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And we could go through all of these. But to do our chi-squared test, we would have said what would be the expected value of each of these data points if we assumed that the null hypothesis was true, that there was no association between foot and hand length. So to help us do that, I'm gonna make a total of our columns here and also a total of our rows here. Let me draw a line here so we know what is going on. And so what are the total number of people who had a longer right hand? Well, it's gonna be 11 plus three plus eight, which is 22. The total number of people who had a longer left hand is two plus nine plus 14, which is 25. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
Let me draw a line here so we know what is going on. And so what are the total number of people who had a longer right hand? Well, it's gonna be 11 plus three plus eight, which is 22. The total number of people who had a longer left hand is two plus nine plus 14, which is 25. And then the total number of people whose hands had the same length, 12 plus 13 plus 28, 25 plus 28, that is 53. And then if I were to total this column, 22 plus 25 is 47 plus 53, we get 100 right over here. And then if we total the number of people who had a longer right foot, 11 plus two plus 12, that's 13 plus 12, that is 25. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
The total number of people who had a longer left hand is two plus nine plus 14, which is 25. And then the total number of people whose hands had the same length, 12 plus 13 plus 28, 25 plus 28, that is 53. And then if I were to total this column, 22 plus 25 is 47 plus 53, we get 100 right over here. And then if we total the number of people who had a longer right foot, 11 plus two plus 12, that's 13 plus 12, that is 25. Longer left foot, three plus nine plus 13, that's also 25. And then we could either add these up and we would get 50 or we could say, hey, 25 plus 25 plus what is 100? Well, that is going to be equal to 50. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
And then if we total the number of people who had a longer right foot, 11 plus two plus 12, that's 13 plus 12, that is 25. Longer left foot, three plus nine plus 13, that's also 25. And then we could either add these up and we would get 50 or we could say, hey, 25 plus 25 plus what is 100? Well, that is going to be equal to 50. Now, to figure out these expected values, remember, we're going to figure out the expected values assuming that the null hypothesis is true, assuming that these distributions are independent, that feet length and hand length are independent variables. Well, if they are independent, which we are assuming, then our best estimate is that 22% have a longer right hand and our best estimate is that 25% have a longer right foot. And so out of 100, you would expect 0.22 times 0.25 times 100 to have a longer right hand and foot. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
Well, that is going to be equal to 50. Now, to figure out these expected values, remember, we're going to figure out the expected values assuming that the null hypothesis is true, assuming that these distributions are independent, that feet length and hand length are independent variables. Well, if they are independent, which we are assuming, then our best estimate is that 22% have a longer right hand and our best estimate is that 25% have a longer right foot. And so out of 100, you would expect 0.22 times 0.25 times 100 to have a longer right hand and foot. I'm just multiplying the probabilities, which you would do if these were independent variables. And so 0.22 times 0.25, let's see, 1 1⁄4 of 22 is 5 1⁄2, so this is going to be equal to 5.5. Now, what number would you expect to have a longer right hand but a longer left foot? | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
And so out of 100, you would expect 0.22 times 0.25 times 100 to have a longer right hand and foot. I'm just multiplying the probabilities, which you would do if these were independent variables. And so 0.22 times 0.25, let's see, 1 1⁄4 of 22 is 5 1⁄2, so this is going to be equal to 5.5. Now, what number would you expect to have a longer right hand but a longer left foot? So that would be 0.22 times 0.25 times 100. Well, we already calculated what that would be. That would be 5.5. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
Now, what number would you expect to have a longer right hand but a longer left foot? So that would be 0.22 times 0.25 times 100. Well, we already calculated what that would be. That would be 5.5. And then to figure out the expected number that it would have a longer right hand but both feet would be the same length, we could multiply 22 out of 100 times 50 out of 100 times 100, which is going to be 1⁄2 of 22, which is equal to 11. And we can keep going. This value right over here would be 0.25 times 0.25 times 100, 25 times 25 is 625, so that would be 6.25. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
That would be 5.5. And then to figure out the expected number that it would have a longer right hand but both feet would be the same length, we could multiply 22 out of 100 times 50 out of 100 times 100, which is going to be 1⁄2 of 22, which is equal to 11. And we can keep going. This value right over here would be 0.25 times 0.25 times 100, 25 times 25 is 625, so that would be 6.25. This value right over here would be 0.25 times 0.25 times 100, which is again 6.25. And then this value right over here, couple of ways we can get it. We can multiply 0.25 times 50 times 100, which would get us to 12.5. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
This value right over here would be 0.25 times 0.25 times 100, 25 times 25 is 625, so that would be 6.25. This value right over here would be 0.25 times 0.25 times 100, which is again 6.25. And then this value right over here, couple of ways we can get it. We can multiply 0.25 times 50 times 100, which would get us to 12.5. Or we could have said this plus this plus this has to equal 25, so this would be 12.5. And now this expected value we can figure out because 5.5 plus 6.25 plus this is going to equal 25. So let's see, 5.5 plus 6.25 is 11.75. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
We can multiply 0.25 times 50 times 100, which would get us to 12.5. Or we could have said this plus this plus this has to equal 25, so this would be 12.5. And now this expected value we can figure out because 5.5 plus 6.25 plus this is going to equal 25. So let's see, 5.5 plus 6.25 is 11.75. 11.75 plus 13.25 is equal to 25. Same thing over here. This would be 13.25, because this is 11.75 plus 13.25 is equal to 25. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
So let's see, 5.5 plus 6.25 is 11.75. 11.75 plus 13.25 is equal to 25. Same thing over here. This would be 13.25, because this is 11.75 plus 13.25 is equal to 25. If we add these two together, we get 26.5. 26.5 plus what is equal to 53? Would be equal to another 26.5. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
This would be 13.25, because this is 11.75 plus 13.25 is equal to 25. If we add these two together, we get 26.5. 26.5 plus what is equal to 53? Would be equal to another 26.5. Now once you figure out all of your expected values, that's a good time to test your conditions. The first condition is that you took a random sample, so let's assume we had done that. The second condition is that your expected value for any of the data points has to be at least equal to five. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
Would be equal to another 26.5. Now once you figure out all of your expected values, that's a good time to test your conditions. The first condition is that you took a random sample, so let's assume we had done that. The second condition is that your expected value for any of the data points has to be at least equal to five. And we can see that all of our expected values are at least equal to five. The actual data points we got do not have to be equal to five. So it's okay that we got a two here because the expected value here is five or larger. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
The second condition is that your expected value for any of the data points has to be at least equal to five. And we can see that all of our expected values are at least equal to five. The actual data points we got do not have to be equal to five. So it's okay that we got a two here because the expected value here is five or larger. And then the last condition is the independence condition that either we are sampling with replacement or that we have to feel comfortable that our sample size is no more than 10% of the population. So let's assume that that happened as well. So assuming we met all of those conditions, we are ready to calculate our chi-squared statistic. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
So it's okay that we got a two here because the expected value here is five or larger. And then the last condition is the independence condition that either we are sampling with replacement or that we have to feel comfortable that our sample size is no more than 10% of the population. So let's assume that that happened as well. So assuming we met all of those conditions, we are ready to calculate our chi-squared statistic. And so what we're going to do is for every data point, we're gonna find the difference between the data point, 11 minus the expected, minus 5.5, squared over the expected. So I did that one. Now I'll do this one. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
So assuming we met all of those conditions, we are ready to calculate our chi-squared statistic. And so what we're going to do is for every data point, we're gonna find the difference between the data point, 11 minus the expected, minus 5.5, squared over the expected. So I did that one. Now I'll do this one. So plus three minus 5.5 squared over 5.5 plus, and I'll do this one, eight minus 11 squared over 11. Then I'll do this one. Two minus 6.25 squared over 6.25. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
Now I'll do this one. So plus three minus 5.5 squared over 5.5 plus, and I'll do this one, eight minus 11 squared over 11. Then I'll do this one. Two minus 6.25 squared over 6.25. And I'll keep doing it. I'm gonna do it for all nine of these data points. And I actually calculated this ahead of time to save some time. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
Two minus 6.25 squared over 6.25. And I'll keep doing it. I'm gonna do it for all nine of these data points. And I actually calculated this ahead of time to save some time. And so if you do this for all nine of the data points, you're going to get a chi-squared statistic of 11.942. Now before we calculate the p-value, we're gonna have to think about what are our degrees of freedom. Now we have a three by three table here. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
And I actually calculated this ahead of time to save some time. And so if you do this for all nine of the data points, you're going to get a chi-squared statistic of 11.942. Now before we calculate the p-value, we're gonna have to think about what are our degrees of freedom. Now we have a three by three table here. So one way to think about it, it's the number of rows minus one times the number of columns minus one. This is two times two, which is equal to four. Another way to think about it is if you know four of these data points and you know the totals, then you could figure out the other five data points. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
Now we have a three by three table here. So one way to think about it, it's the number of rows minus one times the number of columns minus one. This is two times two, which is equal to four. Another way to think about it is if you know four of these data points and you know the totals, then you could figure out the other five data points. And so now we are ready to calculate a p-value. And you could do that using a calculator and you could do that using a chi-squared table. But let's say we did it using a calculator. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
Another way to think about it is if you know four of these data points and you know the totals, then you could figure out the other five data points. And so now we are ready to calculate a p-value. And you could do that using a calculator and you could do that using a chi-squared table. But let's say we did it using a calculator. And we get a p-value of 0.018. And just to remind ourselves what this is, this is the probability of getting a chi-squared statistic at least this large or larger. And so next, we do what we always do with hypothesis testing. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
But let's say we did it using a calculator. And we get a p-value of 0.018. And just to remind ourselves what this is, this is the probability of getting a chi-squared statistic at least this large or larger. And so next, we do what we always do with hypothesis testing. We compare this to our significance level. And we actually should have set our significance level from the beginning. So let's just assume that when we set up our hypotheses here, we also said that we want a significance level of 0.05. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
And so next, we do what we always do with hypothesis testing. We compare this to our significance level. And we actually should have set our significance level from the beginning. So let's just assume that when we set up our hypotheses here, we also said that we want a significance level of 0.05. You really should do this before you calculate all of this. But then you compare your p-value to your significance level. And we see that this p-value is a good bit less than our significance level. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
So let's just assume that when we set up our hypotheses here, we also said that we want a significance level of 0.05. You really should do this before you calculate all of this. But then you compare your p-value to your significance level. And we see that this p-value is a good bit less than our significance level. And so one way to think about it is we got all these expected values, assuming that the null hypothesis was true. But the probability of getting a result this extreme or more extreme is less than 2%, which is lower than our significance level. And so this will lead us to reject our null hypothesis. | Chi-square test for association (independence) AP Statistics Khan Academy.mp3 |
The specific test they use has a false positive rate of 2% and a false negative rate of 1%. Suppose that 5% of all their applicants are actually using illegal drugs and we randomly select an applicant. Given the applicant tests positive, what is the probability that they are actually on drugs? So let's work through this together. So first, let's just make sure we understand what they're telling us. So there is this drug test for the job applicants and then the test has a false positive rate of 2%. What does that mean? | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
So let's work through this together. So first, let's just make sure we understand what they're telling us. So there is this drug test for the job applicants and then the test has a false positive rate of 2%. What does that mean? That means that in 2% of the cases when it should have read negative, that the person didn't do the drugs, it actually read positive. It is a false positive. It should have read negative, but it read positive. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
What does that mean? That means that in 2% of the cases when it should have read negative, that the person didn't do the drugs, it actually read positive. It is a false positive. It should have read negative, but it read positive. Another way to think about it, if someone did not do drugs and you take this test, there's a 2% chance saying that you did do the illegal drugs. They also say that there is a false negative rate of 1%. What does that mean? | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
It should have read negative, but it read positive. Another way to think about it, if someone did not do drugs and you take this test, there's a 2% chance saying that you did do the illegal drugs. They also say that there is a false negative rate of 1%. What does that mean? That means that 1% of the time, if someone did actually take the illegal drugs, it'll say that they didn't. It is falsely giving a negative result when it should have given a positive one. And then they say that 5% of all their applicants are actually using illegal drugs. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
What does that mean? That means that 1% of the time, if someone did actually take the illegal drugs, it'll say that they didn't. It is falsely giving a negative result when it should have given a positive one. And then they say that 5% of all their applicants are actually using illegal drugs. So there's several ways that we can think about it. One of the easiest ways to conceptualize, let's just make up a large number of applicants and I'll use a number where it's fairly straightforward to do the mathematics. So let's say that we start off with 10,000 applicants. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
And then they say that 5% of all their applicants are actually using illegal drugs. So there's several ways that we can think about it. One of the easiest ways to conceptualize, let's just make up a large number of applicants and I'll use a number where it's fairly straightforward to do the mathematics. So let's say that we start off with 10,000 applicants. And so I will both talk in absolute numbers, and I just made this number up. It could have been 1,000, it could have been 100,000, but I like this number because it's easy to do the math. It's better than saying 9,785. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
So let's say that we start off with 10,000 applicants. And so I will both talk in absolute numbers, and I just made this number up. It could have been 1,000, it could have been 100,000, but I like this number because it's easy to do the math. It's better than saying 9,785. And so this is also going to be 100% of the applicants. Now they give us some crucial information here. They tell us that 5% of all their applicants are actually using illegal drugs. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
It's better than saying 9,785. And so this is also going to be 100% of the applicants. Now they give us some crucial information here. They tell us that 5% of all their applicants are actually using illegal drugs. So we can immediately break this 10,000 group into the ones that are doing the drugs and the ones that are not. So 5% are actually on the drugs. 95% are not on the drugs. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
They tell us that 5% of all their applicants are actually using illegal drugs. So we can immediately break this 10,000 group into the ones that are doing the drugs and the ones that are not. So 5% are actually on the drugs. 95% are not on the drugs. So what's 5% of 10,000? So that would be 500. So 500 on drugs. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
95% are not on the drugs. So what's 5% of 10,000? So that would be 500. So 500 on drugs. On drugs. And so once again, this is 5% of our original population. And then how many are not on drugs? | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
So 500 on drugs. On drugs. And so once again, this is 5% of our original population. And then how many are not on drugs? Well, 9,500 not. Not on drugs. And once again, this is 95% of our group of applicants. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
And then how many are not on drugs? Well, 9,500 not. Not on drugs. And once again, this is 95% of our group of applicants. So now let's administer the test. So what is going to happen when we administer the test to the people who are on drugs? Well, the test ideally would give a positive result. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
And once again, this is 95% of our group of applicants. So now let's administer the test. So what is going to happen when we administer the test to the people who are on drugs? Well, the test ideally would give a positive result. It would say positive for all of them, but we know that it's not a perfect test. It's going to give negative for some of them. It will falsely give a negative result for some of them. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
Well, the test ideally would give a positive result. It would say positive for all of them, but we know that it's not a perfect test. It's going to give negative for some of them. It will falsely give a negative result for some of them. And we know that because it has a false negative rate of 1%. And so of these 500, 99% is going to get the correct result in that they're going to test positive. So what is 99% of 500? | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
It will falsely give a negative result for some of them. And we know that because it has a false negative rate of 1%. And so of these 500, 99% is going to get the correct result in that they're going to test positive. So what is 99% of 500? Well, let's see, that would be 495. 495 are going to test positive. I will just use a positive right over there. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
So what is 99% of 500? Well, let's see, that would be 495. 495 are going to test positive. I will just use a positive right over there. And then we're going to have 1%, 1%, which is five, are going to test negative. They are going to falsely test negative. This is the false negative rate. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
I will just use a positive right over there. And then we're going to have 1%, 1%, which is five, are going to test negative. They are going to falsely test negative. This is the false negative rate. And so if we say what percent of our original applicant pool is on drugs and tests positive? Well, 495 over 10,000, this is 4.95%. What percent is of the original applicant pool that is on drugs but tests negative for drugs? | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
This is the false negative rate. And so if we say what percent of our original applicant pool is on drugs and tests positive? Well, 495 over 10,000, this is 4.95%. What percent is of the original applicant pool that is on drugs but tests negative for drugs? The test says that, hey, they're not taking drugs. Well, this is going to be five out of 10,000, which is 0.05%. Another way that you can get these percentages, if you take 5% and multiply by 1%, you're going to get 0.05%, 5 hundredths of a percent. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
What percent is of the original applicant pool that is on drugs but tests negative for drugs? The test says that, hey, they're not taking drugs. Well, this is going to be five out of 10,000, which is 0.05%. Another way that you can get these percentages, if you take 5% and multiply by 1%, you're going to get 0.05%, 5 hundredths of a percent. If you take 5% and multiply by 99%, you're going to get 4.95%. Now let's keep going. Now let's go to the folks who aren't taking the drugs. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
Another way that you can get these percentages, if you take 5% and multiply by 1%, you're going to get 0.05%, 5 hundredths of a percent. If you take 5% and multiply by 99%, you're going to get 4.95%. Now let's keep going. Now let's go to the folks who aren't taking the drugs. And this is where the false positive rate is going to come into effect. So we have a false positive rate of 2%. So 2% are going to test positive. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
Now let's go to the folks who aren't taking the drugs. And this is where the false positive rate is going to come into effect. So we have a false positive rate of 2%. So 2% are going to test positive. What's 2% of 9,500? It's 190 would test positive, even though they're not on drugs. This is the false positive rate. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
So 2% are going to test positive. What's 2% of 9,500? It's 190 would test positive, even though they're not on drugs. This is the false positive rate. So they are testing positive. And then the other 98% will correctly come out negative. And so the other 98%, so 9,500 minus 190, that's going to be 9,310 will correctly test negative. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
This is the false positive rate. So they are testing positive. And then the other 98% will correctly come out negative. And so the other 98%, so 9,500 minus 190, that's going to be 9,310 will correctly test negative. Now what percent of the original applicant pool is this? Well, 190 is 1.9%. And we could calculate it by 190 over 10,000, or you could just say 2% of 95% is 1.9%. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
And so the other 98%, so 9,500 minus 190, that's going to be 9,310 will correctly test negative. Now what percent of the original applicant pool is this? Well, 190 is 1.9%. And we could calculate it by 190 over 10,000, or you could just say 2% of 95% is 1.9%. Once again, multiply the path along the tree. What percent is 9,310? Well, that is going to be 93.10%. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
And we could calculate it by 190 over 10,000, or you could just say 2% of 95% is 1.9%. Once again, multiply the path along the tree. What percent is 9,310? Well, that is going to be 93.10%. You could say this is 9,310 over 10,000, or you can multiply by the path on our probability tree here. 95% times 98% gets us to 93.10%. But now I think we are ready to answer the question. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
Well, that is going to be 93.10%. You could say this is 9,310 over 10,000, or you can multiply by the path on our probability tree here. 95% times 98% gets us to 93.10%. But now I think we are ready to answer the question. Given that the applicant tests positive, what is the probability that they are actually on drugs? So let's look at the first part. Given the applicant tests positive. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
But now I think we are ready to answer the question. Given that the applicant tests positive, what is the probability that they are actually on drugs? So let's look at the first part. Given the applicant tests positive. So which applicants actually tested positive? You have these 495 here tested positive, correctly tested positive, and then you have these 190 right over here incorrectly tested positive, but they did test positive. So how many tested positive? | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
Given the applicant tests positive. So which applicants actually tested positive? You have these 495 here tested positive, correctly tested positive, and then you have these 190 right over here incorrectly tested positive, but they did test positive. So how many tested positive? Well, we have 495 plus 190 tested positive. That's the total number that tested positive. And then which of them were actually on the drugs? | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
So how many tested positive? Well, we have 495 plus 190 tested positive. That's the total number that tested positive. And then which of them were actually on the drugs? Well, of the ones that tested positive, 495 were actually on the drugs. We have 495 divided by 495 plus 190 is equal to 0.7226, so we could say approximately 72%. Approximately 72%. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
And then which of them were actually on the drugs? Well, of the ones that tested positive, 495 were actually on the drugs. We have 495 divided by 495 plus 190 is equal to 0.7226, so we could say approximately 72%. Approximately 72%. Now this is really interesting. Given the applicant tests positive, what is the probability that they are actually on drugs? When you look at these false positive and false negative rates, they seem quite low. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
Approximately 72%. Now this is really interesting. Given the applicant tests positive, what is the probability that they are actually on drugs? When you look at these false positive and false negative rates, they seem quite low. But now when we actually did the calculation, the probability that someone's actually on drugs, it's high, but it's not that high. It's not like if someone were to test positive that you'd say, oh, they are definitely taking the drugs. And you could also get to this result just by using the percentages. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
When you look at these false positive and false negative rates, they seem quite low. But now when we actually did the calculation, the probability that someone's actually on drugs, it's high, but it's not that high. It's not like if someone were to test positive that you'd say, oh, they are definitely taking the drugs. And you could also get to this result just by using the percentages. For example, you could think in terms of what percentage of the original applicants end up testing positive? Well, that's 4.95% plus 1.9%. 4.95, we'll just do it in terms of percent, plus 1.9%, and of them, what percentage were actually on the drugs? | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
And you could also get to this result just by using the percentages. For example, you could think in terms of what percentage of the original applicants end up testing positive? Well, that's 4.95% plus 1.9%. 4.95, we'll just do it in terms of percent, plus 1.9%, and of them, what percentage were actually on the drugs? Well, that was the 4.95%. And notice, this would give you the exact same result. Now there's an interesting takeaway here. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
4.95, we'll just do it in terms of percent, plus 1.9%, and of them, what percentage were actually on the drugs? Well, that was the 4.95%. And notice, this would give you the exact same result. Now there's an interesting takeaway here. Because this is saying, of the people that test positive, 72% are actually on the drugs. You could think about it the other way around. Of the people who test positive, 4.95 plus 1.90, what percentage aren't on drugs? | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
Now there's an interesting takeaway here. Because this is saying, of the people that test positive, 72% are actually on the drugs. You could think about it the other way around. Of the people who test positive, 4.95 plus 1.90, what percentage aren't on drugs? Well, that was 1.90. And this comes out to be approximately 28%. 100% minus 72%. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
Of the people who test positive, 4.95 plus 1.90, what percentage aren't on drugs? Well, that was 1.90. And this comes out to be approximately 28%. 100% minus 72%. And so, if we were in a court of law, and let's say the prosecuting attorney, let's say I got tested positive for drugs, and the prosecuting attorney says, look, this test is very good. It only has a false positive rate of 2%. Sal, and Sal tested positive, he is probably taking drugs. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
100% minus 72%. And so, if we were in a court of law, and let's say the prosecuting attorney, let's say I got tested positive for drugs, and the prosecuting attorney says, look, this test is very good. It only has a false positive rate of 2%. Sal, and Sal tested positive, he is probably taking drugs. A jury who doesn't really understand this well, or go through the trouble that we just did, might say, oh yeah, Sal probably took the drugs. But when we look at this, even if I test positive using this test, there's a 28% chance that I'm not taking drugs, that I was just in this false positive group. And the reason why this number is a good bit larger than this number is because when we looked at the original division between those who take drugs and don't take drugs, most don't take the illegal drugs. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
Sal, and Sal tested positive, he is probably taking drugs. A jury who doesn't really understand this well, or go through the trouble that we just did, might say, oh yeah, Sal probably took the drugs. But when we look at this, even if I test positive using this test, there's a 28% chance that I'm not taking drugs, that I was just in this false positive group. And the reason why this number is a good bit larger than this number is because when we looked at the original division between those who take drugs and don't take drugs, most don't take the illegal drugs. And so 2% of this larger group of the ones that don't take the drugs, well, this is actually a fairly large number relative to the percentage that do take the drugs and test positive. So I will leave you there. This is fascinating, not just for this particular case, but you will see analysis like this all the time when we're looking at whether a certain medication is effective or a certain procedure is effective. | Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3 |
So in a lot of what we're doing in this inferential statistics, we're trying to figure out what is the probability of getting a certain sample mean. So what we've been doing, especially when we have a large sample size, so let me just draw a sampling distribution here, so let's say we have a sampling distribution of the sample mean right here. It has some assumed mean value and some standard deviation. And what we want to do is any result that we get, let's say we get some sample mean out here, we want to figure out the probability of getting a result at least as extreme as this, so you can either figure out the probability of getting a result below this and subtract that from 1, or just figure out this area right over there. And to do that, we've been figuring out how many standard deviations above the mean we actually are. And the way we figure that out is we take our sample mean, we subtract from that our mean itself, what we assume the mean should be, or maybe we don't know what this is, and then we divide that by the standard deviation of the sampling distribution. This is how many standard deviations we are above the mean, that is that distance right over there. | Z-statistics vs. T-statistics Inferential statistics Probability and Statistics Khan Academy.mp3 |
And what we want to do is any result that we get, let's say we get some sample mean out here, we want to figure out the probability of getting a result at least as extreme as this, so you can either figure out the probability of getting a result below this and subtract that from 1, or just figure out this area right over there. And to do that, we've been figuring out how many standard deviations above the mean we actually are. And the way we figure that out is we take our sample mean, we subtract from that our mean itself, what we assume the mean should be, or maybe we don't know what this is, and then we divide that by the standard deviation of the sampling distribution. This is how many standard deviations we are above the mean, that is that distance right over there. Now, we usually don't know what this is either. We normally don't know what that is either. And the central limit theorem told us that, assuming that we have a sufficient sample size, this thing right here, this thing is going to be the same thing as the standard deviation of our population divided by the square root of our sample size. | Z-statistics vs. T-statistics Inferential statistics Probability and Statistics Khan Academy.mp3 |
This is how many standard deviations we are above the mean, that is that distance right over there. Now, we usually don't know what this is either. We normally don't know what that is either. And the central limit theorem told us that, assuming that we have a sufficient sample size, this thing right here, this thing is going to be the same thing as the standard deviation of our population divided by the square root of our sample size. So this thing right over here can be rewritten as our sample mean minus the mean of our sampling distribution of the sample mean divided by this thing right here, divided by our population mean divided by the square root of our sample size. And this is essentially our best sense of how many standard deviations away from the actual mean we are. And this thing right here, we've learned it before, is a z-score. | Z-statistics vs. T-statistics Inferential statistics Probability and Statistics Khan Academy.mp3 |
And the central limit theorem told us that, assuming that we have a sufficient sample size, this thing right here, this thing is going to be the same thing as the standard deviation of our population divided by the square root of our sample size. So this thing right over here can be rewritten as our sample mean minus the mean of our sampling distribution of the sample mean divided by this thing right here, divided by our population mean divided by the square root of our sample size. And this is essentially our best sense of how many standard deviations away from the actual mean we are. And this thing right here, we've learned it before, is a z-score. Or when we're dealing with an actual statistic, when it's derived from the sample mean statistic, we call this a z statistic. And then we could look it up in a z-table or in a normal distribution table to say, what's the probability of getting a value of this z or greater? So that would give us that probability. | Z-statistics vs. T-statistics Inferential statistics Probability and Statistics Khan Academy.mp3 |
And this thing right here, we've learned it before, is a z-score. Or when we're dealing with an actual statistic, when it's derived from the sample mean statistic, we call this a z statistic. And then we could look it up in a z-table or in a normal distribution table to say, what's the probability of getting a value of this z or greater? So that would give us that probability. So what's the probability of getting that extreme of a result? Now, normally when we've done this in the last few videos, we also do not know what the standard deviation of the population is. So in order to approximate that, we say that the z-score is approximately, or the z-statistic is approximately going to be, so let me just write the numerator over again. | Z-statistics vs. T-statistics Inferential statistics Probability and Statistics Khan Academy.mp3 |
So that would give us that probability. So what's the probability of getting that extreme of a result? Now, normally when we've done this in the last few videos, we also do not know what the standard deviation of the population is. So in order to approximate that, we say that the z-score is approximately, or the z-statistic is approximately going to be, so let me just write the numerator over again. Over, we estimate this using our sample standard deviation. We estimate it using our sample standard deviation. And this is OK if our sample size is greater than 30. | Z-statistics vs. T-statistics Inferential statistics Probability and Statistics Khan Academy.mp3 |
So in order to approximate that, we say that the z-score is approximately, or the z-statistic is approximately going to be, so let me just write the numerator over again. Over, we estimate this using our sample standard deviation. We estimate it using our sample standard deviation. And this is OK if our sample size is greater than 30. Or another way to think about it is this will be normally distributed if our sample size is greater than 30. Even this approximation will be approximately normally distributed. Now, if your sample size is less than 30, especially if it's a good bit less than 30, all of a sudden this expression will not be normally distributed. | Z-statistics vs. T-statistics Inferential statistics Probability and Statistics Khan Academy.mp3 |
And this is OK if our sample size is greater than 30. Or another way to think about it is this will be normally distributed if our sample size is greater than 30. Even this approximation will be approximately normally distributed. Now, if your sample size is less than 30, especially if it's a good bit less than 30, all of a sudden this expression will not be normally distributed. So let me rewrite the expression over here. Sample mean minus the mean of your sampling distribution of the sample mean divided by your sample standard deviation over the square root of your sample size. We just said if this thing is well over 30, or at least 30, then this value right here, this statistic, is going to be normally distributed. | Z-statistics vs. T-statistics Inferential statistics Probability and Statistics Khan Academy.mp3 |
Now, if your sample size is less than 30, especially if it's a good bit less than 30, all of a sudden this expression will not be normally distributed. So let me rewrite the expression over here. Sample mean minus the mean of your sampling distribution of the sample mean divided by your sample standard deviation over the square root of your sample size. We just said if this thing is well over 30, or at least 30, then this value right here, this statistic, is going to be normally distributed. If it's not, if this is small, then this is going to have a t distribution. And then you're going to do the exact same thing you did here, but now you would assume that the bell is no longer a normal distribution. So in this example, it was normal. | Z-statistics vs. T-statistics Inferential statistics Probability and Statistics Khan Academy.mp3 |
We just said if this thing is well over 30, or at least 30, then this value right here, this statistic, is going to be normally distributed. If it's not, if this is small, then this is going to have a t distribution. And then you're going to do the exact same thing you did here, but now you would assume that the bell is no longer a normal distribution. So in this example, it was normal. All of the z's are normally distributed. Over here in a t distribution, and this will actually be a normalized t distribution right here, because we subtracted out the mean. So in a normalized t distribution, you're going to have a mean of 0. | Z-statistics vs. T-statistics Inferential statistics Probability and Statistics Khan Academy.mp3 |
So in this example, it was normal. All of the z's are normally distributed. Over here in a t distribution, and this will actually be a normalized t distribution right here, because we subtracted out the mean. So in a normalized t distribution, you're going to have a mean of 0. And what you're going to do is you want to figure out the probability of getting a t value at least this extreme. So this is your t value you would get, and then you essentially figure out the area under the curve right over there. And so a very easy rule of thumb is calculate this quantity either way. | Z-statistics vs. T-statistics Inferential statistics Probability and Statistics Khan Academy.mp3 |
So in a normalized t distribution, you're going to have a mean of 0. And what you're going to do is you want to figure out the probability of getting a t value at least this extreme. So this is your t value you would get, and then you essentially figure out the area under the curve right over there. And so a very easy rule of thumb is calculate this quantity either way. Calculate this quantity either way. If you have more than 30 samples, if your sample size is more than 30, your sample standard deviation is going to be a good approximator for your population standard deviation, and so this whole thing is going to be approximately normally distributed. And so you can use a z table to figure out the probability of getting a result at least that extreme. | Z-statistics vs. T-statistics Inferential statistics Probability and Statistics Khan Academy.mp3 |
And so a very easy rule of thumb is calculate this quantity either way. Calculate this quantity either way. If you have more than 30 samples, if your sample size is more than 30, your sample standard deviation is going to be a good approximator for your population standard deviation, and so this whole thing is going to be approximately normally distributed. And so you can use a z table to figure out the probability of getting a result at least that extreme. If your sample size is small, then this statistic, this quantity, this is going to have a t distribution. And then you're going to have to use a t table to figure out the probability of getting a t value at least this extreme. And we're going to see this in an example a couple of videos from now. | Z-statistics vs. T-statistics Inferential statistics Probability and Statistics Khan Academy.mp3 |
One day he decided to gather data about the distance in miles that people commuted to get to his restaurant. People reported the following distances traveled. So here are all the distances traveled. He wants to create a graph that helps him understand the spread of distances, this is a key word, the spread of distances, and the median distance, and the median distance that people traveled, or that people travel. What kind of graph should he create? So the answer of what kind of graph he should create, that might be a little bit more straightforward than the actual creation of the graph, which we will also do. But he's trying to visualize the spread of information, and at the same time, he wants the median. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
He wants to create a graph that helps him understand the spread of distances, this is a key word, the spread of distances, and the median distance, and the median distance that people traveled, or that people travel. What kind of graph should he create? So the answer of what kind of graph he should create, that might be a little bit more straightforward than the actual creation of the graph, which we will also do. But he's trying to visualize the spread of information, and at the same time, he wants the median. So what graph captures both of that information? Well, a box and whisker plot. So let's actually try to draw a box and whisker plot. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
But he's trying to visualize the spread of information, and at the same time, he wants the median. So what graph captures both of that information? Well, a box and whisker plot. So let's actually try to draw a box and whisker plot. And to do that, we need to come up with the median, and we'll also see the median of the two halves of the data as well. And whenever we're trying to take the median of something, it's really helpful to order our data. So let's start off by attempting to order our data. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
So let's actually try to draw a box and whisker plot. And to do that, we need to come up with the median, and we'll also see the median of the two halves of the data as well. And whenever we're trying to take the median of something, it's really helpful to order our data. So let's start off by attempting to order our data. So what is the smallest number here? Well, let's see, there's one two, so let me mark it off. And then we have another two, another two, so we got all the twos. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
So let's start off by attempting to order our data. So what is the smallest number here? Well, let's see, there's one two, so let me mark it off. And then we have another two, another two, so we got all the twos. Then we have this three. Then we have this three. I think we got all the threes. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
And then we have another two, another two, so we got all the twos. Then we have this three. Then we have this three. I think we got all the threes. Then we have that four. Then we have this four. Do we have any fives? | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
I think we got all the threes. Then we have that four. Then we have this four. Do we have any fives? No. Do we have any sixes? Yep, we have that six. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
Do we have any fives? No. Do we have any sixes? Yep, we have that six. And that looks like the only six. Any sevens? Yep, we have this seven right over here. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
Yep, we have that six. And that looks like the only six. Any sevens? Yep, we have this seven right over here. And I just realized that I missed this one, so let me put the one at the beginning of our set. So I got that one right over there. Actually, there's two ones. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
Yep, we have this seven right over here. And I just realized that I missed this one, so let me put the one at the beginning of our set. So I got that one right over there. Actually, there's two ones. I missed both of them. So both of those ones are right over there. So I have ones, twos, threes, fours, no fives. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
Actually, there's two ones. I missed both of them. So both of those ones are right over there. So I have ones, twos, threes, fours, no fives. This is one six. There was one seven. There's one eight right over here. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
So I have ones, twos, threes, fours, no fives. This is one six. There was one seven. There's one eight right over here. And then let's see, any nines? No nines. Any tens? | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
There's one eight right over here. And then let's see, any nines? No nines. Any tens? Yep, there's a 10. Any 11s? We have an 11 right over there. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
Any tens? Yep, there's a 10. Any 11s? We have an 11 right over there. Any 12s? Nope. 13, 14, then we have a 15, then we have a 20, and then a 22. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
We have an 11 right over there. Any 12s? Nope. 13, 14, then we have a 15, then we have a 20, and then a 22. So we've ordered all our data. Now it should be relatively straightforward to find the middle of our data, the median. So how many data points do we have? | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
13, 14, then we have a 15, then we have a 20, and then a 22. So we've ordered all our data. Now it should be relatively straightforward to find the middle of our data, the median. So how many data points do we have? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17. So the middle number is going to be a number that has eight numbers larger than it and eight numbers smaller than it. So let's think about it. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
So how many data points do we have? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17. So the middle number is going to be a number that has eight numbers larger than it and eight numbers smaller than it. So let's think about it. 1, 2, 3, 4, 5, 6, 7, 8. So the number 6 here is larger than 8 of the values. And if I did the calculations right, it should be smaller than 8 of the values. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
So let's think about it. 1, 2, 3, 4, 5, 6, 7, 8. So the number 6 here is larger than 8 of the values. And if I did the calculations right, it should be smaller than 8 of the values. 1, 2, 3, 4, 5, 6, 7, 8. So it is indeed the median. Now when we take a box and whisker, when we're trying to construct a box and whisker plot, the convention is, OK, we have our median. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
And if I did the calculations right, it should be smaller than 8 of the values. 1, 2, 3, 4, 5, 6, 7, 8. So it is indeed the median. Now when we take a box and whisker, when we're trying to construct a box and whisker plot, the convention is, OK, we have our median. And it's essentially dividing our data into two sets. Now let's take the median of each of those sets. And the convention is to take our median out and have the sets that are left over. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
Now when we take a box and whisker, when we're trying to construct a box and whisker plot, the convention is, OK, we have our median. And it's essentially dividing our data into two sets. Now let's take the median of each of those sets. And the convention is to take our median out and have the sets that are left over. Sometimes people leave it in. But the standard convention, take this median out, and now look separately at this set and look separately at this set. So if we look at this first, the bottom half of our numbers, essentially, what's the median of these numbers? | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
And the convention is to take our median out and have the sets that are left over. Sometimes people leave it in. But the standard convention, take this median out, and now look separately at this set and look separately at this set. So if we look at this first, the bottom half of our numbers, essentially, what's the median of these numbers? Well, we have 1, 2, 3, 4, 5, 6, 7, 8 data points. So we're actually going to have two middle numbers. So the two middle numbers are this 2 and this 3. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
So if we look at this first, the bottom half of our numbers, essentially, what's the median of these numbers? Well, we have 1, 2, 3, 4, 5, 6, 7, 8 data points. So we're actually going to have two middle numbers. So the two middle numbers are this 2 and this 3. Three numbers less than these two, three numbers greater than it. And so when we're looking for a median, you have two middle numbers. We take the mean of these two numbers. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
So the two middle numbers are this 2 and this 3. Three numbers less than these two, three numbers greater than it. And so when we're looking for a median, you have two middle numbers. We take the mean of these two numbers. So halfway in between 2 and 3 is 2.5. Or you could say 2 plus 3 is 5 divided by 2 is 2.5. So here we have a median of this bottom half of 2.5. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
We take the mean of these two numbers. So halfway in between 2 and 3 is 2.5. Or you could say 2 plus 3 is 5 divided by 2 is 2.5. So here we have a median of this bottom half of 2.5. And then the middle of the top half, once again, we have 8 data points. So our middle two numbers are going to be this 11 and this 14. And so if we want to take the mean of these two numbers, 11 plus 14 is 25. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
So here we have a median of this bottom half of 2.5. And then the middle of the top half, once again, we have 8 data points. So our middle two numbers are going to be this 11 and this 14. And so if we want to take the mean of these two numbers, 11 plus 14 is 25. Halfway in between the two is 12.5. So 12.5 is exactly halfway between 11 and 14. And now we've figured out all of the information we need to actually plot or actually create or actually draw our box and whisker plot. | Constructing a box and whisker plot Probability and Statistics Khan Academy.mp3 |
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