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So it is the case that all of the students are less than 17 years old. So this is definitely going to be true. The next statement, at least 75% of the students are 10 years old or older. So when you look at this, this feels right because 10 is the value, that is at the beginning of the second quartile. This is the second quartile right over there. And actually, let me do this in a different color. So this is the second quartile. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So when you look at this, this feels right because 10 is the value, that is at the beginning of the second quartile. This is the second quartile right over there. And actually, let me do this in a different color. So this is the second quartile. So 25% of the value of the numbers are in the second, or roughly, sometimes it's not exactly, so approximately, I'll say roughly, 25% are going to be in the second quartile. Approximately 25% are going to be in the third quartile. And approximately 25% are going to be in the fourth quartile. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So this is the second quartile. So 25% of the value of the numbers are in the second, or roughly, sometimes it's not exactly, so approximately, I'll say roughly, 25% are going to be in the second quartile. Approximately 25% are going to be in the third quartile. And approximately 25% are going to be in the fourth quartile. So it seems reasonable for saying 10 years old or older that this is going to be true. In fact, you could even have a couple of values in the first quartile that are 10. But to make that a little bit more tangible, let's look at some, so I'm feeling good that this is true, but let's look at a few more examples to make this a little bit more concrete. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
And approximately 25% are going to be in the fourth quartile. So it seems reasonable for saying 10 years old or older that this is going to be true. In fact, you could even have a couple of values in the first quartile that are 10. But to make that a little bit more tangible, let's look at some, so I'm feeling good that this is true, but let's look at a few more examples to make this a little bit more concrete. So they don't know, we don't know, based on the information here, exactly how many students are at the party. We'll have to construct some scenarios. So we could do a scenario. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
But to make that a little bit more tangible, let's look at some, so I'm feeling good that this is true, but let's look at a few more examples to make this a little bit more concrete. So they don't know, we don't know, based on the information here, exactly how many students are at the party. We'll have to construct some scenarios. So we could do a scenario. Let's see if we can do, we could do a scenario where, well, let's see, let's see if I can construct something where, let's see, the median is 13. We know that for sure. The median is 13. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So we could do a scenario. Let's see if we can do, we could do a scenario where, well, let's see, let's see if I can construct something where, let's see, the median is 13. We know that for sure. The median is 13. So if I have an odd number, I would have 13 in the middle, just like that. And maybe I have three on each side. And I'm just making that number up. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
The median is 13. So if I have an odd number, I would have 13 in the middle, just like that. And maybe I have three on each side. And I'm just making that number up. I'm just trying to see what I can learn about the different types of data sets that could be described by this box and whiskers plot. So 10 is going to be the middle of the bottom half. So that's 10 right over there. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
And I'm just making that number up. I'm just trying to see what I can learn about the different types of data sets that could be described by this box and whiskers plot. So 10 is going to be the middle of the bottom half. So that's 10 right over there. And 15 is going to be the middle of the top half. That's what this box and whiskers plot is telling us. And they of course tell us what the minimum, the minimum is seven, and they tell us that the maximum is 16. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So that's 10 right over there. And 15 is going to be the middle of the top half. That's what this box and whiskers plot is telling us. And they of course tell us what the minimum, the minimum is seven, and they tell us that the maximum is 16. So we know that's seven, and then that is 16. And then this right over here could be anything. It could be 10, it could be 11, it could be 12, it could be 13. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
And they of course tell us what the minimum, the minimum is seven, and they tell us that the maximum is 16. So we know that's seven, and then that is 16. And then this right over here could be anything. It could be 10, it could be 11, it could be 12, it could be 13. It wouldn't change what these medians are. It wouldn't change this box and whiskers plot. Similarly, this could be 13, it could be 14, it could be 15. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
It could be 10, it could be 11, it could be 12, it could be 13. It wouldn't change what these medians are. It wouldn't change this box and whiskers plot. Similarly, this could be 13, it could be 14, it could be 15. And so any of those values wouldn't change it. And so 75% are 10 or older. Well, this value, in this case, six out of seven are 10 years old or older. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
Similarly, this could be 13, it could be 14, it could be 15. And so any of those values wouldn't change it. And so 75% are 10 or older. Well, this value, in this case, six out of seven are 10 years old or older. And we could try it out with other scenarios where let's try to minimize the number of 10s given this data set. Well, we could do something like, let's say that we have eight. So let's see, one, two, three, four, five, six, seven, eight. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
Well, this value, in this case, six out of seven are 10 years old or older. And we could try it out with other scenarios where let's try to minimize the number of 10s given this data set. Well, we could do something like, let's say that we have eight. So let's see, one, two, three, four, five, six, seven, eight. And so here, we know that the minimum, we know that the minimum is seven, we know that the maximum is 16. We know that the mean of these middle two values, we have an even number now. So the median is going to be the mean of these two values. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So let's see, one, two, three, four, five, six, seven, eight. And so here, we know that the minimum, we know that the minimum is seven, we know that the maximum is 16. We know that the mean of these middle two values, we have an even number now. So the median is going to be the mean of these two values. So it's going to be the mean of this and this is going to be 13. And we know that the mean of, we know that the mean of this and this is going to be 10, and that the mean of this and this is going to be 15. So what could we construct? | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So the median is going to be the mean of these two values. So it's going to be the mean of this and this is going to be 13. And we know that the mean of, we know that the mean of this and this is going to be 10, and that the mean of this and this is going to be 15. So what could we construct? Well, actually, we don't even have to construct to answer this question. We know that this is going to have to be 10 or larger, and then all of these other things are going to be 10 or larger. So this is exactly 75%, exactly 75%, if we assume that this is less than 10, are going to be 10 years old or older. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So what could we construct? Well, actually, we don't even have to construct to answer this question. We know that this is going to have to be 10 or larger, and then all of these other things are going to be 10 or larger. So this is exactly 75%, exactly 75%, if we assume that this is less than 10, are going to be 10 years old or older. So feeling very good, very good, about this one right over here. And actually, just to make this concrete, I'll put in some values here. You know, this could be a nine and an 11. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So this is exactly 75%, exactly 75%, if we assume that this is less than 10, are going to be 10 years old or older. So feeling very good, very good, about this one right over here. And actually, just to make this concrete, I'll put in some values here. You know, this could be a nine and an 11. This could be a 12 and a 14. This could be a 14 and a 16. Or it could be a 15 and a 15. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
You know, this could be a nine and an 11. This could be a 12 and a 14. This could be a 14 and a 16. Or it could be a 15 and a 15. You could think about it in any of those ways. But feeling very good that this is definitely going to be true based on the information given in this plot. Now they say there's only one seven-year-old at the party. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
Or it could be a 15 and a 15. You could think about it in any of those ways. But feeling very good that this is definitely going to be true based on the information given in this plot. Now they say there's only one seven-year-old at the party. One seven-year-old at the party. Well, this first possibility that we looked at, that was the case. There was only one seven-year-old at the party, and there was one 16-year-old at the party. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
Now they say there's only one seven-year-old at the party. One seven-year-old at the party. Well, this first possibility that we looked at, that was the case. There was only one seven-year-old at the party, and there was one 16-year-old at the party. And actually, that was the next statement. There's only one 16-year-old at the party. So both of these seem like we can definitely construct data that's consistent with this box plot, box and whiskers plot, where this is true. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
There was only one seven-year-old at the party, and there was one 16-year-old at the party. And actually, that was the next statement. There's only one 16-year-old at the party. So both of these seem like we can definitely construct data that's consistent with this box plot, box and whiskers plot, where this is true. But could we construct one where it's not true? Well, sure. Let's imagine, let's see, we have our median at 13. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
So both of these seem like we can definitely construct data that's consistent with this box plot, box and whiskers plot, where this is true. But could we construct one where it's not true? Well, sure. Let's imagine, let's see, we have our median at 13. Median at 13. And then we have, let's see, one, two, three, four, five. This is going to be the 10, the median of this bottom half. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
Let's imagine, let's see, we have our median at 13. Median at 13. And then we have, let's see, one, two, three, four, five. This is going to be the 10, the median of this bottom half. This is going to be 15. This is going to be 7. This is going to be 16. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
This is going to be the 10, the median of this bottom half. This is going to be 15. This is going to be 7. This is going to be 16. Well, this could also be 7. It doesn't have to be. It could be 7, 8, 9, or 10. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
This is going to be 16. Well, this could also be 7. It doesn't have to be. It could be 7, 8, 9, or 10. This could also be 16. It doesn't have to be. It could be 15 as well. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
It could be 7, 8, 9, or 10. This could also be 16. It doesn't have to be. It could be 15 as well. But just like that, I've constructed a data set. And these could be 10, 11, 12, 13. This could be 10, 11, 12, 13. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
It could be 15 as well. But just like that, I've constructed a data set. And these could be 10, 11, 12, 13. This could be 10, 11, 12, 13. This could be 13, 14, 15. This one also could be 13, 14, 15. But the simple thing is, or the basic idea here, I can have a data set where I have multiple 7's and multiple 16's. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
This could be 10, 11, 12, 13. This could be 13, 14, 15. This one also could be 13, 14, 15. But the simple thing is, or the basic idea here, I can have a data set where I have multiple 7's and multiple 16's. Or I could have a data set where I only have one 7, or only one 16. So both of these statements, we just plain don't know. Now the next statement, exactly half the students are older than 13. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
But the simple thing is, or the basic idea here, I can have a data set where I have multiple 7's and multiple 16's. Or I could have a data set where I only have one 7, or only one 16. So both of these statements, we just plain don't know. Now the next statement, exactly half the students are older than 13. Well, if you look at this possibility up here, we saw that 3 out of the 7 are older than 13. So that's not exactly half. 3 7's is not 1 half. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
Now the next statement, exactly half the students are older than 13. Well, if you look at this possibility up here, we saw that 3 out of the 7 are older than 13. So that's not exactly half. 3 7's is not 1 half. But in this one over here, we did see that exactly half are older than 13. In fact, if you're saying exactly half, well, in this one, we're saying that exactly half are older than 13. We have an even number right over here. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
3 7's is not 1 half. But in this one over here, we did see that exactly half are older than 13. In fact, if you're saying exactly half, well, in this one, we're saying that exactly half are older than 13. We have an even number right over here. So it is exactly half. So it's possible that it's true. It's possible that it's not true based on the information given. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
We have an even number right over here. So it is exactly half. So it's possible that it's true. It's possible that it's not true based on the information given. We once again do not know. Anyway, hopefully you found this interesting. The whole point of me doing this is, when you look at statistics, sometimes it's easy to kind of say, OK, I think it roughly means that. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
It's possible that it's not true based on the information given. We once again do not know. Anyway, hopefully you found this interesting. The whole point of me doing this is, when you look at statistics, sometimes it's easy to kind of say, OK, I think it roughly means that. And that's sometimes OK. But it's very important to think about what types of actual statements you can make and what you can't make. And it's very important when you're looking at statistics to say, well, you know what? | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
And it's called the Monty Hall problem because Monty Hall was the game show host in Let's Make a Deal, where they would set up a situation very similar to the Monty Hall problem that we're about to say. So let's say that on the show, you're presented with three curtains. So you're the contestant, this little chef-looking character right over there. You're presented with three curtains. Curtain number one, curtain number two, and curtain number three. And you're told that behind one of these three curtains, there's a fabulous prize, something that you really want, a car or a vacation or some type of large amount of cash. And then behind the other two, and we don't know which they are, there is something that you do not want, a new pet goat or an ostrich or something like that, or a beach ball, something that is not as good as the cash prize. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
You're presented with three curtains. Curtain number one, curtain number two, and curtain number three. And you're told that behind one of these three curtains, there's a fabulous prize, something that you really want, a car or a vacation or some type of large amount of cash. And then behind the other two, and we don't know which they are, there is something that you do not want, a new pet goat or an ostrich or something like that, or a beach ball, something that is not as good as the cash prize. And so your goal is to try to find the cash prize. And they say, guess which one, or which one would you like to select? And so let's say that you select door number one, or curtain number one. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
And then behind the other two, and we don't know which they are, there is something that you do not want, a new pet goat or an ostrich or something like that, or a beach ball, something that is not as good as the cash prize. And so your goal is to try to find the cash prize. And they say, guess which one, or which one would you like to select? And so let's say that you select door number one, or curtain number one. Then the Monty Hall and Let's Make a Deal crew will make it a little bit more interesting. They won't just show you whether or not you won. They'll then show you one of the other two doors, or one of the other two curtains. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
And so let's say that you select door number one, or curtain number one. Then the Monty Hall and Let's Make a Deal crew will make it a little bit more interesting. They won't just show you whether or not you won. They'll then show you one of the other two doors, or one of the other two curtains. And they'll show you one of the other two curtains that does not have the prize. And no matter which one you pick, there will always be at least one other curtain that does not have the prize. There might be two if you picked right, but none of them have the prize. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
They'll then show you one of the other two doors, or one of the other two curtains. And they'll show you one of the other two curtains that does not have the prize. And no matter which one you pick, there will always be at least one other curtain that does not have the prize. There might be two if you picked right, but none of them have the prize. And then they will show it to you. And so let's say that they show you curtain number three, and so curtain number three has the goat. And then they will ask you, do you want to switch to curtain number two? | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
There might be two if you picked right, but none of them have the prize. And then they will show it to you. And so let's say that they show you curtain number three, and so curtain number three has the goat. And then they will ask you, do you want to switch to curtain number two? And the question here is, does it make a difference? Are you better off holding fast, sticking to your guns, staying with the original guess? Are you better off switching to whatever curtain is left? | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
And then they will ask you, do you want to switch to curtain number two? And the question here is, does it make a difference? Are you better off holding fast, sticking to your guns, staying with the original guess? Are you better off switching to whatever curtain is left? Or does it not matter? It's just random probability, and it's not going to make a difference whether you switch or not. So that is the brain teaser. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
Are you better off switching to whatever curtain is left? Or does it not matter? It's just random probability, and it's not going to make a difference whether you switch or not. So that is the brain teaser. Pause the video now. I encourage you to think about it. In the next video, we will start to analyze the solution a little bit deeper, whether it makes any difference at all. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
So that is the brain teaser. Pause the video now. I encourage you to think about it. In the next video, we will start to analyze the solution a little bit deeper, whether it makes any difference at all. So now I've assumed that you've unpaused it, you've thought deeply about it. Perhaps you have an opinion on it. Now let's work it through a little bit. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
In the next video, we will start to analyze the solution a little bit deeper, whether it makes any difference at all. So now I've assumed that you've unpaused it, you've thought deeply about it. Perhaps you have an opinion on it. Now let's work it through a little bit. And at any point, we can extrapolate beyond what I've already talked about. So let's think about the game show from the show's point of view. So the show knows where there's the goat and where there isn't the goat. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
Now let's work it through a little bit. And at any point, we can extrapolate beyond what I've already talked about. So let's think about the game show from the show's point of view. So the show knows where there's the goat and where there isn't the goat. So let's door number one, door number two, and door number three. So let's say that our prize is right over here. So our prize is the car. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
So the show knows where there's the goat and where there isn't the goat. So let's door number one, door number two, and door number three. So let's say that our prize is right over here. So our prize is the car. And that we have a goat over here, goat over here, and over here we also have two goats in this situation. So what are we going to do as the game show? Remember, the contestants don't know this. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
So our prize is the car. And that we have a goat over here, goat over here, and over here we also have two goats in this situation. So what are we going to do as the game show? Remember, the contestants don't know this. We know this. So if the contestant picks door number one right over here, then we can't lift door number two because there's a car back there. We're going to lift door number three and we're going to expose this goat, in which case it probably would be good for the person to switch. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
Remember, the contestants don't know this. We know this. So if the contestant picks door number one right over here, then we can't lift door number two because there's a car back there. We're going to lift door number three and we're going to expose this goat, in which case it probably would be good for the person to switch. If the person picks door number two, then we as the game show can show either door number one or door number three, and then it actually does not make sense for the person to switch. If they picked door number three, then we have to show door number one because we can't pick door number two, and in that case it actually makes a lot of sense for the person to switch. Now, with that out of the way, let's think about the probabilities given the two strategies. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
We're going to lift door number three and we're going to expose this goat, in which case it probably would be good for the person to switch. If the person picks door number two, then we as the game show can show either door number one or door number three, and then it actually does not make sense for the person to switch. If they picked door number three, then we have to show door number one because we can't pick door number two, and in that case it actually makes a lot of sense for the person to switch. Now, with that out of the way, let's think about the probabilities given the two strategies. If you don't switch, or another way to think about this strategy is you always stick to your guns. You always stick to your first guess. Well, in that situation, what is your probability of winning? | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
Now, with that out of the way, let's think about the probabilities given the two strategies. If you don't switch, or another way to think about this strategy is you always stick to your guns. You always stick to your first guess. Well, in that situation, what is your probability of winning? Well, there's three doors. The prize is equally likely to be behind any one of them, so there's three possibilities. One has the outcome that you desire. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
Well, in that situation, what is your probability of winning? Well, there's three doors. The prize is equally likely to be behind any one of them, so there's three possibilities. One has the outcome that you desire. The probability of winning will be 1 3rd if you don't switch. Likewise, your probability of losing, well, there's two ways that you can lose out of three possibilities. It's going to be 2 3rd, and these are the only possibilities, and these add up to one right over here. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
One has the outcome that you desire. The probability of winning will be 1 3rd if you don't switch. Likewise, your probability of losing, well, there's two ways that you can lose out of three possibilities. It's going to be 2 3rd, and these are the only possibilities, and these add up to one right over here. So don't switch 1 3rd chance of winning. Now let's think about the switching situation. So let's say always, when you always switch, let's think about how this might play out. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
It's going to be 2 3rd, and these are the only possibilities, and these add up to one right over here. So don't switch 1 3rd chance of winning. Now let's think about the switching situation. So let's say always, when you always switch, let's think about how this might play out. What is your probability of winning? And before we even think about that, think about how you would win if you always switch. So if you pick wrong the first time, they're going to show you this, and so you should always switch. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
So let's say always, when you always switch, let's think about how this might play out. What is your probability of winning? And before we even think about that, think about how you would win if you always switch. So if you pick wrong the first time, they're going to show you this, and so you should always switch. So if you pick door number 1, they're going to show you door number 3. You should switch. If you picked wrong door number 3, they're going to show you door number 1. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
So if you pick wrong the first time, they're going to show you this, and so you should always switch. So if you pick door number 1, they're going to show you door number 3. You should switch. If you picked wrong door number 3, they're going to show you door number 1. You should switch. So if you picked wrong and switch, you will always win. Let me write this down. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
If you picked wrong door number 3, they're going to show you door number 1. You should switch. So if you picked wrong and switch, you will always win. Let me write this down. And this insight actually came from one of the middle school students in the summer camp that Khan Academy was running. It's actually a fabulous way to think about this. So if you pick wrong, so if it's step 1, so initial pick is wrong. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
Let me write this down. And this insight actually came from one of the middle school students in the summer camp that Khan Academy was running. It's actually a fabulous way to think about this. So if you pick wrong, so if it's step 1, so initial pick is wrong. So you pick one of the two wrong doors, and then in step 2, you always switch. You will land on the car, because if you picked one of the wrong doors, they're going to have to show the other wrong door, and so if you switch, you're going to end up on the right answer. So what is the probability of winning if you always switch? | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
So if you pick wrong, so if it's step 1, so initial pick is wrong. So you pick one of the two wrong doors, and then in step 2, you always switch. You will land on the car, because if you picked one of the wrong doors, they're going to have to show the other wrong door, and so if you switch, you're going to end up on the right answer. So what is the probability of winning if you always switch? Well, it's going to be the probability that you initially picked wrong. Well, what's the probability that you initially picked wrong? Well, there's two out of the three ways to initially pick wrong. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
So what is the probability of winning if you always switch? Well, it's going to be the probability that you initially picked wrong. Well, what's the probability that you initially picked wrong? Well, there's two out of the three ways to initially pick wrong. So you actually have a 2 3rds chance of winning. There's a 2 3rds chance you're going to pick wrong and then switch into the right one. Likewise, what's your probability of losing? | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
Well, there's two out of the three ways to initially pick wrong. So you actually have a 2 3rds chance of winning. There's a 2 3rds chance you're going to pick wrong and then switch into the right one. Likewise, what's your probability of losing? Given that you're always going to switch? Well, the way that you would lose is you pick right, you pick correctly, and step 2, they're going to show one of the two empty or non-prized doors, and then step 3, you're going to switch into the other empty. But either way, you're definitely going to switch, so the only way to lose if you're always going to switch is to pick right the first time. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
Likewise, what's your probability of losing? Given that you're always going to switch? Well, the way that you would lose is you pick right, you pick correctly, and step 2, they're going to show one of the two empty or non-prized doors, and then step 3, you're going to switch into the other empty. But either way, you're definitely going to switch, so the only way to lose if you're always going to switch is to pick right the first time. Well, what's the probability of you picking right the first time? Well, that is only 1 3rd. So you see it here, and it's sometimes counterintuitive, but hopefully this makes sense as to why it isn't. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
But either way, you're definitely going to switch, so the only way to lose if you're always going to switch is to pick right the first time. Well, what's the probability of you picking right the first time? Well, that is only 1 3rd. So you see it here, and it's sometimes counterintuitive, but hopefully this makes sense as to why it isn't. You have a 1 3rd chance of winning if you stick to your guns and a 2 3rds chance of winning if you always switch. Another way to think about it is when you first make your initial pick, there's a 1 3rd chance that it's there, and there's a 2 3rds chance that it's in one of the other two doors, and they're going to empty out one of them, so when you switch, you essentially are capturing that 2 3rd probability, and we see that right there. I hope you enjoyed that. | Probability and the Monty Hall problem Probability and combinatorics Precalculus Khan Academy.mp3 |
Now, let's say you have a hunch that, well, maybe it is skewed towards one letter or another. How could you test this? Well, you could start with a null and alternative hypothesis, and then we can actually do a hypothesis test. So let's say that our null hypothesis is equal distribution, equal distribution of correct choices, correct choices. Or another way of thinking about it is A would be correct 25% of the time, B would be correct 25% of the time, C would be correct 25% of the time, and D would be correct 25% of the time. Now, what would be our alternative hypothesis? Well, our alternative hypothesis would be not equal distribution, not equal distribution. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
So let's say that our null hypothesis is equal distribution, equal distribution of correct choices, correct choices. Or another way of thinking about it is A would be correct 25% of the time, B would be correct 25% of the time, C would be correct 25% of the time, and D would be correct 25% of the time. Now, what would be our alternative hypothesis? Well, our alternative hypothesis would be not equal distribution, not equal distribution. Now, how are we going to actually test this? Well, we've seen this show before, at least the beginnings of the show. You have the population of all of your potential items here, and you could take a sample. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
Well, our alternative hypothesis would be not equal distribution, not equal distribution. Now, how are we going to actually test this? Well, we've seen this show before, at least the beginnings of the show. You have the population of all of your potential items here, and you could take a sample. And so let's say we take a sample of 100 items. So n is equal to 100. And let's write down the data that we get when we look at that sample. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
You have the population of all of your potential items here, and you could take a sample. And so let's say we take a sample of 100 items. So n is equal to 100. And let's write down the data that we get when we look at that sample. So this is the correct choice, correct choice. And then this would be the expected number that you would expect. And then this is the actual number. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
And let's write down the data that we get when we look at that sample. So this is the correct choice, correct choice. And then this would be the expected number that you would expect. And then this is the actual number. And if this doesn't make sense yet, we'll see it in a second. So there's four different choices. A, B, C, D. In a sample of 100, remember, in any hypothesis test, we start assuming that the null hypothesis is true. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
And then this is the actual number. And if this doesn't make sense yet, we'll see it in a second. So there's four different choices. A, B, C, D. In a sample of 100, remember, in any hypothesis test, we start assuming that the null hypothesis is true. So the expected number where A is the correct choice would be 25% of this 100. So you'd expect 25 times the A to be the correct choice, 25 times B to be the correct choice, 25 times C to be the correct choice, and 25 times D to be the correct choice. But let's say our actual results, when we look at these 100 items, we get that A is the correct choice 20 times, B is the correct choice 20 times, C is the correct choice 25 times, and D is the correct choice 35 times. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
A, B, C, D. In a sample of 100, remember, in any hypothesis test, we start assuming that the null hypothesis is true. So the expected number where A is the correct choice would be 25% of this 100. So you'd expect 25 times the A to be the correct choice, 25 times B to be the correct choice, 25 times C to be the correct choice, and 25 times D to be the correct choice. But let's say our actual results, when we look at these 100 items, we get that A is the correct choice 20 times, B is the correct choice 20 times, C is the correct choice 25 times, and D is the correct choice 35 times. So if you just look at this, you say, hey, maybe there's a higher frequency of D. But maybe you say, well, this is just a sample, and just a random chance, it might have just gotten more Ds than not. There's some probability of getting this result, even assuming that the null hypothesis is true. And that's the goal of these hypothesis tests. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
But let's say our actual results, when we look at these 100 items, we get that A is the correct choice 20 times, B is the correct choice 20 times, C is the correct choice 25 times, and D is the correct choice 35 times. So if you just look at this, you say, hey, maybe there's a higher frequency of D. But maybe you say, well, this is just a sample, and just a random chance, it might have just gotten more Ds than not. There's some probability of getting this result, even assuming that the null hypothesis is true. And that's the goal of these hypothesis tests. They say, what's the probability of getting a result at least this extreme? And if that probability is below some threshold, then we tend to reject the null hypothesis and accept an alternative. And those thresholds you have seen before, we've seen these significance levels. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
And that's the goal of these hypothesis tests. They say, what's the probability of getting a result at least this extreme? And if that probability is below some threshold, then we tend to reject the null hypothesis and accept an alternative. And those thresholds you have seen before, we've seen these significance levels. Let's say we set a significance level of 5%, 0.05. So if the probability of getting this result, or something even more different than what's expected, is less than the significance level, then we'd reject the null hypothesis. But this all leads to one really interesting question. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
And those thresholds you have seen before, we've seen these significance levels. Let's say we set a significance level of 5%, 0.05. So if the probability of getting this result, or something even more different than what's expected, is less than the significance level, then we'd reject the null hypothesis. But this all leads to one really interesting question. How do we calculate a probability of getting a result this extreme or more extreme? How do we even measure that? And this is where we're going to introduce a new statistic, and also for many of you, a new Greek letter. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
But this all leads to one really interesting question. How do we calculate a probability of getting a result this extreme or more extreme? How do we even measure that? And this is where we're going to introduce a new statistic, and also for many of you, a new Greek letter. And that is the capital Greek letter chi, which might look like an X to you, but it's a little bit curvier, and you could look up more on that. You kind of curve that part of the X. But it's a chi, not an X. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
And this is where we're going to introduce a new statistic, and also for many of you, a new Greek letter. And that is the capital Greek letter chi, which might look like an X to you, but it's a little bit curvier, and you could look up more on that. You kind of curve that part of the X. But it's a chi, not an X. And the statistic is called chi squared. And it's a way of taking the difference between the actual and the expected, and translating that into a number. And the chi squared distribution is well, I really should say distributions, are well studied. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
But it's a chi, not an X. And the statistic is called chi squared. And it's a way of taking the difference between the actual and the expected, and translating that into a number. And the chi squared distribution is well, I really should say distributions, are well studied. And we can use that to figure out what is the probability of getting a result this extreme or more extreme? And if that's lower than our significance level, we reject the null hypothesis, and it suggests the alternative. But how do we calculate the chi squared statistic here? | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
And the chi squared distribution is well, I really should say distributions, are well studied. And we can use that to figure out what is the probability of getting a result this extreme or more extreme? And if that's lower than our significance level, we reject the null hypothesis, and it suggests the alternative. But how do we calculate the chi squared statistic here? Well, it's reasonably intuitive. What we do is, for each of these categories, in this case, it's for each of these choices, we look at the difference between the actual and the expected. So for choice A, we'd say 20 is the actual minus the expected. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
But how do we calculate the chi squared statistic here? Well, it's reasonably intuitive. What we do is, for each of these categories, in this case, it's for each of these choices, we look at the difference between the actual and the expected. So for choice A, we'd say 20 is the actual minus the expected. And then we're going to square that. And then we're going to divide by what was expected. And then we're gonna do that for choice B. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
So for choice A, we'd say 20 is the actual minus the expected. And then we're going to square that. And then we're going to divide by what was expected. And then we're gonna do that for choice B. So we're going to say the actual was 20, expected is 25, so 20 minus 25 squared over the expected over 25. Plus, then we do that for choice C, 25 minus 25, we know where that one will end up, squared over the expected over 25. And then finally, for choice D, which is going to get us 35 minus 25 squared, all of that over 25. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
And then we're gonna do that for choice B. So we're going to say the actual was 20, expected is 25, so 20 minus 25 squared over the expected over 25. Plus, then we do that for choice C, 25 minus 25, we know where that one will end up, squared over the expected over 25. And then finally, for choice D, which is going to get us 35 minus 25 squared, all of that over 25. And we are now, let's see, if we calculate this, this is going to be negative five squared, so that's going to be 25. This is going to be 25. This is going to be zero. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
And then finally, for choice D, which is going to get us 35 minus 25 squared, all of that over 25. And we are now, let's see, if we calculate this, this is going to be negative five squared, so that's going to be 25. This is going to be 25. This is going to be zero. 35 minus 25 is 10 squared, that is 100. So this is one plus one plus zero plus four. So our chi squared statistic in this example came out nice and clean, this won't always be the case, at six. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
This is going to be zero. 35 minus 25 is 10 squared, that is 100. So this is one plus one plus zero plus four. So our chi squared statistic in this example came out nice and clean, this won't always be the case, at six. So what do we make of this? Well, what we can do is then look at a chi squared distribution for the appropriate degrees of freedom, and we'll talk about that in a second, and say what is the probability of getting a chi squared statistic six or larger? And to understand what a chi squared distribution even looks like, these are multiple chi squared distributions for different values for the degrees of freedom. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
So our chi squared statistic in this example came out nice and clean, this won't always be the case, at six. So what do we make of this? Well, what we can do is then look at a chi squared distribution for the appropriate degrees of freedom, and we'll talk about that in a second, and say what is the probability of getting a chi squared statistic six or larger? And to understand what a chi squared distribution even looks like, these are multiple chi squared distributions for different values for the degrees of freedom. And to calculate the degrees of freedom, you look at the number of categories. In this case, we have four categories, and you subtract one. Now that makes a lot of sense, because if you knew how many A's, B's, and C's there are, if you knew the proportions, even the assumed proportions, you can always calculate the fourth one. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
And to understand what a chi squared distribution even looks like, these are multiple chi squared distributions for different values for the degrees of freedom. And to calculate the degrees of freedom, you look at the number of categories. In this case, we have four categories, and you subtract one. Now that makes a lot of sense, because if you knew how many A's, B's, and C's there are, if you knew the proportions, even the assumed proportions, you can always calculate the fourth one. That's why it is four minus one degrees of freedom. So in this case, our degrees of freedom are going to be equal to three. Over here, sometimes you'll see it described as k. So k is equal to three. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
Now that makes a lot of sense, because if you knew how many A's, B's, and C's there are, if you knew the proportions, even the assumed proportions, you can always calculate the fourth one. That's why it is four minus one degrees of freedom. So in this case, our degrees of freedom are going to be equal to three. Over here, sometimes you'll see it described as k. So k is equal to three. So if we look at, that's that little light blue, so we're looking at this chi squared distribution where the degree of freedom is three, and we wanna figure out what is the probability of getting a chi squared statistic that is six or greater? So we would be looking at this area right over here. And you could figure it out using a calculator, or if you're taking some type of a test, like an AP statistics exam, for example, you could use their tables they give you. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
Over here, sometimes you'll see it described as k. So k is equal to three. So if we look at, that's that little light blue, so we're looking at this chi squared distribution where the degree of freedom is three, and we wanna figure out what is the probability of getting a chi squared statistic that is six or greater? So we would be looking at this area right over here. And you could figure it out using a calculator, or if you're taking some type of a test, like an AP statistics exam, for example, you could use their tables they give you. And so a table like this could be quite useful. Remember, we're dealing with a situation where we have three degrees of freedom. We have four categories, so four minus one is three. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
And you could figure it out using a calculator, or if you're taking some type of a test, like an AP statistics exam, for example, you could use their tables they give you. And so a table like this could be quite useful. Remember, we're dealing with a situation where we have three degrees of freedom. We have four categories, so four minus one is three. And we got a chi squared value. Our chi squared statistic was six. So this right over here tells us the probability of getting a 6.25 or greater for a chi squared value is 10%. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
We have four categories, so four minus one is three. And we got a chi squared value. Our chi squared statistic was six. So this right over here tells us the probability of getting a 6.25 or greater for a chi squared value is 10%. If we go back to this chart, we just learned that this probability from 6.25 and up, when we have three degrees of freedom, that this right over here is 10%. Well, if that's 10%, then the probability, the probability of getting a chi squared value greater than or equal to six is going to be greater than 10%, greater than 10%. And we could also view this as our p-value. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
So this right over here tells us the probability of getting a 6.25 or greater for a chi squared value is 10%. If we go back to this chart, we just learned that this probability from 6.25 and up, when we have three degrees of freedom, that this right over here is 10%. Well, if that's 10%, then the probability, the probability of getting a chi squared value greater than or equal to six is going to be greater than 10%, greater than 10%. And we could also view this as our p-value. And so if our probability, assuming the null hypothesis, is greater than 10%, well, it's definitely going to be greater than our significance level. And because of that, we will fail to reject, fail to reject. And so this is an example of, even though in your sample you just happened to get more Ds, the probability of getting a result at least as extreme as what you saw is going to be a little bit over 10%. | Chi-square statistic for hypothesis testing AP Statistics Khan Academy.mp3 |
So this is a histogram here. And in each bucket it tells us the number of guests that are in that age bucket. So we don't have any guests that are under the age of 20. We have a reasonable number between 20 and 30. We have a lot of guests in that bucket between 30 and 40, reasonable number between 40 and 50, and then as we get older we have fewer and fewer guests. So just when you look at something like this, a distribution like this, something might pop out at you. It kind of looks like, if you were to imagine this were an armadillo, this would be the body of the armadillo, and then what we see to the right kind of looks like the tail of the armadillo. | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
We have a reasonable number between 20 and 30. We have a lot of guests in that bucket between 30 and 40, reasonable number between 40 and 50, and then as we get older we have fewer and fewer guests. So just when you look at something like this, a distribution like this, something might pop out at you. It kind of looks like, if you were to imagine this were an armadillo, this would be the body of the armadillo, and then what we see to the right kind of looks like the tail of the armadillo. And we actually use those types of words to describe distributions. So this distribution right over here, it looks like it has a tail to the right. It doesn't have a tail to the left. | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
It kind of looks like, if you were to imagine this were an armadillo, this would be the body of the armadillo, and then what we see to the right kind of looks like the tail of the armadillo. And we actually use those types of words to describe distributions. So this distribution right over here, it looks like it has a tail to the right. It doesn't have a tail to the left. In fact, we have no one under the age of 20. But here when we have a few people between 60 and 70, even fewer between 70 and 80, even fewer between 80 and 90, and if it just kind of keeps going like this, this is a tail and it's on the right side. It's a right-tailed distribution. | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
It doesn't have a tail to the left. In fact, we have no one under the age of 20. But here when we have a few people between 60 and 70, even fewer between 70 and 80, even fewer between 80 and 90, and if it just kind of keeps going like this, this is a tail and it's on the right side. It's a right-tailed distribution. So I'd call this distribution right-tailed. And I'm using Khan Academy exercises because it's a good way to see a lot of examples, and frankly, you should too because it'll help you test your knowledge. But it's not left-tailed. | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
It's a right-tailed distribution. So I'd call this distribution right-tailed. And I'm using Khan Academy exercises because it's a good way to see a lot of examples, and frankly, you should too because it'll help you test your knowledge. But it's not left-tailed. Left-tailed we would see a tail going like that. And frankly, if you're left-tailed and right-tailed, you're likely to be approximately symmetrical. Remember, symmetry, you define a line of symmetry, and one type of symmetry is one where if you were to, where both sides of that line of symmetry are kind of mirror images of each other. | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
But it's not left-tailed. Left-tailed we would see a tail going like that. And frankly, if you're left-tailed and right-tailed, you're likely to be approximately symmetrical. Remember, symmetry, you define a line of symmetry, and one type of symmetry is one where if you were to, where both sides of that line of symmetry are kind of mirror images of each other. You could fold over the line of symmetry and they'll roughly meet. And this one does not meet that because if you were to say, hey, maybe there's a line of symmetry here and you were trying to fold this over, it wouldn't match up. The two sides would not match up. | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
Remember, symmetry, you define a line of symmetry, and one type of symmetry is one where if you were to, where both sides of that line of symmetry are kind of mirror images of each other. You could fold over the line of symmetry and they'll roughly meet. And this one does not meet that because if you were to say, hey, maybe there's a line of symmetry here and you were trying to fold this over, it wouldn't match up. The two sides would not match up. So I feel good saying that it is right-tailed. So let's see. Retirement of age of each guest. | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
The two sides would not match up. So I feel good saying that it is right-tailed. So let's see. Retirement of age of each guest. Well, yeah, these names aren't that great, but let's actually see what they're saying. They're saying by age, they're telling us the number of guests. So this is the number of guests at a Logan-assisted living. | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
Retirement of age of each guest. Well, yeah, these names aren't that great, but let's actually see what they're saying. They're saying by age, they're telling us the number of guests. So this is the number of guests at a Logan-assisted living. So we have a lot of guests that are between 60 and 70 years old, or reasonable that are between 50 and 60, or 70 or 80. And this distribution actually looks pretty symmetrical. If I were to draw a line of symmetry right down here, right at around an age of, you know, I guess the line would be right at an age of 65. | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
So this is the number of guests at a Logan-assisted living. So we have a lot of guests that are between 60 and 70 years old, or reasonable that are between 50 and 60, or 70 or 80. And this distribution actually looks pretty symmetrical. If I were to draw a line of symmetry right down here, right at around an age of, you know, I guess the line would be right at an age of 65. I guess you could say, oh, this is a bucket for ages 60 to 70. Then you could flip it over and it looks pretty symmetrical. Not exactly. | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
If I were to draw a line of symmetry right down here, right at around an age of, you know, I guess the line would be right at an age of 65. I guess you could say, oh, this is a bucket for ages 60 to 70. Then you could flip it over and it looks pretty symmetrical. Not exactly. This bucket doesn't quite match up to this one, but it's pretty close. These roughly match each other. These roughly match each other. | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
Not exactly. This bucket doesn't quite match up to this one, but it's pretty close. These roughly match each other. These roughly match each other. So I feel good about saying it is approximately symmetrical. Now, just to know what these other words mean, skewed to the left, or skewed to the right. These actually have fairly technical definitions when you get further in statistics, but a, I guess, easier to process version of them are when you have a left tail, you tend to be, when you are left-tailed, you also tend to be skewed to the left. | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
These roughly match each other. So I feel good about saying it is approximately symmetrical. Now, just to know what these other words mean, skewed to the left, or skewed to the right. These actually have fairly technical definitions when you get further in statistics, but a, I guess, easier to process version of them are when you have a left tail, you tend to be, when you are left-tailed, you also tend to be skewed to the left. And when you are right-tailed, you tend to be skewed to the right. Another way to think about skewed to the left is that your mean is to the left of your median in mode. That might not make any sense to you. | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
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