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And so that's how many sample standard deviations is it away from its mean. And so that's the z-score for that x data point, and this is the z-score for the corresponding y data point. How many sample standard deviations is it away from the sample mean? In the real world, you won't have only four pairs, and it will be ... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
In the real world, you won't have only four pairs, and it will be very hard to do it by hand, and we typically use software, computer tools to do it, but it's really valuable to do it by hand to get an intuitive understanding of what's going on here. So in this particular situation, r is going to be equal to one over n... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
So this first pair right over here, so the z-score for this one is going to be one minus how far it is away from the x sample mean divided by the x sample standard deviation, 0.816, that times one, now we're looking at the y variable, the y z-score, so it's one minus three, one minus three over the y sample standard de... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
So before I get a calculator out, let's see if there's some simplifications I can do. Two minus two, that's gonna be zero, zero times anything is zero, so this whole thing is zero. Two minus two is zero, three minus three is zero, is gonna actually be zero times zero, so that whole thing is zero. Let's see, this is goi... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
Let's see, this is going to be one minus two, which is negative one, one minus three is negative two, so this is going to be R is equal to 1 3rd times, negative times negative is positive, and so this is going to be two over 0.816 times 2.160, and then plus three minus two is one, six minus three is three, so plus thre... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
We get an R of, and since everything else goes to the thousandths place, I'll just round to the thousandths place, an R of 0.946, so R is approximately 0.946. So what does this tell us? The correlation coefficient is a measure of how well a line can describe the relationship between X and Y. R is always going to be gre... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
If R is positive one, it means that an upward-sloping line can completely describe the relationship. If R is negative one, it means a downward-sloping line can completely describe the relationship. R anywhere in between says, well, it won't be as good. If R is zero, that means that a line isn't describing the relations... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
If R is zero, that means that a line isn't describing the relationships well at all. Now in our situation here, not to use a pun, in our situation here, our R is pretty close to one, which means that a line can get pretty close to describing the relationship between our Xs and our Ys. So for example, I'm just going to ... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
So the mean of the X is two, mean of the Y is three. We'll study that in more depth in future videos. But let's see, this actually does look like a pretty good line. So let me just draw it right over there. You see that I actually can draw a line that gets pretty close to describing. It isn't perfect. If it went throug... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
So let me just draw it right over there. You see that I actually can draw a line that gets pretty close to describing. It isn't perfect. If it went through every point, then I would have an R of one, but it gets pretty close to describing what is going on. Now the next thing I wanna do is focus on the intuition. What w... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
If it went through every point, then I would have an R of one, but it gets pretty close to describing what is going on. Now the next thing I wanna do is focus on the intuition. What was actually going on here with these Z-scores, and how does taking products of corresponding Z-scores get us this property that I just ta... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
Well, let's draw the sample means here. So the X sample mean is two. This is our X axis here. This is X equals two. And our Y sample mean is three. This is the line Y is equal to three. Now we can also draw the standard deviations. | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
This is X equals two. And our Y sample mean is three. This is the line Y is equal to three. Now we can also draw the standard deviations. This is, let's see, the standard deviation for X is 0.816, so I'll be approximating it. So if I go 0.816 less than our mean, it'll get us someplace around there. So that's one standa... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
Now we can also draw the standard deviations. This is, let's see, the standard deviation for X is 0.816, so I'll be approximating it. So if I go 0.816 less than our mean, it'll get us someplace around there. So that's one standard deviation below the mean. One standard deviation above the mean would put us someplace ri... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
So that's one standard deviation below the mean. One standard deviation above the mean would put us someplace right over here. And if I do the same thing in Y, one standard deviation above the mean, 2.160, so that would be 5.160, so it would put us someplace around there. And one standard deviation below the mean, so l... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
And one standard deviation below the mean, so let's see, we're gonna go, if we took away two, we would go to one, and then we're gonna go take another 0.160, so it's gonna be someplace right around here. So for example, for this first pair, one comma one, what were we doing? Well, we said, all right, how many standard ... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
And that turns out to be negative one over 0.816. That's what we have right over here. That's what this would have calculated. And then how many standard deviations for in the Y direction? And that is our negative two over 2.160. But notice, since both of them were negative, it contributed to the R. This would become a... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
And then how many standard deviations for in the Y direction? And that is our negative two over 2.160. But notice, since both of them were negative, it contributed to the R. This would become a positive value. And so one way to think about it, it might be helping us get closer to the one. If both of them have a negativ... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
And so one way to think about it, it might be helping us get closer to the one. If both of them have a negative Z score, that means that there is a positive correlation between the variables. When one is below the mean, the other is, you could say, similarly below the mean. Now, if you go to the next data point, two co... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
Now, if you go to the next data point, two comma two, right over here, what happened? Well, the X variable was right on the mean. And because of that, that entire term became zero. The X Z score was zero. And so that would have taken away a little bit from our correlation coefficient. The reason why it would take away,... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
The X Z score was zero. And so that would have taken away a little bit from our correlation coefficient. The reason why it would take away, even though it's not negative, you're not contributing to the sum, but you're going to be dividing by a slightly higher value by including that extra pair. If you had a data point ... | Calculating correlation coefficient r AP Statistics Khan Academy.mp3 |
And in deciding which car they give me, they're first going to randomly select the engine type. So the engine will come in two different varieties. It'll either be a four-cylinder engine or a six-cylinder engine. And they're literally just gonna flip a fair coin to decide whether I get a four-cylinder engine or a six-c... | Count outcomes using tree diagram Statistics and probability 7th grade Khan Academy.mp3 |
And they're literally just gonna flip a fair coin to decide whether I get a four-cylinder engine or a six-cylinder engine. Then they're going to pick the color. And there's four different colors that the cars come in. So I'll write color in a neutral color. So you could get a red car. That's not red. Let me do that in ... | Count outcomes using tree diagram Statistics and probability 7th grade Khan Academy.mp3 |
So I'll write color in a neutral color. So you could get a red car. That's not red. Let me do that in an actual red color, closer to red. You could get a red car. You could get a blue car. Blue car. | Count outcomes using tree diagram Statistics and probability 7th grade Khan Academy.mp3 |
Let me do that in an actual red color, closer to red. You could get a red car. You could get a blue car. Blue car. You could get a green car. You could get a green car. Or you could get a white car. | Count outcomes using tree diagram Statistics and probability 7th grade Khan Academy.mp3 |
Blue car. You could get a green car. You could get a green car. Or you could get a white car. Or you could get a white car. And once again, they're gonna randomly, let's say, just pick, they're gonna have red, blue, green, and white in little slips of paper in a bowl, and they're just gonna pick one of them out. So all... | Count outcomes using tree diagram Statistics and probability 7th grade Khan Academy.mp3 |
Or you could get a white car. Or you could get a white car. And once again, they're gonna randomly, let's say, just pick, they're gonna have red, blue, green, and white in little slips of paper in a bowl, and they're just gonna pick one of them out. So all of these are equally likely. So given this, that they're just g... | Count outcomes using tree diagram Statistics and probability 7th grade Khan Academy.mp3 |
So all of these are equally likely. So given this, that they're just gonna flip a coin to pick the engine, and they're also going to, that all of these, the color is all equally likely, I wanna think about the probability of getting a six-cylinder white car. The probability of getting a six-cylinder white car. So I enc... | Count outcomes using tree diagram Statistics and probability 7th grade Khan Academy.mp3 |
So I encourage you to pause the video and think about it on your own. Well, one way to think about this is, well, what are all of the equally likely possible outcomes, and then which of those match six-cylinder white car? Well, first we could think about the engine decision. We're either going to get a four-cylinder en... | Count outcomes using tree diagram Statistics and probability 7th grade Khan Academy.mp3 |
We're either going to get a four-cylinder engine. So the first decision is the engine. You could view it that way. You're gonna get a four-cylinder engine, or you're going to get a six-cylinder engine. Now, if you got a four-cylinder engine, you're either going to get red, blue, blue, green, green, or white, or white. ... | Count outcomes using tree diagram Statistics and probability 7th grade Khan Academy.mp3 |
You're gonna get a four-cylinder engine, or you're going to get a six-cylinder engine. Now, if you got a four-cylinder engine, you're either going to get red, blue, blue, green, green, or white, or white. And if you got a six-cylinder engine, once again, you're either going to get red, blue, I think you see where this ... | Count outcomes using tree diagram Statistics and probability 7th grade Khan Academy.mp3 |
So how many possible outcomes are there? Well, you could just count, you could kind of say the leaves of this tree diagram. One, two, three, four, five, six, seven, eight possible outcomes, and that makes sense. You have two possible engines times four possible colors, two times four, and you see that right here, one g... | Count outcomes using tree diagram Statistics and probability 7th grade Khan Academy.mp3 |
You have two possible engines times four possible colors, two times four, and you see that right here, one group of four, two groups of four. So this outcome right here is a four-cylinder blue car. And this outcome over here is a six-cylinder green car. So there's eight equally possible outcomes, and which outcome matc... | Count outcomes using tree diagram Statistics and probability 7th grade Khan Academy.mp3 |
So there's eight equally possible outcomes, and which outcome matches the one that we, I guess, are hoping for, the white six-cylinder car? Well, that's this one right over here. It's one of eight equally likely events, so we have a 1 8th probability. Now, this wasn't the only way that we could have drawn the T diagram... | Count outcomes using tree diagram Statistics and probability 7th grade Khan Academy.mp3 |
Now, this wasn't the only way that we could have drawn the T diagram. We could have thought about color as the first row of this tree. So we could have said, look, we're either gonna get a, let me do it down here so I have a little more space. We're either gonna get a red, a blue, a, that's not blue, changing colors is... | Count outcomes using tree diagram Statistics and probability 7th grade Khan Academy.mp3 |
We're either gonna get a red, a blue, a, that's not blue, changing colors is the hard part, a blue, a green, man, a green, let me get, a green or a white car, or a white car, and then for each of those colors, I'm either gonna get a four-cylinder or a six-cylinder engine. So it's either gonna be four or a six, either g... | Count outcomes using tree diagram Statistics and probability 7th grade Khan Academy.mp3 |
Which of the following are accurate descriptions of the distribution below? Select all that apply. So the first statement is the distribution has an outlier. So an outlier is a data point that's way off of where the other data points are. It's way larger or way smaller than where all of the other data points seem to be... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
So an outlier is a data point that's way off of where the other data points are. It's way larger or way smaller than where all of the other data points seem to be clustered. And if we look over here, we have a lot of data points between zero and six. And just think about what they're measuring. This is shelf time for e... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
And just think about what they're measuring. This is shelf time for each apple at Gorge's Grocer. So for example, we see there's one, two, three, four, five, six, seven apples that have a shelf life of zero days. So they're about to go bad. You see you have one, two, three, four, five, six, seven, eight apples that are... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
So they're about to go bad. You see you have one, two, three, four, five, six, seven, eight apples that are gonna be good for another day. You have two apples that are gonna be good for another six days. And you have one apple that's gonna be good for 10 days. And this is unusual. This is an outlier here. It has a way ... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
And you have one apple that's gonna be good for 10 days. And this is unusual. This is an outlier here. It has a way larger shelf life than all of the other data. So I would say this definitely does have an outlier. We just have this one data point sitting all the way to the right. Way larger, way more shelf life than e... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
It has a way larger shelf life than all of the other data. So I would say this definitely does have an outlier. We just have this one data point sitting all the way to the right. Way larger, way more shelf life than everything else. So it definitely has an outlier. And this one would be the outlier. The distribution ha... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
Way larger, way more shelf life than everything else. So it definitely has an outlier. And this one would be the outlier. The distribution has a cluster from four to six days. And we indeed do see a cluster from four to six days. A cluster, you can imagine it's a grouping of data that's sitting there, or you have a gro... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
The distribution has a cluster from four to six days. And we indeed do see a cluster from four to six days. A cluster, you can imagine it's a grouping of data that's sitting there, or you have a grouping of apples that have a shelf life between four and six days. And you definitely do see that cluster there. And since ... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
And you definitely do see that cluster there. And since I already selected two things, I'm definitely not gonna select none of the above. So let me check my answer. Let me do a few more of these. Which of the following are accurate descriptions of the distribution below? And once again, we're going to select all that a... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
Let me do a few more of these. Which of the following are accurate descriptions of the distribution below? And once again, we're going to select all that apply. So the distribution has an outlier. So let's see, this distribution, I do have a data point here that's at the high end, and I have another data point here tha... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
So the distribution has an outlier. So let's see, this distribution, I do have a data point here that's at the high end, and I have another data point here that's at the low end. But I don't have any data points that are sitting far above or far below the bulk of the data. If I had a data point that was out here, then ... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
If I had a data point that was out here, then yeah, I would say that was an outlier to the right, or a positive outlier. If I had a data point way to the left off the screen over here, maybe that would be an outlier. But I don't really see any obvious outliers. All of the data, it's pretty clustered together. So I woul... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
All of the data, it's pretty clustered together. So I would not say that the distribution has an outlier. The distribution has a peak at 22 degrees. Yeah, it does indeed look like we have, and let's just look at what we're actually measuring, high temperature each day in Edgton, Iowa in July. So it does indeed look lik... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
Yeah, it does indeed look like we have, and let's just look at what we're actually measuring, high temperature each day in Edgton, Iowa in July. So it does indeed look like we have the most number of days that had a high temperature at 22, most number of days in July had a high temperature at 22 degrees Celsius. So tha... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
If you imagine this is kind of a mountain, this is a peak right here. This is a high point. You have, at least locally, you have the most number of days at 22 degrees Celsius. So I would say it definitely has a peak there. Since I selected something, I'm not gonna select none of the above. Let's do a couple more of the... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
So I would say it definitely has a peak there. Since I selected something, I'm not gonna select none of the above. Let's do a couple more of these. Which of the following are accurate descriptions of the distribution below? So the first one, the distribution has an outlier. So let's see, this number of guests by day at... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
Which of the following are accurate descriptions of the distribution below? So the first one, the distribution has an outlier. So let's see, this number of guests by day at Seth's Sandwich Shop. So let's see, the lowest, they have, so they have no days, no days where he had between zero and 19 guests. No days where he ... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
So let's see, the lowest, they have, so they have no days, no days where he had between zero and 19 guests. No days where he had between 20 and 39 guests. Looks like there's about nine days where he had between 40 and 59 guests. Looks like 20 days where he had between 60 and 79 guests. All the way, it looks like this i... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
Looks like 20 days where he had between 60 and 79 guests. All the way, it looks like this is maybe eight days that he had between 180 and 199 guests. But the question of outliers, there doesn't seem to be any day where he had an unusual number of guests. There's not a day that's way out here where he had like 500 guest... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
There's not a day that's way out here where he had like 500 guests. So I would say this distribution does not have an outlier. The distribution has a cluster from zero to 39 guests. So zero to 39 guests is right over here, zero to 39 guests. And there's no days where he had between zero and 39 guests, either zero to 19... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
So zero to 39 guests is right over here, zero to 39 guests. And there's no days where he had between zero and 39 guests, either zero to 19 or 20 to 39. So there's definitely not a cluster there. I would say that the cluster would be between, were days that had between 40 and 199 guests. Definitely not zero and 39. Ther... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
I would say that the cluster would be between, were days that had between 40 and 199 guests. Definitely not zero and 39. There was no days that were between zero and 39 guests. So I would say none of the above, very confidently. Let's do one more of these. Which of the following are accurate descriptions of the distrib... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
So I would say none of the above, very confidently. Let's do one more of these. Which of the following are accurate descriptions of the distribution below? All right. The distribution has a peak from 12 to 13 points. Let me see what this is measuring or what this data is about. Test scores by student in Ms. | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
All right. The distribution has a peak from 12 to 13 points. Let me see what this is measuring or what this data is about. Test scores by student in Ms. Friend's class. So you had one student who got between a zero and a one on the 20 point scale. So got between, I guess you may be out of 20 questions, got between zero... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
Test scores by student in Ms. Friend's class. So you had one student who got between a zero and a one on the 20 point scale. So got between, I guess you may be out of 20 questions, got between zero and one points. And then you see that those students got between two and three or four and five or six and seven. Then we ... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
So got between, I guess you may be out of 20 questions, got between zero and one points. And then you see that those students got between two and three or four and five or six and seven. Then we have another student who got between eight and nine. Looks like three students got between 10 and 11. And then we keep increa... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
Looks like three students got between 10 and 11. And then we keep increasing. This looks like it's about 12 students got either a 16 or a 17 or something in between, maybe if you could get decimal points on that test. And then it looks like 10 students got from 18 to 19. All right, so this says the distribution has a p... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
And then it looks like 10 students got from 18 to 19. All right, so this says the distribution has a peak from 12 to 13 points. 12 to 13 points. There were five students, but this isn't a peak. If you just go to 14 to 15 points, you have more students. So this is definitely not a peak. If you were looking at this as a ... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
There were five students, but this isn't a peak. If you just go to 14 to 15 points, you have more students. So this is definitely not a peak. If you were looking at this as a mountain of some kind, you definitely wouldn't describe this point as a peak. You would say this distribution has a peak, has the most number of ... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
If you were looking at this as a mountain of some kind, you definitely wouldn't describe this point as a peak. You would say this distribution has a peak, has the most number of students who got between 16 and 17 points. So that's the peak right there, not 12 to 13 points. So I would not select that first choice. The d... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
So I would not select that first choice. The distribution has an outlier. Well, yeah, look at this. You have this outlier. Most of the students scored between eight and 19 points. And then you have this one student who got between zero and one. It's really an outlier. | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
You have this outlier. Most of the students scored between eight and 19 points. And then you have this one student who got between zero and one. It's really an outlier. You even see this, when you look at it visually, it's not even connected to the rest of the distribution. It's way to the left. If something's way to t... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
It's really an outlier. You even see this, when you look at it visually, it's not even connected to the rest of the distribution. It's way to the left. If something's way to the left or way to the right, that's an outlier. If it's unusually low or unusually high. So I would say this distribution definitely does have an... | Examples analyzing clusters, gaps, peaks and outliers for distributions 6h grade Khan Academy.mp3 |
Here, we're gonna think about how we can make inferences from a regression line. And so the idea of statistical inference is new to you, or hypothesis testing, once again, watch those videos as well. But let's say we think there's a positive association between shoe size and height. And so what we might wanna do is, we... | Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3 |
And so what we might wanna do is, we could, here on the horizontal axis, that is shoe size, our sizes could go size one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, and it could keep going up from there. And then on this height, on this axis, our y-axis, this would be height. So one foot, two feet, thr... | Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3 |
And then you could, to see if there's an association, you might take a sample, let's say you take a random sample of 20 people from the population, and in future videos, we'll talk about the conditions necessary for making appropriate inferences. Well, let's say those 20 people are these 20 data points. So there's a yo... | Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3 |
You could do it by hand, but we have computers now to do that for us usually. And the computer could try to fit a regression line. And there's many techniques for doing it, but one typical technique is to try to overall minimize the square distance between these points and that line. And this regression line will have ... | Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3 |
And this regression line will have an equation as any line would have, and we tend to show that as saying y hat, this hat tells us that this is a regression line, is equal to the y-intercept, a, plus the slope times our x variable. So this right over here would be a. Now to be clear, if you took another sample, you mig... | Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3 |
In fact, let's call this y sub one for our first sample, a sub one, b sub one, and this is a sub one. If you were to take another sample of 20 folks, so let's do that, maybe you get one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and then you tried to fit a line to that... | Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3 |
And so every time you take a sample, you are likely to get different results for these values, which are essentially statistics. Remember, statistics are things that we can get from samples and we're trying to estimate true population parameters. Well, what would be the true population parameters we're trying to estima... | Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3 |
Well, imagine a world, imagine a world here that you're able to find out the true linear relationship, or maybe there is some true linear relationship between shoe size and height. You could get it if, theoretically, you could measure every human being on the planet and depending what you define as the population, it c... | Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3 |
You would have billions of data points here for the true population, and then if you were to fit a regression line to that, you could view this as the true population regression line. And so that would be y hat is equal to, and to make it clear that here, the y-intercept and the slope, this would be the true population... | Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3 |
But it's very hard to come up exactly with what alpha and beta are, and so that's why we estimate it with a's and b's based on a sample. Now, what's interesting with this in mind is we can start to make inferences based on our sample. So we know that, for example, b sub two is unlikely to be exactly beta, but how confi... | Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3 |
Or can we create a confidence interval around this statistic in order to have a good sense of where the true parameter might actually be? And the simple answer is yes. And to do so, we'll use the same exact ideas that we did when we made inferences based on proportions or based on means. The way that you can make an in... | Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3 |
The way that you can make an inference, for example, for your true population slope of your regression line, you say, okay, I took a sample. I got this slope right over here, so I'll just call that b two, and then I could create a confidence interval around that. And so that confidence interval is going to be based on ... | Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3 |
In this case, it would be the sample regression line slope. But because we don't know exactly what this is, we can't figure out precisely what this is going to be from a sample, we are going to estimate it with what's known as the standard error of the statistic, and we'll go into more depth in this in future videos. A... | Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3 |
And so based on your confidence level you wanna have, let's say it's 95%, based on the degrees of freedom, which we'll see will come out of how many data points we have, we can figure this out. And from our sample, we can figure this out, and we can figure this out, and then we would have constructed a confidence inter... | Introduction to inference about slope in linear regression AP Statistics Khan Academy.mp3 |
Each time one of you catches a fish, you release it back into the water. Jeremy offers you the choice of two different bets. Bet number one, bet number one. We don't encourage betting, but I guess Jeremy wants to bet. If the next three fish he catches are all sunfish, you will pay him $100. Otherwise, he will pay you $... | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
We don't encourage betting, but I guess Jeremy wants to bet. If the next three fish he catches are all sunfish, you will pay him $100. Otherwise, he will pay you $20. Bet two, if you catch at least two sunfish of the next three fish that you catch, he will pay you $50. Otherwise, you will pay him $25. What is your expe... | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
Bet two, if you catch at least two sunfish of the next three fish that you catch, he will pay you $50. Otherwise, you will pay him $25. What is your expected value from problem, from, what is the expected value from bet one? Round your answer to the nearest cent. And I encourage you to pause this video and try to think... | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
Round your answer to the nearest cent. And I encourage you to pause this video and try to think about it on your own. So let's see, the expected value of bet one. So the expected value of, let's just say bet one, or we'll say bet one is the, let's just define a random variable here just to be a little bit better about ... | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
So the expected value of, let's just say bet one, or we'll say bet one is the, let's just define a random variable here just to be a little bit better about this. So let's say X is equal to, is equal to what you pay, what you, or I guess you could say, because you might get something, what your profit is. Your profit i... | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
From bet one. And it's a random variable. And so the expected value, the expected value of X is going to be equal to, well let's see, what's the probability, it's going to be 100, it's going to be negative $100 times the probability that he catches three fish. So the probability that Jeremy, Jeremy catches three sunfis... | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
So the probability that Jeremy, Jeremy catches three sunfish, the next three, three, the next three, the next three fishy catches are going to be sunfish, times $100, times, or I should say, well you're going to pay that, so since you're paying it, we'll put it as negative 100, because we're saying that this is your ex... | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
In that situation, he'll pay you $20. You get $20 there. So the important thing is, is to figure out the probability that Jeremy catches three sunfish. Well the sunfish are 10 out of the 20 fish, so at any given time he's trying to catch fish, there's a 10 in 20 chance, or you could say a 1 1 2 probability that's going... | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
Well the sunfish are 10 out of the 20 fish, so at any given time he's trying to catch fish, there's a 10 in 20 chance, or you could say a 1 1 2 probability that's going to be a sunfish. So the probability that you get three sunfish in a row is going to be 1 1 2, times 1 1 2, times 1 1 2. And they put the fish back in, ... | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
If he wasn't putting the fish back in, then the second sunfish, you would have a nine out of 20 chance of the second one being a sunfish. But in this case, they keep replacing the fish every time they catch it. So there's a 1 8th chance that Jeremy catches three sunfish. So this right over here is 1 8th. And one minus ... | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
So this right over here is 1 8th. And one minus 1 8th, this is 7 8ths. 7 8ths. So you have a 1 8th chance of paying $100, and a 7 8ths chance of getting $20. And so this gets us two. This gets us two. So you're expected, I guess you could say profit here. | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
So you have a 1 8th chance of paying $100, and a 7 8ths chance of getting $20. And so this gets us two. This gets us two. So you're expected, I guess you could say profit here. There's a 1 8th chance, 1 8th probability, that you lose $100 here. So times negative 100. But then there is a 7 8th chance that you get, I'll ... | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
So you're expected, I guess you could say profit here. There's a 1 8th chance, 1 8th probability, that you lose $100 here. So times negative 100. But then there is a 7 8th chance that you get, I'll just put parentheses here to make it a little clearer. I think the order of operations on the calculator would have taken ... | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
But then there is a 7 8th chance that you get, I'll just put parentheses here to make it a little clearer. I think the order of operations on the calculator would have taken care of it, but I'll just do it just so that it looks the same. 7 8ths, there's a 7 8th chance that you get $20. And so your expected payoff here ... | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
And so your expected payoff here is positive $5. So your expected payoff here is equal to $5. So this is your expected value from bet one. Now let's think about bet two. Bet two. If you catch at least two sunfish of the next three fish you catch, he will pay you 50. Otherwise, you will pay him 25. | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
Now let's think about bet two. Bet two. If you catch at least two sunfish of the next three fish you catch, he will pay you 50. Otherwise, you will pay him 25. So let's think about the probability of catching at least two sunfish of the next three fish that you catch. Now there's a bunch of ways to think about this, bu... | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
Otherwise, you will pay him 25. So let's think about the probability of catching at least two sunfish of the next three fish that you catch. Now there's a bunch of ways to think about this, but since there's only kind of three times that you're trying to catch the fish, and there's only one of two outcomes, you could a... | Expected value while fishing Probability and Statistics Khan Academy.mp3 |
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