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So the population is all of the seniors at the school. That's the population, all of the seniors. And they sampled 100 of them. So the 100 seniors that they talked to, that is the sample. That is the sample. So they tell us, identify the population and the sample in the setting. So let's just see which of these choices... | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
So the 100 seniors that they talked to, that is the sample. That is the sample. So they tell us, identify the population and the sample in the setting. So let's just see which of these choices actually match up to what I just said. And like always, I encourage you to pause the video and see if you can work through it o... | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
So let's just see which of these choices actually match up to what I just said. And like always, I encourage you to pause the video and see if you can work through it on your own. So the population is all high school seniors in the world. The sample is all of the seniors at Riverview High. No, this is not right. We're ... | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
The sample is all of the seniors at Riverview High. No, this is not right. We're not trying to figure out, we're not trying to get an indication of how all of the high school seniors in the world feel about the food at Riverview High School. We're trying to get an indication of how the seniors at Riverview High School ... | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
We're trying to get an indication of how the seniors at Riverview High School feel about the lunch at the school's cafeteria. So they did a sample of 100 of them. So this is definitely not going to be, let me cross this one out. The population is all students at Riverview High. The sample is all of the seniors at River... | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
The population is all students at Riverview High. The sample is all of the seniors at Riverview High. Well, they clearly didn't sample all of the seniors. They sample 100 of the seniors. So this isn't gonna be right either. Let's hope that the third choice is working out. The population is all seniors at Riverview High... | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
They sample 100 of the seniors. So this isn't gonna be right either. Let's hope that the third choice is working out. The population is all seniors at Riverview High. The sample is the 100 seniors surveyed. Yep, that's exactly what we talked here. We're trying to get an indication about how all of the seniors at Riverv... | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
The population is all seniors at Riverview High. The sample is the 100 seniors surveyed. Yep, that's exactly what we talked here. We're trying to get an indication about how all of the seniors at Riverview High feel about the food, the lunch offerings. We probably think it's impractical, or the administrators feel it's... | Identifying a sample and population Study design AP Statistics Khan Academy.mp3 |
He asks his listeners to visit his website and participate in the poll. The poll shows that 89% of about 200 respondents love his show. What is the most concerning source of bias in this scenario? And like always, pause this video and see if you can figure it out on your own and then we'll work through it together. Let... | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
And like always, pause this video and see if you can figure it out on your own and then we'll work through it together. Let's think about what's going on. He has this population of listeners right over here. 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 a... | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
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... | 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 sam... | 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 fa... | 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... | 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 i... | 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 ... | 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 overestima... | 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 truthful... | 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... | 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... | 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 undercover... | 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 d... | 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 co... | 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 fi... | 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, a... | 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 the... | 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 concerni... | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
She randomly assigned people to exercise for 30 or 60 minutes, then measured their temperatures. The 18 people who exercised for 30 minutes had a mean temperature, so this is the sample mean for that sample of 18 folks, of 38.3 degrees Celsius with a standard deviation, this is a sample standard deviation for those 18 ... | Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3 |
Assume that the conditions for inference have been met and that Kylie will use the conservative degrees of freedom from the smaller sample size. Which of the following is a 90% confidence interval for the difference in mean body temperature after exercising for the two amounts of time? So pause this video and see if yo... | Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3 |
All right, now let's work through this together. So in previous videos, we talked about the general form of our confidence interval, our T interval, which we're going to use because we're dealing with means and we're dealing with the differences in means. And so our T interval is going to have the form, our difference ... | Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3 |
And that is going to be, I think I have enough space here to do it, that is going to be the sample standard deviation of the 60 minute group squared over the sample size of the 60 minute group plus the sample standard deviation of the 30 minute group squared divided by the sample size of the 30 minute group. And so we ... | Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3 |
Now how do we figure that out? Well, we can use our 90% confidence level that we care about, this 90% confidence interval, but if we're looking up things on a T table, we also need to know our degrees of freedom. And it says here that Kylie will use the conservative degrees of freedom. And that means that she will look... | Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3 |
And that means that she will look at each of those samples, so one has a sample size of 18, one has a sample size of 24, whichever is lower, she will use one less than that as her degrees of freedom. 18 is clearly less, lower than 24, so the degrees of freedom in this situation is 18 or are 18 minus one, so 17. And so ... | Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3 |
So our confidence level, 90%, and then our degrees of freedom, 17, so that is that row. The 90% confidence level is this column. And so that gives us our critical T value of 1.74. So going back here, this is going to be plus or minus 1.74 times the square root, times the square root. What's our sample standard deviatio... | Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3 |
So going back here, this is going to be plus or minus 1.74 times the square root, times the square root. What's our sample standard deviation for the 60-minute group? Well, they give it right over here, 0.29, and we're gonna have to square that, divided by the sample size for the 60-minute group. So let's see, the 24 p... | Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3 |
So let's see, the 24 people who exercised for 60 minutes, so divided by 24, plus the sample standard deviation for the 30-minute group, so that's 0.27, 0.27 squared, divided by the sample size for the 30-minute group, divided by 18. And we're done, and we can look down at the choices. Let's see, they all got the first ... | Calculating confidence interval for difference of means AP Statistics Khan Academy.mp3 |
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 ... | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
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. Yo... | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
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 t... | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
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 abl... | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
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... | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
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 alternativ... | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
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... | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
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... | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
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. This isn't enough evidence to conclud... | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
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 hypothes... | Comparing P-value from t statistic to significance level AP Statistics Khan Academy.mp3 |
Nutritionists measured the sugar content in grams for 32 drinks at Starbucks. A cumulative relative frequency graph, let me underline that, a cumulative relative frequency graph for the data is shown below. So they have different, on the horizontal axis, different amounts of sugar in grams, and then we have the cumulat... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
So let's just make sure we understand how to read this. This is saying that zero, or 0%, of the drinks have a sugar content, have no sugar content. This right over here, this data point, this looks like it's at the.5 grams, and then this looks like it's at 0.1. This says that 0.1, or I guess we would say 10%, of the dr... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
This says that 0.1, or I guess we would say 10%, of the drinks that Starbucks offers has five grams of sugar or less. This data point tells us that 100% of the drinks at Starbucks has 50 grams of sugar or less. The cumulative relative frequency, that's why we, for each of these points, we say this is the frequency that... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
As we add more sugar, we're going to see, we're gonna see a larger proportion or a larger relative frequency has that much sugar or less. So let's read the first question. An iced coffee has 15 grams of sugar. Estimate the percentile of this drink to the nearest whole percent. So iced coffee has 15 grams of sugar, whic... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
Estimate the percentile of this drink to the nearest whole percent. So iced coffee has 15 grams of sugar, which would be right over here. And so let's estimate the percentile. So we can see they actually have a data point right over here. And we can see that 20%, or 0.2, 20% of the drinks that Starbucks offers has 15 g... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
So we can see they actually have a data point right over here. And we can see that 20%, or 0.2, 20% of the drinks that Starbucks offers has 15 grams of sugar or less. So the percentile of this drink, if I were to estimate it, it looks like it's the relative frequency, 0.2, has that much sugar or less, and so this perce... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
Once again, another way to think about it, to read this, you could convert these to percentages. You could say that 20% has this much sugar or less, 15 grams of sugar or less, so an iced coffee is in the 20th percentile. Let's do another question. So here we are asked to estimate the median of the distribution of drink... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
So here we are asked to estimate the median of the distribution of drinks. Hint, think about the 50th percentile. So the median, if you were to line up all of the drinks, you would take the middle drink. And so you could view that as, well, what drink is exactly at the 50th percentile? So now let's look at the 50th per... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
And so you could view that as, well, what drink is exactly at the 50th percentile? So now let's look at the 50th percentile would be a cumulative relative frequency of 0.5, which would be right over here on our vertical axis. Another way to think about it is 0.5, or 50% of the drinks are going, if we go to this point r... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
We see that we are right at, looks like this is 25 grams. So one way to interpret this is 50% of the drinks have less than, or have 25 grams of sugar or less. So this looks like a pretty good estimate for the median, for the middle data point. So the median is approximately 25 grams, that half of the drinks have 25 gra... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
So the median is approximately 25 grams, that half of the drinks have 25 grams or less of sugar. Let's do one more based on the same data set. So here we're asked, what is the best estimate for the interquartile range of the distribution of drinks? So the interquartile range, we wanna figure out, well, what's sitting a... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
So the interquartile range, we wanna figure out, well, what's sitting at the 25th percentile? And we wanna think about what's at the 75th percentile. And then we want to take the difference. That's what the interquartile range is. So let's do that. So first, the 25th percentile, we'd wanna look at the cumulative relati... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
That's what the interquartile range is. So let's do that. So first, the 25th percentile, we'd wanna look at the cumulative relative frequency. So 25th, this would be 30th, so 25th would be right around here. And so it looks like the 25th percentile is, that looks like about, I don't know. And we're estimating here, so ... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
So 25th, this would be 30th, so 25th would be right around here. And so it looks like the 25th percentile is, that looks like about, I don't know. And we're estimating here, so that looks like it's about, this would be 15, looks like I would say maybe 18 grams. So approximately 18 grams. Once again, one way to think ab... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
So approximately 18 grams. Once again, one way to think about it is, 25% of the drinks have 18 grams of sugar or less. And let's look at the 75th percentile. So this is 70th, 75th would be right over there. Actually, I can draw a straighter line than that. I have a line tool here. So 75th percentile would put me right ... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
So this is 70th, 75th would be right over there. Actually, I can draw a straighter line than that. I have a line tool here. So 75th percentile would put me right over there. I don't know, that looks like, well, I'll go with 39 grams, roughly 39 grams. And so what's the difference between these two? Well, the difference... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
So 75th percentile would put me right over there. I don't know, that looks like, well, I'll go with 39 grams, roughly 39 grams. And so what's the difference between these two? Well, the difference between these two, it looks like it's about 21 grams. So our interquartile range, our estimate of our interquartile range, ... | Analyzing a cumulative relative frequency graph AP Statistics Khan Academy.mp3 |
She wants to test whether this holds true for teachers in her state. So she is going to take a random sample of these teachers and see what percent of them are members of a union. Let P represent the proportion of teachers in her state that are members of a union. Write an appropriate set of hypotheses for her signific... | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
Write an appropriate set of hypotheses for her significance test. So pause this video and see if you can do that. All right, now let's do it together. So what we wanna do for this significance test is set up a null hypothesis and an alternative hypothesis. Now, your null hypothesis is the hypothesis that, hey, there's ... | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
So what we wanna do for this significance test is set up a null hypothesis and an alternative hypothesis. Now, your null hypothesis is the hypothesis that, hey, there's no news here. It's what you would expect it to be. And so if you read a report saying that 49% of teachers in the United States were members of labor u... | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
And so if you read a report saying that 49% of teachers in the United States were members of labor unions, well, then it would be reasonable to say that the null hypothesis, the no news here, is that the same percentage of teachers in her state are members of a labor union. So that percentage, that proportion is P. So ... | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
Well, the alternative is that the proportion in her state is not 49%. This is the thing that, hey, there would be news here. There'd be something interesting to report. There's something different about her state. And how would she use this? Well, she would take a sample of teachers in her state, figure out the sample ... | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
There's something different about her state. And how would she use this? Well, she would take a sample of teachers in her state, figure out the sample proportion, figure out the probability of getting that sample proportion if we were to assume that the null hypothesis is true. If that probability is lower than a thres... | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
If that probability is lower than a threshold, which she should have said ahead of time, her significance level, then she would reject the null hypothesis, which would suggest the alternative. Let's do another example here. According to a very large poll in 2015, about 90% of homes in California had access to the inter... | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
Market researchers want to test if that proportion is now higher. So they take a random sample of 1,000 homes in California and find that 920, or 92% of homes sampled, have access to the internet. Let P represent the proportion of homes in California that have access to the internet. Write an appropriate set of hypothe... | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
Write an appropriate set of hypotheses for their significance test. So once again, pause this video and see if you can figure it out. So once again, we want to have a null hypothesis and we want to have an alternative hypothesis. The null hypothesis is the, hey, there's no news here. And so that would say that, you kno... | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
The null hypothesis is the, hey, there's no news here. And so that would say that, you know, it's kind of the status quo, that the proportion of people who have internet is still the same as the last study, is still the same at 90%. Or I could write 90%, or I could write 0.9 right over here. Now some of you might have ... | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
Now some of you might have been tempted to put 92% there. But it's very important to realize, 92% is the sample proportion, that's the sample statistic. When we're writing these hypotheses, this is about, these are hypotheses about the true parameter. What is the proportion of, the true proportion of homes in Californi... | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
What is the proportion of, the true proportion of homes in California that now have the internet? And so this is about the true proportion. And so the alternative here is that it's now greater than 90%, or I could say it's greater than 0.9. I could have written 90% or 0.9 here. And so they really, in this question, the... | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
I could have written 90% or 0.9 here. And so they really, in this question, they wrote this to kind of distract you, to make you think, oh, maybe I have to incorporate this 92% somehow. And once again, how will they use these hypotheses? Well, they will take this sample in which they got 92% of homes samples had access... | Constructing hypotheses for a significance test about a proportion AP Statistics Khan Academy.mp3 |
Here is computer output from a least squares regression analysis for using fertility rate to predict life expectancy. Use this model to predict the life expectancy of a country whose fertility rate is two babies per woman. And you can round your answer to the nearest whole number of years. So pause this number and see ... | Generalizabilty of survey results example AP Statistics Khan Academy.mp3 |
So pause this number and see if you can do it. You might need to use a calculator. All right, now let's do this together. So in general, this computer output is actually giving us a lot of data, more than we need actually to do this prediction. But it's giving us the data we need to know the equation for a regression l... | Generalizabilty of survey results example AP Statistics Khan Academy.mp3 |
So in general, this computer output is actually giving us a lot of data, more than we need actually to do this prediction. But it's giving us the data we need to know the equation for a regression line. So the general form of a regression line, a linear regression line, would be our estimate, and that little hat means ... | Generalizabilty of survey results example AP Statistics Khan Academy.mp3 |
Now in this situation, we're using fertility to predict life expectancy. Or let me circle all of life expectancy. So the thing that we're trying to predict, that is y, life expectancy, and fertility is the thing that we're using to predict that. So that is going to be our x right over there. Now what are a and b? Well,... | Generalizabilty of survey results example AP Statistics Khan Academy.mp3 |
So that is going to be our x right over there. Now what are a and b? Well, our computer output gives us that. It's these numbers right over here. Our constant coefficient right over here, this is a, and our slope is going to be negative 5.97. You could view it as the coefficient on fertility. Remember, this right over ... | Generalizabilty of survey results example AP Statistics Khan Academy.mp3 |
It's these numbers right over here. Our constant coefficient right over here, this is a, and our slope is going to be negative 5.97. You could view it as the coefficient on fertility. Remember, this right over here is fertility. You could even rewrite this as our estimated life expectancy, estimated life expectancy, I ... | Generalizabilty of survey results example AP Statistics Khan Academy.mp3 |
Remember, this right over here is fertility. You could even rewrite this as our estimated life expectancy, estimated life expectancy, I could put a little hat on it to show this is estimated life expectancy, is going to be equal to 89.70 minus 5.97 times fertility, times fertility rate. I'll just call it, say, Fert and... | Generalizabilty of survey results example AP Statistics Khan Academy.mp3 |
Notice, this is the coefficient on fertility and then this is the constant coefficient. We could view that right over there. And now we can use this to estimate the life expectancy of a country whose fertility rate is two babies per woman. For fertility, you just put a two here and then you get your estimated life expe... | Generalizabilty of survey results example AP Statistics Khan Academy.mp3 |
For fertility, you just put a two here and then you get your estimated life expectancy. So what's that going to be? We can get out a calculator. So we can say 5.97 times two is equal to that and then we wanna subtract that from, so I'll put a negative there, and add that to 89.7 is equal to, and we wanna round to the n... | Generalizabilty of survey results example AP Statistics Khan Academy.mp3 |
So we can say 5.97 times two is equal to that and then we wanna subtract that from, so I'll put a negative there, and add that to 89.7 is equal to, and we wanna round to the nearest whole number of years, so that's approximately 78 years. So this is approximately 78 years. And we're done. And just to be clear, what eve... | Generalizabilty of survey results example AP Statistics Khan Academy.mp3 |
And just to be clear, what even happened here is that Nkechi, she did a regression. On the x-axis is fertility. Fertility. On the y-axis is, let's call it L period dot E period. That's our y-axis. Took 10 data points, one, two, three, four, five, six, seven, eight, nine, 10, put a regression line on, tried to fit a reg... | Generalizabilty of survey results example AP Statistics Khan Academy.mp3 |
Here's a simulation created by Khan Academy user TETF. I can assume that's pronounced Ted-eff. And what it allows us to do is give us an intuition as to why we divide by n minus one when we calculate our sample variance and why that gives us an unbiased estimate of population variance. So the way this starts off, and I... | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
So the way this starts off, and I encourage you to go try this out yourself, is that you can construct a distribution. It says build a population by clicking in the blue area. So here, we're actually creating a population. So we're creating, every time I click, it increases the population size. So let me just, and I'm ... | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
So we're creating, every time I click, it increases the population size. So let me just, and I'm just randomly doing this, and I encourage you to go onto this scratch pad. It's on the Khan Academy Computer Science, and try to do it yourself. So here, we are, I can stop at some point. So I've constructed a population. I... | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
So here, we are, I can stop at some point. So I've constructed a population. I can throw out some random points up here. So this is our population. And as you saw while I was doing that, I was calculating parameters for the population. It was calculating the population mean at 204.09, and also the population standard d... | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
So this is our population. And as you saw while I was doing that, I was calculating parameters for the population. It was calculating the population mean at 204.09, and also the population standard deviation, which is derived from the population variance. This is the square root of the population variance, and it's at ... | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
This is the square root of the population variance, and it's at 63.8. It was also plotting the population variance down here. You see it's 63.8, which is the standard deviation. And it's a little harder to see, but it says squared. These are these numbers squared. So this is essentially 63.8 is the population, 63.8 squ... | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
And it's a little harder to see, but it says squared. These are these numbers squared. So this is essentially 63.8 is the population, 63.8 squared is the population variance. So that's interesting by itself, but it really doesn't tell us a lot so far about why we divide by n minus one. And this is the interesting part.... | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
So that's interesting by itself, but it really doesn't tell us a lot so far about why we divide by n minus one. And this is the interesting part. We can now start to take samples, and we can decide what sample size we wanna do. I'll start with really small samples, so the smallest possible sample that makes any sense. ... | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
I'll start with really small samples, so the smallest possible sample that makes any sense. So I'm gonna start with really small samples. And what they're going to do, what the simulation is going to do, is every time I take a sample, it's going to calculate the variance. So the numerator is going to be the sum of each... | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
So the numerator is going to be the sum of each of my data points in my sample minus my sample mean, and I'm gonna square it. And then it's going to divide it by n plus a. So, and it's going to vary a. It's going to divide it by anywhere between n plus negative three, so n minus three, all the way to n plus a. And we'r... | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
It's going to divide it by anywhere between n plus negative three, so n minus three, all the way to n plus a. And we're gonna do it many, many, many, many times. We're gonna essentially take the mean of those variances for any a, and figure out which gives us the best estimate. So if I just generate one sample right ov... | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
So if I just generate one sample right over there, well, we see, when we, when our, we see kind of this curve. When we have high values of a, we are underestimating. When we have lower values of a, we are overestimating the population variance. But that was just for one, that was just for one sample, not really that me... | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
But that was just for one, that was just for one sample, not really that meaningful. It's one sample of size two. Let's generate a bunch of samples, and then average them over many of them. And you see, when you look at many, many, many, many, many samples, something interesting is happening. When you look at the mean ... | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
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