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We don't just add up all of the data points. We then have to divide by the number of data points there are. So we then have to divide, we then have to divide by the number of data points that there actually are. So this might look like very fancy notation, but it's really just saying, add up your data points and divide by the number of data points you have. And this capital Greek letter, sigma, literally means sum, sum all of the x i's from x sub one all the way to x sub n, and then divide by the number of data points you have. Now let's think about how we would denote the same thing, but instead of for the sample mean, doing it for the population mean. So the population mean, they will denote it with mu. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
So this might look like very fancy notation, but it's really just saying, add up your data points and divide by the number of data points you have. And this capital Greek letter, sigma, literally means sum, sum all of the x i's from x sub one all the way to x sub n, and then divide by the number of data points you have. Now let's think about how we would denote the same thing, but instead of for the sample mean, doing it for the population mean. So the population mean, they will denote it with mu. We already talked about that. And here, once again, you're gonna take the sum, but this time it's going to be the sum of all of the elements in your population. So your x sub i's, and you'll still start at i equals one, but it usually gets denoted that, hey, you're taking the whole population. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
So the population mean, they will denote it with mu. We already talked about that. And here, once again, you're gonna take the sum, but this time it's going to be the sum of all of the elements in your population. So your x sub i's, and you'll still start at i equals one, but it usually gets denoted that, hey, you're taking the whole population. So they'll often put a capital N right over here to somehow denote that this is a bigger number than maybe this smaller n. But once again, we are not done. We have to divide by the number of data points that we are actually summing. And so this, once again, is the same thing as x sub one plus x sub two plus x sub three all the way to x sub capital N, all of that divided by our capital N. And once again, in this situation, we found this practical, we found this impractical. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
So to figure out this probability, a good place to start is just to think about all of the different possible ways that we can flip three coins. So we could get all tails. Tails, tails, tails. We could get tails, tails, heads. We could get tails, heads, tails. We could get tails, heads, heads. We could get heads, tails, tails. | Example Probability through counting outcomes Precalculus Khan Academy.mp3 |
We could get tails, tails, heads. We could get tails, heads, tails. We could get tails, heads, heads. We could get heads, tails, tails. We could get heads, tails, heads. We could get heads, heads, tails. And then we could get all heads. | Example Probability through counting outcomes Precalculus Khan Academy.mp3 |
We could get heads, tails, tails. We could get heads, tails, heads. We could get heads, heads, tails. And then we could get all heads. So there are 1, 2, 3, 4, 5, 6, 7, 8 possible outcomes. Now, how many of the outcomes involve flipping exactly two heads? Let's see, that's all tails. | Example Probability through counting outcomes Precalculus Khan Academy.mp3 |
And then we could get all heads. So there are 1, 2, 3, 4, 5, 6, 7, 8 possible outcomes. Now, how many of the outcomes involve flipping exactly two heads? Let's see, that's all tails. That's one head, one head. This has two heads right there. Two heads. | Example Probability through counting outcomes Precalculus Khan Academy.mp3 |
Let's see, that's all tails. That's one head, one head. This has two heads right there. Two heads. That's one head. This is two heads right over there. Then this is two heads right over here. | Example Probability through counting outcomes Precalculus Khan Academy.mp3 |
Two heads. That's one head. This is two heads right over there. Then this is two heads right over here. And then this is three heads, so that doesn't count. So there are three outcomes where we, with exactly two heads. So, let me spell heads properly. | Example Probability through counting outcomes Precalculus Khan Academy.mp3 |
Then this is two heads right over here. And then this is three heads, so that doesn't count. So there are three outcomes where we, with exactly two heads. So, let me spell heads properly. Two heads. So the probability of flipping of exactly two heads, and the word exactly is important, because if you didn't say exactly, then maybe three heads, well, you flip two heads, so you have to say exactly two heads. So you don't include the situation where you get three heads. | Example Probability through counting outcomes Precalculus Khan Academy.mp3 |
So, let me spell heads properly. Two heads. So the probability of flipping of exactly two heads, and the word exactly is important, because if you didn't say exactly, then maybe three heads, well, you flip two heads, so you have to say exactly two heads. So you don't include the situation where you get three heads. So the probability of flipping exactly two heads is equal to the three outcomes with two heads divided by the eight possible outcomes, or 3 8ths. So it is equal to 3 8ths. And we are done. | Example Probability through counting outcomes Precalculus Khan Academy.mp3 |
And so I went to the insurance company and I said, I want to get a $1 million policy. And what I'm actually getting a quote on is a term life policy, which is really, I just care about the next 20 years. After those 20 years, hopefully I can pay off my mortgage and there'll be money saved up. And hopefully my kids would kind of at least have maybe gotten to college or I would have saved up enough money for college. So that's why I'm willing to do a term life policy. The other option is to do a whole life policy, where you could pay a certain amount per year for the rest of your life. And at any point you die, you get the $1 million. | Term life insurance and death probability Finance & Capital Markets Khan Academy.mp3 |
And hopefully my kids would kind of at least have maybe gotten to college or I would have saved up enough money for college. So that's why I'm willing to do a term life policy. The other option is to do a whole life policy, where you could pay a certain amount per year for the rest of your life. And at any point you die, you get the $1 million. In a term life, I'm only going to pay $500 per year for the next 20 years. If at any point over those 20 years I die, my family gets $1 million. At the 21st year, I have to get a new policy. | Term life insurance and death probability Finance & Capital Markets Khan Academy.mp3 |
And at any point you die, you get the $1 million. In a term life, I'm only going to pay $500 per year for the next 20 years. If at any point over those 20 years I die, my family gets $1 million. At the 21st year, I have to get a new policy. And since I'm going to be older and I'd have a higher chance of dying at that point, then it's probably going to be more expensive for me to get insurance. But I really am just worried about the next 20 years. But what I want to do in this video is think about, given these numbers that have been quoted to me by the insurance company, what do they think that my odds of dying are over the next 20 years? | Term life insurance and death probability Finance & Capital Markets Khan Academy.mp3 |
At the 21st year, I have to get a new policy. And since I'm going to be older and I'd have a higher chance of dying at that point, then it's probably going to be more expensive for me to get insurance. But I really am just worried about the next 20 years. But what I want to do in this video is think about, given these numbers that have been quoted to me by the insurance company, what do they think that my odds of dying are over the next 20 years? So what I want to think about is the probability of Sal's death in 20 years, based on what the people at the insurance company are telling me. Or at least, what's the maximum probability of my death in order for them to make money? And the way to think about it, or one way to think about it, kind of a back of the envelope way, is to think about what's the total premiums they're getting over the life of this policy, divided by how much they're insuring me for. | Term life insurance and death probability Finance & Capital Markets Khan Academy.mp3 |
But what I want to do in this video is think about, given these numbers that have been quoted to me by the insurance company, what do they think that my odds of dying are over the next 20 years? So what I want to think about is the probability of Sal's death in 20 years, based on what the people at the insurance company are telling me. Or at least, what's the maximum probability of my death in order for them to make money? And the way to think about it, or one way to think about it, kind of a back of the envelope way, is to think about what's the total premiums they're getting over the life of this policy, divided by how much they're insuring me for. So they're getting $500 times 20 years is equal to $10,000 over the life of this policy. And they're insuring me for $1 million. So they're getting, let's see, those zeros cancel out, this zero cancels out. | Term life insurance and death probability Finance & Capital Markets Khan Academy.mp3 |
And the way to think about it, or one way to think about it, kind of a back of the envelope way, is to think about what's the total premiums they're getting over the life of this policy, divided by how much they're insuring me for. So they're getting $500 times 20 years is equal to $10,000 over the life of this policy. And they're insuring me for $1 million. So they're getting, let's see, those zeros cancel out, this zero cancels out. They're getting, over the life of the policy, $1 in premiums for every $100 in insurance. Or another way to think about it, let's say that there were 100 Sal's, 134-year-olds looking to get 20-year-term life insurance. And they insured all of them. | Term life insurance and death probability Finance & Capital Markets Khan Academy.mp3 |
So they're getting, let's see, those zeros cancel out, this zero cancels out. They're getting, over the life of the policy, $1 in premiums for every $100 in insurance. Or another way to think about it, let's say that there were 100 Sal's, 134-year-olds looking to get 20-year-term life insurance. And they insured all of them. So if you multiplied this times 100, if you multiplied this by 100, they would get $100 in premiums. $100 in premiums. This is the case where you have 100 Sal's, or 100 people who are pretty similar to me. | Term life insurance and death probability Finance & Capital Markets Khan Academy.mp3 |
And they insured all of them. So if you multiplied this times 100, if you multiplied this by 100, they would get $100 in premiums. $100 in premiums. This is the case where you have 100 Sal's, or 100 people who are pretty similar to me. 100 Sal's. They would get $100 in premium. And the only way that they could make money is if, at most, one of those Sal's, or really just break even, if at most one of those Sal's were to die. | Term life insurance and death probability Finance & Capital Markets Khan Academy.mp3 |
This is the case where you have 100 Sal's, or 100 people who are pretty similar to me. 100 Sal's. They would get $100 in premium. And the only way that they could make money is if, at most, one of those Sal's, or really just break even, if at most one of those Sal's were to die. So break even if only one Sal dies. I don't like talking about this. It's a little bit morbid. | Term life insurance and death probability Finance & Capital Markets Khan Academy.mp3 |
And the only way that they could make money is if, at most, one of those Sal's, or really just break even, if at most one of those Sal's were to die. So break even if only one Sal dies. I don't like talking about this. It's a little bit morbid. So one way to think about it, they're getting $1 in premium for $100 insurance. Or if they had 100 Sal's, they would get $100 in premium. And the only way they would break even if only one of those Sal's dies. | Term life insurance and death probability Finance & Capital Markets Khan Academy.mp3 |
It's a little bit morbid. So one way to think about it, they're getting $1 in premium for $100 insurance. Or if they had 100 Sal's, they would get $100 in premium. And the only way they would break even if only one of those Sal's dies. So what they're really saying is that the only way they can break even is if the probability of Sal dying in the next 20 years is less than or equal to 1 in 100. And this is an insurance company. They're trying to make money. | Term life insurance and death probability Finance & Capital Markets Khan Academy.mp3 |
And the only way they would break even if only one of those Sal's dies. So what they're really saying is that the only way they can break even is if the probability of Sal dying in the next 20 years is less than or equal to 1 in 100. And this is an insurance company. They're trying to make money. So they're probably giving these numbers because they think the probability of me dying is a good, maybe it's 1 in 200, or it's 1 in 300, something lower. So that they can insure, one way to think about it, they could insure more Sal's for every $100 in premium they have to pay out. But either way, it's a back of the envelope way of thinking about it. | Term life insurance and death probability Finance & Capital Markets Khan Academy.mp3 |
Lucio wants to test whether playing violent video games makes people more violent. He asks his friends whether they play violent video games and whether they have been in a fight in the last month. He recorded the results in the table shown below. Fill in the table to show the fraction of each group of students who have been in a fight. Then decide whether there's an association between violent video games and getting in a fight amongst Lucio's friends. So let's see what they're doing here. So students who play video games, fractions who have been in a fight, fraction who haven't. | Video games and violence Statistical studies Probability and Statistics Khan Academy.mp3 |
Fill in the table to show the fraction of each group of students who have been in a fight. Then decide whether there's an association between violent video games and getting in a fight amongst Lucio's friends. So let's see what they're doing here. So students who play video games, fractions who have been in a fight, fraction who haven't. Students who don't play violent video games, fraction who have been in a fight, fraction who haven't. So let's answer the first part of this. Students who play violent video games. | Video games and violence Statistical studies Probability and Statistics Khan Academy.mp3 |
So students who play video games, fractions who have been in a fight, fraction who haven't. Students who don't play violent video games, fraction who have been in a fight, fraction who haven't. So let's answer the first part of this. Students who play violent video games. So let's look at those students. So the students who play violent video games, it looks like Ellen plays violent video games. Actually, let me just focus on the data that we care about. | Video games and violence Statistical studies Probability and Statistics Khan Academy.mp3 |
Students who play violent video games. So let's look at those students. So the students who play violent video games, it looks like Ellen plays violent video games. Actually, let me just focus on the data that we care about. So Ellen, so let's look at all the people who play violent video games. So let's see. This column is violent video games, so we have a yes here. | Video games and violence Statistical studies Probability and Statistics Khan Academy.mp3 |
Actually, let me just focus on the data that we care about. So Ellen, so let's look at all the people who play violent video games. So let's see. This column is violent video games, so we have a yes here. So we have both of these right over here. And then we have down here. And then that's all of them. | Video games and violence Statistical studies Probability and Statistics Khan Academy.mp3 |
This column is violent video games, so we have a yes here. So we have both of these right over here. And then we have down here. And then that's all of them. There's one, two, three, four, five people who play violent video games. Now what fraction of them have been in a fight? Well, it looks like one out of the five have been in a fight. | Video games and violence Statistical studies Probability and Statistics Khan Academy.mp3 |
And then that's all of them. There's one, two, three, four, five people who play violent video games. Now what fraction of them have been in a fight? Well, it looks like one out of the five have been in a fight. The rest of them have not been in a fight. So we could say 1 5th. Let's just write that down. | Video games and violence Statistical studies Probability and Statistics Khan Academy.mp3 |
Well, it looks like one out of the five have been in a fight. The rest of them have not been in a fight. So we could say 1 5th. Let's just write that down. So 1 5th have been in a fight. 1 5th have been in a fight. Fraction who haven't, 4 5ths. | Video games and violence Statistical studies Probability and Statistics Khan Academy.mp3 |
Let's just write that down. So 1 5th have been in a fight. 1 5th have been in a fight. Fraction who haven't, 4 5ths. So that's all these other nos. They play violent video games, but they haven't been in a fight, one, two, three, four, 4 5ths. So students who don't play violent video games. | Video games and violence Statistical studies Probability and Statistics Khan Academy.mp3 |
Fraction who haven't, 4 5ths. So that's all these other nos. They play violent video games, but they haven't been in a fight, one, two, three, four, 4 5ths. So students who don't play violent video games. Well, that's everyone else. And let's see how many data points that is. That is one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15. | Video games and violence Statistical studies Probability and Statistics Khan Academy.mp3 |
So students who don't play violent video games. Well, that's everyone else. And let's see how many data points that is. That is one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15. So there's a total of 15 students. And how many of them have been in a fight? So let's see. | Video games and violence Statistical studies Probability and Statistics Khan Academy.mp3 |
That is one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15. So there's a total of 15 students. And how many of them have been in a fight? So let's see. We have one, and then let's see, one, and two, and three. So three out of the 15 have been in a fight. So three out of 15 is the same thing as one out of five. | Video games and violence Statistical studies Probability and Statistics Khan Academy.mp3 |
So let's see. We have one, and then let's see, one, and two, and three. So three out of the 15 have been in a fight. So three out of 15 is the same thing as one out of five. Those are equivalent fractions. And then the fraction who haven't, well, that's just going to be everyone else. That's going to be 12 out of 15 or four out of five. | Video games and violence Statistical studies Probability and Statistics Khan Academy.mp3 |
So three out of 15 is the same thing as one out of five. Those are equivalent fractions. And then the fraction who haven't, well, that's just going to be everyone else. That's going to be 12 out of 15 or four out of five. So based on Lucio's data, and this wasn't a huge sample size, obviously, he only found five kids who were playing violent video games, and one of them had gotten into a fight. So this isn't a super rigorous study. But at least based on his data, if we're trying to decide whether there's an association between violent games and getting into a fight amongst Lucio's friends, it doesn't seem like there is. | Video games and violence Statistical studies Probability and Statistics Khan Academy.mp3 |
That's going to be 12 out of 15 or four out of five. So based on Lucio's data, and this wasn't a huge sample size, obviously, he only found five kids who were playing violent video games, and one of them had gotten into a fight. So this isn't a super rigorous study. But at least based on his data, if we're trying to decide whether there's an association between violent games and getting into a fight amongst Lucio's friends, it doesn't seem like there is. It seems like relatively, whether or not they play violent video games or not, one fifth of them have been in a fight in the last month. So it really doesn't seem any difference. If this number was, I don't know, four fifths or five fifths or all of them, then I would say, hey, even with Lucio's fairly small sample, I would say, hey, maybe there is some type of a strong association between playing violent video games and fighting. | Video games and violence Statistical studies Probability and Statistics Khan Academy.mp3 |
And as we will see as we build up our understanding of them, not only are they interesting in their own right, but there's a lot of very powerful probability and statistics that we can do based on our understanding of binomial variables. So to make things concrete as quickly as possible, I'll start with a very tangible example of a binomial variable, and then we'll think a little bit more abstractly about what makes it binomial. So let's say that I have a coin. So this is my coin here. Doesn't even have to be a fair coin. Let me just draw this really fast. So that's my coin. | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
So this is my coin here. Doesn't even have to be a fair coin. Let me just draw this really fast. So that's my coin. And let's say on a given flip of that coin, the probability that I get heads is 0.6, and the probability that I get tails, well, it would be one minus 0.6, or 0.4. And what I'm going to do is I'm going to define a random variable X as being equal to the number of heads after 10 flips of my coin. Now what makes this a binomial variable? | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
So that's my coin. And let's say on a given flip of that coin, the probability that I get heads is 0.6, and the probability that I get tails, well, it would be one minus 0.6, or 0.4. And what I'm going to do is I'm going to define a random variable X as being equal to the number of heads after 10 flips of my coin. Now what makes this a binomial variable? Well, one of the first conditions that's often given for a binomial variable is that it's made up of a finite number of independent trials. So it's made up, made up of independent, independent trials. Now what do I mean by independent trials? | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
Now what makes this a binomial variable? Well, one of the first conditions that's often given for a binomial variable is that it's made up of a finite number of independent trials. So it's made up, made up of independent, independent trials. Now what do I mean by independent trials? Well, a trial is each flip of my coin. So a flip is equal to a trial in the language of this statement that I just made. And what do I mean by each flip or each trial being independent? | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
Now what do I mean by independent trials? Well, a trial is each flip of my coin. So a flip is equal to a trial in the language of this statement that I just made. And what do I mean by each flip or each trial being independent? Well, the probability of whether I get heads or tails on each flip are independent of whether I just got heads or tails on some previous flip. So in this case, we are made up of independent trials. So another condition is each trial can be clearly classified as either a success or failure. | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
And what do I mean by each flip or each trial being independent? Well, the probability of whether I get heads or tails on each flip are independent of whether I just got heads or tails on some previous flip. So in this case, we are made up of independent trials. So another condition is each trial can be clearly classified as either a success or failure. Or another way of thinking about it, each trial clearly has one of two discrete outcomes. So each trial, and in the example I'm giving, the flip is a trial, can be classified, can be classified as either success or failure. So in the context of this random variable X, we could define heads as a success because that's what we are happening to count up. | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
So another condition is each trial can be clearly classified as either a success or failure. Or another way of thinking about it, each trial clearly has one of two discrete outcomes. So each trial, and in the example I'm giving, the flip is a trial, can be classified, can be classified as either success or failure. So in the context of this random variable X, we could define heads as a success because that's what we are happening to count up. And so you're either going to have success or failure. You're either gonna have heads or tails on each of these trials. Now another condition for being a binomial variable is that you have a fixed number of trials. | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
So in the context of this random variable X, we could define heads as a success because that's what we are happening to count up. And so you're either going to have success or failure. You're either gonna have heads or tails on each of these trials. Now another condition for being a binomial variable is that you have a fixed number of trials. Fixed number of trials. So in this case, we're saying that we have 10 trials, 10 flips of our coin. And then the last condition is the probability of success, and in this context, success is a heads, on each trial, each trial is constant. | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
Now another condition for being a binomial variable is that you have a fixed number of trials. Fixed number of trials. So in this case, we're saying that we have 10 trials, 10 flips of our coin. And then the last condition is the probability of success, and in this context, success is a heads, on each trial, each trial is constant. And we've already talked about it. On each trial on each flip, the probability of heads is going to stay at 0.6. If for some reason that were to change from trial to trial, maybe if you were to swap the coin and each coin had a different probability, then this would no longer be a binomial variable. | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
And then the last condition is the probability of success, and in this context, success is a heads, on each trial, each trial is constant. And we've already talked about it. On each trial on each flip, the probability of heads is going to stay at 0.6. If for some reason that were to change from trial to trial, maybe if you were to swap the coin and each coin had a different probability, then this would no longer be a binomial variable. And so you might say, okay, that's reasonable. I get why this is a binomial variable. Can you give me an example of something that is not a binomial variable? | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
If for some reason that were to change from trial to trial, maybe if you were to swap the coin and each coin had a different probability, then this would no longer be a binomial variable. And so you might say, okay, that's reasonable. I get why this is a binomial variable. Can you give me an example of something that is not a binomial variable? Well, let's say that I were to define the variable Y, and it's equal to the number of kings after taking two cards from a standard deck of cards, a standard deck, without replacement, without replacement. So you might immediately say, well, this feels like it could be binomial. We have each trial can be classified as either a success or failure. | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
Can you give me an example of something that is not a binomial variable? Well, let's say that I were to define the variable Y, and it's equal to the number of kings after taking two cards from a standard deck of cards, a standard deck, without replacement, without replacement. So you might immediately say, well, this feels like it could be binomial. We have each trial can be classified as either a success or failure. For each trials, when I take a card out, if I get a king, that looks like that would be a success. If I don't get a king, that would be a failure. So it seems to meet that right over there. | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
We have each trial can be classified as either a success or failure. For each trials, when I take a card out, if I get a king, that looks like that would be a success. If I don't get a king, that would be a failure. So it seems to meet that right over there. It has a fixed number of trials. I'm taking two cards out of the deck, so it seems to meet that. But what about these conditions, that it's made up of independent trials, or that the probability of success on each trial is constant? | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
So it seems to meet that right over there. It has a fixed number of trials. I'm taking two cards out of the deck, so it seems to meet that. But what about these conditions, that it's made up of independent trials, or that the probability of success on each trial is constant? Well, if I get a king, the probability of king on the first trial, probability, I say king, on first trial would be equal to, well, out of a deck of 52 cards, you're going to have four kings in it. So the probability of a king on the first trial would be four out of 52. But what about the probability of getting a king on the second, on the second trial? | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
But what about these conditions, that it's made up of independent trials, or that the probability of success on each trial is constant? Well, if I get a king, the probability of king on the first trial, probability, I say king, on first trial would be equal to, well, out of a deck of 52 cards, you're going to have four kings in it. So the probability of a king on the first trial would be four out of 52. But what about the probability of getting a king on the second, on the second trial? What would this be equal to? Well, it depends on what happened on the first trial. If the first trial, you had a king, well, then you would have, so let's see, this would be the situation given first trial, first king. | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
But what about the probability of getting a king on the second, on the second trial? What would this be equal to? Well, it depends on what happened on the first trial. If the first trial, you had a king, well, then you would have, so let's see, this would be the situation given first trial, first king. Well, now, there would be three kings left in a deck of 51 cards. But if you did not get a king on the first trial, now you have four kings in a deck of 51 cards, because remember, we're doing it without replacement. You're just taking that first card, whatever you did, and you're taking it aside. | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
If the first trial, you had a king, well, then you would have, so let's see, this would be the situation given first trial, first king. Well, now, there would be three kings left in a deck of 51 cards. But if you did not get a king on the first trial, now you have four kings in a deck of 51 cards, because remember, we're doing it without replacement. You're just taking that first card, whatever you did, and you're taking it aside. So what's interesting here is this is not made up of independent trials. It does not meet this condition. The probability on your second trial is dependent on what happens on your first trial. | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
You're just taking that first card, whatever you did, and you're taking it aside. So what's interesting here is this is not made up of independent trials. It does not meet this condition. The probability on your second trial is dependent on what happens on your first trial. And another way to think about it is because we aren't replacing each card that we're picking, the probability of success on each trial also is not constant. And so that's why this right over here is not a binomial variable. Now, if y, if we got rid of, without replacement, and if we said we did replace every card after we picked it, then things would be different. | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
The probability on your second trial is dependent on what happens on your first trial. And another way to think about it is because we aren't replacing each card that we're picking, the probability of success on each trial also is not constant. And so that's why this right over here is not a binomial variable. Now, if y, if we got rid of, without replacement, and if we said we did replace every card after we picked it, then things would be different. Then we actually would be looking at a binomial variable. So instead of without replacement, if I just said with replacement, well then, your probability of a king on each trial is going to be four out of 52. You have a finite number of trials. | Binomial variables Random variables AP Statistics Khan Academy.mp3 |
So we have some data here that we can plot on a scatter plot that looks something like that. And so the next question, given that we've been talking a lot about lines of regression or regression lines, is can we fit a regression line to this? Well, if we try to, we might get something that looks like this, or maybe something that looks like this. I'm just eyeballing it. Obviously, we could input it into a computer to try to develop a linear regression model to try to minimize the sum of the squared distances from the points to the line. But you can see it's pretty difficult. And some of you might be saying, well, this looks more like some type of an exponential. | Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3 |
I'm just eyeballing it. Obviously, we could input it into a computer to try to develop a linear regression model to try to minimize the sum of the squared distances from the points to the line. But you can see it's pretty difficult. And some of you might be saying, well, this looks more like some type of an exponential. So maybe we could fit an exponential to it. So it could look something like that. And you wouldn't be wrong. | Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3 |
And some of you might be saying, well, this looks more like some type of an exponential. So maybe we could fit an exponential to it. So it could look something like that. And you wouldn't be wrong. But there is a way that we can apply our tools of linear regression to this data set. And the way we can is instead of plotting x versus y, we can think about x versus the logarithm of y. So this is this exact same data set. | Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3 |
And you wouldn't be wrong. But there is a way that we can apply our tools of linear regression to this data set. And the way we can is instead of plotting x versus y, we can think about x versus the logarithm of y. So this is this exact same data set. You see the x values are the same. But for the y values, I just took the log base 10 of all of these. So 10 to the what power is equal to 2,307.23. | Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3 |
So this is this exact same data set. You see the x values are the same. But for the y values, I just took the log base 10 of all of these. So 10 to the what power is equal to 2,307.23. 10 to the 3.36 power is equal to 2,307.23. I did that for all of these data points. I did it on a spreadsheet. | Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3 |
So 10 to the what power is equal to 2,307.23. 10 to the 3.36 power is equal to 2,307.23. I did that for all of these data points. I did it on a spreadsheet. And if you were to plot all of these, something neat happens. All of a sudden, when we're plotting x versus the log of y or the log of y versus x, all of a sudden it looks linear. Now, be clear. | Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3 |
I did it on a spreadsheet. And if you were to plot all of these, something neat happens. All of a sudden, when we're plotting x versus the log of y or the log of y versus x, all of a sudden it looks linear. Now, be clear. The true relationship between x and y is not linear. It looks like some type of an exponential relationship. But the value of transforming the data, and there's different ways you can do it. | Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3 |
Now, be clear. The true relationship between x and y is not linear. It looks like some type of an exponential relationship. But the value of transforming the data, and there's different ways you can do it. In this case, the value of taking the log of y and thinking about it that way is now we can use our tools of linear regression. Because this data set, you could actually fit a linear regression line to this quite well. You could imagine a line that looks something like this. | Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3 |
But the value of transforming the data, and there's different ways you can do it. In this case, the value of taking the log of y and thinking about it that way is now we can use our tools of linear regression. Because this data set, you could actually fit a linear regression line to this quite well. You could imagine a line that looks something like this. It would fit the data quite well. And the reason why you might wanna do this versus trying to fit an exponential is because we've already developed so many tools around linear regression and hypothesis testing around the slope and confidence intervals. And so this might be the direction you wanna go at. | Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3 |
You could imagine a line that looks something like this. It would fit the data quite well. And the reason why you might wanna do this versus trying to fit an exponential is because we've already developed so many tools around linear regression and hypothesis testing around the slope and confidence intervals. And so this might be the direction you wanna go at. And what's neat is once you fit a linear regression, it's not difficult to mathematically unwind from your linear model back to an exponential one. So the big takeaway here is is that the tools of linear regression can be useful even when the underlying relationship between x and y are nonlinear. And the way that we do that is by transforming the data. | Transforming nonlinear data More on regression AP Statistics Khan Academy.mp3 |
She randomized 50 workdays between a treatment group and a control group. For each day from the treatment group, she took bus A, and for each day from the control group, she took bus B. Each day, she timed the length of her drive. This is really interesting what she did. It's very important. She randomized the 50 workdays. It's important she did this instead of just kind of waking up in the morning and just deciding on her own which bus to take because humans are infamously bad at being random. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
This is really interesting what she did. It's very important. She randomized the 50 workdays. It's important she did this instead of just kind of waking up in the morning and just deciding on her own which bus to take because humans are infamously bad at being random. Even when we think we're being random, we're actually not that random. She might inadvertently be taking bus A earlier in the week where maybe the commute times are shorter, or maybe she inadvertently takes bus A when the weather is better, when there's less traffic. Remember, there's a natural tendency for human beings to want to confirm their hypothesis. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
It's important she did this instead of just kind of waking up in the morning and just deciding on her own which bus to take because humans are infamously bad at being random. Even when we think we're being random, we're actually not that random. She might inadvertently be taking bus A earlier in the week where maybe the commute times are shorter, or maybe she inadvertently takes bus A when the weather is better, when there's less traffic. Remember, there's a natural tendency for human beings to want to confirm their hypothesis. If she thinks that bus A is faster, maybe she'll want to pick the days where she'll get data to confirm her hypothesis. It's really important that she randomized the 50 workdays. What I can imagine she did is maybe she wrote each of the workdays, the dates, on a piece of paper. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
Remember, there's a natural tendency for human beings to want to confirm their hypothesis. If she thinks that bus A is faster, maybe she'll want to pick the days where she'll get data to confirm her hypothesis. It's really important that she randomized the 50 workdays. What I can imagine she did is maybe she wrote each of the workdays, the dates, on a piece of paper. She would have 50 pieces of paper, and then she turned them all upside down, or maybe she closed her eyes, and then she moved them all over her table. Then with her eyes closed, she randomly moved them to either the left or the right of the table. If they move to the left of the table, then those are the days she'll take bus A. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
What I can imagine she did is maybe she wrote each of the workdays, the dates, on a piece of paper. She would have 50 pieces of paper, and then she turned them all upside down, or maybe she closed her eyes, and then she moved them all over her table. Then with her eyes closed, she randomly moved them to either the left or the right of the table. If they move to the left of the table, then those are the days she'll take bus A. If she moves them to the right of the table, those are the days she takes bus B. That's how she can make sure that this is truly random. All right. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
If they move to the left of the table, then those are the days she'll take bus A. If she moves them to the right of the table, those are the days she takes bus B. That's how she can make sure that this is truly random. All right. Then they tell us, the results, this is important, the results of the experiment show that the median travel duration for bus A is eight minutes less than the median travel duration for bus B. Or one way to think about it, if we said the treatment group median minus the control group median, what would we get? Well, the treatment group is eight minutes less than the control group. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
All right. Then they tell us, the results, this is important, the results of the experiment show that the median travel duration for bus A is eight minutes less than the median travel duration for bus B. Or one way to think about it, if we said the treatment group median minus the control group median, what would we get? Well, the treatment group is eight minutes less than the control group. This is A, this is B. If this is eight less than this, then this is going to be equal to negative eight. This is just another way of restating what I have underlined right over here. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
Well, the treatment group is eight minutes less than the control group. This is A, this is B. If this is eight less than this, then this is going to be equal to negative eight. This is just another way of restating what I have underlined right over here. Someone's car alarm went off. I hope you're not hearing that. Anyway, I'll try to pay attention while it's going off. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
This is just another way of restating what I have underlined right over here. Someone's car alarm went off. I hope you're not hearing that. Anyway, I'll try to pay attention while it's going off. To test whether the results could be explained by random chance, she created the table below, which summarizes the results of 1,000 re-randomizations of the data, with differences between medians rounded to the nearest five minutes. What is going on over here? You might say, well, look, she got her result that she wanted to get. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
Anyway, I'll try to pay attention while it's going off. To test whether the results could be explained by random chance, she created the table below, which summarizes the results of 1,000 re-randomizations of the data, with differences between medians rounded to the nearest five minutes. What is going on over here? You might say, well, look, she got her result that she wanted to get. She sees this data seems to confirm that bus A gets her to work faster. What's all this other business with re-randomizations she's doing? Well, the important thing to realize is, and she realizes this, is that she might have just gotten this data that I underlined by random chance. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
You might say, well, look, she got her result that she wanted to get. She sees this data seems to confirm that bus A gets her to work faster. What's all this other business with re-randomizations she's doing? Well, the important thing to realize is, and she realizes this, is that she might have just gotten this data that I underlined by random chance. There's some chance maybe A and B are completely similar in terms of how long they take in reality, and she just happened to pick bus A on days where bus A got to work faster. Maybe bus B is faster, but she just happened to take bus A on the days that it was faster, the days that just happened to have less traffic. What she's doing here is she re-randomized the data, and she wants to see, well, with all this re-randomized data, out of these 1,000 re-randomizations, what fraction of them do I get a result like this? | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
Well, the important thing to realize is, and she realizes this, is that she might have just gotten this data that I underlined by random chance. There's some chance maybe A and B are completely similar in terms of how long they take in reality, and she just happened to pick bus A on days where bus A got to work faster. Maybe bus B is faster, but she just happened to take bus A on the days that it was faster, the days that just happened to have less traffic. What she's doing here is she re-randomized the data, and she wants to see, well, with all this re-randomized data, out of these 1,000 re-randomizations, what fraction of them do I get a result like this? Do I get a result where A is eight minutes or more faster, I guess, or you could say that the median travel duration for bus A is eight minutes less or even less than that, than the median travel for bus B. If it was nine minutes less, or 10 minutes less, or 15 minutes less, those are all the interesting ones. Those are the ones that confirm our hypothesis that bus A gets to work faster. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
What she's doing here is she re-randomized the data, and she wants to see, well, with all this re-randomized data, out of these 1,000 re-randomizations, what fraction of them do I get a result like this? Do I get a result where A is eight minutes or more faster, I guess, or you could say that the median travel duration for bus A is eight minutes less or even less than that, than the median travel for bus B. If it was nine minutes less, or 10 minutes less, or 15 minutes less, those are all the interesting ones. Those are the ones that confirm our hypothesis that bus A gets to work faster. Let's look at this table. It's not below. It's actually to the right. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
Those are the ones that confirm our hypothesis that bus A gets to work faster. Let's look at this table. It's not below. It's actually to the right. Let's just remind ourselves what she did here because the first time you try to process this, it can seem a little bit daunting. In her experiment, let me write this down, experiment, the car alarm outside, which you probably, hopefully, are not hearing. It's actually a surprisingly pleasant-sounding car alarm. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
It's actually to the right. Let's just remind ourselves what she did here because the first time you try to process this, it can seem a little bit daunting. In her experiment, let me write this down, experiment, the car alarm outside, which you probably, hopefully, are not hearing. It's actually a surprisingly pleasant-sounding car alarm. It sounds like a slightly obnoxious bird. Anyway, her experiment is, the way I described it, 25 days she would take bus A, 25 days she took bus B, and she would record all the travel times. Let's say that I just have 25 data points in each column. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
It's actually a surprisingly pleasant-sounding car alarm. It sounds like a slightly obnoxious bird. Anyway, her experiment is, the way I described it, 25 days she would take bus A, 25 days she took bus B, and she would record all the travel times. Let's say that I just have 25 data points in each column. Let's say you get 12 minutes, 20 minutes, 25 minutes, and you just keep going. There's 25 data points. Let's just say that there are 12 data points less than 20 minutes and 12 data points more than 20 minutes. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
Let's say that I just have 25 data points in each column. Let's say you get 12 minutes, 20 minutes, 25 minutes, and you just keep going. There's 25 data points. Let's just say that there are 12 data points less than 20 minutes and 12 data points more than 20 minutes. In this circumstance, her median time for bus A would be 20 minutes. I just made this number up. In order for this to be eight minutes less than the median time for bus B, the median for bus B, we would have to be 28. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
Let's just say that there are 12 data points less than 20 minutes and 12 data points more than 20 minutes. In this circumstance, her median time for bus A would be 20 minutes. I just made this number up. In order for this to be eight minutes less than the median time for bus B, the median for bus B, we would have to be 28. Maybe you have data points here. Maybe this is 18. You have 12 more that are less than 28. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
In order for this to be eight minutes less than the median time for bus B, the median for bus B, we would have to be 28. Maybe you have data points here. Maybe this is 18. You have 12 more that are less than 28. Then you have 12 more that are greater than 28. The median time for bus B would be 28. Once again, I just made this data up. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
You have 12 more that are less than 28. Then you have 12 more that are greater than 28. The median time for bus B would be 28. Once again, I just made this data up. If you took treatment group median, and I'll just write TGM for short, TGM minus control group median, what do you get? 20 minus 28 is negative eight. This is the actual results of, these are theoretical or potential results, hypothetical results, for her actual experiment. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
Once again, I just made this data up. If you took treatment group median, and I'll just write TGM for short, TGM minus control group median, what do you get? 20 minus 28 is negative eight. This is the actual results of, these are theoretical or potential results, hypothetical results, for her actual experiment. What's all this business over here? What she did is she took these times and she said, you know what, let's just imagine a world where I could have gotten any of these times randomly on either bus. She just randomly resorted them between A and B. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
This is the actual results of, these are theoretical or potential results, hypothetical results, for her actual experiment. What's all this business over here? What she did is she took these times and she said, you know what, let's just imagine a world where I could have gotten any of these times randomly on either bus. She just randomly resorted them between A and B. She did that a thousand times. The first time, the second time, the third time, and she does this 1,000 times. I'm assuming she used some type of a computer program to do it. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
She just randomly resorted them between A and B. She did that a thousand times. The first time, the second time, the third time, and she does this 1,000 times. I'm assuming she used some type of a computer program to do it. Each time, once again, she just took the data that she had and she just rearranged it. She just reshuffled it. Maybe A on one day, maybe it gets this 18, maybe it gets the 25, maybe it gets a 30. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
I'm assuming she used some type of a computer program to do it. Each time, once again, she just took the data that she had and she just rearranged it. She just reshuffled it. Maybe A on one day, maybe it gets this 18, maybe it gets the 25, maybe it gets a 30. Once again, so we've got the 18, the 25, and the 30, and maybe B gets the, and she's reshuffling all this other data points that I just have with dots, and maybe B, let's see, she got the 18, 25, and 30, maybe 12, 20, and 28. In this circumstance, this random reshuffling, and she keeps doing it over and over again, in this random reshuffling, the treatment group median minus the control group median is going to be what? It's going to be equal to positive five. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
Maybe A on one day, maybe it gets this 18, maybe it gets the 25, maybe it gets a 30. Once again, so we've got the 18, the 25, and the 30, and maybe B gets the, and she's reshuffling all this other data points that I just have with dots, and maybe B, let's see, she got the 18, 25, and 30, maybe 12, 20, and 28. In this circumstance, this random reshuffling, and she keeps doing it over and over again, in this random reshuffling, the treatment group median minus the control group median is going to be what? It's going to be equal to positive five. In this random shuffling, this hypothetical scenario, bus A's median would have been five minutes more, longer than bus B's. If she gets this result with this random resorting, this would have been, and this is actually a, she would have had a column here for five, and then she would have notched, put one notch right over here, but it looks like she classified things, or maybe she didn't even get the data, but she classified them by multiples of two. But then if she got this again, then she would have put a two here, and then she should have said, okay, in how many of these random reshufflings am I getting a scenario where there's a five-minute difference, or where the treatment group is five minutes longer? | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
It's going to be equal to positive five. In this random shuffling, this hypothetical scenario, bus A's median would have been five minutes more, longer than bus B's. If she gets this result with this random resorting, this would have been, and this is actually a, she would have had a column here for five, and then she would have notched, put one notch right over here, but it looks like she classified things, or maybe she didn't even get the data, but she classified them by multiples of two. But then if she got this again, then she would have put a two here, and then she should have said, okay, in how many of these random reshufflings am I getting a scenario where there's a five-minute difference, or where the treatment group is five minutes longer? So what is this saying? So for example, this is saying that 18 out of the 1,000 reshufflings, which she just randomly reshuffled the data, 18 out of those 1,000 times, she found a scenario where her treatment group median was 10 minutes longer than her control group, where bus A's median was, this hypothetical re-randomization, where the treatment group is 10 minutes slower than the control group. There were 159 times where the treatment group, once again, in her random reshuffling, these aren't based on observations, these are random reshufflings, there's 159 times where her treatment group is four minutes slower than her control group. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
But then if she got this again, then she would have put a two here, and then she should have said, okay, in how many of these random reshufflings am I getting a scenario where there's a five-minute difference, or where the treatment group is five minutes longer? So what is this saying? So for example, this is saying that 18 out of the 1,000 reshufflings, which she just randomly reshuffled the data, 18 out of those 1,000 times, she found a scenario where her treatment group median was 10 minutes longer than her control group, where bus A's median was, this hypothetical re-randomization, where the treatment group is 10 minutes slower than the control group. There were 159 times where the treatment group, once again, in her random reshuffling, these aren't based on observations, these are random reshufflings, there's 159 times where her treatment group is four minutes slower than her control group. So the whole reason for doing this is she says, okay, what's the probability of getting a result like this or better? And I say better is, you know, I guess one that even more confirms her hypothesis, that the treatment group is faster than the control group. Well, this scenario is this one right over here, and then another one where the treatment group is even faster is this right over here. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
There were 159 times where the treatment group, once again, in her random reshuffling, these aren't based on observations, these are random reshufflings, there's 159 times where her treatment group is four minutes slower than her control group. So the whole reason for doing this is she says, okay, what's the probability of getting a result like this or better? And I say better is, you know, I guess one that even more confirms her hypothesis, that the treatment group is faster than the control group. Well, this scenario is this one right over here, and then another one where the treatment group is even faster is this right over here. Here the treatment group median is 10 less than the control group median. So in how many of these scenarios out of the 1,000 is this occurring? Well, this one occurs 85 times, this one occurs eight, so if you add these two together, 93 out of the 1,000 times out of her re-randomization, or I guess you could say 9.3% of the time, if the data, 9.3% of the randomized, in the 1,000 re-randomizations, 9.3% of the time she got data that was as validating of a hypothesis or more than the actual experiment. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
Well, this scenario is this one right over here, and then another one where the treatment group is even faster is this right over here. Here the treatment group median is 10 less than the control group median. So in how many of these scenarios out of the 1,000 is this occurring? Well, this one occurs 85 times, this one occurs eight, so if you add these two together, 93 out of the 1,000 times out of her re-randomization, or I guess you could say 9.3% of the time, if the data, 9.3% of the randomized, in the 1,000 re-randomizations, 9.3% of the time she got data that was as validating of a hypothesis or more than the actual experiment. So one way to think about this is the probability of randomly getting the results from her experiment, or better results from her experiment, are 9.3%. So they're low, it's a reasonably low probability that this happened purely by chance. Now, a question is, well, what's the threshold? | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
Well, this one occurs 85 times, this one occurs eight, so if you add these two together, 93 out of the 1,000 times out of her re-randomization, or I guess you could say 9.3% of the time, if the data, 9.3% of the randomized, in the 1,000 re-randomizations, 9.3% of the time she got data that was as validating of a hypothesis or more than the actual experiment. So one way to think about this is the probability of randomly getting the results from her experiment, or better results from her experiment, are 9.3%. So they're low, it's a reasonably low probability that this happened purely by chance. Now, a question is, well, what's the threshold? If it was a 50%, you say, okay, this was, you know, very likely to happen by chance. If this was a 25%, you're like, okay, it's less likely to happen by chance, but it could happen. 9.3%, I mean, it's roughly 10%, you know, for every 10 people who do an experiment like she did, one person, even if it was random, one person would get data like this. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
Now, a question is, well, what's the threshold? If it was a 50%, you say, okay, this was, you know, very likely to happen by chance. If this was a 25%, you're like, okay, it's less likely to happen by chance, but it could happen. 9.3%, I mean, it's roughly 10%, you know, for every 10 people who do an experiment like she did, one person, even if it was random, one person would get data like this. So what typically happens among statisticians is they draw a threshold, and the threshold for statistical significance is usually 5%. So one way to think about the probability of her getting this result by chance, or this result, or a more extreme result, one that more confirms her hypothesis by chance, is 9.3%. Now, if your cutoff for significance is 5%, if you say, okay, this has to be 5% or less, then you say, okay, this is not statistically significant. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
9.3%, I mean, it's roughly 10%, you know, for every 10 people who do an experiment like she did, one person, even if it was random, one person would get data like this. So what typically happens among statisticians is they draw a threshold, and the threshold for statistical significance is usually 5%. So one way to think about the probability of her getting this result by chance, or this result, or a more extreme result, one that more confirms her hypothesis by chance, is 9.3%. Now, if your cutoff for significance is 5%, if you say, okay, this has to be 5% or less, then you say, okay, this is not statistically significant. There's a more than a 5% chance that I could have gotten this result purely through random chance. Now, once again, that just depends on where you have that threshold. So when we go back, I think we've already answered the final question. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
Now, if your cutoff for significance is 5%, if you say, okay, this has to be 5% or less, then you say, okay, this is not statistically significant. There's a more than a 5% chance that I could have gotten this result purely through random chance. Now, once again, that just depends on where you have that threshold. So when we go back, I think we've already answered the final question. According to the simulations, what is the probability of the treatment group's median being lower than the control group's median by eight minutes or more? Which, once again, eight minutes or more, that would be negative eight and negative 10, and we just figured that out. That was 93 out of the 1,000 re-randomizations, so it's a 9.3% chance. | Statistical significance on bus speeds Probability and Statistics Khan Academy.mp3 |
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