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I'll assume that the number of listeners is more than 200. And he says, hey, I wanna find a sample and I can't ask all of my listeners. Who knows, maybe he has 10,000 listeners. They don't tell us that. But let's say there's 10,000 listeners here and he says, well, I wanna get an indication of what percent like my show. So I need a sample. But instead of taking a truly random sample, he asks them to volunteer. | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
They don't tell us that. But let's say there's 10,000 listeners here and he says, well, I wanna get an indication of what percent like my show. So I need a sample. But instead of taking a truly random sample, he asks them to volunteer. He asks his listeners to visit his website. So that's classic volunteer response sampling. This is non-random because who decides to go to his website and listen to what he just said and maybe even has access to a computer? | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
But instead of taking a truly random sample, he asks them to volunteer. He asks his listeners to visit his website. So that's classic volunteer response sampling. This is non-random because who decides to go to his website and listen to what he just said and maybe even has access to a computer? That's not random. In fact, the people more likely to do that, so these are the people out of the 10,000. So these are the 200 responses here who decide to do it. | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
This is non-random because who decides to go to his website and listen to what he just said and maybe even has access to a computer? That's not random. In fact, the people more likely to do that, so these are the people out of the 10,000. So these are the 200 responses here who decide to do it. These are more likely to be the people who already like David or like to listen to what he tells them to do. The people, the listeners who are not into David or don't wanna do what he tells them to do, well, they're unlikely to be in the say, oh, I'm not really into David and I don't like him telling me what to do, but hey, I'm gonna go to his website anyway. I'm gonna fill out that poll. | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
So these are the 200 responses here who decide to do it. These are more likely to be the people who already like David or like to listen to what he tells them to do. The people, the listeners who are not into David or don't wanna do what he tells them to do, well, they're unlikely to be in the say, oh, I'm not really into David and I don't like him telling me what to do, but hey, I'm gonna go to his website anyway. I'm gonna fill out that poll. That's less likely. Or you might get extreme. So people who really don't like him might say, I'm gonna definitely go there. | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
I'm gonna fill out that poll. That's less likely. Or you might get extreme. So people who really don't like him might say, I'm gonna definitely go there. But in this case, I would say that it's more likely your fans are gonna do what you ask them to do and go to your website and spend time on your website. And because of that, that 89% is probably an overestimate. 89% is probably an overestimate of the number of listeners who really love his show, because you're more likely to get the ones who love him to show up and fill out that actual survey. | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
So people who really don't like him might say, I'm gonna definitely go there. But in this case, I would say that it's more likely your fans are gonna do what you ask them to do and go to your website and spend time on your website. And because of that, that 89% is probably an overestimate. 89% is probably an overestimate of the number of listeners who really love his show, because you're more likely to get the ones who love him to show up and fill out that actual survey. Now, these other forms of bias, response bias. This is when you're asking something that people don't necessarily wanna answer truthfully or the way that it's phrased, it might make someone respond, you say, in a biased way. Classic examples of this are like, have you lied to your parents in the past week? | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
89% is probably an overestimate of the number of listeners who really love his show, because you're more likely to get the ones who love him to show up and fill out that actual survey. Now, these other forms of bias, response bias. This is when you're asking something that people don't necessarily wanna answer truthfully or the way that it's phrased, it might make someone respond, you say, in a biased way. Classic examples of this are like, have you lied to your parents in the past week? Or have you ever cheated on your spouse? Or something like that. Or have you smoked? | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
Classic examples of this are like, have you lied to your parents in the past week? Or have you ever cheated on your spouse? Or something like that. Or have you smoked? Any of these things that people might not wanna answer completely truthfully or they might be hiding from the world and might not just wanna answer that truthfully on a survey. And so you're gonna have response bias. But that's not the case right over here. | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
Or have you smoked? Any of these things that people might not wanna answer completely truthfully or they might be hiding from the world and might not just wanna answer that truthfully on a survey. And so you're gonna have response bias. But that's not the case right over here. And undercoverage is when the way that you're sampling, you're definitely missing out on an important constituency. You know, voluntary response, we're likely missing out on some important constituencies, on some people who might not be into going to your website. But undercoverage is where it's a little bit more clear that that is happening. | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
But that's not the case right over here. And undercoverage is when the way that you're sampling, you're definitely missing out on an important constituency. You know, voluntary response, we're likely missing out on some important constituencies, on some people who might not be into going to your website. But undercoverage is where it's a little bit more clear that that is happening. Now let's do another case. Let's do another case, maybe an alternate reality, where David's trying to figure this out again. He's still hosting a podcast, and he's still curious how much his listeners like his show, but he tries to take a different sample. | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
But undercoverage is where it's a little bit more clear that that is happening. Now let's do another case. Let's do another case, maybe an alternate reality, where David's trying to figure this out again. He's still hosting a podcast, and he's still curious how much his listeners like his show, but he tries to take a different sample. He decides, in this case, to poll the next 100 listeners who send him fan emails. They don't all respond, but 94 out of the 97 listeners polled say they loved his show. What is the most concerning source of bias in this scenario? | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
He's still hosting a podcast, and he's still curious how much his listeners like his show, but he tries to take a different sample. He decides, in this case, to poll the next 100 listeners who send him fan emails. They don't all respond, but 94 out of the 97 listeners polled say they loved his show. What is the most concerning source of bias in this scenario? Well, this is a classic, hey, I have a group, I have a sample sitting in front of me, it's in my inbox on my email, let me just go to them. Isn't that convenient? So this is classic convenience sample. | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
What is the most concerning source of bias in this scenario? Well, this is a classic, hey, I have a group, I have a sample sitting in front of me, it's in my inbox on my email, let me just go to them. Isn't that convenient? So this is classic convenience sample. And this isn't just like, hey, you know, these are the first 100 people to walk through the door, and there's, you know, a lot of times you could argue why that might be not so random, but these are the next 100 listeners who sent him fan emails. So this is convenience sampling, and the sample that you happen to use out of convenience is one that's going to be very skewed to liking you. So once again, this is overestimating, overestimating the percent, the percent that love his show. | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
So this is classic convenience sample. And this isn't just like, hey, you know, these are the first 100 people to walk through the door, and there's, you know, a lot of times you could argue why that might be not so random, but these are the next 100 listeners who sent him fan emails. So this is convenience sampling, and the sample that you happen to use out of convenience is one that's going to be very skewed to liking you. So once again, this is overestimating, overestimating the percent, the percent that love his show. Now, nonresponse is when you ask a certain number of people to fill out a survey, or to answer a questionnaire, and for some reason, some percent do not fill it out, and you're like, well, who were those people? Maybe they would have said something important, and maybe their viewpoint is not properly represented in the overall number that actually did fill it out. And there is some nonresponse going on here. | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
So once again, this is overestimating, overestimating the percent, the percent that love his show. Now, nonresponse is when you ask a certain number of people to fill out a survey, or to answer a questionnaire, and for some reason, some percent do not fill it out, and you're like, well, who were those people? Maybe they would have said something important, and maybe their viewpoint is not properly represented in the overall number that actually did fill it out. And there is some nonresponse going on here. He asks 100 people who sent fan emails to fill out the survey, to say whether they love it or not, 97 fill it out, so there were three people who did not fill out the survey. So there is some nonresponse going on that would be a source of bias, but it's not the most concerning. You know, right over here, they're asking us, fill out the most concerning source of bias, and the convenience sampling is definitely the biggest deal here. | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
And there is some nonresponse going on here. He asks 100 people who sent fan emails to fill out the survey, to say whether they love it or not, 97 fill it out, so there were three people who did not fill out the survey. So there is some nonresponse going on that would be a source of bias, but it's not the most concerning. You know, right over here, they're asking us, fill out the most concerning source of bias, and the convenience sampling is definitely the biggest deal here. There were three people who didn't respond, but that's not as big of a deal. Voluntary response sampling, well, he didn't ask people, like in the last example, like, hey, if you can go here and fill it out, I guess there is actually, actually, no, take that back. There is a little bit of voluntary response here, where he goes to these 100 people and he asks them to respond, and so you have the 97 people who choose to respond. | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
So we see this first example. A statistician recorded the length of each of Pixar's first 14 films. The statistician made a dot plot. Each dot is a film, a histogram, and a box plot to display the running time data. Which display could be used to find the median? To find the median. All right, so let's look at these displays. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
Each dot is a film, a histogram, and a box plot to display the running time data. Which display could be used to find the median? To find the median. All right, so let's look at these displays. So over here we see the 14, this is the dot plot. We have a dot for each of the 14 films. So one film had a running time of 81 minutes. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
All right, so let's look at these displays. So over here we see the 14, this is the dot plot. We have a dot for each of the 14 films. So one film had a running time of 81 minutes. We see that there. One film had a running time of 92. One had a running time of 93. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
So one film had a running time of 81 minutes. We see that there. One film had a running time of 92. One had a running time of 93. We see one had a running time of 95. We see two had running times of 96 minutes. And so on and so forth. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
One had a running time of 93. We see one had a running time of 95. We see two had running times of 96 minutes. And so on and so forth. So I claim that I could use this to figure out the median because I could make a list of all of the running times of the films, I could order them, and then I could find the middle value. I could literally make a list. I could write down 81 and then write down 92, then write down 93, then write down 95, then I could write down 96 twice, and then I could write down 98, then I could write down 100. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
And so on and so forth. So I claim that I could use this to figure out the median because I could make a list of all of the running times of the films, I could order them, and then I could find the middle value. I could literally make a list. I could write down 81 and then write down 92, then write down 93, then write down 95, then I could write down 96 twice, and then I could write down 98, then I could write down 100. I see where you, I think you see where this is going. I could write out the entire list and then I could find the middle value. So the dot plot I could definitely use to find the median. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
I could write down 81 and then write down 92, then write down 93, then write down 95, then I could write down 96 twice, and then I could write down 98, then I could write down 100. I see where you, I think you see where this is going. I could write out the entire list and then I could find the middle value. So the dot plot I could definitely use to find the median. Now what about the histogram? This is the histogram right over here. And the key here is for median, to figure out a median I just need to figure out a list of numbers. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
So the dot plot I could definitely use to find the median. Now what about the histogram? This is the histogram right over here. And the key here is for median, to figure out a median I just need to figure out a list of numbers. I need to figure out a list of numbers. So here, I don't know, you know, they say I have one film that's between 80 and 85, but I don't know its exact running time. Its running time might have been 81 minutes. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
And the key here is for median, to figure out a median I just need to figure out a list of numbers. I need to figure out a list of numbers. So here, I don't know, you know, they say I have one film that's between 80 and 85, but I don't know its exact running time. Its running time might have been 81 minutes. Its running time might have been 84 minutes. So I don't know here, and so I can't really make a list of the running times of the films and find the middle value. So I don't think I'm gonna be able to do it using the histogram. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
Its running time might have been 81 minutes. Its running time might have been 84 minutes. So I don't know here, and so I can't really make a list of the running times of the films and find the middle value. So I don't think I'm gonna be able to do it using the histogram. Now, with the box plot right over here, so I'm not gonna click histogram, with the box plot over here, I might not be able to make a list of all the values, but the box plot explicitly tells us what the median is. This middle line in the middle of the box that tells us the median is, what is this? This median is, if this is 100, this is 99. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
So I don't think I'm gonna be able to do it using the histogram. Now, with the box plot right over here, so I'm not gonna click histogram, with the box plot over here, I might not be able to make a list of all the values, but the box plot explicitly tells us what the median is. This middle line in the middle of the box that tells us the median is, what is this? This median is, if this is 100, this is 99. So this is 95, 96, 97, 98, 99. It explicitly tells us the median is 99. This is actually the easiest for calculating the median. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
This median is, if this is 100, this is 99. So this is 95, 96, 97, 98, 99. It explicitly tells us the median is 99. This is actually the easiest for calculating the median. So I'll go with the box plot. So the histogram is of no use to me if I want to calculate the median. Let's do a couple more of these. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
This is actually the easiest for calculating the median. So I'll go with the box plot. So the histogram is of no use to me if I want to calculate the median. Let's do a couple more of these. Nam owns a used car lot. He checked the odometers of the cars and recorded how far they had driven. He then created both a histogram and a box plot to display the same data. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
Let's do a couple more of these. Nam owns a used car lot. He checked the odometers of the cars and recorded how far they had driven. He then created both a histogram and a box plot to display the same data. Both diagrams are shown below. Which display can be used to find how many vehicles had driven more than 200,000 kilometers? So how many vehicles had driven more than 200,000 kilometers? | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
He then created both a histogram and a box plot to display the same data. Both diagrams are shown below. Which display can be used to find how many vehicles had driven more than 200,000 kilometers? So how many vehicles had driven more than 200,000 kilometers? So it looks like here in this histogram, I have three vehicles that were between 200 and 250, and then I have two vehicles that are between 250 and 300. So it looks pretty clear that I have five vehicles. Three that had a mileage between 200,000 and 250,000, and then I had two that had mileage between 250,000 and 300,000. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
So how many vehicles had driven more than 200,000 kilometers? So it looks like here in this histogram, I have three vehicles that were between 200 and 250, and then I have two vehicles that are between 250 and 300. So it looks pretty clear that I have five vehicles. Three that had a mileage between 200,000 and 250,000, and then I had two that had mileage between 250,000 and 300,000. So I'm able to answer the question. Five vehicles had a mileage more than 200,000. And so I would say that the histogram is pretty useful. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
Three that had a mileage between 200,000 and 250,000, and then I had two that had mileage between 250,000 and 300,000. So I'm able to answer the question. Five vehicles had a mileage more than 200,000. And so I would say that the histogram is pretty useful. But let's verify that the box plot isn't so useful. So I want to know how many vehicles had a mileage more than 200,000. Well, I know that if I have a mileage more than 200,000, I'm going to be in the fourth quartile, but I don't know how many values I have sitting there in the fourth quartile, just looking at this data over here. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
And so I would say that the histogram is pretty useful. But let's verify that the box plot isn't so useful. So I want to know how many vehicles had a mileage more than 200,000. Well, I know that if I have a mileage more than 200,000, I'm going to be in the fourth quartile, but I don't know how many values I have sitting there in the fourth quartile, just looking at this data over here. So that's not going to be useful for answering that question. Let's look at the second question. Which display can be used to find the median distance was approximately 140,000 kilometers? | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
Well, I know that if I have a mileage more than 200,000, I'm going to be in the fourth quartile, but I don't know how many values I have sitting there in the fourth quartile, just looking at this data over here. So that's not going to be useful for answering that question. Let's look at the second question. Which display can be used to find the median distance was approximately 140,000 kilometers? Well, to calculate the median, you essentially want to be able to list all of the numbers and then find the middle number. And over here, I can't list all of the numbers. I know that there's three values that are between zero and 50,000 kilometers, but I don't know what they are. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
Which display can be used to find the median distance was approximately 140,000 kilometers? Well, to calculate the median, you essentially want to be able to list all of the numbers and then find the middle number. And over here, I can't list all of the numbers. I know that there's three values that are between zero and 50,000 kilometers, but I don't know what they are. It could be 10,000, 10,000, 10,000. It could be 10,000, 15,000, and 40,000. I don't know what they are. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
I know that there's three values that are between zero and 50,000 kilometers, but I don't know what they are. It could be 10,000, 10,000, 10,000. It could be 10,000, 15,000, and 40,000. I don't know what they are. And so if I can't list all of these things and put them in order, I really am going to have trouble finding the middle value, the middle values going to be in this range right around here, but I don't know exactly what it's going to be. The histogram is not useful because it's throwing all the values into these buckets. While on the box plot, it explicitly, it directly tells me the median value. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
I don't know what they are. And so if I can't list all of these things and put them in order, I really am going to have trouble finding the middle value, the middle values going to be in this range right around here, but I don't know exactly what it's going to be. The histogram is not useful because it's throwing all the values into these buckets. While on the box plot, it explicitly, it directly tells me the median value. This line right over here, the middle of the box, this tells us the median value. And we see that the median value here, this is 140,000 kilometers. This is 100, 110, 120, 130, 140,000 kilometers is the median mileage for the cars. | Comparing dot plots, histograms, and box plots Data and statistics 6th grade Khan Academy.mp3 |
He filled a sample of 20 drinks to test his null hypothesis, which is the actual population mean for how much drink there was in the drinks, per drink is 530 milliliters, versus his alternative hypothesis is that the population mean is not 530 milliliters, where mu is the mean filling amount. The drinks in the sample contained a mean amount of 528 milliliters with a standard deviation of four milliliters. These results produced a test statistic of t is equal to negative 2.236 and a p-value of approximately 0.038. Assuming the conditions for inference were met, what is an appropriate conclusion at the alpha equals 0.05 significance level? And they give us some choices here. And like always, I encourage you to pause this video and see if you can figure it out on your own. All right, so now let's work through this together. | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
Assuming the conditions for inference were met, what is an appropriate conclusion at the alpha equals 0.05 significance level? And they give us some choices here. And like always, I encourage you to pause this video and see if you can figure it out on your own. All right, so now let's work through this together. So let's just remind ourselves what's going on. So you have some population of drinks, and we care about the true population mean. You have a null hypothesis around it that the true mean is 530 milliliters, but then there's the alternative hypothesis that it's not 530 milliliters. | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
All right, so now let's work through this together. So let's just remind ourselves what's going on. So you have some population of drinks, and we care about the true population mean. You have a null hypothesis around it that the true mean is 530 milliliters, but then there's the alternative hypothesis that it's not 530 milliliters. So to test your null hypothesis, you take a sample. In this case, we had a sample of 20 drinks. And using that sample, you calculate a sample mean, and then you also calculate a sample standard deviation. | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
You have a null hypothesis around it that the true mean is 530 milliliters, but then there's the alternative hypothesis that it's not 530 milliliters. So to test your null hypothesis, you take a sample. In this case, we had a sample of 20 drinks. And using that sample, you calculate a sample mean, and then you also calculate a sample standard deviation. They tell us these things right over here. And then using this information and actually our sample size, you're able to calculate a t-statistic. You're able to calculate a t-statistic, and then using that t-statistic, you are able to calculate a p-value. | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
And using that sample, you calculate a sample mean, and then you also calculate a sample standard deviation. They tell us these things right over here. And then using this information and actually our sample size, you're able to calculate a t-statistic. You're able to calculate a t-statistic, and then using that t-statistic, you are able to calculate a p-value. And the p-value is what is the probability of getting a result at least this extreme if we assume that the null hypothesis is true? And if that probability is lower than our significance level, then we say, hey, that's a very low probability. We're going to reject our null hypothesis, which would suggest our alternative. | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
You're able to calculate a t-statistic, and then using that t-statistic, you are able to calculate a p-value. And the p-value is what is the probability of getting a result at least this extreme if we assume that the null hypothesis is true? And if that probability is lower than our significance level, then we say, hey, that's a very low probability. We're going to reject our null hypothesis, which would suggest our alternative. So the key to this question is just to compare this p-value right over here to our significance level. And as we see, the p-value, 0.038, is indeed less than 0.05. And so because of this, we would reject the null hypothesis. | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
We're going to reject our null hypothesis, which would suggest our alternative. So the key to this question is just to compare this p-value right over here to our significance level. And as we see, the p-value, 0.038, is indeed less than 0.05. And so because of this, we would reject the null hypothesis. We would reject the null hypothesis, which would suggest the alternative, that the true mean is something different than 530 mL. And so if we look at our choices here, so the first choice says reject the null hypothesis. This is strong evidence that the mean filling amount is different than 530 mL. | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
And so because of this, we would reject the null hypothesis. We would reject the null hypothesis, which would suggest the alternative, that the true mean is something different than 530 mL. And so if we look at our choices here, so the first choice says reject the null hypothesis. This is strong evidence that the mean filling amount is different than 530 mL. Yeah, that one looks good. This suggests, this is strong evidence. This suggests the alternative hypothesis, which is that right over there. | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
This is strong evidence that the mean filling amount is different than 530 mL. Yeah, that one looks good. This suggests, this is strong evidence. This suggests the alternative hypothesis, which is that right over there. But let's read the other one, just to make sure that they don't make sense. So this is rejecting the null hypothesis. That looks true so far. | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
This suggests the alternative hypothesis, which is that right over there. But let's read the other one, just to make sure that they don't make sense. So this is rejecting the null hypothesis. That looks true so far. This isn't enough evidence to conclude that the mean filling amount is different than 530 mL. No, not, the first one is definitely much stronger. Fail to reject the null hypothesis. | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
That looks true so far. This isn't enough evidence to conclude that the mean filling amount is different than 530 mL. No, not, the first one is definitely much stronger. Fail to reject the null hypothesis. No, we are rejecting the null hypothesis because our p-value is lower than our significance level. Fail to reject. We'd rule that one out as well. | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
So we have the men, and the men, some proportion are going to vote for the candidate. We'll call that P1. So this is the proportion that will vote for the candidate. And the rest of the men will not vote for the candidate. So 1 minus P1 will not vote for the candidate. And then for the women, you're going to see something similar. So this is the women right over here. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
And the rest of the men will not vote for the candidate. So 1 minus P1 will not vote for the candidate. And then for the women, you're going to see something similar. So this is the women right over here. And some proportion will vote for the candidate. We don't know if it's the same as P1. We don't know if it's the same as the men. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
So this is the women right over here. And some proportion will vote for the candidate. We don't know if it's the same as P1. We don't know if it's the same as the men. So we'll call it P2. And then the rest of the women will not vote for the candidate. 1 minus P2. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
We don't know if it's the same as the men. So we'll call it P2. And then the rest of the women will not vote for the candidate. 1 minus P2. So the not voting are 0s. The ones that are voting are 1s. And these are both Bernoulli distributions. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
1 minus P2. So the not voting are 0s. The ones that are voting are 1s. And these are both Bernoulli distributions. And we know, just because this will be useful later on, we know that the means of this distribution are the same as the proportion that will vote for it. So you can see the mean of the men, or the proportion of the men that will vote. So we'll call that mean 1 is equal to P1. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
And these are both Bernoulli distributions. And we know, just because this will be useful later on, we know that the means of this distribution are the same as the proportion that will vote for it. So you can see the mean of the men, or the proportion of the men that will vote. So we'll call that mean 1 is equal to P1. And the variance here, the variance of this first distribution, I should have done everything in yellow actually. I'll do everything in yellow. So the mean of this distribution is P1. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
So we'll call that mean 1 is equal to P1. And the variance here, the variance of this first distribution, I should have done everything in yellow actually. I'll do everything in yellow. So the mean of this distribution is P1. The variance of this distribution, we'll call that variance 1, is just these two proportions multiplied by each other. So it's P1 times 1 minus P1. And we saw this many, many videos ago when we learned about Bernoulli distributions. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
So the mean of this distribution is P1. The variance of this distribution, we'll call that variance 1, is just these two proportions multiplied by each other. So it's P1 times 1 minus P1. And we saw this many, many videos ago when we learned about Bernoulli distributions. And we're going to see the exact same thing with the women. The mean of this Bernoulli distribution is going to be P2. And then the variance of this Bernoulli distribution is going to be these two proportions multiplied. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
And we saw this many, many videos ago when we learned about Bernoulli distributions. And we're going to see the exact same thing with the women. The mean of this Bernoulli distribution is going to be P2. And then the variance of this Bernoulli distribution is going to be these two proportions multiplied. So P2 times 1 minus P2. Now, what I want to do, and I think I said this at the beginning of the video, is I want to figure out if there's a meaningful difference between the way that the men will vote and the women will vote. I want to figure out is this meaningful? | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
And then the variance of this Bernoulli distribution is going to be these two proportions multiplied. So P2 times 1 minus P2. Now, what I want to do, and I think I said this at the beginning of the video, is I want to figure out if there's a meaningful difference between the way that the men will vote and the women will vote. I want to figure out is this meaningful? So is there a meaningful difference here? And what we're going to do in this video is try to come up with a 95% confidence interval for this parameter. This difference of parameters is still a parameter. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
I want to figure out is this meaningful? So is there a meaningful difference here? And what we're going to do in this video is try to come up with a 95% confidence interval for this parameter. This difference of parameters is still a parameter. We don't know what the true difference of these two population parameters are, or these two population proportions, but we're going to try to come up with a 95% confidence interval for that difference. And the way we do that, we go out and we find 1,000 likely men to vote, or men likely to vote, and 1,000 women likely to vote. So let's write this down. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
This difference of parameters is still a parameter. We don't know what the true difference of these two population parameters are, or these two population proportions, but we're going to try to come up with a 95% confidence interval for that difference. And the way we do that, we go out and we find 1,000 likely men to vote, or men likely to vote, and 1,000 women likely to vote. So let's write this down. So we get 1,000 men. When we survey the 1,000 men, let's say 642 say that they will vote for the candidate, so they are 1s. And then the remainder, we could even, what would it be, 358? | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
So let's write this down. So we get 1,000 men. When we survey the 1,000 men, let's say 642 say that they will vote for the candidate, so they are 1s. And then the remainder, we could even, what would it be, 358? Well, I'll just say the remainder. So the rest are 0s. And we do the same thing with women. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
And then the remainder, we could even, what would it be, 358? Well, I'll just say the remainder. So the rest are 0s. And we do the same thing with women. We survey 1,000 women who are likely to vote, but we survey them randomly. And let's say 591 say that they will vote for the candidate, and the rest say that they will not vote for the candidate. So just here, based on our sample proportions or our sample means, it looks like there is a difference, but we still have to come up with our confidence interval. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
And we do the same thing with women. We survey 1,000 women who are likely to vote, but we survey them randomly. And let's say 591 say that they will vote for the candidate, and the rest say that they will not vote for the candidate. So just here, based on our sample proportions or our sample means, it looks like there is a difference, but we still have to come up with our confidence interval. And let's just make sure we understand what we just did. We just found, so we could figure out a sample proportion over here for the men, so the sample proportion here for the men, which is really just the sample mean of this sample right over here. We have 642 1s, the rest are 0, so we have 642 in the numerator. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
So just here, based on our sample proportions or our sample means, it looks like there is a difference, but we still have to come up with our confidence interval. And let's just make sure we understand what we just did. We just found, so we could figure out a sample proportion over here for the men, so the sample proportion here for the men, which is really just the sample mean of this sample right over here. We have 642 1s, the rest are 0, so we have 642 in the numerator. We have 1,000 samples. 642 divided by 1,000 is 0.642. So you could view this as a sample mean or as a sample proportion. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
We have 642 1s, the rest are 0, so we have 642 in the numerator. We have 1,000 samples. 642 divided by 1,000 is 0.642. So you could view this as a sample mean or as a sample proportion. And if you do the same thing for the women, the sample proportion is going to be 0.591, or you could even just view this as the sample mean of the sample of 1,000 women. We're the ones voting for it, there's 1, the rest are 0. And just to visualize it properly, let me draw the sampling distribution for the sample proportion. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
So you could view this as a sample mean or as a sample proportion. And if you do the same thing for the women, the sample proportion is going to be 0.591, or you could even just view this as the sample mean of the sample of 1,000 women. We're the ones voting for it, there's 1, the rest are 0. And just to visualize it properly, let me draw the sampling distribution for the sample proportion. So the sampling distribution, we have a large sample size. We have a large sample size here, and especially because the proportions that we're dealing with aren't close to 1 or 0, and we have a large sample size, the sampling distribution will be approximately normal. So this sampling distribution, let me write this, the sampling distribution of the sample proportion, so it's going to have some mean over here, so the mean of the sampling distribution of the sample proportion, and we've seen it multiple times. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
And just to visualize it properly, let me draw the sampling distribution for the sample proportion. So the sampling distribution, we have a large sample size. We have a large sample size here, and especially because the proportions that we're dealing with aren't close to 1 or 0, and we have a large sample size, the sampling distribution will be approximately normal. So this sampling distribution, let me write this, the sampling distribution of the sample proportion, so it's going to have some mean over here, so the mean of the sampling distribution of the sample proportion, and we've seen it multiple times. It's going to be the same thing as the mean of the population, and the mean of the population is actually the true population proportion. So this is going to be equal to P1. This is something that we don't know about. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
So this sampling distribution, let me write this, the sampling distribution of the sample proportion, so it's going to have some mean over here, so the mean of the sampling distribution of the sample proportion, and we've seen it multiple times. It's going to be the same thing as the mean of the population, and the mean of the population is actually the true population proportion. So this is going to be equal to P1. This is something that we don't know about. And then the variance of this, and we've seen this several times already, the variance of this distribution, I have to put a 1 here, we're dealing with the men, the variance of this distribution by the central limit theorem is going to be the variance of this distribution up here, which is P1 times 1 minus P1, over our sample size, over 1,000. And we can do the exact same thing for the women. The exact same thing for the women. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
This is something that we don't know about. And then the variance of this, and we've seen this several times already, the variance of this distribution, I have to put a 1 here, we're dealing with the men, the variance of this distribution by the central limit theorem is going to be the variance of this distribution up here, which is P1 times 1 minus P1, over our sample size, over 1,000. And we can do the exact same thing for the women. The exact same thing for the women. So this is the sampling distribution. Let me write it over here. This is the sampling distribution. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
The exact same thing for the women. So this is the sampling distribution. Let me write it over here. This is the sampling distribution. This is for P2 bar, or this sample mean over here, let me put a 1 over here. Remember, this is for all four of the men, and then this over here is all for the women. Can't forget those 2's over there. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
This is the sampling distribution. This is for P2 bar, or this sample mean over here, let me put a 1 over here. Remember, this is for all four of the men, and then this over here is all for the women. Can't forget those 2's over there. So this distribution is going to have some mean. Let me draw it right over here. So mu sub P2 with a bar over it, so the mean of the sampling distribution for this sample proportion for the women, which is going to be the same thing as the mean of the population, which we already saw is going to be equal to P2. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
Can't forget those 2's over there. So this distribution is going to have some mean. Let me draw it right over here. So mu sub P2 with a bar over it, so the mean of the sampling distribution for this sample proportion for the women, which is going to be the same thing as the mean of the population, which we already saw is going to be equal to P2. And then the variance for this distribution, for this sampling distribution over here, is going to be P2, is going to be this variance, this variance over here, divided by our sample size. So P2 times 1 minus P2, all of that over N. Now, our whole goal is to get a 95% confidence interval for that. And so what we're going to do is we're going to think about the sampling distribution not for this, and not the sampling distribution for this, but we're going to think about the sampling distribution for the difference of this sample, this sample proportion, and this sample proportion. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
So mu sub P2 with a bar over it, so the mean of the sampling distribution for this sample proportion for the women, which is going to be the same thing as the mean of the population, which we already saw is going to be equal to P2. And then the variance for this distribution, for this sampling distribution over here, is going to be P2, is going to be this variance, this variance over here, divided by our sample size. So P2 times 1 minus P2, all of that over N. Now, our whole goal is to get a 95% confidence interval for that. And so what we're going to do is we're going to think about the sampling distribution not for this, and not the sampling distribution for this, but we're going to think about the sampling distribution for the difference of this sample, this sample proportion, and this sample proportion. We've seen it already. This is really, you know, we're talking about proportions, but it's really the same exact ideas that we did when we just compared sample means generally. So let's look at that. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
And so what we're going to do is we're going to think about the sampling distribution not for this, and not the sampling distribution for this, but we're going to think about the sampling distribution for the difference of this sample, this sample proportion, and this sample proportion. We've seen it already. This is really, you know, we're talking about proportions, but it's really the same exact ideas that we did when we just compared sample means generally. So let's look at that. Let's look at that distribution. Let's look at this distribution. And just to be clear, when we got this sample mean here, this sample proportion, we just sampled it. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
So let's look at that. Let's look at that distribution. Let's look at this distribution. And just to be clear, when we got this sample mean here, this sample proportion, we just sampled it. You could view it as taking a sample from this distribution over here. When we got this sample proportion, it was like taking a sample from this over here. We took 591, or we took 1,000 samples from this when we took their mean, where it's equivalent to taking a sample from the sampling distribution. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
And just to be clear, when we got this sample mean here, this sample proportion, we just sampled it. You could view it as taking a sample from this distribution over here. When we got this sample proportion, it was like taking a sample from this over here. We took 591, or we took 1,000 samples from this when we took their mean, where it's equivalent to taking a sample from the sampling distribution. Now, this distribution over here is going to be the distribution of all of the differences of the sampling proportions, or of the sample proportions. And so it will look like this. It will have some mean value. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
We took 591, or we took 1,000 samples from this when we took their mean, where it's equivalent to taking a sample from the sampling distribution. Now, this distribution over here is going to be the distribution of all of the differences of the sampling proportions, or of the sample proportions. And so it will look like this. It will have some mean value. I should do this in a different color. I'll do it in green. Yellow and blue make green. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
It will have some mean value. I should do this in a different color. I'll do it in green. Yellow and blue make green. Let me do this one in green. So I'll call this the sampling distribution. This is the sampling distribution of this statistic, of P1 minus P2. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
Yellow and blue make green. Let me do this one in green. So I'll call this the sampling distribution. This is the sampling distribution of this statistic, of P1 minus P2. And so it has some mean over here. It has some mean, the mean of the sample of P1 minus the sample mean, or the sample proportion of P2. And we know from things that we've done in the last several videos, that this is going to be the exact same thing as this mean minus this mean, which is the exact same thing as P1 minus P2. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
This is the sampling distribution of this statistic, of P1 minus P2. And so it has some mean over here. It has some mean, the mean of the sample of P1 minus the sample mean, or the sample proportion of P2. And we know from things that we've done in the last several videos, that this is going to be the exact same thing as this mean minus this mean, which is the exact same thing as P1 minus P2. So this is going to be equal to P1 minus P2. And the variance of this distribution, so the variance of this distribution, P1 minus P2, just like this, is going to be the sum of the variances of these two distributions. So it's going to be this thing over here. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
And we know from things that we've done in the last several videos, that this is going to be the exact same thing as this mean minus this mean, which is the exact same thing as P1 minus P2. So this is going to be equal to P1 minus P2. And the variance of this distribution, so the variance of this distribution, P1 minus P2, just like this, is going to be the sum of the variances of these two distributions. So it's going to be this thing over here. I'll just copy and paste it. It's going to be that thing over there, plus this variance over here. There's no radical sign because we're not taking the standard deviation. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
So it's going to be this thing over here. I'll just copy and paste it. It's going to be that thing over there, plus this variance over here. There's no radical sign because we're not taking the standard deviation. We're focused on the variance right now. So plus this thing right over here. So let me copy and let me paste it. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
There's no radical sign because we're not taking the standard deviation. We're focused on the variance right now. So plus this thing right over here. So let me copy and let me paste it. So plus this thing right over here. So that's going to be the variance. This is going to be the variance. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
So let me copy and let me paste it. So plus this thing right over here. So that's going to be the variance. This is going to be the variance. And if you want the standard deviation, you can literally just get rid of this. You're taking the square root of both sides. So you take the square root of the variance, you get the standard deviation. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
This is going to be the variance. And if you want the standard deviation, you can literally just get rid of this. You're taking the square root of both sides. So you take the square root of the variance, you get the standard deviation. That's why I got rid of that to the second power. And you want to take a square root of the right-hand side, just like that. Now, all I did right now is just to kind of conceptually set things up in our brain. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
So you take the square root of the variance, you get the standard deviation. That's why I got rid of that to the second power. And you want to take a square root of the right-hand side, just like that. Now, all I did right now is just to kind of conceptually set things up in our brain. What we now need to do is actually tackle the confidence interval. We actually need to come up with a 95% confidence interval for P1 minus P2, or a 95% confidence interval for this mean right over here. And because I'm trying to make my best effort not to make videos too long, I'll do part two in the next video where we actually solve the confidence interval. | Comparing population proportions 1 Probability and Statistics Khan Academy.mp3 |
It's a little unusual looking. It looks more like a triangle than our standard density curves, but it's valid. Which of the following statements are true? Choose all answers that apply. The mean of the density curve is less than the median. Pause this video and see if you can figure out whether that's true. Well, we don't know exactly where the mean and median are just by looking at this, but remember, the median is going to be the value for which the area to the right and the left are going to be equal. | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
Choose all answers that apply. The mean of the density curve is less than the median. Pause this video and see if you can figure out whether that's true. Well, we don't know exactly where the mean and median are just by looking at this, but remember, the median is going to be the value for which the area to the right and the left are going to be equal. So I would guess the median is going to be someplace like that. So that's my guess, my approximation. That is the median. | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
Well, we don't know exactly where the mean and median are just by looking at this, but remember, the median is going to be the value for which the area to the right and the left are going to be equal. So I would guess the median is going to be someplace like that. So that's my guess, my approximation. That is the median. And because our distribution goes off further to the left than it does to the right, you could view this as something of a tail, it's reasonable to say that this is left skewed, left skewed. And generally speaking, if a distribution is left skewed, the mean is to the left of the median. So because it is left skewed, the mean might be someplace like right over there. | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
That is the median. And because our distribution goes off further to the left than it does to the right, you could view this as something of a tail, it's reasonable to say that this is left skewed, left skewed. And generally speaking, if a distribution is left skewed, the mean is to the left of the median. So because it is left skewed, the mean might be someplace like right over there. Another way to even think about the mean is that the mean would be the balance point where you'd place a fulcrum if this were a mass. And you might say, why doesn't that happen at the median? Well, remember, even when you're balancing something, a smaller weight that is far away from the fulcrum can balance out a heavier weight that is closer into the fulcrum. | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
So because it is left skewed, the mean might be someplace like right over there. Another way to even think about the mean is that the mean would be the balance point where you'd place a fulcrum if this were a mass. And you might say, why doesn't that happen at the median? Well, remember, even when you're balancing something, a smaller weight that is far away from the fulcrum can balance out a heavier weight that is closer into the fulcrum. So in terms of this first one, the mean of the density curve is less than the median, in this case, or you could say to the left of the median, we can consider this to be true. Now, what about the median of the density curve is three? Well, I already approximated where the median might be, saying, hey, this area looks roughly comparable to this area. | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
Well, remember, even when you're balancing something, a smaller weight that is far away from the fulcrum can balance out a heavier weight that is closer into the fulcrum. So in terms of this first one, the mean of the density curve is less than the median, in this case, or you could say to the left of the median, we can consider this to be true. Now, what about the median of the density curve is three? Well, I already approximated where the median might be, saying, hey, this area looks roughly comparable to this area. The median definitely, I might not be right there, but the median is definitely not going to be three. This area right over here is for sure smaller than this area right over here. So we can rule that out. | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
Well, I already approximated where the median might be, saying, hey, this area looks roughly comparable to this area. The median definitely, I might not be right there, but the median is definitely not going to be three. This area right over here is for sure smaller than this area right over here. So we can rule that out. The area underneath the density curve is one. Pause this video. Is that true? | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
So we can rule that out. The area underneath the density curve is one. Pause this video. Is that true? Yes, this is true. The area underneath any density curve is going to be one. If we look at the total area under the curve, it's always going to be one. | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
Is that true? Yes, this is true. The area underneath any density curve is going to be one. If we look at the total area under the curve, it's always going to be one. So we answered this question. I'll leave you with one extra question that we can actually figure out from the information they've given us. What is the height of this point of this density curve right over here? | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
If we look at the total area under the curve, it's always going to be one. So we answered this question. I'll leave you with one extra question that we can actually figure out from the information they've given us. What is the height of this point of this density curve right over here? What is this value? What is this height going to be? See if you can pause this video and figure it out. | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
What is the height of this point of this density curve right over here? What is this value? What is this height going to be? See if you can pause this video and figure it out. And I'll give you a hint. The hint is this third statement. The area under the density curve is one. | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
See if you can pause this video and figure it out. And I'll give you a hint. The hint is this third statement. The area under the density curve is one. All right, now let's try to work through it together. If we call this height h, we know how to find the area of a triangle. It's 1 1⁄2 base times height. | Density curve worked example Modeling data distributions AP Statistics Khan Academy.mp3 |
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