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It seems like it kind of approximates this trend, although it doesn't seem like a great trend. And if we ignore these two points right over here, we could do something like, maybe something like that. But we can't just ignore points like that, so I would say that there's actually no good line of best fit here. So let m... | Estimating the line of best fit exercise Regression Probability and Statistics Khan Academy.mp3 |
So let me check my answer. Let's try a couple more of these. Find the line of best fit. Well, this feels very similar. It really feels like there's no, I mean, I could do that, but then I'm ignoring these two points. I could do something like that, then I'd be ignoring these points. So I'd also say no good best fit lin... | Estimating the line of best fit exercise Regression Probability and Statistics Khan Academy.mp3 |
Well, this feels very similar. It really feels like there's no, I mean, I could do that, but then I'm ignoring these two points. I could do something like that, then I'd be ignoring these points. So I'd also say no good best fit line exists. So let's try one more. So here, it looks like there's very clearly this trend,... | Estimating the line of best fit exercise Regression Probability and Statistics Khan Academy.mp3 |
So I'd also say no good best fit line exists. So let's try one more. So here, it looks like there's very clearly this trend, and I could try to fit it a little bit better than it's fit right now. So it feels like something like that fits this trend line quite well. I could maybe drop this down a little bit, something l... | Estimating the line of best fit exercise Regression Probability and Statistics Khan Academy.mp3 |
Let's say that you have a cherry pie store, and you've noticed that there is variability in the number of cherries on each pie that you sell. Some pies might have over 100 cherries, while other pies might have fewer than 50 cherries. So what you're curious about is what is the distribution? How many of the different ty... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
How many of the different types of pies do you have? How many pies do you have that have a lot of cherries? How many pies do you have that have very few cherries? How many pies are in between? And so to do that, you set up a histogram. What you do is you take each pie in your store. Let's see if I can draw a pie of som... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
How many pies are in between? And so to do that, you set up a histogram. What you do is you take each pie in your store. Let's see if I can draw a pie of some kind. It's a cherry pie. I don't know if this is an adequate drawing of a pie. But you take each of the pies in your store, and you count the number of cherries ... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
Let's see if I can draw a pie of some kind. It's a cherry pie. I don't know if this is an adequate drawing of a pie. But you take each of the pies in your store, and you count the number of cherries on it. So this pie right over here is one, two, three, four, five, six, seven, eight, nine, ten. Let's see, you keep coun... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
But you take each of the pies in your store, and you count the number of cherries on it. So this pie right over here is one, two, three, four, five, six, seven, eight, nine, ten. Let's see, you keep counting, and let's say it has 32 cherries. And you do that for every pie. And then you created buckets, because you don'... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
And you do that for every pie. And then you created buckets, because you don't want to create just a graph of how many have exactly 32. You just want to get a general sense of things. So you create buckets of 30. You say, how many pies have between zero and 29 cherries? How many pies have between 30 and 59, including 3... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
So you create buckets of 30. You say, how many pies have between zero and 29 cherries? How many pies have between 30 and 59, including 30 and 59? How many pies have at least 60 and at most 89 cherries? How many pies have at least 90 and at most 119? And then how many pies have at least 120 and at most 149? And you know... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
How many pies have at least 60 and at most 89 cherries? How many pies have at least 90 and at most 119? And then how many pies have at least 120 and at most 149? And you know that you don't have any pies that have more than 149 cherries. So this should account for everything. And then you count them. So for example, yo... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
And you know that you don't have any pies that have more than 149 cherries. So this should account for everything. And then you count them. So for example, you say, okay, five pies have 30 to 59 cherries. And so we create a histogram, or you create a histogram, and you make this magenta bar go up to five. So that's how... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
So for example, you say, okay, five pies have 30 to 59 cherries. And so we create a histogram, or you create a histogram, and you make this magenta bar go up to five. So that's how you would construct this histogram. That's what this pies at different cherry levels histogram is telling us. So now that we know how to co... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
That's what this pies at different cherry levels histogram is telling us. So now that we know how to construct it, let's see if we can interpret it based on the information given in the histogram. So the first question is, based on just this information, can you figure out the total number of pies in your store, assumi... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
And I encourage you to pause the video and try to figure it out on your own. Well, what's the total number of pies? Well, let's see. There's five pies that have more than, or 30 or more, have at least 30 cherries, but no more than 59. You have eight pies in this blue bucket. You have four pies in this green bucket. And... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
There's five pies that have more than, or 30 or more, have at least 30 cherries, but no more than 59. You have eight pies in this blue bucket. You have four pies in this green bucket. And then you have, what is this, five, no, this is three pies that have at least 120, but no more than 149 cherries. And this accounts f... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
And then you have, what is this, five, no, this is three pies that have at least 120, but no more than 149 cherries. And this accounts for all of the pies. So the total number of pies you have at this store are five plus eight plus four plus three, which is what? Five plus eight is 13, plus four is 17, plus three is 20... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
Five plus eight is 13, plus four is 17, plus three is 20. So there are 20 pies in this store. But then you can ask more nuanced questions. What if you wanted to know the number of pies with more than 60 cherries? The number of pies with more than 60. So number of pies with, I'll say, let's say 60 or more. 60 or more, 6... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
What if you wanted to know the number of pies with more than 60 cherries? The number of pies with more than 60. So number of pies with, I'll say, let's say 60 or more. 60 or more, 60 or more cherries. So let's think about it. Well, this magenta bar doesn't apply because these all have less than 60, but all of these oth... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
60 or more, 60 or more cherries. So let's think about it. Well, this magenta bar doesn't apply because these all have less than 60, but all of these other bars are counting pies that have 60 or more cherries. This is 60 to 89, this is 90 to 119, this is 120 to 149. So it's going to be these eight cherries that are, sor... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
This is 60 to 89, this is 90 to 119, this is 120 to 149. So it's going to be these eight cherries that are, sorry, these eight pies that are in this bucket plus these four pies, plus these three pies. So it is going to be essentially everything but this first bucket. Everything but all the pies except for these five pi... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
Everything but all the pies except for these five pies have 60 or more cherries. So it should be five less than 20, and so let's see, eight plus four is 12, plus three is 15, which is five less than 20. So using this histogram, we can answer a really interesting question. We can say, well, wait, how many more pies do w... | How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3 |
This one has a population of 383, and then it calculates the parameters for that population directly from it. The mean is 10.9, the variance is 25.5. And then it uses that population and samples from it, and it does samples of size two, three, four, five, all the way up to 10, and it keeps sampling from it, calculates ... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
And you can actually click on each of these and zoom in to really be able to study these graphs in detail. So I've already taken a screenshot of this and put it on my little doodle pad, so it can really delve into some of the math and the intuition of what this is actually showing us. So here I took a screenshot, and y... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
And down here in this chart, he plots the population mean right here at 10.6, right over there, and you see that the population variance is at 36.8, and right here he plots that right over here, 36.8. So this first chart on the bottom left tells us a couple of interesting things. And just to be clear, this is the biase... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
This is the biased sample variance. So he's calculating it. That is being calculated for each of our data points, so starting with our first data point in each of our samples, going to our nth data point in the sample. You're taking that data point, subtracting out the sample mean, squaring it, and then dividing the wh... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
You're taking that data point, subtracting out the sample mean, squaring it, and then dividing the whole thing not by n minus one, but by lowercase n. And this tells us several interesting things. The first thing it shows us is that the cases where we are significantly underestimating the sample variance, when we're ge... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
The cases where the mean is way far off from the sample mean, it seems like you're much more likely to underestimate the sample variance in those situations. The other thing that might pop out at you is the realization that the pinker dots are the ones for smaller sample size, while the bluer dots are the ones of a lar... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
That most of the bluish or the purplish dots are focused right in the middle right over here, that they are giving us better estimates. There are some red ones here, and that's why it gives us that purplish color. But out here on these tails, it's almost purely some of these red. Every now and then, by happenstance, yo... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
Every now and then, by happenstance, you get a little blue one. But this is disproportionately far more red, which really makes sense. When you have a smaller sample size, you're more likely to get a sample mean that is a bad estimate of the population mean, that's far from the population mean, and you're more likely t... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
Now this next chart really gets to the meat, really gets to the meat of the issue, because what it's telling us is that for each of these sample sizes, so this right over here, for sample size two, if we keep taking sample size two and we keep calculating the by sample variances and dividing that by the population vari... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
And so we can come up with a general theme that's happening. When we use the biased estimate, when we use the biased estimate, we're not approaching the population variance, we're approaching n minus one, let me write this down, we're approaching n minus one over n times the population variance. When n was two, this ap... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
When n is three, this is 2 3rds. When n is four, this is 3 4ths. So this is giving us a biased estimate. So how would we unbiased this? Well, if we really wanna get our best estimate of the true population variance, not n minus one over n times the population variance, we would wanna multiply, let me do this in a color... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
So how would we unbiased this? Well, if we really wanna get our best estimate of the true population variance, not n minus one over n times the population variance, we would wanna multiply, let me do this in a color I haven't used yet, we would wanna multiply times n over n minus one. We would wanna multiply n over n m... | Simulation showing bias in sample variance Probability and Statistics Khan Academy.mp3 |
What proportion of laptop prices are between $624 and $768? So let's think about what they are asking. So we have a normal distribution for the prices. So it would look something like this. And this is just my hand-drawn sketch of a normal distribution. So it would look something like this. It should be symmetric. | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
So it would look something like this. And this is just my hand-drawn sketch of a normal distribution. So it would look something like this. It should be symmetric. So I'm making it as symmetric as I can hand-draw it. And we have the mean right in the center. So the mean would be right there. | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
It should be symmetric. So I'm making it as symmetric as I can hand-draw it. And we have the mean right in the center. So the mean would be right there. And that is $750. They also tell us that we have a standard deviation of $60. So that means one standard deviation above the mean would be roughly right over here. | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
So the mean would be right there. And that is $750. They also tell us that we have a standard deviation of $60. So that means one standard deviation above the mean would be roughly right over here. And that'd be 750 plus 60. So that would be $810. One standard deviation below the mean would put us right about there. | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
So that means one standard deviation above the mean would be roughly right over here. And that'd be 750 plus 60. So that would be $810. One standard deviation below the mean would put us right about there. And that would be 750 minus $60, which would be $690. And then they tell us what proportion of laptop prices are b... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
One standard deviation below the mean would put us right about there. And that would be 750 minus $60, which would be $690. And then they tell us what proportion of laptop prices are between $624 and $768? So the lower bound, $624, that's going to actually be more than another standard deviation less. So that's going t... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
So the lower bound, $624, that's going to actually be more than another standard deviation less. So that's going to be right around here. So that is $624. And $768 would put us right at about there. And once again, this is just a hand-drawn sketch. But that is 768. And so what proportion are between those two values? | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
And $768 would put us right at about there. And once again, this is just a hand-drawn sketch. But that is 768. And so what proportion are between those two values? We want to find essentially the area under this distribution between these two values. The way we are going to approach it, we're going to figure out the z-... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
And so what proportion are between those two values? We want to find essentially the area under this distribution between these two values. The way we are going to approach it, we're going to figure out the z-score for 768. It's going to be positive because it's above the mean. And then we're going to use a z-table to ... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
It's going to be positive because it's above the mean. And then we're going to use a z-table to figure out what proportion is below 768. So essentially we're going to figure out this entire area. We're even gonna figure out the stuff that's below 624. That's what that z-table will give us. And then we'll figure out the... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
We're even gonna figure out the stuff that's below 624. That's what that z-table will give us. And then we'll figure out the z-score for 624. That will be negative two point something. And we will use a z-table again to figure out the proportion that is less than that. And so then we can subtract this red area from the... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
That will be negative two point something. And we will use a z-table again to figure out the proportion that is less than that. And so then we can subtract this red area from the proportion that is less than 768 to get this area in between. So let's do that. Let's figure out first the z-score for 768. And then we'll do... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
So let's do that. Let's figure out first the z-score for 768. And then we'll do it for 624. The z-score for 768, I'll write it like that, is going to be 768 minus 750 over the standard deviation, over 60. So this is going to be equal to 18 over 60, which is the same thing as six over, let's see if we divide the numerat... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
The z-score for 768, I'll write it like that, is going to be 768 minus 750 over the standard deviation, over 60. So this is going to be equal to 18 over 60, which is the same thing as six over, let's see if we divide the numerator and denominator by three, 6 20ths, and this is the same thing as 0.30. So that is the z-s... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
Let's figure out what proportion is less than that. For that, we take out a z-table. Get our z-table. And let's see, we want to get 0.30. So this is zero. And so this is 0.30, this first column, and we've done this in other videos, this goes up until the 10th place for our z-score, and then if we want to get to our 100... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
And let's see, we want to get 0.30. So this is zero. And so this is 0.30, this first column, and we've done this in other videos, this goes up until the 10th place for our z-score, and then if we want to get to our 100th place, that's what these other columns give us. But we're at 0.30, so we're going to be in this row... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
But we're at 0.30, so we're going to be in this row, and our 100th place is right over here, it's a zero. So this is the proportion that is less than $768. So 0.6179. So 0.6179. So now let's do the same exercise, but do it for the proportion that's below $624. The z-score for 624 is going to be equal to 624 minus the m... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
So 0.6179. So now let's do the same exercise, but do it for the proportion that's below $624. The z-score for 624 is going to be equal to 624 minus the mean of 750, all of that over 60. And so what is that going to be? I'll get my calculator out for this one, don't want to make a careless error. 624 minus 750 is equal ... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
And so what is that going to be? I'll get my calculator out for this one, don't want to make a careless error. 624 minus 750 is equal to, 624 minus 750 is equal to, and then divide by 60 is equal to negative 2.1. So that lower bound is 2.1 standard deviations below the mean, or you could say it has a z-score of negativ... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
So that lower bound is 2.1 standard deviations below the mean, or you could say it has a z-score of negative 2.1, is equal to negative 2.1. And so to figure out the proportion that is less than that, this red area right over here, we go back to our z-table. And we'd actually go to the first part of the z-table. So same... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
So same idea, but this starts at negative, a z-score of negative 3.4, 3.4 standard deviations below the mean. But just like we saw before, this is our zero hundredths, one hundredth, two hundredth, so on and so forth. And we want to go to negative 2.1, we could say negative 2.10, just to be precise. So this is going to... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
So this is going to get us, let's see, negative 2.1, there we go. And so we are negative 2.1, and it's negative 2.10, so we have zero hundredths. So we're gonna be right here on our table. So we see the proportion that is less than 624 is.0179, or 0.0179. So 0.0179. So 0.0179. And so if we want to figure out the propor... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
So we see the proportion that is less than 624 is.0179, or 0.0179. So 0.0179. So 0.0179. And so if we want to figure out the proportion that's in between the two, we just subtract this red area from this entire area, the entire proportion that's less than 768, to get what's in between. 0.6179, once again, I know I keep... | Standard normal table for proportion between values AP Statistics Khan Academy.mp3 |
Let me throw a few blue ones in there. And what we're going to concern ourselves in this video are the yellow gumballs. And let's say that we know that the proportion of yellow gumballs over here is p. This right over here is a population, population parameter. Parameter. And for the sake of argument, just to make thin... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
Parameter. And for the sake of argument, just to make things concrete, let's just say that 60% of the gumballs are yellow, or 0.6 of them. Now let's review some things that we have seen before. I'm gonna define our Bernoulli random variable. Let's call this capital Y, which is equal to one if when we take one random gu... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
I'm gonna define our Bernoulli random variable. Let's call this capital Y, which is equal to one if when we take one random gumball out of that machine, we get a, we pick a yellow gumball. And it's equal to zero if when we pick one random gumball out of that machine, we don't pick yellow. So not yellow. From previous v... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
So not yellow. From previous videos, we know some interesting things about this Bernoulli random variable. We know its mean. We know the mean of our Bernoulli random variable is going to be the proportion of yellow balls in this population. So it's going to be equal to P, which in this particular case we know is 0.6. A... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
We know the mean of our Bernoulli random variable is going to be the proportion of yellow balls in this population. So it's going to be equal to P, which in this particular case we know is 0.6. And we know what the standard deviation of this Bernoulli random variable is. It is going to be P times one minus P. Actually,... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
It is going to be P times one minus P. Actually, that's the variance. We want to take the square root of that to get the standard deviation. So in this particular scenario, that's going to be the square root of 0.6 times 0.4. Fair enough. This is all review so far. But now let me define, let me define another random va... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
Fair enough. This is all review so far. But now let me define, let me define another random variable, X, which is equal to the sum of 10 independent, independent trials, trials, trials of Y. Now, we have seen random variables like this before. This is a binomial random variable. Now, what do we know about its mean and ... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
Now, we have seen random variables like this before. This is a binomial random variable. Now, what do we know about its mean and standard deviation? Well, in previous videos, we know that the mean of this binomial random variable is just going to be equal to N times the mean of each of the Bernoulli trials right over h... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
Well, in previous videos, we know that the mean of this binomial random variable is just going to be equal to N times the mean of each of the Bernoulli trials right over here. So N times P, which in this particular situation is going to be, N is 10, we're doing 10 trials, and P is 0.6, which is equal to six. And that m... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
If 60% of the balls here are yellow, and if you were to take a sample, or if you were to take 10 trials, so if you were to take 10 balls one at a time, they have to be independent, so you keep looking at them and then replacing them, but if you took 10, then you would expect that six of them would be yellow. They're no... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
So our standard deviation is equal to, and we have proved this in other videos, it's equal to the square root of N times P times one minus P. Notice, you just put an N right over here under the radical sign, and so this is going to get us, in this particular situation, to 10 times 0.6 times 0.4, and then the square roo... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
But what we're going to do in this video is think about a sampling distribution, and it's going to be the sampling distribution for a sample statistic known as the sample proportion, which we actually talked about when we first introduced sampling distributions. So let's say, so let's just park all of this. This is bac... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
Let's start taking samples of 10, and I didn't pick that randomly. I want to make it reconcile with what we did with our random variable here. And so let's take a sample of 10 gumballs, and let's calculate the proportion that are yellow. And so we will call that our sample proportion, and I might as well just do that i... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
And so we will call that our sample proportion, and I might as well just do that in yellow. So we want to calculate the sample proportion that are yellow. And what is this equivalent to? Well, you could say, well, this is just equivalent to my random variable X. I want to count the number of gumballs that are yellow, a... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
Well, you could say, well, this is just equivalent to my random variable X. I want to count the number of gumballs that are yellow, and then I'm gonna divide it by the sample size. So I'm gonna divide it by N. And in this case, it would be X divided by 10. I know what some of you are thinking. Wait, wait, wait, hold on... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
Wait, wait, wait, hold on for a second. X is sum of 10 independent, independent trials right over here. To be independent, you can't just take 10 gumballs. You have to take them one at a time and then replace them back in order for them to be truly independent. But remember, we have our 10% rule. We have our 10% rule, ... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
You have to take them one at a time and then replace them back in order for them to be truly independent. But remember, we have our 10% rule. We have our 10% rule, which tells us that if a sample is less than or equal to 10% of the population, then you can treat each of the gumballs in this situation as being independe... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
So let's just say for the sake of argument that there are 10,000 gumballs in here. And so we can feel pretty good that these samples, that each of the things in the sample are independent of each other by our 10% rule. And so each of these are, each of these 10 gumballs, what we see, they are going to be independent. I... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
I'm gonna put them in quotes by the 10% rule. And so in that situation, we can make this claim or we can feel good that this claim is roughly true. And so let's say for that first sample that we do, our sample proportion is equal to 0.3. So three of our gumballs just happened, three of our 10 gumballs just happened to ... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
So three of our gumballs just happened, three of our 10 gumballs just happened to be yellow. Then we do it again. So we take another sample. We calculate our sample proportion, the statistic again. Remember, it's trying to estimate our population parameter. And let's say that time it happens to be seven out of 10. And ... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
We calculate our sample proportion, the statistic again. Remember, it's trying to estimate our population parameter. And let's say that time it happens to be seven out of 10. And we just keep doing that. And if we keep doing that and we plot it on a dot chart or dot distribution, I guess we could say, where our possibl... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
And we just keep doing that. And if we keep doing that and we plot it on a dot chart or dot distribution, I guess we could say, where our possible outcomes, you could have zero out of 10, one out of 10, two, three, four, five, that's so half of them, six, seven, eight, nine, 10. So that would be all of them. And so you... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
And so you could plot, okay, 0.3, one, two, three. That's one scenario where I got zero, where my sample proportion is 0.3. Then 0.7, that's one situation where I got a 0.7. And let's say I were to take another sample of 10 and I were to get 0.7, then you would plot that over here. And if you kept taking samples and ke... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
And let's say I were to take another sample of 10 and I were to get 0.7, then you would plot that over here. And if you kept taking samples and kept calculating these sample proportions and you kept plotting it here, you would get a better and better and better approximation for the sampling distribution of the samplin... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
What is going to be the mean of this sampling distribution? And what is going to be the standard deviation? Well, we can derive that from what we see right over here. The mean of our sampling distribution of our sample proportion is just going to be equal to the mean of our random variable X divided by N. It's just goi... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
The mean of our sampling distribution of our sample proportion is just going to be equal to the mean of our random variable X divided by N. It's just going to be the mean of X divided by N, which is equal to what? Well, the mean of X is N times P. This is N times P. You divided it by N, you're going to get P. And that ... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
And so this is also a good indicator that this is going to be a reasonably unbiased estimator. Now let's think about the standard deviation for our sample proportion. Well, we can just view that as our standard deviation of our binomial random variable X divided by N. So this is going to be equal to the square root of ... | Sampling distribution of sample proportion part 1 AP Statistics Khan Academy.mp3 |
Hashim obtained a random sample of students and noticed a positive linear relationship between their ages and their backpack weights. A 95% confidence interval for the slope of the regression line was 0.39 plus or minus 0.23. Hashim wants to use this interval to test the null hypothesis that the true slope of the popul... | Using a confidence interval to test slope More on regression AP Statistics Khan Academy.mp3 |
Assume that all conditions for inference have been met. So given the information that we just have about what Hashim is doing, what would be his conclusion? Would he reject the null hypothesis, which would suggest the alternative, or would he be unable to reject the null hypothesis? Well, let's just think about this a ... | Using a confidence interval to test slope More on regression AP Statistics Khan Academy.mp3 |
Well, let's just think about this a little bit. We have a 95% confidence interval. Let me write this down. So our 95% confidence, confidence interval, could write it like this, or you could say that it goes from 0.39 minus 0.23, so that'd be 0.16, so it goes from 0.16 until 0.39 plus 0.23 is going to be what? 0.62. Now... | Using a confidence interval to test slope More on regression AP Statistics Khan Academy.mp3 |
So our 95% confidence, confidence interval, could write it like this, or you could say that it goes from 0.39 minus 0.23, so that'd be 0.16, so it goes from 0.16 until 0.39 plus 0.23 is going to be what? 0.62. Now what a 95% confidence interval tells us is that 95% of the time that we take a sample and we construct a 9... | Using a confidence interval to test slope More on regression AP Statistics Khan Academy.mp3 |
But in this hypothesis test, remember, we are assuming that the true population parameter is equal to zero, and that does not overlap with this confidence interval. So assuming, let me write this down, assuming null hypothesis is true, null hypothesis is true, we are in the less than or equal to 5% chance of situations... | Using a confidence interval to test slope More on regression AP Statistics Khan Academy.mp3 |
And that's exactly what's happening here. And this null hypothesis value is nowhere even close to overlapping, it's over 16 hundredths below the low end of this bound. And so because of that, we would reject the null hypothesis. Reject the null hypothesis, which suggests the alternative, which suggests the alternative ... | Using a confidence interval to test slope More on regression AP Statistics Khan Academy.mp3 |
The male and female heights are each normally distributed. We independently, randomly select a man and a woman. What is the probability that the woman is taller than the man? So I encourage you to pause this video and think through it and I'll give you a hint. What if we were to define the random variable m as equal to... | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
So I encourage you to pause this video and think through it and I'll give you a hint. What if we were to define the random variable m as equal to the height of a randomly selected man, height of random man. What if we defined the random variable w to be equal to the height of a random woman, woman, and we defined a thi... | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
So let me call this d for difference. And it is equal to the difference in height between a randomly selected man and a randomly selected woman. So d, the random variable d, is equal to the random variable m minus the random variable w. So the first two are clearly normally distributed. They tell us that right over her... | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
They tell us that right over here. The male and female heights are each normally distributed. And we also know, or you're about to know, that the difference of random variables that are each normally distributed is also going to be normally distributed. So given this, can you think about how to tackle this question? Th... | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
So given this, can you think about how to tackle this question? The probability that the woman is taller than the man. All right, now let's work through this together and to help us visualize, I'll draw the normal distribution curves for these three random variables. So this first one is for the variable m. And so righ... | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
So this first one is for the variable m. And so right here in the middle, that is the mean of m. And we know that this is going to be equal to 178 centimeters. We'll assume everything is in centimeters. We also know that it has a standard deviation of eight centimeters. So for example, if this is one standard deviation... | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
So for example, if this is one standard deviation above, this is one standard deviation below, this point right over here would be eight centimeters more than 178, so that would be 186. And this would be eight centimeters below that, so this would be 170 centimeters. So this is for the random variable m. Now let's thin... | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
And one standard deviation above the mean is going to be six centimeters above the mean. The standard deviation is six, six centimeters. So this would be minus six, is to go to one standard deviation below the mean. Now let's think about the difference between the two, the random variable d. So let me think about this ... | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
Now let's think about the difference between the two, the random variable d. So let me think about this a little bit. The random variable d, the mean of d is going to be equal to the differences in the means of these random variables. So it's going to be equal to the mean of m, the mean of m, minus the mean of w, minus... | Example Analyzing the difference in distributions Random variables AP Statistics Khan Academy.mp3 |
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