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With that out of the way, let's think about your actual question. To answer it, I dug up your stats right over here from NBA.com. We'll use your career stats. We want to compare three free throws to a three pointer. Right over here we have your three pointer percentage. This is in your career. I'll round it to the nearest hundredths. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
We want to compare three free throws to a three pointer. Right over here we have your three pointer percentage. This is in your career. I'll round it to the nearest hundredths. It looks like it's about 33%. Your three point percentage is 33%. Then your free throw percentage, your career free throw percentage, and this is in your career, we'll round to the nearest hundredths. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
I'll round it to the nearest hundredths. It looks like it's about 33%. Your three point percentage is 33%. Then your free throw percentage, your career free throw percentage, and this is in your career, we'll round to the nearest hundredths. We'll round up right over here. That gets us to right at about 75%. Clearly, looking at these numbers right over here, you're much more likely to make a given free throw than making a given three pointer. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
Then your free throw percentage, your career free throw percentage, and this is in your career, we'll round to the nearest hundredths. We'll round up right over here. That gets us to right at about 75%. Clearly, looking at these numbers right over here, you're much more likely to make a given free throw than making a given three pointer. You're more than twice as likely to make a free throw. But that's not what you asked. You asked, what about three free throws in a row? | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
Clearly, looking at these numbers right over here, you're much more likely to make a given free throw than making a given three pointer. You're more than twice as likely to make a free throw. But that's not what you asked. You asked, what about three free throws in a row? We'll do an analysis very similar to the last time when we asked about 10 free throws in a row. Let's think about the first free throw. Free throw number one. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
You asked, what about three free throws in a row? We'll do an analysis very similar to the last time when we asked about 10 free throws in a row. Let's think about the first free throw. Free throw number one. If we imagined a gazillion LeBron Jameses, identical LeBron Jameses, all taking that first free throw, we would expect on average that 75% would make that first free throw. 75% is three-fourths. 75% would make that first free throw. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
Free throw number one. If we imagined a gazillion LeBron Jameses, identical LeBron Jameses, all taking that first free throw, we would expect on average that 75% would make that first free throw. 75% is three-fourths. 75% would make that first free throw. 25% we would expect on average, won't always be the case, but this is what we would expect, 25% would miss that first free throw. Let's go to the second free throw. Free throw number two. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
75% would make that first free throw. 25% we would expect on average, won't always be the case, but this is what we would expect, 25% would miss that first free throw. Let's go to the second free throw. Free throw number two. We only care about the ones that the LeBron Jameses that keep making their free throws. Let's think about of the 75% that made that first one. Some of the 25% might make that second one and then maybe even the third one, but let's just think about the ones that made the first one. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
Free throw number two. We only care about the ones that the LeBron Jameses that keep making their free throws. Let's think about of the 75% that made that first one. Some of the 25% might make that second one and then maybe even the third one, but let's just think about the ones that made the first one. Of the ones that made the first one, we would expect 75% of them to make the second one. 75% of the 75%, that's half of the 75%, that's about 75% of the 75% would make that second free throw and the first free throw. Now we have, this is going to be 75% times 75%. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
Some of the 25% might make that second one and then maybe even the third one, but let's just think about the ones that made the first one. Of the ones that made the first one, we would expect 75% of them to make the second one. 75% of the 75%, that's half of the 75%, that's about 75% of the 75% would make that second free throw and the first free throw. Now we have, this is going to be 75% times 75%. Of course, there's other combinations out here where someone's missed at least one of the free throws. Let's go to the third free throw. Free throw number three. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
Now we have, this is going to be 75% times 75%. Of course, there's other combinations out here where someone's missed at least one of the free throws. Let's go to the third free throw. Free throw number three. What percentage of these LeBron Jameses right here will make the third free throw? 75% of these will make the third free throw. 75% of this number, let me just draw it visually, that's half, that's about 75% of that number. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
Free throw number three. What percentage of these LeBron Jameses right here will make the third free throw? 75% of these will make the third free throw. 75% of this number, let me just draw it visually, that's half, that's about 75% of that number. They will make the third one as well. This is 75% of this number, which is 75% of 75%. This is how many LeBron Jameses are going to make all three of the free throws. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
75% of this number, let me just draw it visually, that's half, that's about 75% of that number. They will make the third one as well. This is 75% of this number, which is 75% of 75%. This is how many LeBron Jameses are going to make all three of the free throws. Once again, we can write this as, we can either multiply it out or we can just write this as 75% to the third power, which is the same thing as 75%, literally means 75 per 100, same thing as 75 over 100 to the third power, which is the same thing as 0.75 to the third power. Let's calculate it, get the calculator out. Actually, let me show you, we get the same result. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
This is how many LeBron Jameses are going to make all three of the free throws. Once again, we can write this as, we can either multiply it out or we can just write this as 75% to the third power, which is the same thing as 75%, literally means 75 per 100, same thing as 75 over 100 to the third power, which is the same thing as 0.75 to the third power. Let's calculate it, get the calculator out. Actually, let me show you, we get the same result. We can write 0.75 times 0.75. On this calculator, that little snowflake looking thing, it means multiplication, times 0.75. Then we get 0.42, I'll round to the nearest hundredths. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
Actually, let me show you, we get the same result. We can write 0.75 times 0.75. On this calculator, that little snowflake looking thing, it means multiplication, times 0.75. Then we get 0.42, I'll round to the nearest hundredths. That's the same thing we would get if we got 0.75 to the third power. Once again, 0.42. Let me write that. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
Then we get 0.42, I'll round to the nearest hundredths. That's the same thing we would get if we got 0.75 to the third power. Once again, 0.42. Let me write that. This gets us to approximately 0.42, which is the same thing as 42%. Your probability of making three free throws in a row is 42%, which is still higher than making one three-pointer. I'll leave you there, but I want the people who are watching this video to think about what would happen if the numbers were different, or maybe look up some NBA players or maybe some college players, and figure out and compare the probability of making three free throws to a three-pointer, and see if you can find any players where their probability of making a three-pointer is actually higher than making three free throws in a row. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
Accident frequency, and I'm just making this up. And I could just show these data points, maybe for some kind of statistical survey, that when the age is this, whatever number this is, maybe this is 20 years old, this is the accident frequency, and it could be a number of accidents per hundred. And that when the age is 21 years old, this is the frequency. And so these data scientists or statisticians went and plotted all of these in this scatter plot. This is often known as bivariate data, which is a very fancy way of saying, hey, you're plotting things that take two variables into consideration, and you're trying to see whether there's a pattern with how they relate. And what we're going to do in this video is think about, well, can we try to fit a line? Does it look like there's a linear or nonlinear relationship between the variables on the different axes? | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
And so these data scientists or statisticians went and plotted all of these in this scatter plot. This is often known as bivariate data, which is a very fancy way of saying, hey, you're plotting things that take two variables into consideration, and you're trying to see whether there's a pattern with how they relate. And what we're going to do in this video is think about, well, can we try to fit a line? Does it look like there's a linear or nonlinear relationship between the variables on the different axes? How strong is that variable? Is it a positive, is it a negative relationship? And then we'll think about this idea of outliers. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
Does it look like there's a linear or nonlinear relationship between the variables on the different axes? How strong is that variable? Is it a positive, is it a negative relationship? And then we'll think about this idea of outliers. So let's just first think about whether there's a linear or nonlinear relationship. And I'll get my little ruler tool out here. So this data right over here, it looks like I could put a line through it that gets pretty close through the data. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
And then we'll think about this idea of outliers. So let's just first think about whether there's a linear or nonlinear relationship. And I'll get my little ruler tool out here. So this data right over here, it looks like I could put a line through it that gets pretty close through the data. You're not gonna, it's very unlikely you're gonna be able to go through all of the data points, but you can try to get a line, and I'm just doing this. There's more numerical, more precise ways of doing this, but I'm just eyeballing it right over here. And it looks like I could plot a line that looks something like that that goes roughly through the data. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
So this data right over here, it looks like I could put a line through it that gets pretty close through the data. You're not gonna, it's very unlikely you're gonna be able to go through all of the data points, but you can try to get a line, and I'm just doing this. There's more numerical, more precise ways of doing this, but I'm just eyeballing it right over here. And it looks like I could plot a line that looks something like that that goes roughly through the data. So this looks pretty linear. So I would call this a linear relationship. And since as we increase one variable, it looks like the other variable decreases. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
And it looks like I could plot a line that looks something like that that goes roughly through the data. So this looks pretty linear. So I would call this a linear relationship. And since as we increase one variable, it looks like the other variable decreases. This is a downward-sloping line. I would say this is a negative, this is a negative linear relationship. But this one looks pretty strong. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
And since as we increase one variable, it looks like the other variable decreases. This is a downward-sloping line. I would say this is a negative, this is a negative linear relationship. But this one looks pretty strong. So because the dots aren't that far from my line, this one gets a little bit further, but it's not, you know, there's not some dots way out there. So most of them are pretty close to the line. So I'll call this a negative, reasonably strong linear relationship. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
But this one looks pretty strong. So because the dots aren't that far from my line, this one gets a little bit further, but it's not, you know, there's not some dots way out there. So most of them are pretty close to the line. So I'll call this a negative, reasonably strong linear relationship. Negative, strong. I call reasonably, I ought to say strong, but reasonably strong linear, linear relationship between these two variables. Now let's look at this one, and pause this video and think about what this one would be for you. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
So I'll call this a negative, reasonably strong linear relationship. Negative, strong. I call reasonably, I ought to say strong, but reasonably strong linear, linear relationship between these two variables. Now let's look at this one, and pause this video and think about what this one would be for you. Well, let's see. I'll get my ruler tool out again. It looks like I can try to put a line. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
Now let's look at this one, and pause this video and think about what this one would be for you. Well, let's see. I'll get my ruler tool out again. It looks like I can try to put a line. It looks like generally speaking, as one variable increases, the other variable increases as well. So something like this goes through the data and approximates the direction. And this looks positive. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
It looks like I can try to put a line. It looks like generally speaking, as one variable increases, the other variable increases as well. So something like this goes through the data and approximates the direction. And this looks positive. As one variable increases, the other variable increases, roughly. So this is a positive relationship. But this is weak. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
And this looks positive. As one variable increases, the other variable increases, roughly. So this is a positive relationship. But this is weak. A lot of the data is off, well off of the line. So positive, weak. But I'd say this is still linear. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
But this is weak. A lot of the data is off, well off of the line. So positive, weak. But I'd say this is still linear. It seems that as we increase one, the other one increases at roughly the same rate, although these data points are all over the place. So I would still call this linear. Now there's also this notion of outliers. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
But I'd say this is still linear. It seems that as we increase one, the other one increases at roughly the same rate, although these data points are all over the place. So I would still call this linear. Now there's also this notion of outliers. If I said, hey, this line is trying to describe the data, well, we have some data that is fairly off the line. So for example, even though we're saying it's a positive, weak, linear relationship, this one over here is reasonably high on the vertical variable, but it's low on the horizontal variable. And so this one right over here is an outlier. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
Now there's also this notion of outliers. If I said, hey, this line is trying to describe the data, well, we have some data that is fairly off the line. So for example, even though we're saying it's a positive, weak, linear relationship, this one over here is reasonably high on the vertical variable, but it's low on the horizontal variable. And so this one right over here is an outlier. It's quite far away from the line. You could view that as an outlier. And this is a little bit subjective. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
And so this one right over here is an outlier. It's quite far away from the line. You could view that as an outlier. And this is a little bit subjective. Outliers, well, what looks pretty far from the rest of the data? This could also be an outlier. Let me label these. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
And this is a little bit subjective. Outliers, well, what looks pretty far from the rest of the data? This could also be an outlier. Let me label these. Outlier. Now pause the video and see if you can think about this one. Is this positive or negative? | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
Let me label these. Outlier. Now pause the video and see if you can think about this one. Is this positive or negative? Is it linear, non-linear? Is it strong or weak? I'll get my ruler tool out here. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
Is this positive or negative? Is it linear, non-linear? Is it strong or weak? I'll get my ruler tool out here. So this goes here. It seems like I can fit a line pretty well to this. So I could fit, maybe I'll do the line in purple. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
I'll get my ruler tool out here. So this goes here. It seems like I can fit a line pretty well to this. So I could fit, maybe I'll do the line in purple. I could fit a line that looks like that. And so this one looks like it's positive. As one variable increases, the other one does for these data points. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
So I could fit, maybe I'll do the line in purple. I could fit a line that looks like that. And so this one looks like it's positive. As one variable increases, the other one does for these data points. So it's a positive. I'd say this is pretty strong. The dots are pretty close to the line. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
As one variable increases, the other one does for these data points. So it's a positive. I'd say this is pretty strong. The dots are pretty close to the line. It really does look like a little bit of a fat line if you just look at the dots. So positive, strong, linear, linear relationship. And none of these data points are really strong outliers. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
The dots are pretty close to the line. It really does look like a little bit of a fat line if you just look at the dots. So positive, strong, linear, linear relationship. And none of these data points are really strong outliers. This one's a little bit further out. But they're all pretty close to the line and seem to describe that trend roughly. All right, now let's look at this data right over here. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
And none of these data points are really strong outliers. This one's a little bit further out. But they're all pretty close to the line and seem to describe that trend roughly. All right, now let's look at this data right over here. So let me get my line tool out again. So it looks like I can fit a line. So it looks like it's a positive relationship. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
All right, now let's look at this data right over here. So let me get my line tool out again. So it looks like I can fit a line. So it looks like it's a positive relationship. The line would be upward sloping. It would look something like this. And once again, I'm eyeballing it. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
So it looks like it's a positive relationship. The line would be upward sloping. It would look something like this. And once again, I'm eyeballing it. You can use computers and other methods to actually find a more precise line that minimizes the collective distance to all of the points. But it looks like there is a positive. But I would say this one is a weak linear relationship because you have a lot of points that are far off the line. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
And once again, I'm eyeballing it. You can use computers and other methods to actually find a more precise line that minimizes the collective distance to all of the points. But it looks like there is a positive. But I would say this one is a weak linear relationship because you have a lot of points that are far off the line. So not so strong. So I would call this a positive, weak, linear relationship. And there's a lot of outliers here. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
But I would say this one is a weak linear relationship because you have a lot of points that are far off the line. So not so strong. So I would call this a positive, weak, linear relationship. And there's a lot of outliers here. You know, this one over here is pretty far out. Now let's look at this one. Pause this video and think about is it positive, negative? | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
And there's a lot of outliers here. You know, this one over here is pretty far out. Now let's look at this one. Pause this video and think about is it positive, negative? Is it strong or weak? Is this linear, nonlinear? Well, the first thing we wanna do is just think about it in linear, nonlinear. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
Pause this video and think about is it positive, negative? Is it strong or weak? Is this linear, nonlinear? Well, the first thing we wanna do is just think about it in linear, nonlinear. I could try to put a line on it. But if I try to put a line on it, it's actually quite difficult. If I try to do a line like this, notice everything is kind of bending away from the line. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
Well, the first thing we wanna do is just think about it in linear, nonlinear. I could try to put a line on it. But if I try to put a line on it, it's actually quite difficult. If I try to do a line like this, notice everything is kind of bending away from the line. It looks like generally as one variable increases, the other variable decreases. But they're not doing it in a linear fashion. It looks like there's some other type of curve at play. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
If I try to do a line like this, notice everything is kind of bending away from the line. It looks like generally as one variable increases, the other variable decreases. But they're not doing it in a linear fashion. It looks like there's some other type of curve at play. So I could try to do a fancier curve that looks something like this. And this seems to fit the data a lot better. So this one I would describe as nonlinear. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
It looks like there's some other type of curve at play. So I could try to do a fancier curve that looks something like this. And this seems to fit the data a lot better. So this one I would describe as nonlinear. And it is a negative relationship. As one variable increases, the other variable decreases. So this is a negative, I would say reasonably strong nonlinear relationship. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
So this one I would describe as nonlinear. And it is a negative relationship. As one variable increases, the other variable decreases. So this is a negative, I would say reasonably strong nonlinear relationship. Pretty strong. Pretty strong. Once again, this is subjective. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
So this is a negative, I would say reasonably strong nonlinear relationship. Pretty strong. Pretty strong. Once again, this is subjective. So I'll say negative, reasonably strong, nonlinear relationship. And maybe you could call this one an outlier, but it's not that far. And I might even be able to fit a curve that gets a little bit closer to that. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
Once again, this is subjective. So I'll say negative, reasonably strong, nonlinear relationship. And maybe you could call this one an outlier, but it's not that far. And I might even be able to fit a curve that gets a little bit closer to that. Once again, I'm eyeballing this. Now let's do this last one. So this one looks like a negative linear relationship to me. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
And I might even be able to fit a curve that gets a little bit closer to that. Once again, I'm eyeballing this. Now let's do this last one. So this one looks like a negative linear relationship to me. A fairly strong negative linear relationship, although there's some outliers. So let me draw this line. So that seems to fit the data pretty good. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
So this one looks like a negative linear relationship to me. A fairly strong negative linear relationship, although there's some outliers. So let me draw this line. So that seems to fit the data pretty good. So this is a negative, reasonably strong, reasonably strong linear relationship. But these are very clear outliers. These are well away from the data or from the cluster of where most of the points are. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
So that seems to fit the data pretty good. So this is a negative, reasonably strong, reasonably strong linear relationship. But these are very clear outliers. These are well away from the data or from the cluster of where most of the points are. So with some significant, with at least these two significant outliers here. So hopefully this makes you a little bit familiar with some of this terminology. And it's important to keep in mind, this is a little bit subjective. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
These are well away from the data or from the cluster of where most of the points are. So with some significant, with at least these two significant outliers here. So hopefully this makes you a little bit familiar with some of this terminology. And it's important to keep in mind, this is a little bit subjective. There'll be some cases that are more obvious than others. So for, and oftentimes you wanna make a comparison. That this is a stronger linear, positive linear relationship than this one is, right over here, because you can see most of the data is closer to the line. | Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3 |
A set of middle school students' heights are normally distributed with a mean of 150 centimeters and a standard deviation of 20 centimeters. Darnell is a middle school student with a height of 161.4 centimeters. What proportion of student heights are lower than Darnell's height? So let's think about what they are asking. So they're saying that heights are normally distributed. So it would have a shape that looks something like that. That's my hand-drawn version of it. | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
So let's think about what they are asking. So they're saying that heights are normally distributed. So it would have a shape that looks something like that. That's my hand-drawn version of it. There's a mean of 150 centimeters. So right over here, that would be 150 centimeters. They tell us that there's a standard deviation of 20 centimeters, and Darnell has a height of 161.4 centimeters. | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
That's my hand-drawn version of it. There's a mean of 150 centimeters. So right over here, that would be 150 centimeters. They tell us that there's a standard deviation of 20 centimeters, and Darnell has a height of 161.4 centimeters. So Darnell is above the mean. So let's say he is right over here, and I'm not drawing it exactly, but you get the idea. That is 161.4 centimeters. | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
They tell us that there's a standard deviation of 20 centimeters, and Darnell has a height of 161.4 centimeters. So Darnell is above the mean. So let's say he is right over here, and I'm not drawing it exactly, but you get the idea. That is 161.4 centimeters. And we want to figure out what proportion of students' heights are lower than Darnell's height. So we want to figure out what is the area under the normal curve right over here. That will give us the proportion that are lower than Darnell's height. | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
That is 161.4 centimeters. And we want to figure out what proportion of students' heights are lower than Darnell's height. So we want to figure out what is the area under the normal curve right over here. That will give us the proportion that are lower than Darnell's height. And I'll give you a hint on how to do this. We need to think about how many standard deviations above the mean is Darnell. And we can do that because they tell us what the standard deviation is, and we know the difference between Darnell's height and the mean height. | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
That will give us the proportion that are lower than Darnell's height. And I'll give you a hint on how to do this. We need to think about how many standard deviations above the mean is Darnell. And we can do that because they tell us what the standard deviation is, and we know the difference between Darnell's height and the mean height. And then once we know how many standard deviations he is above the mean, that's our z-score, we can look at a z-table that tell us what proportion is less than that amount in a normal distribution. So let's do that. So I have my TI-84 emulator right over here. | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
And we can do that because they tell us what the standard deviation is, and we know the difference between Darnell's height and the mean height. And then once we know how many standard deviations he is above the mean, that's our z-score, we can look at a z-table that tell us what proportion is less than that amount in a normal distribution. So let's do that. So I have my TI-84 emulator right over here. And let's see, Darnell is 161.4 centimeters, 161.4. Now the mean is 150, minus 150, is equal to, we could have done that in our head, 11.4 centimeters. Now how many standard deviations is that above the mean? | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
So I have my TI-84 emulator right over here. And let's see, Darnell is 161.4 centimeters, 161.4. Now the mean is 150, minus 150, is equal to, we could have done that in our head, 11.4 centimeters. Now how many standard deviations is that above the mean? Well they tell us that a standard deviation in this case for this distribution is 20 centimeters. So we'll take 11.4 divided by 20. So we will just take our previous answer. | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
Now how many standard deviations is that above the mean? Well they tell us that a standard deviation in this case for this distribution is 20 centimeters. So we'll take 11.4 divided by 20. So we will just take our previous answer. So this just means our previous answer divided by 20 centimeters. And that gets us 0.57. So we can say that this is 0.57 standard deviations 0.57 standard deviations, deviations above the mean. | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
So we will just take our previous answer. So this just means our previous answer divided by 20 centimeters. And that gets us 0.57. So we can say that this is 0.57 standard deviations 0.57 standard deviations, deviations above the mean. Now why is that useful? Well you could take this z-score right over here and look at a z-table to figure out what proportion is less than 0.57 standard deviations above the mean. So let's get a z-table over here. | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
So we can say that this is 0.57 standard deviations 0.57 standard deviations, deviations above the mean. Now why is that useful? Well you could take this z-score right over here and look at a z-table to figure out what proportion is less than 0.57 standard deviations above the mean. So let's get a z-table over here. So what we're going to do is we're gonna look up this z-score on this table. And the way that you do it, this first column, each row tells us our z-score up until the tenths place. And then each of these columns after that tell us which hundredths we're in. | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
So let's get a z-table over here. So what we're going to do is we're gonna look up this z-score on this table. And the way that you do it, this first column, each row tells us our z-score up until the tenths place. And then each of these columns after that tell us which hundredths we're in. So 0.57, the tenths place is right over here. So we're going to be in this row. And then our hundredths place is this seven. | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
And then each of these columns after that tell us which hundredths we're in. So 0.57, the tenths place is right over here. So we're going to be in this row. And then our hundredths place is this seven. So we'll look right over here. So 0.57, this tells us the proportion that is lower than 0.57 standard deviations above the mean. And so it is 0.7157, or another way to think about it is, if the heights are truly normally distributed, 71.57% of the students would have a height less than Darnell's. | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
And then our hundredths place is this seven. So we'll look right over here. So 0.57, this tells us the proportion that is lower than 0.57 standard deviations above the mean. And so it is 0.7157, or another way to think about it is, if the heights are truly normally distributed, 71.57% of the students would have a height less than Darnell's. But the answer to this question, what proportion of students' heights are lower than Darnell's height? Well, that would be 0.7157. And they want our answer to four decimal places, which is exactly what we have done. | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
Well ideally, we would go to the entire population of likely voters right over here. Let's say there's 100,000 likely voters and we would ask every one of them, who do you support? And from that, we would be able to get the population proportion. Which would be, this is the proportion that support candidate A. But it might not be realistic. In fact, it definitely will not be realistic to ask all 100,000 people. So instead, we do the thing that we tend to do in statistics is that we sample this population and we calculate a statistic from that sample in order to estimate this parameter. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
Which would be, this is the proportion that support candidate A. But it might not be realistic. In fact, it definitely will not be realistic to ask all 100,000 people. So instead, we do the thing that we tend to do in statistics is that we sample this population and we calculate a statistic from that sample in order to estimate this parameter. So let's say we take a sample right over here. So this sample size, let's say n equals 100 and we calculate the sample proportion that support candidate A. So out of the 100, let's say that 54 say that they're going to support candidate A. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So instead, we do the thing that we tend to do in statistics is that we sample this population and we calculate a statistic from that sample in order to estimate this parameter. So let's say we take a sample right over here. So this sample size, let's say n equals 100 and we calculate the sample proportion that support candidate A. So out of the 100, let's say that 54 say that they're going to support candidate A. So the sample proportion here is 0.54. And just to appreciate that we're not always going to get 0.54, there could have been a situation where we sampled a different 100 and we would have maybe gotten a different sample proportion, maybe in that one we got 0.58. And we already have the tools in statistics to think about this, the distribution of the possible sample proportions we could get. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So out of the 100, let's say that 54 say that they're going to support candidate A. So the sample proportion here is 0.54. And just to appreciate that we're not always going to get 0.54, there could have been a situation where we sampled a different 100 and we would have maybe gotten a different sample proportion, maybe in that one we got 0.58. And we already have the tools in statistics to think about this, the distribution of the possible sample proportions we could get. We've talked about it when we thought about sampling distributions. So you could have the sampling distribution of the sample proportions, of the sample proportions, and it's going to, this distribution is going to be specific to what our sample size is for n is equal to 100. And so we can describe the possible sample proportions we could get and their likelihoods with this sampling distribution. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
And we already have the tools in statistics to think about this, the distribution of the possible sample proportions we could get. We've talked about it when we thought about sampling distributions. So you could have the sampling distribution of the sample proportions, of the sample proportions, and it's going to, this distribution is going to be specific to what our sample size is for n is equal to 100. And so we can describe the possible sample proportions we could get and their likelihoods with this sampling distribution. So let me do that. So it would look something like this. Because our sample size is so much smaller than the population, it's way less than 10%, we can assume that each person we're asking that it's approximately independent. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
And so we can describe the possible sample proportions we could get and their likelihoods with this sampling distribution. So let me do that. So it would look something like this. Because our sample size is so much smaller than the population, it's way less than 10%, we can assume that each person we're asking that it's approximately independent. Also, if we make the assumption that the true proportion isn't too close to zero or not too close to one, then we can say that, well look, this sampling distribution is roughly going to be normal. So we'll have a normal, this kind of bell curve shape. And we know a lot about the sampling distribution of the sample proportions. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
Because our sample size is so much smaller than the population, it's way less than 10%, we can assume that each person we're asking that it's approximately independent. Also, if we make the assumption that the true proportion isn't too close to zero or not too close to one, then we can say that, well look, this sampling distribution is roughly going to be normal. So we'll have a normal, this kind of bell curve shape. And we know a lot about the sampling distribution of the sample proportions. We know already, for example, and if this is foreign to you, I encourage you to watch the videos on this on Khan Academy, that the mean of this sampling distribution is going to be the actual population proportion. And we also know what the standard deviation of this is going to be. So let me, that's maybe, that's one standard deviation, this is two standard deviations, that's three standard deviations. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
And we know a lot about the sampling distribution of the sample proportions. We know already, for example, and if this is foreign to you, I encourage you to watch the videos on this on Khan Academy, that the mean of this sampling distribution is going to be the actual population proportion. And we also know what the standard deviation of this is going to be. So let me, that's maybe, that's one standard deviation, this is two standard deviations, that's three standard deviations. Above the mean, that's one standard deviation, two standard deviations, three standard deviations below the mean. So this distance, let me do this in a different color. This standard deviation right over here, which we denote as the standard deviation of the sample proportions for this sampling distribution. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So let me, that's maybe, that's one standard deviation, this is two standard deviations, that's three standard deviations. Above the mean, that's one standard deviation, two standard deviations, three standard deviations below the mean. So this distance, let me do this in a different color. This standard deviation right over here, which we denote as the standard deviation of the sample proportions for this sampling distribution. This is, we've already seen the formula there. It's the square root of p times one minus p, where p is once again our population proportion, divided by our sample size. That's why it's specific for n equals 100 here. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
This standard deviation right over here, which we denote as the standard deviation of the sample proportions for this sampling distribution. This is, we've already seen the formula there. It's the square root of p times one minus p, where p is once again our population proportion, divided by our sample size. That's why it's specific for n equals 100 here. And so in this first scenario, and let's just focus on this one right over here. When we took a sample size of n equals 100 and we got the sample proportion of 0.54, we could have gotten all sorts of outcomes here. Maybe 0.54 is right over here. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
That's why it's specific for n equals 100 here. And so in this first scenario, and let's just focus on this one right over here. When we took a sample size of n equals 100 and we got the sample proportion of 0.54, we could have gotten all sorts of outcomes here. Maybe 0.54 is right over here. Maybe 0.54 is right over here. And the reason why I have this uncertainty is we actually don't know what the real population parameter is, what the real population proportion is. But let me ask you maybe a slightly easier question. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
Maybe 0.54 is right over here. Maybe 0.54 is right over here. And the reason why I have this uncertainty is we actually don't know what the real population parameter is, what the real population proportion is. But let me ask you maybe a slightly easier question. What is, what is the probability, probability that our sample proportion of 0.54 is within, is within two times two standard deviations of p? Pause the video and think about that. Well, that's just saying, look, if I'm gonna take a sample and calculate the sample proportion right over here, what's the probability that I'm within two standard deviations of the mean? | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
But let me ask you maybe a slightly easier question. What is, what is the probability, probability that our sample proportion of 0.54 is within, is within two times two standard deviations of p? Pause the video and think about that. Well, that's just saying, look, if I'm gonna take a sample and calculate the sample proportion right over here, what's the probability that I'm within two standard deviations of the mean? Well, that's essentially going to be this area right over here. And we know from studying normal curves that approximately 95% of the area is within two standard deviations. So this is approximately 95%. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
Well, that's just saying, look, if I'm gonna take a sample and calculate the sample proportion right over here, what's the probability that I'm within two standard deviations of the mean? Well, that's essentially going to be this area right over here. And we know from studying normal curves that approximately 95% of the area is within two standard deviations. So this is approximately 95%. 95% of the time that I take a sample size of 100 and I calculate this sample proportion, 95% of the time I'm going to be within two standard deviations. But if you take this statement, you can actually construct another statement that starts to feel a little bit more, I guess we could say, inferential. We could say there, there is a 95% probability that the population proportion, p, is within two standard deviations, two standard deviations of p hat, which is equal to 0.54. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So this is approximately 95%. 95% of the time that I take a sample size of 100 and I calculate this sample proportion, 95% of the time I'm going to be within two standard deviations. But if you take this statement, you can actually construct another statement that starts to feel a little bit more, I guess we could say, inferential. We could say there, there is a 95% probability that the population proportion, p, is within two standard deviations, two standard deviations of p hat, which is equal to 0.54. Pause this video. Appreciate that these two are equivalent statements. If there's a 95% chance that our sample proportion is within two standard deviations of the true proportion, well, that's equivalent to saying that there's a 95% chance that our true proportion is within two standard deviations of our sample proportion. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
We could say there, there is a 95% probability that the population proportion, p, is within two standard deviations, two standard deviations of p hat, which is equal to 0.54. Pause this video. Appreciate that these two are equivalent statements. If there's a 95% chance that our sample proportion is within two standard deviations of the true proportion, well, that's equivalent to saying that there's a 95% chance that our true proportion is within two standard deviations of our sample proportion. And this is really, really interesting because if we were able to figure out what this value is, well, then we would be able to create what you could call a confidence interval. Now, you immediately might be seeing a problem here. In order to calculate this, our standard deviation of this distribution, we have to know our population parameter. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
If there's a 95% chance that our sample proportion is within two standard deviations of the true proportion, well, that's equivalent to saying that there's a 95% chance that our true proportion is within two standard deviations of our sample proportion. And this is really, really interesting because if we were able to figure out what this value is, well, then we would be able to create what you could call a confidence interval. Now, you immediately might be seeing a problem here. In order to calculate this, our standard deviation of this distribution, we have to know our population parameter. So pause this video and think about what we would do instead if we don't know what p is here, if we don't know our population proportion, do we have something that we could use as an estimate for our population proportion? Well, yes, we calculated p hat already. We calculated our sample proportion. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
In order to calculate this, our standard deviation of this distribution, we have to know our population parameter. So pause this video and think about what we would do instead if we don't know what p is here, if we don't know our population proportion, do we have something that we could use as an estimate for our population proportion? Well, yes, we calculated p hat already. We calculated our sample proportion. And so a new statistic that we could define is the standard error. The standard error of our sample proportions. And we can define that as being equal to, since we don't know the population proportion, we're gonna use a sample proportion. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
We calculated our sample proportion. And so a new statistic that we could define is the standard error. The standard error of our sample proportions. And we can define that as being equal to, since we don't know the population proportion, we're gonna use a sample proportion. P hat times one minus p hat, all of that over n. In this case, of course, n is 100. We do know that. And it actually turns out, I'm not gonna prove it in this video, that this actually is an unbiased estimator for this right over here. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
And we can define that as being equal to, since we don't know the population proportion, we're gonna use a sample proportion. P hat times one minus p hat, all of that over n. In this case, of course, n is 100. We do know that. And it actually turns out, I'm not gonna prove it in this video, that this actually is an unbiased estimator for this right over here. So this is going to be equal to 0.54 times one minus 0.54. So it's 0.46. All of that over 100. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
And it actually turns out, I'm not gonna prove it in this video, that this actually is an unbiased estimator for this right over here. So this is going to be equal to 0.54 times one minus 0.54. So it's 0.46. All of that over 100. So we have the square root of 0.54 times 0.46 divided by 100, close my parentheses, Enter. So if we're around to the nearest hundredth, it's going to be actually, even around to the nearest thousandth, it's gonna be approximately five hundredths. So this is going to be, this is approximately 0.05. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
All of that over 100. So we have the square root of 0.54 times 0.46 divided by 100, close my parentheses, Enter. So if we're around to the nearest hundredth, it's going to be actually, even around to the nearest thousandth, it's gonna be approximately five hundredths. So this is going to be, this is approximately 0.05. So another way to say all of these things is, instead, we don't know exactly this, but now we have an estimate for it. So we can now say, with 95% confidence, and that will often be known as our confidence level right over here, with 95% confidence, between, between, and so we'd wanna go two standard errors below our sample proportion that we just happened to calculate. So that would be 0.54 minus two times five hundredths. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So this is going to be, this is approximately 0.05. So another way to say all of these things is, instead, we don't know exactly this, but now we have an estimate for it. So we can now say, with 95% confidence, and that will often be known as our confidence level right over here, with 95% confidence, between, between, and so we'd wanna go two standard errors below our sample proportion that we just happened to calculate. So that would be 0.54 minus two times five hundredths. So that would be 0.54 minus 10 hundredths, which would be 0.44. And we'd also wanna go two standard errors above the sample proportion. So that would be that plus 10 hundredths, and 0.64 of voters, of voters, support, support A. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So that would be 0.54 minus two times five hundredths. So that would be 0.54 minus 10 hundredths, which would be 0.44. And we'd also wanna go two standard errors above the sample proportion. So that would be that plus 10 hundredths, and 0.64 of voters, of voters, support, support A. And so this interval that we have right over here, from 0.44 to 0.64, this will be known as our confidence interval. Confidence interval. And this will change, not just in the starting point and the end point, but it will change the actual length of our confidence interval, will change depending on what sample proportion we happen to pick for that sample of 100. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So that would be that plus 10 hundredths, and 0.64 of voters, of voters, support, support A. And so this interval that we have right over here, from 0.44 to 0.64, this will be known as our confidence interval. Confidence interval. And this will change, not just in the starting point and the end point, but it will change the actual length of our confidence interval, will change depending on what sample proportion we happen to pick for that sample of 100. A related idea to the confidence interval is this notion of margin of error. Margin of error. And for this particular case, for this particular sample, our margin of error, because we care about 95% confidence, so that would be two standard errors. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
And this will change, not just in the starting point and the end point, but it will change the actual length of our confidence interval, will change depending on what sample proportion we happen to pick for that sample of 100. A related idea to the confidence interval is this notion of margin of error. Margin of error. And for this particular case, for this particular sample, our margin of error, because we care about 95% confidence, so that would be two standard errors. So our margin of error here is two times our standard error, which is be 0.1 or 0.10. And so we're going one margin of error above our sample proportion, right over here, and one margin of error below our sample proportion, right over here, to define our confidence interval. And as I mentioned, this margin of error is not going to be fixed every time we take a sample. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
And for this particular case, for this particular sample, our margin of error, because we care about 95% confidence, so that would be two standard errors. So our margin of error here is two times our standard error, which is be 0.1 or 0.10. And so we're going one margin of error above our sample proportion, right over here, and one margin of error below our sample proportion, right over here, to define our confidence interval. And as I mentioned, this margin of error is not going to be fixed every time we take a sample. Depending on what our sample proportion is, it's going to affect our margin of error, because that is calculated essentially with the standard error. Another interpretation of this is that the method that we used to get this interval right over here, the method that we used to get this confidence interval, when we use it over and over, it will produce intervals, and the intervals won't always be the same, it's gonna be dependent on our sample proportion, but it will produce intervals which include the true proportion, which we might not know and often don't know. It'll include the true proportion 95% of the time. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
And as I mentioned, this margin of error is not going to be fixed every time we take a sample. Depending on what our sample proportion is, it's going to affect our margin of error, because that is calculated essentially with the standard error. Another interpretation of this is that the method that we used to get this interval right over here, the method that we used to get this confidence interval, when we use it over and over, it will produce intervals, and the intervals won't always be the same, it's gonna be dependent on our sample proportion, but it will produce intervals which include the true proportion, which we might not know and often don't know. It'll include the true proportion 95% of the time. I'll cover that intuition more in future videos. We'll see how the interval changes, how the margin of error changes, but when you do this calculation over and over and over again, 95% of the time, your true proportion is going to be contained in whatever interval you happen to calculate that time. Now another interesting question is, well, what if you wanted to tighten up the intervals on average? | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
It'll include the true proportion 95% of the time. I'll cover that intuition more in future videos. We'll see how the interval changes, how the margin of error changes, but when you do this calculation over and over and over again, 95% of the time, your true proportion is going to be contained in whatever interval you happen to calculate that time. Now another interesting question is, well, what if you wanted to tighten up the intervals on average? How would you do that? Well, if you wanted to lower your margin of error, the best way to lower the margin of error is if you increase this denominator right over here, and increasing that denominator means increasing the sample size. And so one thing that you will often see when people are talking about election coverage is, well, we need to sample more people in order to get a lower margin of error. | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So let's see, we have a bunch of data points. And we want to find a line that at least shows the trend in the data. And this one seems a little difficult, because if we ignore these three points down here, maybe we could do a line that looks something like this. 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 me check my answer. | Estimating the line of best fit exercise Regression Probability and Statistics Khan Academy.mp3 |
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