we could take each of these terms. What's going to be the variance of the sample proportion from plant A? Well, if every time we looked at one of the cars, we looked at it and then we put it back into the mix. So if we were sampling with replacement, which means that each of our observations are independent of th... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
So if we were sampling with replacement, which means that each of our observations are independent of the other ones, we have a formula. We know that this variance would be the population proportion of plant A times one minus the population proportion of plant A divided by the number that we sampled from plant A. Now, ... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
We just took 200 at a time and looked at them. We didn't take one at a time and replace it and do that 200 times. But we also know that this is a pretty good approximation even when you are not sampling with replacement. If your sample is less than 10% of the population and 200 is less than 10% of 3000. So this is a pr... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
If your sample is less than 10% of the population and 200 is less than 10% of 3000. So this is a pretty good approximation, what you would use in a first year statistics class. And of course, we can use the same logic. This is going to be equal to the population proportion of plant B times one minus the population prop... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
This is going to be equal to the population proportion of plant B times one minus the population proportion in plant B, all of that over your sample size from plant B. And we know all of these things. We know that your population proportion in plant A is 8% or 0.08. One minus that is 0.92. We're taking samples of 200 a... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
One minus that is 0.92. We're taking samples of 200 at a time from plant A. And then in plant B, we know the population proportion they told us is 6% or 0.06. One minus that is 0.94. And then the sample size from plant B is also going to be 200. It's going to be 200. We get 0.08 times 0.92 divided by 200. | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
One minus that is 0.94. And then the sample size from plant B is also going to be 200. It's going to be 200. We get 0.08 times 0.92 divided by 200. And then plus, let's open parentheses here. We get 0.06 times 0.94 divided by 200. And then actually let me close the parentheses. | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
We get 0.08 times 0.92 divided by 200. And then plus, let's open parentheses here. We get 0.06 times 0.94 divided by 200. And then actually let me close the parentheses. And that equals this business. So 0.00065. So 0.00065. | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
And then actually let me close the parentheses. And that equals this business. So 0.00065. So 0.00065. And then from this, we can figure out what the standard deviation is going to be. The standard deviation of the difference between our sample proportions is going to be just the square root of this. It's going to be t... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
So 0.00065. And then from this, we can figure out what the standard deviation is going to be. The standard deviation of the difference between our sample proportions is going to be just the square root of this. It's going to be the square root of 0.00065. And that is approximately equal to, let's just take the square r... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
It's going to be the square root of 0.00065. And that is approximately equal to, let's just take the square root, and we get this 0.025. 0.025. And there you have it. We have thought about the standard deviation. And then last but not least, let's think about the shape. So just as a review, we just have to remind ourse... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
And there you have it. We have thought about the standard deviation. And then last but not least, let's think about the shape. So just as a review, we just have to remind ourselves that the distribution of each sample proportion is going to be normal as long as we expect at least 10 successes and 10 failures. Well, let... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
So just as a review, we just have to remind ourselves that the distribution of each sample proportion is going to be normal as long as we expect at least 10 successes and 10 failures. Well, let's look at each of these. How many successes do you expect? Or where success would actually be a defect. But let's think about ... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
Or where success would actually be a defect. But let's think about this. 8% of, in each case, of a sample of 200, that's going to be 16. So you would expect 16 defects. And then you would expect 200 minus 16, which is a lot larger than 10, of no defects. So both of those are greater than or equal to 10. And then if you... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
So you would expect 16 defects. And then you would expect 200 minus 16, which is a lot larger than 10, of no defects. So both of those are greater than or equal to 10. And then if you did the same thing for plant B, you get the same idea. 6% of 200 is 12. And then if you say the ones that have no defects, that's 200 mi... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
And then if you did the same thing for plant B, you get the same idea. 6% of 200 is 12. And then if you say the ones that have no defects, that's 200 minus 12, which is way more than 10, especially in that latter case. But in every situation, we expect to have at least 10 successes and 10 failures. And so we can assume... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
But in every situation, we expect to have at least 10 successes and 10 failures. And so we can assume that the distributions of each of these are going to be normal. And we also know that the difference of two normally distributed variables is also normal so long as they pass that large count condition that we just tal... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
And so let's draw what this distribution might look like. It might look something like this. It's going to be a normal distribution where you have a mean right over here of, I'll do that in that same color, a mean of 0.02. You can definitely take on negative values because there's some situations in which your sample p... | Sampling distribution of the difference in sample proportions AP Statistics Khan Academy.mp3 |
You can definitely take on negative values because there's some situations in which your sample proportion from plant B actually could be larger just by random chance than it is from plant A. So you can definitely take on negative values. But if I wanted to show where zero is, maybe zero is right over here. So we could... | Sampling distribution of the difference in sample proportions AP Statistics 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 al... | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
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