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Yellow and getting a cube. And so what we've just done here, and you could play with the numbers, the numbers I just used as an example right here to make things a little bit concrete, but you can see this is a generalizable thing. If we have the probability of one condition or another condition, so let me rewrite it. The probability, and I'll just write a little bit more generally here. This gives us an interesting idea. The probability of getting one condition of an object being a member of set A or a member of set B is equal to the probability that it is a member of set A plus the probability that it is a member of set B minus the probability that it is a member of both. And this is a really useful result. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
The probability, and I'll just write a little bit more generally here. This gives us an interesting idea. The probability of getting one condition of an object being a member of set A or a member of set B is equal to the probability that it is a member of set A plus the probability that it is a member of set B minus the probability that it is a member of both. And this is a really useful result. I think sometimes it's called the addition rule of probability, but I wanted to show you it's a completely common sense thing. The reason why you can't just add these two probabilities is because they might have some overlap. There's a probability of getting both. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
And this is a really useful result. I think sometimes it's called the addition rule of probability, but I wanted to show you it's a completely common sense thing. The reason why you can't just add these two probabilities is because they might have some overlap. There's a probability of getting both. And if you just added both of these, you would be double counting that overlap, which we've already seen earlier in this video. So you have to subtract one version of the overlap out so you are not double counting it. And I'll throw one other idea out. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
There's a probability of getting both. And if you just added both of these, you would be double counting that overlap, which we've already seen earlier in this video. So you have to subtract one version of the overlap out so you are not double counting it. And I'll throw one other idea out. Sometimes you have possibilities that have no overlap. So let's say this is a set of all possibilities. And let's say this is a set that meets condition A. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
And I'll throw one other idea out. Sometimes you have possibilities that have no overlap. So let's say this is a set of all possibilities. And let's say this is a set that meets condition A. Let's say that this is the set that meets condition B. So in this situation, there is no overlap. Nothing is a member of both set A and B. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
And let's say this is a set that meets condition A. Let's say that this is the set that meets condition B. So in this situation, there is no overlap. Nothing is a member of both set A and B. So in this situation, the probability of A and B is 0. There is no overlap. And these type of conditions, or these two events, are called mutually exclusive. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
Nothing is a member of both set A and B. So in this situation, the probability of A and B is 0. There is no overlap. And these type of conditions, or these two events, are called mutually exclusive. So if events are mutually exclusive, that means that they both cannot happen at the same time. There's no event that meets both of these conditions. And if things are mutually exclusive, then you can say the probability of A or B is a probability of A plus B, because this thing is 0. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
And these type of conditions, or these two events, are called mutually exclusive. So if events are mutually exclusive, that means that they both cannot happen at the same time. There's no event that meets both of these conditions. And if things are mutually exclusive, then you can say the probability of A or B is a probability of A plus B, because this thing is 0. But if things are not mutually exclusive, you would have to subtract out the overlap. Probably the best way to think about it is to just always realize that you have to subtract out the overlap. And obviously, if something is mutually exclusive, the probability of getting A and B is going to be 0. | Addition rule for probability Probability and Statistics Khan Academy.mp3 |
Maybe it's the mean of a population, the mean height of all the people in the city. And we've determined that it's unpractical, or there's no way for us to know the true population parameter, but we could try to estimate it by taking a sample size. So we take n samples, and then we calculate a statistic based on that. We've also seen that not only can we calculate the statistic which is trying to estimate this parameter, but we can construct a confidence interval about that statistic based on some confidence level. And so that confidence interval would look something like this. It would be the value of the statistic that we have just calculated, plus or minus some margin of error. And so we'll often say this critical value, z, and this will be based on the number of standard deviations we wanna go above and below that statistic. | Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3 |
We've also seen that not only can we calculate the statistic which is trying to estimate this parameter, but we can construct a confidence interval about that statistic based on some confidence level. And so that confidence interval would look something like this. It would be the value of the statistic that we have just calculated, plus or minus some margin of error. And so we'll often say this critical value, z, and this will be based on the number of standard deviations we wanna go above and below that statistic. And so then we'll multiply that times the standard deviation of the sampling distribution for that statistic. Now what we'll see is we often don't know this. To know this, you oftentimes even need to know this parameter. | Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3 |
And so we'll often say this critical value, z, and this will be based on the number of standard deviations we wanna go above and below that statistic. And so then we'll multiply that times the standard deviation of the sampling distribution for that statistic. Now what we'll see is we often don't know this. To know this, you oftentimes even need to know this parameter. For example, in the situation where the parameter that we're trying to estimate and construct confidence intervals for is say the population proportion, what percentage of the population supports a certain candidate? Well, in that world, the statistic is the sample proportion. So we would have the sample proportion, plus or minus z star, times, well, we can't calculate this unless we know the population proportion. | Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3 |
To know this, you oftentimes even need to know this parameter. For example, in the situation where the parameter that we're trying to estimate and construct confidence intervals for is say the population proportion, what percentage of the population supports a certain candidate? Well, in that world, the statistic is the sample proportion. So we would have the sample proportion, plus or minus z star, times, well, we can't calculate this unless we know the population proportion. So instead, we estimate this with the standard error of the statistic, which in this case is p hat times one minus p hat, the sample proportion times one minus the sample proportion over our sample size. If the parameter we're trying to estimate is the population mean, then our statistic is going to be the sample mean. So in that scenario, we are going to be looking at our statistic is our sample mean, plus or minus z star. | Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3 |
So we would have the sample proportion, plus or minus z star, times, well, we can't calculate this unless we know the population proportion. So instead, we estimate this with the standard error of the statistic, which in this case is p hat times one minus p hat, the sample proportion times one minus the sample proportion over our sample size. If the parameter we're trying to estimate is the population mean, then our statistic is going to be the sample mean. So in that scenario, we are going to be looking at our statistic is our sample mean, plus or minus z star. Now if we knew the standard deviation of this population, we would know what the standard deviation of the sampling distribution of our statistic is. It would be equal to the standard deviation of our population times the square root of our sample size. But we often will not know this. | Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3 |
So in that scenario, we are going to be looking at our statistic is our sample mean, plus or minus z star. Now if we knew the standard deviation of this population, we would know what the standard deviation of the sampling distribution of our statistic is. It would be equal to the standard deviation of our population times the square root of our sample size. But we often will not know this. In fact, it's very unusual to know this. And so sometimes you will say, okay, if we don't know this, let's just figure out the sample standard deviation of our sample here. So instead, we'll say, okay, let's take our sample mean, plus or minus z star, times the sample standard deviation of our sample, which we can calculate, divided by the square root of n. Now this might seem pretty good if we're trying to construct a confidence interval for our sample, for our mean. | Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3 |
But we often will not know this. In fact, it's very unusual to know this. And so sometimes you will say, okay, if we don't know this, let's just figure out the sample standard deviation of our sample here. So instead, we'll say, okay, let's take our sample mean, plus or minus z star, times the sample standard deviation of our sample, which we can calculate, divided by the square root of n. Now this might seem pretty good if we're trying to construct a confidence interval for our sample, for our mean. But it turns out that this is not, not so good. Because it turns out that this right over here is going to actually underestimate the actual interval, the true margin of error you need for your confidence level. And so that's why statisticians have invented another statistic. | Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3 |
So instead, we'll say, okay, let's take our sample mean, plus or minus z star, times the sample standard deviation of our sample, which we can calculate, divided by the square root of n. Now this might seem pretty good if we're trying to construct a confidence interval for our sample, for our mean. But it turns out that this is not, not so good. Because it turns out that this right over here is going to actually underestimate the actual interval, the true margin of error you need for your confidence level. And so that's why statisticians have invented another statistic. Instead of using z, they call it t. Instead of using a z table, they use a t table. And we're going to see this in future videos. And so if you are actually trying to construct a confidence interval for a sample mean, and you don't know the true standard deviation of your population, which is normally the case, instead of doing this, what we're going to do is, we're gonna take our sample mean, plus or minus, and our critical value, we'll call that t star, times our sample standard deviation, which we can calculate, divided by the square root of n. And so the real functional difference is that this actually is going to give us the confidence interval that actually has the level of confidence that we want. | Introduction to t statistics Confidence intervals AP Statistics Khan Academy.mp3 |
For a senior project, Richard is researching how much money a college graduate can expect to earn based on his or her major. He finds the following interesting facts. Basketball superstar Michael Jordan was a geology major at the University of North Carolina. There were only three civil engineering majors from the University of Montana. They all took the exact same job at the same company, earning the same salary. Of the 35 finance majors from Wesleyan University, 32 got high-paying consulting jobs, and the other three were unemployed. For geology majors from the University of North Carolina, the median income will likely be, and we have some options here, less than, equal to, or greater than the mean. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
There were only three civil engineering majors from the University of Montana. They all took the exact same job at the same company, earning the same salary. Of the 35 finance majors from Wesleyan University, 32 got high-paying consulting jobs, and the other three were unemployed. For geology majors from the University of North Carolina, the median income will likely be, and we have some options here, less than, equal to, or greater than the mean. And then we have to answer the same question for civil engineering majors from Montana. The median income, oh actually, this is, we're both about median. The median income will be, and we compare it against the mean, and then for finance majors from Wesleyan, we're gonna compare the median income to the mean. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
For geology majors from the University of North Carolina, the median income will likely be, and we have some options here, less than, equal to, or greater than the mean. And then we have to answer the same question for civil engineering majors from Montana. The median income, oh actually, this is, we're both about median. The median income will be, and we compare it against the mean, and then for finance majors from Wesleyan, we're gonna compare the median income to the mean. So to visualize this a little bit more, I've copy and pasted this exact same problem onto my scratch pad, so here it is. I can now write on this. So let's think about each of these. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
The median income will be, and we compare it against the mean, and then for finance majors from Wesleyan, we're gonna compare the median income to the mean. So to visualize this a little bit more, I've copy and pasted this exact same problem onto my scratch pad, so here it is. I can now write on this. So let's think about each of these. For geology majors from UNC, the median income will likely be, how will that compare to the mean? Well, what do they tell us about UNC? They tell us that Michael Jordan was a geology major at the University of North Carolina. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
So let's think about each of these. For geology majors from UNC, the median income will likely be, how will that compare to the mean? Well, what do they tell us about UNC? They tell us that Michael Jordan was a geology major at the University of North Carolina. So what will the distribution of salaries probably look like? So if we're thinking about the University of North Carolina, it probably looks something like this. And I'm gonna do a very rough, a very rough distribution right over here. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
They tell us that Michael Jordan was a geology major at the University of North Carolina. So what will the distribution of salaries probably look like? So if we're thinking about the University of North Carolina, it probably looks something like this. And I'm gonna do a very rough, a very rough distribution right over here. And let's say that this salary, this would be a salary of zero, and let's say that this is a salary of, I don't know, let me put a salary of 50K here. I'll do this in thousands. Let's say this is 100,000 right over here. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
And I'm gonna do a very rough, a very rough distribution right over here. And let's say that this salary, this would be a salary of zero, and let's say that this is a salary of, I don't know, let me put a salary of 50K here. I'll do this in thousands. Let's say this is 100,000 right over here. And then you have Michael Jordan, who is, actually I'll do a little gap here because he's so far up. I don't know exactly what he was making, but it was definitely in the tens of millions of dollars a year. So Michael Jordan is way, way, way, way up here. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
Let's say this is 100,000 right over here. And then you have Michael Jordan, who is, actually I'll do a little gap here because he's so far up. I don't know exactly what he was making, but it was definitely in the tens of millions of dollars a year. So Michael Jordan is way, way, way, way up here. So if you were to make a histogram or a plot of all of the salaries, you could say, okay, well, you know, maybe we have, if you put all of the folks from geology majors at University of North Carolina, well, there's probably, especially right when they graduated, there's probably, you know, one, two, three, I could keep doing it. A bunch of people, maybe making 50K, maybe some people making a little bit more, maybe some people up here, maybe some people there, some people there, some people there, like there. Maybe someone's making 100K, maybe someone, so maybe it's a couple of people up there. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
So Michael Jordan is way, way, way, way up here. So if you were to make a histogram or a plot of all of the salaries, you could say, okay, well, you know, maybe we have, if you put all of the folks from geology majors at University of North Carolina, well, there's probably, especially right when they graduated, there's probably, you know, one, two, three, I could keep doing it. A bunch of people, maybe making 50K, maybe some people making a little bit more, maybe some people up here, maybe some people there, some people there, some people there, like there. Maybe someone's making 100K, maybe someone, so maybe it's a couple of people up there. Maybe someone isn't making anything. Maybe they weren't able to find a job. And then, of course, you have Michael Jordan up here making $10 million or $20 million or something like that. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
Maybe someone's making 100K, maybe someone, so maybe it's a couple of people up there. Maybe someone isn't making anything. Maybe they weren't able to find a job. And then, of course, you have Michael Jordan up here making $10 million or $20 million or something like that. So when you have a situation like this, where you have this outlier of Michael Jordan, it's going to put, one way I think about it, it kind of tugs on the mean. It wouldn't affect the median, because remember, the median is the middle value. So it doesn't matter how high this number is. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
And then, of course, you have Michael Jordan up here making $10 million or $20 million or something like that. So when you have a situation like this, where you have this outlier of Michael Jordan, it's going to put, one way I think about it, it kind of tugs on the mean. It wouldn't affect the median, because remember, the median is the middle value. So it doesn't matter how high this number is. You could make this a trillion dollars. It's not going to change what the middle value is. The middle value is still going to be the same middle value. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
So it doesn't matter how high this number is. You could make this a trillion dollars. It's not going to change what the middle value is. The middle value is still going to be the same middle value. You could move this anywhere around in this range. It's just not going to change the median. But the mean will change. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
The middle value is still going to be the same middle value. You could move this anywhere around in this range. It's just not going to change the median. But the mean will change. If this becomes really, really astronomically high, it will distort the actual mean here. Actually, it would distort it a good bit. So for geology majors from UNC, the median income is going to be lower than the mean, because Michael Jordan is pulling the mean up. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
But the mean will change. If this becomes really, really astronomically high, it will distort the actual mean here. Actually, it would distort it a good bit. So for geology majors from UNC, the median income is going to be lower than the mean, because Michael Jordan is pulling the mean up. So let me fill that in. So for geology majors from UNC, the median will be less than the mean. Now let's think about the other ones. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
So for geology majors from UNC, the median income is going to be lower than the mean, because Michael Jordan is pulling the mean up. So let me fill that in. So for geology majors from UNC, the median will be less than the mean. Now let's think about the other ones. For civil engineering majors from Montana, the median income will be blank the mean. Well, they tell us there were only three civil engineering majors from the University of Montana. They all took the exact same job at the same company, earning the same salary. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
Now let's think about the other ones. For civil engineering majors from Montana, the median income will be blank the mean. Well, they tell us there were only three civil engineering majors from the University of Montana. They all took the exact same job at the same company, earning the same salary. So let's say all three of them earned $50,000. Let's say that's their salary. So if you were to calculate the mean, it would be 50 plus 50 plus 50 over three, which of course is 50. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
They all took the exact same job at the same company, earning the same salary. So let's say all three of them earned $50,000. Let's say that's their salary. So if you were to calculate the mean, it would be 50 plus 50 plus 50 over three, which of course is 50. That would be the mean. If you wanted the median, you list the salaries in order, and then you take the middle one. Well, the middle one is 50. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
So if you were to calculate the mean, it would be 50 plus 50 plus 50 over three, which of course is 50. That would be the mean. If you wanted the median, you list the salaries in order, and then you take the middle one. Well, the middle one is 50. So in this case, the median is equal to the mean. So let's fill that in. Median is equal to the mean. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
Well, the middle one is 50. So in this case, the median is equal to the mean. So let's fill that in. Median is equal to the mean. And then finally, let me go back to my scratch pad. Whoops, let me go back to my scratch pad here. For finance majors from Wesleyan, the median income will be blank the mean. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
Median is equal to the mean. And then finally, let me go back to my scratch pad. Whoops, let me go back to my scratch pad here. For finance majors from Wesleyan, the median income will be blank the mean. So let's think about this distribution here. So here, we have out of the 35, 32 got high-paying consulting jobs. So let's say that they were making six figures. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
For finance majors from Wesleyan, the median income will be blank the mean. So let's think about this distribution here. So here, we have out of the 35, 32 got high-paying consulting jobs. So let's say that they were making six figures. So the distribution might look something like this, where if this is zero, and let's say this is 50K, and let's say that this right over here is $100,000 a year. So 32 got high-paying consulting jobs. You might have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
So let's say that they were making six figures. So the distribution might look something like this, where if this is zero, and let's say this is 50K, and let's say that this right over here is $100,000 a year. So 32 got high-paying consulting jobs. You might have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32. So the distribution for the people who got the jobs might look something like that. But there were three people who were unemployed. So let's say they got no income. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
You might have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32. So the distribution for the people who got the jobs might look something like that. But there were three people who were unemployed. So let's say they got no income. So you have one, two, three. So this is now, you have three outliers, like the Michael Jordan situation, but instead of them being very high, they are very low. So they're going to pull the mean lower. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
So let's say they got no income. So you have one, two, three. So this is now, you have three outliers, like the Michael Jordan situation, but instead of them being very high, they are very low. So they're going to pull the mean lower. They're not going to, if these were zero, or if these were 50, or if these were over here, they're not going to affect the median. The middle number is still going to be the same. But they are going to pull down the mean. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
So they're going to pull the mean lower. They're not going to, if these were zero, or if these were 50, or if these were over here, they're not going to affect the median. The middle number is still going to be the same. But they are going to pull down the mean. So here, I would say that the median income will likely be higher, will likely be greater than the mean, because the mean is going to get pulled down by these outliers, these three people not making anything. So let's fill that out. For finance majors from Wesleyan, the median income will likely be greater than the mean. | Means and medians of different distributions Probability and Statistics Khan Academy.mp3 |
I can assume that's pronounced Ted-eff. And what it allows us to do is give us an intuition as to why we divide by n minus one when we calculate our sample variance and why that gives us an unbiased estimate of population variance. So the way this starts off, and I encourage you to go try this out yourself, is that you can construct a distribution. It says build a population by clicking in the blue area. So here, we're actually creating a population. So we're creating, every time I click, it increases the population size. So let me just, and I'm just randomly doing this, and I encourage you to go onto this scratch pad. | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
It says build a population by clicking in the blue area. So here, we're actually creating a population. So we're creating, every time I click, it increases the population size. So let me just, and I'm just randomly doing this, and I encourage you to go onto this scratch pad. It's on the Khan Academy Computer Science, and try to do it yourself. So here, we are, I can stop at some point. So I've constructed a population. | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
So let me just, and I'm just randomly doing this, and I encourage you to go onto this scratch pad. It's on the Khan Academy Computer Science, and try to do it yourself. So here, we are, I can stop at some point. So I've constructed a population. I can throw out some random points up here. So this is our population. And as you saw while I was doing that, I was calculating parameters for the population. | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
So I've constructed a population. I can throw out some random points up here. So this is our population. And as you saw while I was doing that, I was calculating parameters for the population. It was calculating the population mean at 204.09, and also the population standard deviation, which is derived from the population variance. This is the square root of the population variance, and it's at 63.8. It was also plotting the population variance down here. | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
And as you saw while I was doing that, I was calculating parameters for the population. It was calculating the population mean at 204.09, and also the population standard deviation, which is derived from the population variance. This is the square root of the population variance, and it's at 63.8. It was also plotting the population variance down here. You see it's 63.8, which is the standard deviation. And it's a little harder to see, but it says squared. These are these numbers squared. | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
It was also plotting the population variance down here. You see it's 63.8, which is the standard deviation. And it's a little harder to see, but it says squared. These are these numbers squared. So this is essentially 63.8 is the population, 63.8 squared is the population variance. So that's interesting by itself, but it really doesn't tell us a lot so far about why we divide by n minus one. And this is the interesting part. | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
These are these numbers squared. So this is essentially 63.8 is the population, 63.8 squared is the population variance. So that's interesting by itself, but it really doesn't tell us a lot so far about why we divide by n minus one. And this is the interesting part. We can now start to take samples, and we can decide what sample size we wanna do. I'll start with really small samples, so the smallest possible sample that makes any sense. So I'm gonna start with really small samples. | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
And this is the interesting part. We can now start to take samples, and we can decide what sample size we wanna do. I'll start with really small samples, so the smallest possible sample that makes any sense. So I'm gonna start with really small samples. And what they're going to do, what the simulation is going to do, is every time I take a sample, it's going to calculate the variance. So the numerator is going to be the sum of each of my data points in my sample minus my sample mean, and I'm gonna square it. And then it's going to divide it by n plus a. | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
So I'm gonna start with really small samples. And what they're going to do, what the simulation is going to do, is every time I take a sample, it's going to calculate the variance. So the numerator is going to be the sum of each of my data points in my sample minus my sample mean, and I'm gonna square it. And then it's going to divide it by n plus a. So, and it's going to vary a. It's going to divide it by anywhere between n plus negative three, so n minus three, all the way to n plus a. And we're gonna do it many, many, many, many times. | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
And then it's going to divide it by n plus a. So, and it's going to vary a. It's going to divide it by anywhere between n plus negative three, so n minus three, all the way to n plus a. And we're gonna do it many, many, many, many times. We're gonna essentially take the mean of those variances for any a, and figure out which gives us the best estimate. So if I just generate one sample right over there, well, we see, when we, when our, we see kind of this curve. When we have high values of a, we are underestimating. | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
And we're gonna do it many, many, many, many times. We're gonna essentially take the mean of those variances for any a, and figure out which gives us the best estimate. So if I just generate one sample right over there, well, we see, when we, when our, we see kind of this curve. When we have high values of a, we are underestimating. When we have lower values of a, we are overestimating the population variance. But that was just for one, that was just for one sample, not really that meaningful. It's one sample of size two. | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
When we have high values of a, we are underestimating. When we have lower values of a, we are overestimating the population variance. But that was just for one, that was just for one sample, not really that meaningful. It's one sample of size two. Let's generate a bunch of samples, and then average them over many of them. And you see, when you look at many, many, many, many, many samples, something interesting is happening. When you look at the mean of those samples, when you average together those curves from all of those samples, you see that our best estimate is when a is pretty close to negative one, is when this is n plus negative one, or n minus one. | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
It's one sample of size two. Let's generate a bunch of samples, and then average them over many of them. And you see, when you look at many, many, many, many, many samples, something interesting is happening. When you look at the mean of those samples, when you average together those curves from all of those samples, you see that our best estimate is when a is pretty close to negative one, is when this is n plus negative one, or n minus one. Anything less than negative one, if we did negative n minus 1.05, or n minus 1.5, we start overestimating the variance. Anything less than negative one, so if we have n plus zero, if we divide by n, or if we have n plus.05, or whatever it might be, we start underestimating, we start underestimating the population variance. And you can do this for samples of different sizes. | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
When you look at the mean of those samples, when you average together those curves from all of those samples, you see that our best estimate is when a is pretty close to negative one, is when this is n plus negative one, or n minus one. Anything less than negative one, if we did negative n minus 1.05, or n minus 1.5, we start overestimating the variance. Anything less than negative one, so if we have n plus zero, if we divide by n, or if we have n plus.05, or whatever it might be, we start underestimating, we start underestimating the population variance. And you can do this for samples of different sizes. Let me try a sample size six. And here you go, once again, as I press, I'm just keeping generate sample pressed down. As we generate more and more and more samples, and for all of the a's, we essentially take the average across those samples for the variance, depending on how we calculate it, you'll see that, once again, our best estimate is pretty darn close, is pretty darn close to negative one. | Simulation providing evidence that (n-1) gives us unbiased estimate Khan Academy.mp3 |
In a local teaching district, a technology grant is available to teachers in order to install a cluster of four computers in their classroom. From the 6,250 teachers in the district, 250 were randomly selected and asked if they felt that computers were an essential teaching tool for their classroom. Of those selected, 142 teachers felt that computers were an essential teaching tool. Then they asked us, calculate a 99% confidence interval for the proportion of teachers who felt that computers are an essential teaching tool. Let's think about the entire population. We weren't able to survey all of them, but the entire population, some of them fall into the bucket and we'll define that as 1. They thought it was a good tool. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
Then they asked us, calculate a 99% confidence interval for the proportion of teachers who felt that computers are an essential teaching tool. Let's think about the entire population. We weren't able to survey all of them, but the entire population, some of them fall into the bucket and we'll define that as 1. They thought it was a good tool. They thought that the computers were a good tool. We'll just define a zero value as a teacher that says not good. Some proportion of the total teachers think that it is a good learning tool. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
They thought it was a good tool. They thought that the computers were a good tool. We'll just define a zero value as a teacher that says not good. Some proportion of the total teachers think that it is a good learning tool. That proportion is P. The rest of them think it's a bad learning tool. 1 minus P. We have a Bernoulli distribution right over here. We know that the mean of this distribution, or the expected value of this distribution, is actually going to be P. It's actually going to be a value. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
Some proportion of the total teachers think that it is a good learning tool. That proportion is P. The rest of them think it's a bad learning tool. 1 minus P. We have a Bernoulli distribution right over here. We know that the mean of this distribution, or the expected value of this distribution, is actually going to be P. It's actually going to be a value. It's neither 0 or 1, so not an actual value that you could actually get out of a teacher if you were to ask them. They cannot say something in between good and not good, but the actual expected value is something in between. It is P. What we do is, we're taking a sample of those 250 teachers, and we got that 142 felt that computers were an essential teaching tool. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
We know that the mean of this distribution, or the expected value of this distribution, is actually going to be P. It's actually going to be a value. It's neither 0 or 1, so not an actual value that you could actually get out of a teacher if you were to ask them. They cannot say something in between good and not good, but the actual expected value is something in between. It is P. What we do is, we're taking a sample of those 250 teachers, and we got that 142 felt that computers were an essential teaching tool. In our survey, we had 250 sampled, and we got 142 said that it is good. We'll say that this is a 1. We got 142 1's, or we sampled 1 142 times from this distribution. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
It is P. What we do is, we're taking a sample of those 250 teachers, and we got that 142 felt that computers were an essential teaching tool. In our survey, we had 250 sampled, and we got 142 said that it is good. We'll say that this is a 1. We got 142 1's, or we sampled 1 142 times from this distribution. Then the rest of the times, what's left over? There's another 108 who said that it's not good. 108 said not good, or you could view them as you were sampling a 0. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
We got 142 1's, or we sampled 1 142 times from this distribution. Then the rest of the times, what's left over? There's another 108 who said that it's not good. 108 said not good, or you could view them as you were sampling a 0. 108 plus 142 is 250. What is our sample mean here? We have 1 times 142 plus 0 times 108 divided by our total number of samples, divided by 250. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
108 said not good, or you could view them as you were sampling a 0. 108 plus 142 is 250. What is our sample mean here? We have 1 times 142 plus 0 times 108 divided by our total number of samples, divided by 250. It is equal to 142 over 250. You could even view this as the sample proportion of teachers who thought that the computers were a good teaching tool. Let me get a calculator out to calculate this. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
We have 1 times 142 plus 0 times 108 divided by our total number of samples, divided by 250. It is equal to 142 over 250. You could even view this as the sample proportion of teachers who thought that the computers were a good teaching tool. Let me get a calculator out to calculate this. We have 142 divided by 250 is equal to 0.568. Our sample proportion is 0.568, or 56.8% either one. 0.568. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
Let me get a calculator out to calculate this. We have 142 divided by 250 is equal to 0.568. Our sample proportion is 0.568, or 56.8% either one. 0.568. Let's also figure out our sample variance, because we can use it later for building our confidence interval. Our sample variance here, we're going to take the weighted sum of the squared differences from the mean and divide by this minus 1. We can get the best estimator of the true variance. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
0.568. Let's also figure out our sample variance, because we can use it later for building our confidence interval. Our sample variance here, we're going to take the weighted sum of the squared differences from the mean and divide by this minus 1. We can get the best estimator of the true variance. It's 1 times 142 samples that were 1 minus 0.568 away from our sample mean. We're this far from the sample mean 142 times, and we're going to square those distances. Plus the other 108 times, we got a 0, so we were 0 minus 0.568 away from the sample mean. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
We can get the best estimator of the true variance. It's 1 times 142 samples that were 1 minus 0.568 away from our sample mean. We're this far from the sample mean 142 times, and we're going to square those distances. Plus the other 108 times, we got a 0, so we were 0 minus 0.568 away from the sample mean. Then we are going to divide that by the total number of samples minus 1. That minus 1 is our adjuster so that we don't underestimate. 250 minus 1, let's get our calculator out again. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
Plus the other 108 times, we got a 0, so we were 0 minus 0.568 away from the sample mean. Then we are going to divide that by the total number of samples minus 1. That minus 1 is our adjuster so that we don't underestimate. 250 minus 1, let's get our calculator out again. We have 142 times 1 minus 0.568 squared plus 108 times 0 minus 0.568 squared, and then all of that divided by 250 minus 1 is 249. Our sample variance is 0.246. If you were to take the square root of that, our actual sample standard deviation is 0.496. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
250 minus 1, let's get our calculator out again. We have 142 times 1 minus 0.568 squared plus 108 times 0 minus 0.568 squared, and then all of that divided by 250 minus 1 is 249. Our sample variance is 0.246. If you were to take the square root of that, our actual sample standard deviation is 0.496. I'll just round that up to 0.50. That is our sample standard deviation. Let's think of it this way. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
If you were to take the square root of that, our actual sample standard deviation is 0.496. I'll just round that up to 0.50. That is our sample standard deviation. Let's think of it this way. We are sampling from some sampling distribution of the sample mean. It looks like this over here. It looks like that over there. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
Let's think of it this way. We are sampling from some sampling distribution of the sample mean. It looks like this over here. It looks like that over there. It has some mean. The sampling distribution of the sample mean is actually going to be the same thing as this mean over here. It's going to be the same mean value, which is the same thing as our population proportion. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
It looks like that over there. It has some mean. The sampling distribution of the sample mean is actually going to be the same thing as this mean over here. It's going to be the same mean value, which is the same thing as our population proportion. We've seen this multiple times. The sampling distribution's standard deviation, so the standard deviation of the sampling distribution, we could view that as one standard deviation right over there. The standard deviation of the sampling distribution we've seen multiple times is equal to the standard deviation of our original population divided by the square root of the number of samples. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
It's going to be the same mean value, which is the same thing as our population proportion. We've seen this multiple times. The sampling distribution's standard deviation, so the standard deviation of the sampling distribution, we could view that as one standard deviation right over there. The standard deviation of the sampling distribution we've seen multiple times is equal to the standard deviation of our original population divided by the square root of the number of samples. We do not know this right over here. We do not know the actual standard deviation in our population. Our best estimate of that, and that's why we call it confident, we're confident that the real population proportion is going to be in this interval. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
The standard deviation of the sampling distribution we've seen multiple times is equal to the standard deviation of our original population divided by the square root of the number of samples. We do not know this right over here. We do not know the actual standard deviation in our population. Our best estimate of that, and that's why we call it confident, we're confident that the real population proportion is going to be in this interval. We're confident, but we're not 100% sure because we're going to estimate this over here. If we're estimating this, we're really estimating that over there. If this can be estimated, it's going to be estimated by this sample standard deviation. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
Our best estimate of that, and that's why we call it confident, we're confident that the real population proportion is going to be in this interval. We're confident, but we're not 100% sure because we're going to estimate this over here. If we're estimating this, we're really estimating that over there. If this can be estimated, it's going to be estimated by this sample standard deviation. Then we can say this is going to be approximately, or if we didn't get a weird, completely skewed sample, it actually might not even be approximately if we just had a really strange sample. Maybe we should write confident that. Confident that the standard deviation of our sampling distribution is going to be around, instead of using this, we can use our standard deviation of our sample, our sample standard deviation. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
If this can be estimated, it's going to be estimated by this sample standard deviation. Then we can say this is going to be approximately, or if we didn't get a weird, completely skewed sample, it actually might not even be approximately if we just had a really strange sample. Maybe we should write confident that. Confident that the standard deviation of our sampling distribution is going to be around, instead of using this, we can use our standard deviation of our sample, our sample standard deviation. 0.50 divided by the square root of 250, and what's that going to be? That is going to be, so we have this value right over here, actually I don't have to round it, divided by the square root of 250. We get 0.031, so this is equal to 0.031 over here. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
Confident that the standard deviation of our sampling distribution is going to be around, instead of using this, we can use our standard deviation of our sample, our sample standard deviation. 0.50 divided by the square root of 250, and what's that going to be? That is going to be, so we have this value right over here, actually I don't have to round it, divided by the square root of 250. We get 0.031, so this is equal to 0.031 over here. That's one standard deviation. Now, they want a 99% confidence interval. The way I think about it is, if I randomly pick a sample from the sampling distribution, what's the 99% chance, or how many, let me think of it this way, how many standard deviations away from the mean do we have to be that we can be 99% confident that any sample from the sampling distribution will be in that interval? | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
We get 0.031, so this is equal to 0.031 over here. That's one standard deviation. Now, they want a 99% confidence interval. The way I think about it is, if I randomly pick a sample from the sampling distribution, what's the 99% chance, or how many, let me think of it this way, how many standard deviations away from the mean do we have to be that we can be 99% confident that any sample from the sampling distribution will be in that interval? Another way to think about it, think about how many standard deviations we need to be away from the mean. We're going to be a certain number of standard deviations away from the mean, such that any sample, any mean that we sample from here, any sample from this distribution has a 99% chance of being with plus or minus that many standard deviations. It might be from there to there. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
The way I think about it is, if I randomly pick a sample from the sampling distribution, what's the 99% chance, or how many, let me think of it this way, how many standard deviations away from the mean do we have to be that we can be 99% confident that any sample from the sampling distribution will be in that interval? Another way to think about it, think about how many standard deviations we need to be away from the mean. We're going to be a certain number of standard deviations away from the mean, such that any sample, any mean that we sample from here, any sample from this distribution has a 99% chance of being with plus or minus that many standard deviations. It might be from there to there. That's what we want. We want a 99% chance that if we pick a sample from the sampling distribution of the sample mean, it will be within this many standard deviations of the actual mean. To figure that out, let's look at an actual z-table. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
It might be from there to there. That's what we want. We want a 99% chance that if we pick a sample from the sampling distribution of the sample mean, it will be within this many standard deviations of the actual mean. To figure that out, let's look at an actual z-table. We want 99% confidence. Another way to think about it, if we want 99% confidence, if we just look at the upper half right over here, that orange area should be 0.475, because if this is 0.475, then this other part is going to be 0.475. We want to get to 99%, so it's not going to be 0.475. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
To figure that out, let's look at an actual z-table. We want 99% confidence. Another way to think about it, if we want 99% confidence, if we just look at the upper half right over here, that orange area should be 0.475, because if this is 0.475, then this other part is going to be 0.475. We want to get to 99%, so it's not going to be 0.475. We're going to have to go to 0.495 if we want 99% confidence. This area has to be 0.495 over here, because if that is, that over here will also be, so that their sum will be 99% of the area. This is 0.495, this value on the z-table right here will have to be 0.5, because all of this area, if you include all of this, is going to be 0.5. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
We want to get to 99%, so it's not going to be 0.475. We're going to have to go to 0.495 if we want 99% confidence. This area has to be 0.495 over here, because if that is, that over here will also be, so that their sum will be 99% of the area. This is 0.495, this value on the z-table right here will have to be 0.5, because all of this area, if you include all of this, is going to be 0.5. It's going to be 0.5 plus 0.495. It's going to be 0.995. Let me make sure I got that right, 0.995. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
This is 0.495, this value on the z-table right here will have to be 0.5, because all of this area, if you include all of this, is going to be 0.5. It's going to be 0.5 plus 0.495. It's going to be 0.995. Let me make sure I got that right, 0.995. Let's look at our z-table. Where do we get 0.995 on our z-table? 0.995 is pretty close, just to have a little error. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
Let me make sure I got that right, 0.995. Let's look at our z-table. Where do we get 0.995 on our z-table? 0.995 is pretty close, just to have a little error. It will be right over here. This is 0.9951. Another way to think about it is 99, so this value right here gives us the whole cumulative area, up to that, up to our mean. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
0.995 is pretty close, just to have a little error. It will be right over here. This is 0.9951. Another way to think about it is 99, so this value right here gives us the whole cumulative area, up to that, up to our mean. If you look at the entire distribution like this, this is the mean right over here. This tells us that at 2.5 standard deviations above the mean, so this is 2.5 times the standard deviation of the sampling distribution, if you look at this whole area over here, if you look at the z-table, is going to be 0.9951, which tells us that just this area right over here is going to be 0.4951, which tells us that this area plus a symmetric area of that many standard deviations below the mean, if you combine them, 0.4951 times 2 gets us to 99.2. This whole area right here is 99.992. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
Another way to think about it is 99, so this value right here gives us the whole cumulative area, up to that, up to our mean. If you look at the entire distribution like this, this is the mean right over here. This tells us that at 2.5 standard deviations above the mean, so this is 2.5 times the standard deviation of the sampling distribution, if you look at this whole area over here, if you look at the z-table, is going to be 0.9951, which tells us that just this area right over here is going to be 0.4951, which tells us that this area plus a symmetric area of that many standard deviations below the mean, if you combine them, 0.4951 times 2 gets us to 99.2. This whole area right here is 99.992. If we look at the area of 2.5 standard deviations above and below the mean, let me be careful, this isn't just 2.5, we have to add another digit of precision. This is 2.5, and the next digit of precision is given by this column over here. We have to look all the way up into the second to last column, and we have to add a digit of 8 here. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
This whole area right here is 99.992. If we look at the area of 2.5 standard deviations above and below the mean, let me be careful, this isn't just 2.5, we have to add another digit of precision. This is 2.5, and the next digit of precision is given by this column over here. We have to look all the way up into the second to last column, and we have to add a digit of 8 here. This is 2.58 standard deviations. We have 2.5 over here, and then we get the next digit, 8, from the column. 2.58 standard deviations above and below the standard deviation encompasses a little over 99% of the total probability. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
We have to look all the way up into the second to last column, and we have to add a digit of 8 here. This is 2.58 standard deviations. We have 2.5 over here, and then we get the next digit, 8, from the column. 2.58 standard deviations above and below the standard deviation encompasses a little over 99% of the total probability. There's a little over 99% chance that any sample mean that I select from the sampling distribution of the sample mean will fall within this much of the standard deviation. Let me put it this way. There is a 99.2% chance. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
2.58 standard deviations above and below the standard deviation encompasses a little over 99% of the total probability. There's a little over 99% chance that any sample mean that I select from the sampling distribution of the sample mean will fall within this much of the standard deviation. Let me put it this way. There is a 99.2% chance. If you multiply this times 2, you get 0.99. You get 0.9902. Let's say roughly 99% chance that any sample that a random sample mean is within 2.58 standard deviations of the sampling mean of our population mean, of the mean of the sampling distribution of the sampling mean, which is the same thing as our actual population mean, which is the same thing as our population proportion, so of p. We know what this value is right here. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
There is a 99.2% chance. If you multiply this times 2, you get 0.99. You get 0.9902. Let's say roughly 99% chance that any sample that a random sample mean is within 2.58 standard deviations of the sampling mean of our population mean, of the mean of the sampling distribution of the sampling mean, which is the same thing as our actual population mean, which is the same thing as our population proportion, so of p. We know what this value is right here. At least we have a decent estimate for this value. We don't know exactly what this is, but our best estimate for this value is this over here. We could rewrite this, and we could say that we are confident because we are really using an estimator to get this value over here. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
Let's say roughly 99% chance that any sample that a random sample mean is within 2.58 standard deviations of the sampling mean of our population mean, of the mean of the sampling distribution of the sampling mean, which is the same thing as our actual population mean, which is the same thing as our population proportion, so of p. We know what this value is right here. At least we have a decent estimate for this value. We don't know exactly what this is, but our best estimate for this value is this over here. We could rewrite this, and we could say that we are confident because we are really using an estimator to get this value over here. We are confident that there is a 99% chance that a random X, a random sample mean, is within, and let's figure out this value right here using a calculator. It is 2.58 times our best estimate of the standard deviation of the sampling distribution, so times 0.031 is equal to 0.0, well, let's just round this up because it's so close to 0.08, is within 0.08 of the population proportion, or you could say that you are confident that the population proportion is within 0.08 of your sample mean. That's the exact same statement. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
We could rewrite this, and we could say that we are confident because we are really using an estimator to get this value over here. We are confident that there is a 99% chance that a random X, a random sample mean, is within, and let's figure out this value right here using a calculator. It is 2.58 times our best estimate of the standard deviation of the sampling distribution, so times 0.031 is equal to 0.0, well, let's just round this up because it's so close to 0.08, is within 0.08 of the population proportion, or you could say that you are confident that the population proportion is within 0.08 of your sample mean. That's the exact same statement. If we want our confidence interval, our actual number that we got for there, our actual sample mean, we got was 0.568. We could replace this, and actually let me do it. I can delete this right here, let me clear it. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
That's the exact same statement. If we want our confidence interval, our actual number that we got for there, our actual sample mean, we got was 0.568. We could replace this, and actually let me do it. I can delete this right here, let me clear it. I can replace this, because we actually did take a sample, so I can replace this with 0.568. We could be confident that there is a 99% chance that 0.568 is within 0.08 of the actual sample, of the population proportion, which is the same thing as the population mean, which is the same thing as the mean of the sampling distribution of the sample mean, so forth and so on. Just to make it clear, we can actually swap these two. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
I can delete this right here, let me clear it. I can replace this, because we actually did take a sample, so I can replace this with 0.568. We could be confident that there is a 99% chance that 0.568 is within 0.08 of the actual sample, of the population proportion, which is the same thing as the population mean, which is the same thing as the mean of the sampling distribution of the sample mean, so forth and so on. Just to make it clear, we can actually swap these two. It wouldn't change the meaning. If this is within 0.08 of that, then that is within 0.08 of this. Let me switch this up a little bit. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
Just to make it clear, we can actually swap these two. It wouldn't change the meaning. If this is within 0.08 of that, then that is within 0.08 of this. Let me switch this up a little bit. We could put a P is within of 0.568. Now, linguistically, it sounds a little bit more like a confidence interval. We are confident that there is a 99% chance that P is within 0.08 of the sample mean of 0.568. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
Let me switch this up a little bit. We could put a P is within of 0.568. Now, linguistically, it sounds a little bit more like a confidence interval. We are confident that there is a 99% chance that P is within 0.08 of the sample mean of 0.568. What would be our confidence interval? It would be 0.568 plus or minus 0.08. What would that be? | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
We are confident that there is a 99% chance that P is within 0.08 of the sample mean of 0.568. What would be our confidence interval? It would be 0.568 plus or minus 0.08. What would that be? If you add 0.08 to this right over here, at the upper end, you're going to have 0.648, and at the lower end of our range, so this is the upper end, the lower end, if we subtract 8 from this, we get 0.488. We are 99% confident that the true population proportion is between these two numbers. Or another way, that the true percentage of teachers who think those computers are good ideas is between. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
What would that be? If you add 0.08 to this right over here, at the upper end, you're going to have 0.648, and at the lower end of our range, so this is the upper end, the lower end, if we subtract 8 from this, we get 0.488. We are 99% confident that the true population proportion is between these two numbers. Or another way, that the true percentage of teachers who think those computers are good ideas is between. We're 99% confident, we're confident that there's a 99% chance that the true percentage of teachers that like the computers is between 48.8% and 64.8%. Now, that we answered the first part of the question. The second part, how could the survey be changed to narrow the confidence interval, but to maintain the 99% confidence interval? | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
Or another way, that the true percentage of teachers who think those computers are good ideas is between. We're 99% confident, we're confident that there's a 99% chance that the true percentage of teachers that like the computers is between 48.8% and 64.8%. Now, that we answered the first part of the question. The second part, how could the survey be changed to narrow the confidence interval, but to maintain the 99% confidence interval? Well, you could just take more samples. If you take more samples, then our estimate of the standard deviation of this distribution will go down, because this denominator will be higher. If that denominator is higher, then this whole thing will go down. | Confidence interval example Inferential statistics Probability and Statistics Khan Academy.mp3 |
Liliana runs a cake decorating business for which 10% of her orders come over the telephone. Let C be the number of cake orders Liliana receives in a month until she first gets an order over the telephone. Assume the method of placing each cake order is independent. So C, if we assume a few things, is a classic geometric random variable. What tells us that? Well, the giveaway is that we're gonna keep doing these independent trials where the probability of success is constant and there's a clear success. A telephone order in this case is a success. | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
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