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The times in the finals vary noticeably more than the times in the semifinals. That does look to be true. We see in the semifinals a lot of the times were clumped up right around here, at 53.3 seconds and 53.5 seconds. The high time isn't as high as this time. The low time isn't as low there. The final round is definitely, definitely, they vary noticeably more. Individually, the swimmers all swam faster in the finals than they did in the semifinals. | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
The high time isn't as high as this time. The low time isn't as low there. The final round is definitely, definitely, they vary noticeably more. Individually, the swimmers all swam faster in the finals than they did in the semifinals. That's not true. Whoever this was, clearly they were one of these data points up here. This data point took more time than all of these data points. | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
Let's say that we have a random variable x. Maybe it represents the height of a randomly selected person walking out of the mall or something like that. And right over here, we have its probability distribution. And I've drawn it as a bell curve, as a normal distribution right over here, but it could have many other distributions, but for the visualization's sake, it's a normal one in this example. And I've also drawn the mean of this distribution right over here, and I've also drawn one standard deviation above the mean and one standard deviation below the mean. What we're going to do in this video is think about how does this distribution, and in particular, how does the mean and the standard deviation get affected if we were to add to this random variable or if we were to scale this random variable. So let's first think about what would happen if we have another random variable, which is equal to, let's call this random variable y, which is equal to whatever the random variable x is, and we're going to add a constant. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
And I've drawn it as a bell curve, as a normal distribution right over here, but it could have many other distributions, but for the visualization's sake, it's a normal one in this example. And I've also drawn the mean of this distribution right over here, and I've also drawn one standard deviation above the mean and one standard deviation below the mean. What we're going to do in this video is think about how does this distribution, and in particular, how does the mean and the standard deviation get affected if we were to add to this random variable or if we were to scale this random variable. So let's first think about what would happen if we have another random variable, which is equal to, let's call this random variable y, which is equal to whatever the random variable x is, and we're going to add a constant. So let's say we add, so we're gonna add some constant here. I'll do a lowercase k. This is not a random variable. This is a constant. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
So let's first think about what would happen if we have another random variable, which is equal to, let's call this random variable y, which is equal to whatever the random variable x is, and we're going to add a constant. So let's say we add, so we're gonna add some constant here. I'll do a lowercase k. This is not a random variable. This is a constant. It could be the number 10. So if these are random heights of people walking out of the mall, well, you're just gonna add 10 inches to their height for some reason. Maybe you wanna figure out, well, the distribution of people's heights with helmets on or plumed hats or whatever it might be, how would that affect, how would the mean of y and the standard deviation of y relate to x? | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
This is a constant. It could be the number 10. So if these are random heights of people walking out of the mall, well, you're just gonna add 10 inches to their height for some reason. Maybe you wanna figure out, well, the distribution of people's heights with helmets on or plumed hats or whatever it might be, how would that affect, how would the mean of y and the standard deviation of y relate to x? So we could visualize that. So what the distribution of y would look like, so instead of this, instead of the center of the distribution, instead of the mean here being right at this point, it's going to be shifted up by k. In fact, we can shift, the entire distribution would be shifted to the right by k in this example. And maybe k is quite large. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
Maybe you wanna figure out, well, the distribution of people's heights with helmets on or plumed hats or whatever it might be, how would that affect, how would the mean of y and the standard deviation of y relate to x? So we could visualize that. So what the distribution of y would look like, so instead of this, instead of the center of the distribution, instead of the mean here being right at this point, it's going to be shifted up by k. In fact, we can shift, the entire distribution would be shifted to the right by k in this example. And maybe k is quite large. Maybe it looks something like that. This is my distribution for my random variable y here. And you can see that the distribution has just shifted to the right by k. So we have moved to the right by k. We would have moved to the left if k was negative or if we were subtracting k. And so this clearly changes the mean. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
And maybe k is quite large. Maybe it looks something like that. This is my distribution for my random variable y here. And you can see that the distribution has just shifted to the right by k. So we have moved to the right by k. We would have moved to the left if k was negative or if we were subtracting k. And so this clearly changes the mean. The mean is going to now be k larger. So we can write that down. We can say that the mean of our random variable y is equal to the mean of x, the mean of x of our random variable x plus k, plus k. You see that right over here. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
And you can see that the distribution has just shifted to the right by k. So we have moved to the right by k. We would have moved to the left if k was negative or if we were subtracting k. And so this clearly changes the mean. The mean is going to now be k larger. So we can write that down. We can say that the mean of our random variable y is equal to the mean of x, the mean of x of our random variable x plus k, plus k. You see that right over here. But has the standard deviation changed? Well, remember, standard deviation is a way of measuring typical spread from the mean, and that won't change. So for our random variable x, this length right over here is one standard deviation. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
We can say that the mean of our random variable y is equal to the mean of x, the mean of x of our random variable x plus k, plus k. You see that right over here. But has the standard deviation changed? Well, remember, standard deviation is a way of measuring typical spread from the mean, and that won't change. So for our random variable x, this length right over here is one standard deviation. Well, that's also going to be the same as one standard deviation here. This is one standard deviation here. This is going to be the same as our standard deviation for our random variable y. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
So for our random variable x, this length right over here is one standard deviation. Well, that's also going to be the same as one standard deviation here. This is one standard deviation here. This is going to be the same as our standard deviation for our random variable y. And so we can say the standard deviation of y, of our random variable y, is equal to the standard deviation of our random variable x. So if you just add to a random variable, it would change the mean, but not the standard deviation. You see it visually here. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
This is going to be the same as our standard deviation for our random variable y. And so we can say the standard deviation of y, of our random variable y, is equal to the standard deviation of our random variable x. So if you just add to a random variable, it would change the mean, but not the standard deviation. You see it visually here. Now, what if you were to scale a random variable? So what if I have another random variable? I don't know, let's call it z. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
You see it visually here. Now, what if you were to scale a random variable? So what if I have another random variable? I don't know, let's call it z. And let's say z is equal to some constant, some constant times x. And so remember, this isn't, the k is not a random variable, it's just gonna be a number. It could be, say, the number two. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
I don't know, let's call it z. And let's say z is equal to some constant, some constant times x. And so remember, this isn't, the k is not a random variable, it's just gonna be a number. It could be, say, the number two. Well, let's think about what would happen. So let me redraw the distribution for our random variable x. So let's see, if k were two, what would happen is, is this distribution would be scaled out, it would be stretched out by two, and since the area always has to be one, it would actually be flattened down by a scale of two as well, so it still has the same area. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
It could be, say, the number two. Well, let's think about what would happen. So let me redraw the distribution for our random variable x. So let's see, if k were two, what would happen is, is this distribution would be scaled out, it would be stretched out by two, and since the area always has to be one, it would actually be flattened down by a scale of two as well, so it still has the same area. So I can do that with my little drawing tool here. Let me try to, first I'm going to stretch it out by, whoops, first, actually, I'll make it shorter by a factor of two, but more importantly, it is going to be stretched out by a factor of two. So let me align the axes here so that we can appreciate this. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
So let's see, if k were two, what would happen is, is this distribution would be scaled out, it would be stretched out by two, and since the area always has to be one, it would actually be flattened down by a scale of two as well, so it still has the same area. So I can do that with my little drawing tool here. Let me try to, first I'm going to stretch it out by, whoops, first, actually, I'll make it shorter by a factor of two, but more importantly, it is going to be stretched out by a factor of two. So let me align the axes here so that we can appreciate this. So it's going to look something like this. It's going to look something like this when you scale the random variable. This is what the distribution of our random variable z is going to look like. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
So let me align the axes here so that we can appreciate this. So it's going to look something like this. It's going to look something like this when you scale the random variable. This is what the distribution of our random variable z is going to look like. I'll do it in the z's color so that it's clear. And so you can see two things. One, the mean for sure shifted. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
This is what the distribution of our random variable z is going to look like. I'll do it in the z's color so that it's clear. And so you can see two things. One, the mean for sure shifted. The mean here for sure got pushed out. It definitely got scaled up. But also, we see that the standard deviations got scaled, that the standard deviation right over here of z, that this has been scaled, it actually turns out that it's been scaled by a factor of k. So this is going to be equal to k times the standard deviation of our random variable x. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
One, the mean for sure shifted. The mean here for sure got pushed out. It definitely got scaled up. But also, we see that the standard deviations got scaled, that the standard deviation right over here of z, that this has been scaled, it actually turns out that it's been scaled by a factor of k. So this is going to be equal to k times the standard deviation of our random variable x. And it turns out that our mean right over here, so let me write that too, that our mean of our random variable z is going to be equal to, that's also going to be scaled up, times, or it's gonna be k times the mean of our random variable x. So the big takeaways here, if you have one random variable that's constructed by adding a constant to another random variable, it's going to shift the mean by that constant, but it's not going to affect the standard deviation. If you try to scale, if you multiply one random variable to get another one by some constant, then that's going to affect both the standard deviation, it's gonna scale that, and it's going to affect the mean. | Impact of transforming (scaling and shifting) random variables AP Statistics Khan Academy.mp3 |
So they took a sample of 200 residents to test the null hypothesis, is that the unemployment rate is the same as the national one versus the alternative hypothesis, which is that the unemployment rate is not the same as the national, where P is the proportion of residents in the town that are unemployed. The sample included 22 residents who were unemployed. Assuming that the conditions for inference have been met, and so that's the random, normal, and independence conditions that we've talked about in previous videos, identify the correct test statistic for this significance test. So let me just, I like to rewrite everything just to make sure I've understood what's going on. We have a null hypothesis that the true proportion of unemployed people in our town, that's what this P represents, is the same as the national unemployment. And remember, a null hypothesis tends to be the no news here, nothing to report, so to speak. And we have our alternative hypothesis that no, the true unemployment in this town is different, is different than 8%. | Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3 |
So let me just, I like to rewrite everything just to make sure I've understood what's going on. We have a null hypothesis that the true proportion of unemployed people in our town, that's what this P represents, is the same as the national unemployment. And remember, a null hypothesis tends to be the no news here, nothing to report, so to speak. And we have our alternative hypothesis that no, the true unemployment in this town is different, is different than 8%. And so what we would do is we would set some type of a significance level. We would assume that the mayor of the town sets it. Let's say he sets or she sets a significance level of 0.5. | Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3 |
And we have our alternative hypothesis that no, the true unemployment in this town is different, is different than 8%. And so what we would do is we would set some type of a significance level. We would assume that the mayor of the town sets it. Let's say he sets or she sets a significance level of 0.5. And then what we wanna do is conduct the experiment. So this is the entire population of the town. They take a sample of 200 people. | Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3 |
Let's say he sets or she sets a significance level of 0.5. And then what we wanna do is conduct the experiment. So this is the entire population of the town. They take a sample of 200 people. So this is our sample. N is equal to 200. Since it met the independence condition, we'll assume that this is less than 10% of the population. | Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3 |
They take a sample of 200 people. So this is our sample. N is equal to 200. Since it met the independence condition, we'll assume that this is less than 10% of the population. And we calculate a sample statistic here. And it would be, since we care about the true population proportion, the sample statistic we would care about is the sample proportion. And we figure out that it is 22 out of the 200 people in the sample are unemployed. | Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3 |
Since it met the independence condition, we'll assume that this is less than 10% of the population. And we calculate a sample statistic here. And it would be, since we care about the true population proportion, the sample statistic we would care about is the sample proportion. And we figure out that it is 22 out of the 200 people in the sample are unemployed. So this is 0.11. Now the next step is, assuming the null hypothesis is true, what is the probability of getting a result this far away or further from the assumed population proportion? And if that probability is lower than alpha, then we would reject the null hypothesis, which would suggest the alternative. | Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3 |
And we figure out that it is 22 out of the 200 people in the sample are unemployed. So this is 0.11. Now the next step is, assuming the null hypothesis is true, what is the probability of getting a result this far away or further from the assumed population proportion? And if that probability is lower than alpha, then we would reject the null hypothesis, which would suggest the alternative. But how do you figure out this probability? Well, one way to think about it is we could say how many standard deviations away from the true proportion, the assumed proportion, is it? And then we could say, what's the probability of getting that many standard deviations or further from the true proportion? | Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3 |
And if that probability is lower than alpha, then we would reject the null hypothesis, which would suggest the alternative. But how do you figure out this probability? Well, one way to think about it is we could say how many standard deviations away from the true proportion, the assumed proportion, is it? And then we could say, what's the probability of getting that many standard deviations or further from the true proportion? We could use a z-table to do that. And so what we wanna do is figure out the number of standard deviations. And so that would be a z-statistic. | Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3 |
And then we could say, what's the probability of getting that many standard deviations or further from the true proportion? We could use a z-table to do that. And so what we wanna do is figure out the number of standard deviations. And so that would be a z-statistic. And so how do we figure it out? Well, we can figure out the difference between the sample proportion here and the assumed population proportion. So that would be 0.11 minus 0.08 divided by the standard deviation of the sampling distribution of the sample proportions. | Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3 |
And so that would be a z-statistic. And so how do we figure it out? Well, we can figure out the difference between the sample proportion here and the assumed population proportion. So that would be 0.11 minus 0.08 divided by the standard deviation of the sampling distribution of the sample proportions. And we can figure that out. Remember, all that is is, and sometimes we say, well, we don't know what the population proportion is. But here we're assuming a population proportion. | Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3 |
So that would be 0.11 minus 0.08 divided by the standard deviation of the sampling distribution of the sample proportions. And we can figure that out. Remember, all that is is, and sometimes we say, well, we don't know what the population proportion is. But here we're assuming a population proportion. So we're assuming it is 0.08. And then we'll multiply that times one minus 0.08. So we'll multiply that times 0.92. | Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3 |
But here we're assuming a population proportion. So we're assuming it is 0.08. And then we'll multiply that times one minus 0.08. So we'll multiply that times 0.92. And this comes straight from, we've seen it in previous videos, the standard deviation of the sampling distribution of sample proportions. And then you divide that by n, which is 200 right over here. And we could get a calculator out to figure this out. | Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3 |
So we'll multiply that times 0.92. And this comes straight from, we've seen it in previous videos, the standard deviation of the sampling distribution of sample proportions. And then you divide that by n, which is 200 right over here. And we could get a calculator out to figure this out. But this will give us some value, which just says how many standard deviations away from 0.08 is 0.11. And then we could use a z-table to figure out what's the probability of getting that far or further from the true proportion. And then that will give us our p-value, which we can compare to significance level. | Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3 |
And we could get a calculator out to figure this out. But this will give us some value, which just says how many standard deviations away from 0.08 is 0.11. And then we could use a z-table to figure out what's the probability of getting that far or further from the true proportion. And then that will give us our p-value, which we can compare to significance level. Sometimes you will see a formula that looks something like this. And you say, hey, look, you have your sample proportion. You find the difference between that and the assumed proportion in the null hypothesis. | Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3 |
And then that will give us our p-value, which we can compare to significance level. Sometimes you will see a formula that looks something like this. And you say, hey, look, you have your sample proportion. You find the difference between that and the assumed proportion in the null hypothesis. That's what this little zero says. Now, this is the assumed population proportion from the null hypothesis. And you divide that by the standard deviation, the assumed standard deviation of the sampling distribution of the sample proportions. | Calculating a z statistic in a test about a proportion AP Statistics Khan Academy.mp3 |
The government created the following stem and leaf plot, showing the number of turtles at each major zoo in the country. How many zoos have fewer than 46 turtles? So what the stem and leaf plot does is it gives us the first digit in each number, and essentially you could say it called us the tens place, and then it gives us the ones place. So there was only one zoo that had four turtles, so you could view this as zero, zero, four, or four turtles. Then there's, let's see, so everything here, the tens place is a one, so this number right over here is really an 11, this is a 14, this right over here would be a 16, that's a 16, and so forth and so on, this would be a 17, 18. All of this, this is 23, this is 23, this is 26, because we have our tens place right over here, this is the first digit. So let's go ahead and answer the question, how many zoos had fewer than 46 turtles? | Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So there was only one zoo that had four turtles, so you could view this as zero, zero, four, or four turtles. Then there's, let's see, so everything here, the tens place is a one, so this number right over here is really an 11, this is a 14, this right over here would be a 16, that's a 16, and so forth and so on, this would be a 17, 18. All of this, this is 23, this is 23, this is 26, because we have our tens place right over here, this is the first digit. So let's go ahead and answer the question, how many zoos had fewer than 46 turtles? So there are no zoos that had 40 anything turtles, and so all of these zoos here, so all of these had 30 something turtles, this, these had 20 something turtles, these have, in the teens, this has single digits. So it's literally as many zoos as we have listed here. So one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17. | Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So let's go ahead and answer the question, how many zoos had fewer than 46 turtles? So there are no zoos that had 40 anything turtles, and so all of these zoos here, so all of these had 30 something turtles, this, these had 20 something turtles, these have, in the teens, this has single digits. So it's literally as many zoos as we have listed here. So one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17. 17 zoos have fewer than 46 turtles. Let's do another one. The buyer for a chain of supermarkets created the following stem and leaf plot showing the number of coconuts at each of the stores. | Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17. 17 zoos have fewer than 46 turtles. Let's do another one. The buyer for a chain of supermarkets created the following stem and leaf plot showing the number of coconuts at each of the stores. What was the smallest number of coconuts at any one grocery store? So the buyer for a chain of supermarkets created the following stem of leaf plots showing the number of coconuts at each of the stores. So at any one grocery store, the smallest number, well, that's this one right over here. | Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
The buyer for a chain of supermarkets created the following stem and leaf plot showing the number of coconuts at each of the stores. What was the smallest number of coconuts at any one grocery store? So the buyer for a chain of supermarkets created the following stem of leaf plots showing the number of coconuts at each of the stores. So at any one grocery store, the smallest number, well, that's this one right over here. And remember, it's not two, we have our tens places right over here, it's a one. So this right over here represents 12 coconuts at that store. So we'll put 12 right over here. | Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So at any one grocery store, the smallest number, well, that's this one right over here. And remember, it's not two, we have our tens places right over here, it's a one. So this right over here represents 12 coconuts at that store. So we'll put 12 right over here. Let's try out another one. A statistician for a chain of department stores created the following stem and leaf plot showing the number of watches at each of the stores. How many department stores have exactly seven watches? | Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So we'll put 12 right over here. Let's try out another one. A statistician for a chain of department stores created the following stem and leaf plot showing the number of watches at each of the stores. How many department stores have exactly seven watches? Well, that's only this one right over here, zero, seven, watches, this one right, this, this, and that one are not seven. This is representing 17 because it's in the row with one at the beginning. This right over here represents 27 because it's in the row with the two at the beginning. | Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
How many department stores have exactly seven watches? Well, that's only this one right over here, zero, seven, watches, this one right, this, this, and that one are not seven. This is representing 17 because it's in the row with one at the beginning. This right over here represents 27 because it's in the row with the two at the beginning. So there's only one store that has exactly seven watches. Let's do one more, this is kind of fun. A zookeeper created the following stem and leaf plot showing the number of tigers at each major zoo in the country. | Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
This right over here represents 27 because it's in the row with the two at the beginning. So there's only one store that has exactly seven watches. Let's do one more, this is kind of fun. A zookeeper created the following stem and leaf plot showing the number of tigers at each major zoo in the country. How many zoos have more than 24 tigers? So we can ignore the zeros and the teens and we get into the 20s. This is 25, so that meets the criteria, and then you go to 28, 29. | Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
A zookeeper created the following stem and leaf plot showing the number of tigers at each major zoo in the country. How many zoos have more than 24 tigers? So we can ignore the zeros and the teens and we get into the 20s. This is 25, so that meets the criteria, and then you go to 28, 29. So all of these, all of these in the 30s, and all of these right over here, this three zero, this doesn't mean zero tigers, this is 30 tigers. This is 40 tigers. So we count one, two, three, four, five, six, seven, eight, nine, nine zoos have more than 24 tigers. | Reading stem and leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
And I suggested that, hey, why don't you visualize this, draw, graph this probability distribution, this binomial probability distribution. And when I thought about it, I said, well, you know, I too would enjoy graphing it, and we might as well do it together, because whenever you graph these things, it makes it very visual, and kind of the shape of a binomial distribution like this. So let's do that. So let me maybe move over to the right a little bit. I really just need to be able to keep track of these things right over here. So let me draw some lines. So let me, if I were to just draw one line there, and then another line here, and then we have the different percentages. | Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3 |
So let me maybe move over to the right a little bit. I really just need to be able to keep track of these things right over here. So let me draw some lines. So let me, if I were to just draw one line there, and then another line here, and then we have the different percentages. So let's do that. See, the highest one is a little over 30%, 32%. So maybe we'll go as high as 40% here. | Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3 |
So let me, if I were to just draw one line there, and then another line here, and then we have the different percentages. So let's do that. See, the highest one is a little over 30%, 32%. So maybe we'll go as high as 40% here. 40%, and then this would be 20%. 20, that looks about halfway, 20%. This would be 10%. | Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3 |
So maybe we'll go as high as 40% here. 40%, and then this would be 20%. 20, that looks about halfway, 20%. This would be 10%. 10%, and this would be 30%. 30%. And then in this axis, let's do the different values that the random variable could take on. | Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3 |
This would be 10%. 10%, and this would be 30%. 30%. And then in this axis, let's do the different values that the random variable could take on. So the random variable taking on the value zero. The random variable taking on the value one. Zero, one. | Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3 |
And then in this axis, let's do the different values that the random variable could take on. So the random variable taking on the value zero. The random variable taking on the value one. Zero, one. The random variable taking on two. Two, we're almost there. Let's see, three. | Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3 |
Zero, one. The random variable taking on two. Two, we're almost there. Let's see, three. And then four. Four, and then five. Five, and then finally six. | Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3 |
Let's see, three. And then four. Four, and then five. Five, and then finally six. Take x equals six. And then six, and now let's just graph all of these. So this first one, 0.1%, well that's barely gonna register on this graph right here, so I'll just kind of give it a little bit of a showing right over there. | Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3 |
Five, and then finally six. Take x equals six. And then six, and now let's just graph all of these. So this first one, 0.1%, well that's barely gonna register on this graph right here, so I'll just kind of give it a little bit of a showing right over there. Actually, let me do it in that green color. So let me make sure, so in that green color, you're gonna have just a little bit of a showing. One as well is kind of barely a showing, so it shows up a little bit more. | Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3 |
So this first one, 0.1%, well that's barely gonna register on this graph right here, so I'll just kind of give it a little bit of a showing right over there. Actually, let me do it in that green color. So let me make sure, so in that green color, you're gonna have just a little bit of a showing. One as well is kind of barely a showing, so it shows up a little bit more. So let me draw it like that. That is 1% right over there. Now two is 6%, which on this scale is gonna be about, it's about that high. | Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3 |
One as well is kind of barely a showing, so it shows up a little bit more. So let me draw it like that. That is 1% right over there. Now two is 6%, which on this scale is gonna be about, it's about that high. So we draw it like that. So that is two, so that is 6%, right there. X equaling three, 18.5% shot of that happening. | Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3 |
Now two is 6%, which on this scale is gonna be about, it's about that high. So we draw it like that. So that is two, so that is 6%, right there. X equaling three, 18.5% shot of that happening. So 18.5, it gets us right about, it's a hand-drawn chart, or histogram, so you have to bear with me. So it's roughly there. And then four was 32.4%. | Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3 |
X equaling three, 18.5% shot of that happening. So 18.5, it gets us right about, it's a hand-drawn chart, or histogram, so you have to bear with me. So it's roughly there. And then four was 32.4%. So that is up here. So 32.4% is right, looks like that. So let me shade that in. | Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3 |
And then four was 32.4%. So that is up here. So 32.4% is right, looks like that. So let me shade that in. 32.4%, and then five was 30.3%. So 30.3%, and slightly lower, just like that. And it looks like this. | Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3 |
So let me shade that in. 32.4%, and then five was 30.3%. So 30.3%, and slightly lower, just like that. And it looks like this. 30.3%, and finally six is 11.8%. So really this whole video was just an exercise in making a histogram, but it's useful because to actually visualize what the distribution looks like, and what's really interesting is to think about, well, how does this change as you change the free throw percentage, or as you change the number of shots you take, how does this change this binomial distribution? And you could do that on a spreadsheet and actually see how that all works out. | Graphing basketball binomial distribution Probability and Statistics Khan Academy.mp3 |
They give us this, as they say, the two-way table of column relative frequencies. So for example, this column right over here is men. The column total is one, or you could say 100%. And we could see that 0.42 of the men, or 42% of the men, voted for Obama. We can see 52% of the men, or 0.52 of the men, voted for Romney. And we can see that the other, that the neither Obama, or 6% went for neither Obama or, nor Romney. And for women, 52% went for Obama, 43% went for Romney, 5% went for other. | Interpreting two-way tables Data and modeling 8th grade Khan Academy.mp3 |
And we could see that 0.42 of the men, or 42% of the men, voted for Obama. We can see 52% of the men, or 0.52 of the men, voted for Romney. And we can see that the other, that the neither Obama, or 6% went for neither Obama or, nor Romney. And for women, 52% went for Obama, 43% went for Romney, 5% went for other. And then these, this 52 plus 43 plus five will add up to 100% of the women. During the 2012 United States presidential election, were male voters more likely to vote for Romney than female voters? So let's see. | Interpreting two-way tables Data and modeling 8th grade Khan Academy.mp3 |
And for women, 52% went for Obama, 43% went for Romney, 5% went for other. And then these, this 52 plus 43 plus five will add up to 100% of the women. During the 2012 United States presidential election, were male voters more likely to vote for Romney than female voters? So let's see. If we, there's a couple of ways you could think about it. Well, actually, let's do it this way. Male voters, if you were a man, 52% of them voted for Romney, while for the women, 43% of them voted for Romney. | Interpreting two-way tables Data and modeling 8th grade Khan Academy.mp3 |
So let's see. If we, there's a couple of ways you could think about it. Well, actually, let's do it this way. Male voters, if you were a man, 52% of them voted for Romney, while for the women, 43% of them voted for Romney. So a man was more likely. There's a, if you randomly picked a man who voted, there was a 52% chance they voted for Romney, while if you randomly picked a woman, there was a 43%, a woman who voted, there was a 43% chance that she voted for Romney. So yes, male voters were more likely to vote for Romney than female voters. | Interpreting two-way tables Data and modeling 8th grade Khan Academy.mp3 |
Consider the density curve below. And so we have a density curve that describes the probability distribution for a continuous random variable. This random variable can take on values from one to five and has an equal probability of taking on any of these values from one to five. Find the probability that x is less than four. So x can go from one to four, there's no probability that it'll be less than one. So we know the entire area under the density curve is going to be one. So if we can find the fraction of the area that meets our criteria, then we know the answer to the question. | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
Find the probability that x is less than four. So x can go from one to four, there's no probability that it'll be less than one. So we know the entire area under the density curve is going to be one. So if we can find the fraction of the area that meets our criteria, then we know the answer to the question. So what we're going to look at is we want to go from one to four. The reason why I know we can start at one is there's no probability, there's zero chances that I'll get a value less than one. We see that from the density curve. | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
So if we can find the fraction of the area that meets our criteria, then we know the answer to the question. So what we're going to look at is we want to go from one to four. The reason why I know we can start at one is there's no probability, there's zero chances that I'll get a value less than one. We see that from the density curve. And so we just need to think about what is the area here? What is this area right over here? Well, this is just a rectangle where the height is 0.25 and the width is one, two, three. | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
We see that from the density curve. And so we just need to think about what is the area here? What is this area right over here? Well, this is just a rectangle where the height is 0.25 and the width is one, two, three. So our area is going to be 0.25 times three, which is equal to 0.75. So the probability that x is less than four is 0.75 or you could say it's a 75% probability. Let's do another one of these with a slightly more involved density curve. | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
Well, this is just a rectangle where the height is 0.25 and the width is one, two, three. So our area is going to be 0.25 times three, which is equal to 0.75. So the probability that x is less than four is 0.75 or you could say it's a 75% probability. Let's do another one of these with a slightly more involved density curve. A set of middle school students' heights are normally distributed with a mean of 150 centimeters and a standard deviation of 20 centimeters. Let h be the height of a randomly selected student from this set. Find and interpret the probability that h, that the height of a randomly selected student from the set is greater than 170 centimeters. | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
Let's do another one of these with a slightly more involved density curve. A set of middle school students' heights are normally distributed with a mean of 150 centimeters and a standard deviation of 20 centimeters. Let h be the height of a randomly selected student from this set. Find and interpret the probability that h, that the height of a randomly selected student from the set is greater than 170 centimeters. So let's first visualize the density curve. It is a normal distribution. They tell us that the mean is 150 centimeters. | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
Find and interpret the probability that h, that the height of a randomly selected student from the set is greater than 170 centimeters. So let's first visualize the density curve. It is a normal distribution. They tell us that the mean is 150 centimeters. So let me draw that. So the mean, that is 150. And they also say that we have a standard deviation of 20 centimeters. | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
They tell us that the mean is 150 centimeters. So let me draw that. So the mean, that is 150. And they also say that we have a standard deviation of 20 centimeters. So 20 centimeters above the mean, one standard deviation above the mean is 170. One standard deviation below the mean is 130. And we want the probability of, if we randomly select from these middle school students, what's the probability that the height is greater than 170? | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
And they also say that we have a standard deviation of 20 centimeters. So 20 centimeters above the mean, one standard deviation above the mean is 170. One standard deviation below the mean is 130. And we want the probability of, if we randomly select from these middle school students, what's the probability that the height is greater than 170? So that's going to be this area under this normal distribution curve. It's going to be that area. So how can we figure that out? | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
And we want the probability of, if we randomly select from these middle school students, what's the probability that the height is greater than 170? So that's going to be this area under this normal distribution curve. It's going to be that area. So how can we figure that out? Well, there's several ways to do it. We know that this is the area above one standard deviation above the mean. You could use a z-table, or you could use some generally useful knowledge about normal distributions. | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
So how can we figure that out? Well, there's several ways to do it. We know that this is the area above one standard deviation above the mean. You could use a z-table, or you could use some generally useful knowledge about normal distributions. And that's that the area between one standard deviation below the mean and one standard deviation above the mean, this area right over here, is roughly 68%. It's closer to 68.2%. For our purposes, 68 will work fine. | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
You could use a z-table, or you could use some generally useful knowledge about normal distributions. And that's that the area between one standard deviation below the mean and one standard deviation above the mean, this area right over here, is roughly 68%. It's closer to 68.2%. For our purposes, 68 will work fine. And so if we're looking at just from the mean to one standard deviation above the mean, it would be half of that. So this is going to be approximately 34%. Now, we also know that for a normal distribution, the area below the mean is going to be 50%. | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
For our purposes, 68 will work fine. And so if we're looking at just from the mean to one standard deviation above the mean, it would be half of that. So this is going to be approximately 34%. Now, we also know that for a normal distribution, the area below the mean is going to be 50%. So we know all of that is 50%. And so the combined area below 170, below one standard deviation above the mean, is going to be 84%, or approximately 84%. And so that helps us figure out what is the area above one standard deviation above the mean, which will answer our question. | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
Now, we also know that for a normal distribution, the area below the mean is going to be 50%. So we know all of that is 50%. And so the combined area below 170, below one standard deviation above the mean, is going to be 84%, or approximately 84%. And so that helps us figure out what is the area above one standard deviation above the mean, which will answer our question. The entire area under this density curve, under any density curve, is going to be equal to one. And so if the entire area is one, this green area is 84%, or 0.84, well, then we just subtract that from one to get this blue area. So this is going to be one minus 0.84, or I'll say approximately. | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
And so that helps us figure out what is the area above one standard deviation above the mean, which will answer our question. The entire area under this density curve, under any density curve, is going to be equal to one. And so if the entire area is one, this green area is 84%, or 0.84, well, then we just subtract that from one to get this blue area. So this is going to be one minus 0.84, or I'll say approximately. And so that's going to be approximately 0.16. If you want a slightly more precise value, you could use a z-table. The area below one standard deviation above the mean will be closer to about 84.1%, in which case this would be about 15.9%, or 0.159. | Probabilities from density curves Random variables AP Statistics Khan Academy.mp3 |
Exclude the median when computing the quartiles. All right, let's see if we can do this. So we have a bunch of data here, and they say if it helps, you might drag the numbers to put them in a different order so we can drag these numbers around, which is useful because we will want to order them. The order isn't checked with your answer. I'm doing this off of the Khan Academy exercises, so I don't have my drawing tablet here. I just have my mouse and I'm interacting with the exercise, which I encourage you to do too because the best way to learn any of this stuff is to actually practice it, and at Khan Academy we have 150,000 exercises for you to practice with. Anyway, so let's do this. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
The order isn't checked with your answer. I'm doing this off of the Khan Academy exercises, so I don't have my drawing tablet here. I just have my mouse and I'm interacting with the exercise, which I encourage you to do too because the best way to learn any of this stuff is to actually practice it, and at Khan Academy we have 150,000 exercises for you to practice with. Anyway, so let's do this. Let's order this thing so we can figure out the range of numbers. What's the lowest and what's the highest? So let's see, there's a seven here. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
Anyway, so let's do this. Let's order this thing so we can figure out the range of numbers. What's the lowest and what's the highest? So let's see, there's a seven here. Then let's see, we have some eights. We've got some eights going on. And then we have some nines. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
So let's see, there's a seven here. Then let's see, we have some eights. We've got some eights going on. And then we have some nines. Actually, we have a bunch of nines. We have four nines here. We have some nines. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
And then we have some nines. Actually, we have a bunch of nines. We have four nines here. We have some nines. And then let's see, 13 is the largest number. There we go, we've ordered the numbers. So our smallest number is seven. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
We have some nines. And then let's see, 13 is the largest number. There we go, we've ordered the numbers. So our smallest number is seven. And this is what the whiskers are useful for, for helping us figure out the entire range of numbers. Our smallest number is seven. Our largest number is 13. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
So our smallest number is seven. And this is what the whiskers are useful for, for helping us figure out the entire range of numbers. Our smallest number is seven. Our largest number is 13. So we know the range. Now let's plot the median. And this will help us. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
Our largest number is 13. So we know the range. Now let's plot the median. And this will help us. One's getting this center line of our box, but then also we need to do that to figure out what these other lines are that kind of define the box, to define the middle two fourths of our number, of our data, or the middle two quartiles, roughly the middle two quartiles. It depends how some of the numbers work out. But this middle number, this middle line is going to be the median of our entire data set. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
And this will help us. One's getting this center line of our box, but then also we need to do that to figure out what these other lines are that kind of define the box, to define the middle two fourths of our number, of our data, or the middle two quartiles, roughly the middle two quartiles. It depends how some of the numbers work out. But this middle number, this middle line is going to be the median of our entire data set. Now the median is just the middle number. If we sort them in order, median is just the middle number. We have 11 numbers here. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
But this middle number, this middle line is going to be the median of our entire data set. Now the median is just the middle number. If we sort them in order, median is just the middle number. We have 11 numbers here. So the middle one is gonna have five on either side. So it's gonna be this nine. If we had 10 numbers here, if we had an even number of numbers, you actually would have had two middle numbers. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
We have 11 numbers here. So the middle one is gonna have five on either side. So it's gonna be this nine. If we had 10 numbers here, if we had an even number of numbers, you actually would have had two middle numbers. And then to find the median, you would have found the mean of those two. If that last sentence was confusing, watch the videos on Khan Academy on median. And I go into much more detail on that. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
If we had 10 numbers here, if we had an even number of numbers, you actually would have had two middle numbers. And then to find the median, you would have found the mean of those two. If that last sentence was confusing, watch the videos on Khan Academy on median. And I go into much more detail on that. But here I have 11 numbers. So my median is going to be the middle one. It has five larger, five less. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
And I go into much more detail on that. But here I have 11 numbers. So my median is going to be the middle one. It has five larger, five less. It's this nine right over here. If I had my pen tablet, I would circle it. So it's this nine. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
It has five larger, five less. It's this nine right over here. If I had my pen tablet, I would circle it. So it's this nine. That is the median. And now we need to figure out, well, what number is halfway, or let me put it this way, what number is the median of the numbers in this bottom half? And they told us to exclude the median when we compute the quartiles. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
So it's this nine. That is the median. And now we need to figure out, well, what number is halfway, or let me put it this way, what number is the median of the numbers in this bottom half? And they told us to exclude the median when we compute the quartiles. So this was the median. Let's ignore that. So let's look at all the numbers below that. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
And they told us to exclude the median when we compute the quartiles. So this was the median. Let's ignore that. So let's look at all the numbers below that. So this nine, eight, eight, eight, and seven. So we have five numbers. What's the median of these five numbers? | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
So let's look at all the numbers below that. So this nine, eight, eight, eight, and seven. So we have five numbers. What's the median of these five numbers? Well, the median's the middle number. That is eight. So the beginning of our second quartile is gonna be at eight right over there. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
What's the median of these five numbers? Well, the median's the middle number. That is eight. So the beginning of our second quartile is gonna be at eight right over there. And we do the same thing for our third quartile. Remember, this was our median of our entire data set. Let's exclude it. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
So the beginning of our second quartile is gonna be at eight right over there. And we do the same thing for our third quartile. Remember, this was our median of our entire data set. Let's exclude it. Let's look at the top half of the numbers, so to speak. And there's five numbers here in order. So the middle one, the median of this, is 10. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
Let's exclude it. Let's look at the top half of the numbers, so to speak. And there's five numbers here in order. So the middle one, the median of this, is 10. So that's gonna be the top of our second quartile. And just like that, we're done. We have constructed our box-end Whitaker plot. | Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3 |
He asks his listeners to visit his website and participate in the poll. The poll shows that 89% of about 200 respondents love his show. What is the most concerning source of bias in this scenario? And like always, pause this video and see if you can figure it out on your own and then we'll work through it together. Let's think about what's going on. He has this population of listeners right over here. I'll assume that the number of listeners is more than 200. | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
And like always, pause this video and see if you can figure it out on your own and then we'll work through it together. Let's think about what's going on. He has this population of listeners right over here. I'll assume that the number of listeners is more than 200. And he says, hey, I wanna find a sample and I can't ask all of my listeners. Who knows, maybe he has 10,000 listeners. They don't tell us that. | Examples of bias in surveys Study design AP Statistics Khan Academy.mp3 |
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