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Our mean is p, the probability of success. We see that it indeed was, it was 0.6. And we know that our variance is essentially the probability of success times the probability of failure. That's our variance right over there. Probability of success in this example was 0.4, or probability of success was 0.6, probability of failure was 0.4. You multiply the two, you get 0.24, which is exactly what we got in the last example. And if you take its square root for the standard deviation, which is what we do right here, it's 0.49. | Bernoulli distribution mean and variance formulas Probability and Statistics Khan Academy.mp3 |
And in the last video, we actually estimated that using a 95% confidence interval for the difference in the proportion of men and the difference in the proportion of women. What I want to do in this video is just to ask the question more directly, or just do a straight up hypothesis test to see, is there a difference? So we're going to make our null hypothesis. Let me just get a clear screen here. Let's make our null hypothesis no difference between how the men and the women will vote. Or another way of viewing it is that the proportion of men who will vote for the candidate is going to be the same as the proportion of women who are going to vote for the candidate. Or another way you could say that is that the difference, P1 minus P2, the true proportion of men voting for the candidate minus the true population proportion of women voting for the candidate, is going to be 0. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
Let me just get a clear screen here. Let's make our null hypothesis no difference between how the men and the women will vote. Or another way of viewing it is that the proportion of men who will vote for the candidate is going to be the same as the proportion of women who are going to vote for the candidate. Or another way you could say that is that the difference, P1 minus P2, the true proportion of men voting for the candidate minus the true population proportion of women voting for the candidate, is going to be 0. That's our null hypothesis. Our alternative hypothesis is that there is a difference. Our alternative hypothesis is that there is a difference. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
Or another way you could say that is that the difference, P1 minus P2, the true proportion of men voting for the candidate minus the true population proportion of women voting for the candidate, is going to be 0. That's our null hypothesis. Our alternative hypothesis is that there is a difference. Our alternative hypothesis is that there is a difference. Or that P1 does not equal P2. Or that P1 minus P2, the proportion of men voting minus the proportion of women voting, the true population proportions, do not equal 0. And we're going to test this. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
Our alternative hypothesis is that there is a difference. Or that P1 does not equal P2. Or that P1 minus P2, the proportion of men voting minus the proportion of women voting, the true population proportions, do not equal 0. And we're going to test this. We're going to do the hypothesis test with a significance level of 5%. And all that means, and we've done this multiple times, is we are going to assume the null hypothesis. And then assuming the null hypothesis is true, we're going to then figure out the probability of getting our actual difference of our sample proportions. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
And we're going to test this. We're going to do the hypothesis test with a significance level of 5%. And all that means, and we've done this multiple times, is we are going to assume the null hypothesis. And then assuming the null hypothesis is true, we're going to then figure out the probability of getting our actual difference of our sample proportions. So we're going to figure out the probability of actually getting our actual difference between our male sample proportion and our female sample proportion, given the assumption that our null hypothesis is correct. And if this probability is less than 5%, if this probability is less than our significance level, so if the odds of getting these two samples and the difference between those two samples is less than 5%, then we are going to reject the null hypothesis. So how are we going to do this? | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
And then assuming the null hypothesis is true, we're going to then figure out the probability of getting our actual difference of our sample proportions. So we're going to figure out the probability of actually getting our actual difference between our male sample proportion and our female sample proportion, given the assumption that our null hypothesis is correct. And if this probability is less than 5%, if this probability is less than our significance level, so if the odds of getting these two samples and the difference between those two samples is less than 5%, then we are going to reject the null hypothesis. So how are we going to do this? So if we assume the null hypothesis, what does the sampling distribution of this statistic start to look like? Well, the mean, if we assume that P1 and the true population proportions are actually the same between men and women, if P1 and P2 are actually the same, then this right here is going to be 0. This right here is going to be 0. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So how are we going to do this? So if we assume the null hypothesis, what does the sampling distribution of this statistic start to look like? Well, the mean, if we assume that P1 and the true population proportions are actually the same between men and women, if P1 and P2 are actually the same, then this right here is going to be 0. This right here is going to be 0. So what we can do is we can figure out that we got, when we took the proportion of men and we subtracted from that the proportion of women, so this is our sample proportion of men who are going to vote for, at least in our poll, said they would vote for the candidate. This is the proportion of women who said they would vote for the candidate. The difference between the two was 0.051. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
This right here is going to be 0. So what we can do is we can figure out that we got, when we took the proportion of men and we subtracted from that the proportion of women, so this is our sample proportion of men who are going to vote for, at least in our poll, said they would vote for the candidate. This is the proportion of women who said they would vote for the candidate. The difference between the two was 0.051. So what we can do is figure out what's the probability, assuming that the true proportions are equal, that the mean of the sampling distribution of this statistic is actually 0, what's the probability that we get a difference of 0.051? So what's the likelihood that we get something that extreme? And what we're going to do here is just figure out a z score for this, essentially figure out how many standard deviations away from the mean this is. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
The difference between the two was 0.051. So what we can do is figure out what's the probability, assuming that the true proportions are equal, that the mean of the sampling distribution of this statistic is actually 0, what's the probability that we get a difference of 0.051? So what's the likelihood that we get something that extreme? And what we're going to do here is just figure out a z score for this, essentially figure out how many standard deviations away from the mean this is. That would be our z score. And then figure out, is the likelihood of getting a standard deviation or that extreme of a result or that many standard deviations away from the mean, is that likelihood more or less than 5%? If it is less than 5%, we're going to reject the null hypothesis. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
And what we're going to do here is just figure out a z score for this, essentially figure out how many standard deviations away from the mean this is. That would be our z score. And then figure out, is the likelihood of getting a standard deviation or that extreme of a result or that many standard deviations away from the mean, is that likelihood more or less than 5%? If it is less than 5%, we're going to reject the null hypothesis. So let's first of all figure out our z score. So we're assuming the null hypothesis, p1 is equal to p2. Our z score, the number of standard deviations that our actual result is away from the mean, the actual difference that we sampled in the last few videos between the men and the women, was 0.051. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
If it is less than 5%, we're going to reject the null hypothesis. So let's first of all figure out our z score. So we're assuming the null hypothesis, p1 is equal to p2. Our z score, the number of standard deviations that our actual result is away from the mean, the actual difference that we sampled in the last few videos between the men and the women, was 0.051. And from that, we're going to subtract the assumed mean. Remember, we're assuming that these two things are equal. So the mean of this sampling distribution right here is 0. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
Our z score, the number of standard deviations that our actual result is away from the mean, the actual difference that we sampled in the last few videos between the men and the women, was 0.051. And from that, we're going to subtract the assumed mean. Remember, we're assuming that these two things are equal. So the mean of this sampling distribution right here is 0. So we're just going to subtract 0. And then we have to divide this by the standard deviation of the sampling distribution of this statistic right here. So the sampling distribution of this statistic, p1 minus p2. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So the mean of this sampling distribution right here is 0. So we're just going to subtract 0. And then we have to divide this by the standard deviation of the sampling distribution of this statistic right here. So the sampling distribution of this statistic, p1 minus p2. Now what's the standard deviation of the distribution going to be? In the last video, we figured out that we could represent it by this formula over here. But with our null hypothesis, we're assuming that p1 and p2 are the same value. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So the sampling distribution of this statistic, p1 minus p2. Now what's the standard deviation of the distribution going to be? In the last video, we figured out that we could represent it by this formula over here. But with our null hypothesis, we're assuming that p1 and p2 are the same value. So let me rewrite it. So in our last video, and I don't want to confuse the issue, because in the last video I made this approximation over here. So let me write the clean version down here. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
But with our null hypothesis, we're assuming that p1 and p2 are the same value. So let me rewrite it. So in our last video, and I don't want to confuse the issue, because in the last video I made this approximation over here. So let me write the clean version down here. We know that the standard deviation of our sampling distribution of this statistic, of the sample mean of p1 minus the sample proportion or sample mean of p2, is equal to the square root of p1 times 1 minus p1 over 1,000 plus p2 times 1 minus p2 over 1,000. We've seen this in several videos. But in the null hypothesis, we are going to assume that p1 is equal to p2. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So let me write the clean version down here. We know that the standard deviation of our sampling distribution of this statistic, of the sample mean of p1 minus the sample proportion or sample mean of p2, is equal to the square root of p1 times 1 minus p1 over 1,000 plus p2 times 1 minus p2 over 1,000. We've seen this in several videos. But in the null hypothesis, we are going to assume that p1 is equal to p2. That's what we do. We assume the null hypothesis and see the probability of this occurring. So if p1 is equal to p2, we can just represent them as just some true population proportion. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
But in the null hypothesis, we are going to assume that p1 is equal to p2. That's what we do. We assume the null hypothesis and see the probability of this occurring. So if p1 is equal to p2, we can just represent them as just some true population proportion. So we could say that this is going to be equal to, so we could write it like this. The square root of, we can literally just factor out, 1 over 1,000 times p times 1 minus p plus p times 1 minus p, because they're going to be the same value. That's what we're assuming in the null hypothesis. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So if p1 is equal to p2, we can just represent them as just some true population proportion. So we could say that this is going to be equal to, so we could write it like this. The square root of, we can literally just factor out, 1 over 1,000 times p times 1 minus p plus p times 1 minus p, because they're going to be the same value. That's what we're assuming in the null hypothesis. And so this is just 2 of these over here. So this is going to be equal to 2p times 1 minus p, all of that over 1,000. And we're going to take the square root of that. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
That's what we're assuming in the null hypothesis. And so this is just 2 of these over here. So this is going to be equal to 2p times 1 minus p, all of that over 1,000. And we're going to take the square root of that. Now this is the standard deviation, once again, of the distribution of this statistic right over here. The sample mean of the sample proportion for the men minus the sample proportion of the women. Now, we still don't know this. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
And we're going to take the square root of that. Now this is the standard deviation, once again, of the distribution of this statistic right over here. The sample mean of the sample proportion for the men minus the sample proportion of the women. Now, we still don't know this. We still don't know the true proportion. We still don't know what that is. But we can estimate it. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
Now, we still don't know this. We still don't know the true proportion. We still don't know what that is. But we can estimate it. We can estimate it using our samples. And since we're assuming that the men and the women, that there's no difference between them, we can actually view it as a sample size of 2,000 to figure out that true proportion. So we can actually substitute it. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
But we can estimate it. We can estimate it using our samples. And since we're assuming that the men and the women, that there's no difference between them, we can actually view it as a sample size of 2,000 to figure out that true proportion. So we can actually substitute it. We can actually substitute this with a sample proportion. And we can pretend like our survey of the men and women is just one huge survey. So you have your sample proportion is going to be equal to, we're surveying a total of 2,000 people. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So we can actually substitute it. We can actually substitute this with a sample proportion. And we can pretend like our survey of the men and women is just one huge survey. So you have your sample proportion is going to be equal to, we're surveying a total of 2,000 people. 1,000 men and 1,000 women. But we're assuming that they're no different. That's what our null hypothesis is all about. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So you have your sample proportion is going to be equal to, we're surveying a total of 2,000 people. 1,000 men and 1,000 women. But we're assuming that they're no different. That's what our null hypothesis is all about. Assuming there's no difference between men and women. And we got, let's go back to our original, we got 642 yeses amongst the men and 591 amongst the women. So we got a total of, I already forgot, 642, 591. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
That's what our null hypothesis is all about. Assuming there's no difference between men and women. And we got, let's go back to our original, we got 642 yeses amongst the men and 591 amongst the women. So we got a total of, I already forgot, 642, 591. So it is 642 plus 591. If you viewed it as just one huge sample of 2,000 people, we got, get the calculator out, we got 642 plus 591 is equal to 1,233 divided by 2,000. Gives us 0.6165. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So we got a total of, I already forgot, 642, 591. So it is 642 plus 591. If you viewed it as just one huge sample of 2,000 people, we got, get the calculator out, we got 642 plus 591 is equal to 1,233 divided by 2,000. Gives us 0.6165. And this is our best estimate of this consistent population proportion that is true of both men and women. Because we are assuming that they are no different. So we can substitute this value in for p to estimate the standard deviation of the sampling distribution of this statistic right over here. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
Gives us 0.6165. And this is our best estimate of this consistent population proportion that is true of both men and women. Because we are assuming that they are no different. So we can substitute this value in for p to estimate the standard deviation of the sampling distribution of this statistic right over here. Assuming that the proportion of men and women are the same. Or the proportion that will vote for the candidate. So let's do that. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So we can substitute this value in for p to estimate the standard deviation of the sampling distribution of this statistic right over here. Assuming that the proportion of men and women are the same. Or the proportion that will vote for the candidate. So let's do that. It's going to be the square root of 2 times p, which is 0.6165 times 1 minus p. Divided by 1,000. Let me make sure I get it. 2 times 0.6165, that's that p right there, times 1 minus p divided by 1,000. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So let's do that. It's going to be the square root of 2 times p, which is 0.6165 times 1 minus p. Divided by 1,000. Let me make sure I get it. 2 times 0.6165, that's that p right there, times 1 minus p divided by 1,000. We're taking the square root of the whole thing. And so we get a standard deviation of 0.0217. Let me write this over here. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
2 times 0.6165, that's that p right there, times 1 minus p divided by 1,000. We're taking the square root of the whole thing. And so we get a standard deviation of 0.0217. Let me write this over here. So this thing right over here is 0.0217. So if we want to figure out our z-score, if we want to figure out how many standard deviations the actual sample that we got of this statistic right over here, if we want to figure out how many standard deviations that is away from our assumed mean, that the assumed mean is that there's no difference, then we just divide 0.051 by this standard deviation right over here. So let's do that. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
Let me write this over here. So this thing right over here is 0.0217. So if we want to figure out our z-score, if we want to figure out how many standard deviations the actual sample that we got of this statistic right over here, if we want to figure out how many standard deviations that is away from our assumed mean, that the assumed mean is that there's no difference, then we just divide 0.051 by this standard deviation right over here. So let's do that. So we have 0.051 divided by this standard deviation. That was our answer up here. So I'll just do divided by our answer. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So let's do that. So we have 0.051 divided by this standard deviation. That was our answer up here. So I'll just do divided by our answer. And we are 2.35 standard deviations away. So our z-score is equal to 2.35. So just to review what we're doing, we're assuming the null hypothesis. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So I'll just do divided by our answer. And we are 2.35 standard deviations away. So our z-score is equal to 2.35. So just to review what we're doing, we're assuming the null hypothesis. There's no difference. If we assume there's no difference, then the sampling distribution of this statistic right here is going to have a mean of 0, and the result that we actually got for this statistic has a z-score of 2.34. Or this is equivalent to being 2.34 standard deviations away from this mean of 0. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So just to review what we're doing, we're assuming the null hypothesis. There's no difference. If we assume there's no difference, then the sampling distribution of this statistic right here is going to have a mean of 0, and the result that we actually got for this statistic has a z-score of 2.34. Or this is equivalent to being 2.34 standard deviations away from this mean of 0. So in order to reject the null hypothesis, that has to be less probable than our significance level. And to see that, let's see what the minimum z-score we need to reject our hypothesis. So let's think about that a second. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
Or this is equivalent to being 2.34 standard deviations away from this mean of 0. So in order to reject the null hypothesis, that has to be less probable than our significance level. And to see that, let's see what the minimum z-score we need to reject our hypothesis. So let's think about that a second. I'll go back to my z-table. We want to have a significance level of 5%, which means the entire area of our rejection, the entire area in which we would reject the null hypothesis, is 5%. This is a two-tailed test. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So let's think about that a second. I'll go back to my z-table. We want to have a significance level of 5%, which means the entire area of our rejection, the entire area in which we would reject the null hypothesis, is 5%. This is a two-tailed test. An extreme event either far above the mean or far below the mean will allow us to reject the hypothesis. So we care about area over here. And over here we would put 2.5%. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
This is a two-tailed test. An extreme event either far above the mean or far below the mean will allow us to reject the hypothesis. So we care about area over here. And over here we would put 2.5%. And over here we would have 2.5%. And we would have 95% in the middle. So we need to find this critical z-score, critical z value, and if our z value is greater than the positive version of this critical z value, then that's less probable than the odds of getting something so extreme is less than 5%. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
And over here we would put 2.5%. And over here we would have 2.5%. And we would have 95% in the middle. So we need to find this critical z-score, critical z value, and if our z value is greater than the positive version of this critical z value, then that's less probable than the odds of getting something so extreme is less than 5%. And we can, assuming the null hypothesis is correct, so then we can reject the null hypothesis. So let's see what this critical z value is. So essentially we want a z value where the entire percentage below it is going to be 97.5%, because then you're going to have 2.5% over here. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So we need to find this critical z-score, critical z value, and if our z value is greater than the positive version of this critical z value, then that's less probable than the odds of getting something so extreme is less than 5%. And we can, assuming the null hypothesis is correct, so then we can reject the null hypothesis. So let's see what this critical z value is. So essentially we want a z value where the entire percentage below it is going to be 97.5%, because then you're going to have 2.5% over here. And we've actually already figured that out. This whole cumulative has to be 97.5%. We did that in the last video. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So essentially we want a z value where the entire percentage below it is going to be 97.5%, because then you're going to have 2.5% over here. And we've actually already figured that out. This whole cumulative has to be 97.5%. We did that in the last video. If you look for that, you get 97.975 right there. It's a z-score of 1.96. I even wrote it over there. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
We did that in the last video. If you look for that, you get 97.975 right there. It's a z-score of 1.96. I even wrote it over there. So this critical z value is 1.96. So what that tells you is there is a 5% chance. So this tells us that there is a 5% chance of sampling a z statistic greater than 1.96, assuming the null hypothesis is correct. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
I even wrote it over there. So this critical z value is 1.96. So what that tells you is there is a 5% chance. So this tells us that there is a 5% chance of sampling a z statistic greater than 1.96, assuming the null hypothesis is correct. Now, we just figured out that we just sampled a z statistic of 2.34, assuming the null hypothesis is correct. So the probability of sampling this, given the null hypothesis is correct, is going to be less than 5%. It is more extreme than this critical z value. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So this tells us that there is a 5% chance of sampling a z statistic greater than 1.96, assuming the null hypothesis is correct. Now, we just figured out that we just sampled a z statistic of 2.34, assuming the null hypothesis is correct. So the probability of sampling this, given the null hypothesis is correct, is going to be less than 5%. It is more extreme than this critical z value. It's going to be out here someplace. And because of that, we can reject the null hypothesis. And sorry for jumping around so much in this video. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
It is more extreme than this critical z value. It's going to be out here someplace. And because of that, we can reject the null hypothesis. And sorry for jumping around so much in this video. I had already written a lot, so I just kind of leveraged what I had already written. But since the odds of getting that, assuming the null hypothesis, are less than 5%, and that was our significance level, we can reject the null hypothesis and say that there is a difference. We don't know 100% sure that there is, but statistically we are in favor of the idea that there is a difference between the proportion of men and the proportion of women who are going to vote for the candidate. | Hypothesis test comparing population proportions Probability and Statistics Khan Academy.mp3 |
So there's a 95% chance that the true mean, and let me put this here, this is also the same thing as the mean of the sampling distribution of the sampling mean, is in that interval. And to do that, let me just throw out a few ideas. What is the probability, that if I take a sample and I were to take a mean of that sample, so the probability that a random sample mean is within two standard deviations of the sampling mean of our sample mean. So what is this probability right over here? Let's just look at our actual distribution. So this is our distribution. This right here is our sampling mean. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
So what is this probability right over here? Let's just look at our actual distribution. So this is our distribution. This right here is our sampling mean. Maybe I should do it in blue because that's the number up here. That's the color up here. This is our sampling mean. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
This right here is our sampling mean. Maybe I should do it in blue because that's the number up here. That's the color up here. This is our sampling mean. And so what is the probability that a random sampling mean is going to be in two standard deviations? Well, a random sampling mean is a sample from this distribution. It is a sample from the sampling distribution of the sample mean. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
This is our sampling mean. And so what is the probability that a random sampling mean is going to be in two standard deviations? Well, a random sampling mean is a sample from this distribution. It is a sample from the sampling distribution of the sample mean. So it's literally, what is the probability of finding a sample within two standard deviations of the mean? That's one standard deviation. That's another standard deviation right over there. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
It is a sample from the sampling distribution of the sample mean. So it's literally, what is the probability of finding a sample within two standard deviations of the mean? That's one standard deviation. That's another standard deviation right over there. And in general, if you haven't committed this to memory already, it's not a bad thing to commit to memory, is that if you have a normal distribution, the probability of taking a sample within two standard deviations is 95, and if you want to get a little bit more accurate, it's 95.4%. But you could say it's roughly 95%. And really that's all that matters because we have this little funny language here called reasonably confident, and we have to estimate the standard deviation anyway. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
That's another standard deviation right over there. And in general, if you haven't committed this to memory already, it's not a bad thing to commit to memory, is that if you have a normal distribution, the probability of taking a sample within two standard deviations is 95, and if you want to get a little bit more accurate, it's 95.4%. But you could say it's roughly 95%. And really that's all that matters because we have this little funny language here called reasonably confident, and we have to estimate the standard deviation anyway. In fact, we could say if we want, I could say it's going to be exactly equal to 95.4%. But in general, two standard deviations, 95%, that's what people equate with each other. Now, this statement is the exact same thing as the probability that the sampling mean, or the mean of the sample, not the sample mean, the probability of the mean of the sampling distribution is within two standard deviations of the sampling distribution of x is also going to be the same number, is also going to be equal to 95.4%. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
And really that's all that matters because we have this little funny language here called reasonably confident, and we have to estimate the standard deviation anyway. In fact, we could say if we want, I could say it's going to be exactly equal to 95.4%. But in general, two standard deviations, 95%, that's what people equate with each other. Now, this statement is the exact same thing as the probability that the sampling mean, or the mean of the sample, not the sample mean, the probability of the mean of the sampling distribution is within two standard deviations of the sampling distribution of x is also going to be the same number, is also going to be equal to 95.4%. These are the exact same statement. If x is within two standard deviations of this, then this, then the mean is within two standard deviations of x. These are just two ways of phrasing the same thing. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
Now, this statement is the exact same thing as the probability that the sampling mean, or the mean of the sample, not the sample mean, the probability of the mean of the sampling distribution is within two standard deviations of the sampling distribution of x is also going to be the same number, is also going to be equal to 95.4%. These are the exact same statement. If x is within two standard deviations of this, then this, then the mean is within two standard deviations of x. These are just two ways of phrasing the same thing. Now, we know that the mean of the sampling distribution is the same thing as the mean of the population distribution, which is the same thing as the parameter p, the proportion of people, or the proportion of the population that is a one. This right here is the same thing as the population mean. This statement right here, we can switch this with p. The probability that p is within two standard deviations of the sampling distribution of x is 95.4%. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
These are just two ways of phrasing the same thing. Now, we know that the mean of the sampling distribution is the same thing as the mean of the population distribution, which is the same thing as the parameter p, the proportion of people, or the proportion of the population that is a one. This right here is the same thing as the population mean. This statement right here, we can switch this with p. The probability that p is within two standard deviations of the sampling distribution of x is 95.4%. Now, we don't know what this number right here is, but we have estimated it. Remember, our best estimate of this is the true standard, or it is the true standard deviation of the population divided by 10. We can estimate the true standard deviation of the population with our sampling standard deviation, which was 0.5, 0.5 divided by 10. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
This statement right here, we can switch this with p. The probability that p is within two standard deviations of the sampling distribution of x is 95.4%. Now, we don't know what this number right here is, but we have estimated it. Remember, our best estimate of this is the true standard, or it is the true standard deviation of the population divided by 10. We can estimate the true standard deviation of the population with our sampling standard deviation, which was 0.5, 0.5 divided by 10. Our best estimate of the standard deviation of the sampling distribution of the sample mean is 0.05. Now, we can say, and I'll switch colors, the probability that the parameter p, the proportion of the population saying one, is within two times, remember, our best estimate of this right here is 0.05, of a sample mean that we take is equal to 95.4%. And so, we could say the probability that p is within two times 0.05 is going to be equal to, 2.0 is going to be 0.10, of our mean is equal to 95. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
We can estimate the true standard deviation of the population with our sampling standard deviation, which was 0.5, 0.5 divided by 10. Our best estimate of the standard deviation of the sampling distribution of the sample mean is 0.05. Now, we can say, and I'll switch colors, the probability that the parameter p, the proportion of the population saying one, is within two times, remember, our best estimate of this right here is 0.05, of a sample mean that we take is equal to 95.4%. And so, we could say the probability that p is within two times 0.05 is going to be equal to, 2.0 is going to be 0.10, of our mean is equal to 95. And actually, let me be a little careful here. I can't say the equal now, because over here, if we knew this, if we knew this parameter of the sampling distribution of the sample mean, we could say that it is 95.4%. We don't know it. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
And so, we could say the probability that p is within two times 0.05 is going to be equal to, 2.0 is going to be 0.10, of our mean is equal to 95. And actually, let me be a little careful here. I can't say the equal now, because over here, if we knew this, if we knew this parameter of the sampling distribution of the sample mean, we could say that it is 95.4%. We don't know it. We are just trying to find our best estimator for it. So actually, what I'm going to do here is actually just say is roughly, and just to show that we don't even have that level of accuracy, I'm going to say roughly 95%. We're reasonably confident that it's about 95% because we're using this estimator that came out of our sample. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
We don't know it. We are just trying to find our best estimator for it. So actually, what I'm going to do here is actually just say is roughly, and just to show that we don't even have that level of accuracy, I'm going to say roughly 95%. We're reasonably confident that it's about 95% because we're using this estimator that came out of our sample. And if the sample is really skewed, this is going to be a really weird number. So this is why we just have to be a little bit more exact about what we're doing. But this is a tool for at least saying how good is our result. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
We're reasonably confident that it's about 95% because we're using this estimator that came out of our sample. And if the sample is really skewed, this is going to be a really weird number. So this is why we just have to be a little bit more exact about what we're doing. But this is a tool for at least saying how good is our result. And so, this is going to be about 95%. Or we could say that the probability that p is within 0.10 of our sample mean that we actually got. So what was the sample mean that we actually got? | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
But this is a tool for at least saying how good is our result. And so, this is going to be about 95%. Or we could say that the probability that p is within 0.10 of our sample mean that we actually got. So what was the sample mean that we actually got? It was 0.43. So if we're within 0.1 of 0.43, that means we are within 0.43 plus or minus 0.1 is also, roughly, we're reasonably confident, it's about 95%. And I want to be very clear. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
So what was the sample mean that we actually got? It was 0.43. So if we're within 0.1 of 0.43, that means we are within 0.43 plus or minus 0.1 is also, roughly, we're reasonably confident, it's about 95%. And I want to be very clear. Everything that I started, all the way from pure and brown to yellow to all this magenta, I'm just restating the same thing inside of this. It became a little bit more loosey-goosey once I went from the exact standard deviation of the sampling distribution to an estimator for it. And that's why this is just becoming, I kind of put the squiggly equal signs there to say, we're reasonably confident. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
And I want to be very clear. Everything that I started, all the way from pure and brown to yellow to all this magenta, I'm just restating the same thing inside of this. It became a little bit more loosey-goosey once I went from the exact standard deviation of the sampling distribution to an estimator for it. And that's why this is just becoming, I kind of put the squiggly equal signs there to say, we're reasonably confident. I even got rid of some of the precision. But we just found our interval. An interval that we can be reasonably confident that there's a 95% probability that p is within that is going to be 0.43 plus or minus 0.1. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
And that's why this is just becoming, I kind of put the squiggly equal signs there to say, we're reasonably confident. I even got rid of some of the precision. But we just found our interval. An interval that we can be reasonably confident that there's a 95% probability that p is within that is going to be 0.43 plus or minus 0.1. Or an interval of, we have a confidence interval. We have a 95% confidence interval. And we could say 0.43 minus 0.1. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
An interval that we can be reasonably confident that there's a 95% probability that p is within that is going to be 0.43 plus or minus 0.1. Or an interval of, we have a confidence interval. We have a 95% confidence interval. And we could say 0.43 minus 0.1. Minus 0.1 is 0.33. If we write that as a percent, we could say 33% 2. And if we add the 0.1, 0.43 plus 0.1, we get 53%. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
And we could say 0.43 minus 0.1. Minus 0.1 is 0.33. If we write that as a percent, we could say 33% 2. And if we add the 0.1, 0.43 plus 0.1, we get 53%. 2, 53%. So we are 95% confident. So we're not saying precisely that the probability of the actual proportion is 95%. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
And if we add the 0.1, 0.43 plus 0.1, we get 53%. 2, 53%. So we are 95% confident. So we're not saying precisely that the probability of the actual proportion is 95%. But we're 95% confident that the true proportion is between 33% and 55%. That p is in this range over here. Or another way, and you'll see this in a lot of surveys that have been done, people will say, we did a survey. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
So we're not saying precisely that the probability of the actual proportion is 95%. But we're 95% confident that the true proportion is between 33% and 55%. That p is in this range over here. Or another way, and you'll see this in a lot of surveys that have been done, people will say, we did a survey. And we got 43% will vote for number one. And number one in this case is candidate B. For candidate B. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
Or another way, and you'll see this in a lot of surveys that have been done, people will say, we did a survey. And we got 43% will vote for number one. And number one in this case is candidate B. For candidate B. And then the other side, since everyone else voted for candidate A, 57% will vote for A. And then they're going to put a margin of error. And you'll see this in any survey that you see on TV. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
For candidate B. And then the other side, since everyone else voted for candidate A, 57% will vote for A. And then they're going to put a margin of error. And you'll see this in any survey that you see on TV. They'll put a margin of error. And the margin of error is just another way of describing this confidence interval. And they'll say that the margin of error in this case is 10%. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
And you'll see this in any survey that you see on TV. They'll put a margin of error. And the margin of error is just another way of describing this confidence interval. And they'll say that the margin of error in this case is 10%. Which means that there's a 95% confidence interval if you go plus or minus 10% from that value right over there. And I really want to emphasize, you can't say with certainty that there is a 95% chance that the true result will be within 10% of this. Because we had to estimate the standard deviation of the sampling mean. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
And they'll say that the margin of error in this case is 10%. Which means that there's a 95% confidence interval if you go plus or minus 10% from that value right over there. And I really want to emphasize, you can't say with certainty that there is a 95% chance that the true result will be within 10% of this. Because we had to estimate the standard deviation of the sampling mean. But this is the best measure we can with the information you have. If you're going to do a survey of 100 people, this is the best kind of confidence that we can get. And this number is actually fairly big. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
Because we had to estimate the standard deviation of the sampling mean. But this is the best measure we can with the information you have. If you're going to do a survey of 100 people, this is the best kind of confidence that we can get. And this number is actually fairly big. So if you were to look at this, you would say, roughly, there's a 95% chance that the true value of this number is between 33% and 53%. So there's actually still a chance that candidate B can win, even though only 43% of your 100 are going to vote for him. If you wanted to make it a little bit more precise, you would want to take more samples. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
And this number is actually fairly big. So if you were to look at this, you would say, roughly, there's a 95% chance that the true value of this number is between 33% and 53%. So there's actually still a chance that candidate B can win, even though only 43% of your 100 are going to vote for him. If you wanted to make it a little bit more precise, you would want to take more samples. You can imagine, instead of taking 100 samples, instead of n being 100, if you made n equal 1,000, then you would take this number over here and divide by the square root of 1,000 instead of the square root of 100. So you'd be dividing by 33 or whatever. And so then your margin of the number, the size of the standard deviation of your sampling distribution, will go down. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
If you wanted to make it a little bit more precise, you would want to take more samples. You can imagine, instead of taking 100 samples, instead of n being 100, if you made n equal 1,000, then you would take this number over here and divide by the square root of 1,000 instead of the square root of 100. So you'd be dividing by 33 or whatever. And so then your margin of the number, the size of the standard deviation of your sampling distribution, will go down. And so the distance of two standard deviations will be a smaller number. And so then you will have a smaller margin of error. And maybe you want to get the margin of error small enough so that you can figure out decisively who's going to win the election. | Margin of error 2 Inferential statistics Probability and Statistics Khan Academy.mp3 |
And these, well, I was going to say that they tend to be integers, but they don't always have to be integers. You have discrete random, and so finite meaning you can't have an infinite number of values for a discrete random variable. Well, then we have the continuous, which can take on an infinite number. And the example I gave for continuous is, let's say, random variable X. And people do tend to use, let me change it a little bit, just so you can see, it can be something other than an X. Let's say I have the random variable capital Y. They do tend to be capital letters. | Probability density functions Probability and Statistics Khan Academy.mp3 |
And the example I gave for continuous is, let's say, random variable X. And people do tend to use, let me change it a little bit, just so you can see, it can be something other than an X. Let's say I have the random variable capital Y. They do tend to be capital letters. Is equal to the exact amount of rain tomorrow. And I say rain because I'm in Northern California. It's actually raining quite hard right now. | Probability density functions Probability and Statistics Khan Academy.mp3 |
They do tend to be capital letters. Is equal to the exact amount of rain tomorrow. And I say rain because I'm in Northern California. It's actually raining quite hard right now. Which, we're short right now, so that's a positive. We've been having a drought, so it's a good thing. But the exact amount of rain tomorrow. | Probability density functions Probability and Statistics Khan Academy.mp3 |
It's actually raining quite hard right now. Which, we're short right now, so that's a positive. We've been having a drought, so it's a good thing. But the exact amount of rain tomorrow. And let's say, I don't know what the actual probability distribution function for this is, but I'll draw one. And then we'll interpret it. Just so you can kind of think about continuous random variables. | Probability density functions Probability and Statistics Khan Academy.mp3 |
But the exact amount of rain tomorrow. And let's say, I don't know what the actual probability distribution function for this is, but I'll draw one. And then we'll interpret it. Just so you can kind of think about continuous random variables. Let me draw its probability distribution, or they call it its probability density function. Let me draw it like this. And let's say that there is, let me think, it looks something like this. | Probability density functions Probability and Statistics Khan Academy.mp3 |
Just so you can kind of think about continuous random variables. Let me draw its probability distribution, or they call it its probability density function. Let me draw it like this. And let's say that there is, let me think, it looks something like this. All right. And then, I don't know what this height is. So this, the x-axis here is the amount of rain. | Probability density functions Probability and Statistics Khan Academy.mp3 |
And let's say that there is, let me think, it looks something like this. All right. And then, I don't know what this height is. So this, the x-axis here is the amount of rain. Where this is 0 inches, this is 1 inch, this is 2 inches, this is 3 inches, 4 inches. And then this is some height. Let's say it peaks out here at, I don't know, let's say this is 0.5. | Probability density functions Probability and Statistics Khan Academy.mp3 |
So this, the x-axis here is the amount of rain. Where this is 0 inches, this is 1 inch, this is 2 inches, this is 3 inches, 4 inches. And then this is some height. Let's say it peaks out here at, I don't know, let's say this is 0.5. So the way to think about it, if you were to look at this and I were to ask you, what is the probability that y, because that's our random variable now, that y is exactly equal to 2 inches? That y is exactly equal to 2 inches. What's the probability of that happening? | Probability density functions Probability and Statistics Khan Academy.mp3 |
Let's say it peaks out here at, I don't know, let's say this is 0.5. So the way to think about it, if you were to look at this and I were to ask you, what is the probability that y, because that's our random variable now, that y is exactly equal to 2 inches? That y is exactly equal to 2 inches. What's the probability of that happening? Well, based on how we thought about the probability distribution functions for the discrete random variable, you'd say, OK, let's see, 2 inches, that's the case we care about right now. Let me go up here. Say, OK, it looks like it's about 0.5. | Probability density functions Probability and Statistics Khan Academy.mp3 |
What's the probability of that happening? Well, based on how we thought about the probability distribution functions for the discrete random variable, you'd say, OK, let's see, 2 inches, that's the case we care about right now. Let me go up here. Say, OK, it looks like it's about 0.5. And you say, well, I don't know, is it a 0.5 chance? And I would say, no, it is not a 0.5 chance. And before we even think about how we would interpret it visually, let's just think about it logically. | Probability density functions Probability and Statistics Khan Academy.mp3 |
Say, OK, it looks like it's about 0.5. And you say, well, I don't know, is it a 0.5 chance? And I would say, no, it is not a 0.5 chance. And before we even think about how we would interpret it visually, let's just think about it logically. What is the probability that tomorrow we have exactly 2 inches of rain? Not 2.01 inches of rain, not 1.99 inches of rain, not 1.9999 inches of rain, not 2.00001 inches of rain, exactly 2 inches of rain. There's not a single extra atom water molecule above the 2 inch mark and not a single water molecule below the 2 inch mark. | Probability density functions Probability and Statistics Khan Academy.mp3 |
And before we even think about how we would interpret it visually, let's just think about it logically. What is the probability that tomorrow we have exactly 2 inches of rain? Not 2.01 inches of rain, not 1.99 inches of rain, not 1.9999 inches of rain, not 2.00001 inches of rain, exactly 2 inches of rain. There's not a single extra atom water molecule above the 2 inch mark and not a single water molecule below the 2 inch mark. It's essentially 0, right? It might not be obvious to you because you probably heard, oh, we had 2 inches of rain last night. But think about the exactly 2 inches, right? | Probability density functions Probability and Statistics Khan Academy.mp3 |
There's not a single extra atom water molecule above the 2 inch mark and not a single water molecule below the 2 inch mark. It's essentially 0, right? It might not be obvious to you because you probably heard, oh, we had 2 inches of rain last night. But think about the exactly 2 inches, right? Normally if it's like 2.01, people will say that's 2. But we're saying, no, this does not count. It can't be 2 inches. | Probability density functions Probability and Statistics Khan Academy.mp3 |
But think about the exactly 2 inches, right? Normally if it's like 2.01, people will say that's 2. But we're saying, no, this does not count. It can't be 2 inches. We want exactly 2. 1.99 does not count. Normally, I mean, our measurements, we don't even have tools that can tell us whether it is exactly 2 inches, right? | Probability density functions Probability and Statistics Khan Academy.mp3 |
It can't be 2 inches. We want exactly 2. 1.99 does not count. Normally, I mean, our measurements, we don't even have tools that can tell us whether it is exactly 2 inches, right? No ruler you can even say is exactly 2 inches long. At some point, just the way we manufacture things, there's going to be an extra atom on it here or there. So the odds of actually anything being exactly a measurement to the exact infinite decimal point is actually 0. | Probability density functions Probability and Statistics Khan Academy.mp3 |
Normally, I mean, our measurements, we don't even have tools that can tell us whether it is exactly 2 inches, right? No ruler you can even say is exactly 2 inches long. At some point, just the way we manufacture things, there's going to be an extra atom on it here or there. So the odds of actually anything being exactly a measurement to the exact infinite decimal point is actually 0. The way you would think about a continuous random variable, you could say, what is the probability that y is almost 2? So if we said that the absolute value of y minus 2 is less than some tolerance, is less than, I don't know, 0.1. And if that doesn't make sense to you, this is essentially just saying that what is the probability that y is greater than 1.9 and less than 2.1? | Probability density functions Probability and Statistics Khan Academy.mp3 |
So the odds of actually anything being exactly a measurement to the exact infinite decimal point is actually 0. The way you would think about a continuous random variable, you could say, what is the probability that y is almost 2? So if we said that the absolute value of y minus 2 is less than some tolerance, is less than, I don't know, 0.1. And if that doesn't make sense to you, this is essentially just saying that what is the probability that y is greater than 1.9 and less than 2.1? These two statements are equivalent. I'll let you think about it a little bit. But now this starts to make a little bit sense. | Probability density functions Probability and Statistics Khan Academy.mp3 |
And if that doesn't make sense to you, this is essentially just saying that what is the probability that y is greater than 1.9 and less than 2.1? These two statements are equivalent. I'll let you think about it a little bit. But now this starts to make a little bit sense. Now we have an interval here. So we want all y's between 1.9 and 2.1. So we are now talking about this whole area. | Probability density functions Probability and Statistics Khan Academy.mp3 |
But now this starts to make a little bit sense. Now we have an interval here. So we want all y's between 1.9 and 2.1. So we are now talking about this whole area. And area is key. So if you want to know the probability of this occurring, you actually want the area under this curve from this point to this point. And for those of you who have studied your calculus, that would essentially be the definite integral of this probability density function from this point to this point. | Probability density functions Probability and Statistics Khan Academy.mp3 |
So we are now talking about this whole area. And area is key. So if you want to know the probability of this occurring, you actually want the area under this curve from this point to this point. And for those of you who have studied your calculus, that would essentially be the definite integral of this probability density function from this point to this point. So from, let me see, I've run out of space down here. So let's say if this graph, let me draw it in a different color, if this line was defined by, I don't know, I'll call it f of x. I could call it p of x or something. The probability of this happening would be equal to the integral, for those of you who have studied calculus, from 1.9 to 2.1 of f of x dx, assuming this is the x-axis. | Probability density functions Probability and Statistics Khan Academy.mp3 |
And for those of you who have studied your calculus, that would essentially be the definite integral of this probability density function from this point to this point. So from, let me see, I've run out of space down here. So let's say if this graph, let me draw it in a different color, if this line was defined by, I don't know, I'll call it f of x. I could call it p of x or something. The probability of this happening would be equal to the integral, for those of you who have studied calculus, from 1.9 to 2.1 of f of x dx, assuming this is the x-axis. So it's a very important thing to realize. Because when a random variable can take on an infinite number of values, or it can take on any value between an interval, to get an exact value, to get exactly 1.999, the probability is actually 0. It's like asking you, what is the area under a curve on just this line? | Probability density functions Probability and Statistics Khan Academy.mp3 |
The probability of this happening would be equal to the integral, for those of you who have studied calculus, from 1.9 to 2.1 of f of x dx, assuming this is the x-axis. So it's a very important thing to realize. Because when a random variable can take on an infinite number of values, or it can take on any value between an interval, to get an exact value, to get exactly 1.999, the probability is actually 0. It's like asking you, what is the area under a curve on just this line? Or even more specifically, it's like asking you, what's the area of a line? An area of a line, if you were to just draw a line, you'd say, well, area is height times base. Well, the height has some dimension, but the base, what's the width of a line? | Probability density functions Probability and Statistics Khan Academy.mp3 |
It's like asking you, what is the area under a curve on just this line? Or even more specifically, it's like asking you, what's the area of a line? An area of a line, if you were to just draw a line, you'd say, well, area is height times base. Well, the height has some dimension, but the base, what's the width of a line? As far as the way we've defined a line, a line has no width, and therefore no area. And it should make intuitive sense that you cannot, the probability of a very super exact thing happening is pretty much 0. That you really have to say, OK, what's the probability that we get close to 2? | Probability density functions Probability and Statistics Khan Academy.mp3 |
Well, the height has some dimension, but the base, what's the width of a line? As far as the way we've defined a line, a line has no width, and therefore no area. And it should make intuitive sense that you cannot, the probability of a very super exact thing happening is pretty much 0. That you really have to say, OK, what's the probability that we get close to 2? And then you can define an area. And if you said, oh, what's the probability that we get someplace between 1 and 3 inches of rain, then of course the probability is much higher. It would be all of this kind of stuff. | Probability density functions Probability and Statistics Khan Academy.mp3 |
That you really have to say, OK, what's the probability that we get close to 2? And then you can define an area. And if you said, oh, what's the probability that we get someplace between 1 and 3 inches of rain, then of course the probability is much higher. It would be all of this kind of stuff. You could also say, what's the probability we have less than 0.1 inches of rain? Then you would go here, and you would calculate, if this was 0.1, you would calculate this area. And you could say, what's the probability that we have more than 4 inches of rain tomorrow? | Probability density functions Probability and Statistics Khan Academy.mp3 |
It would be all of this kind of stuff. You could also say, what's the probability we have less than 0.1 inches of rain? Then you would go here, and you would calculate, if this was 0.1, you would calculate this area. And you could say, what's the probability that we have more than 4 inches of rain tomorrow? Then you would start here, and you would calculate the area on the curve all the way to infinity, if the curve has area all the way to infinity. And hopefully that's not an infinite number, right? Then your probability won't make any sense. | Probability density functions Probability and Statistics Khan Academy.mp3 |
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