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Her test statistic, I can never say that right, was t is equal to negative 1.9. Assume that the conditions for inference were met. What is the approximate p-value for Miriam's test? So pause this video and see if you can figure this out on your own. All right, well I always just like to remind ourselves what's going on here before I just go ahead and calculate the p-value. So there's some data set, some population here, and the null hypothesis is that the true mean is 18. The alternative is that it's less than 18. | Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3 |
So pause this video and see if you can figure this out on your own. All right, well I always just like to remind ourselves what's going on here before I just go ahead and calculate the p-value. So there's some data set, some population here, and the null hypothesis is that the true mean is 18. The alternative is that it's less than 18. So to test that null hypothesis, Miriam takes a sample. Sample size is equal to seven. From that, she would calculate her sample mean and her sample standard deviation. | Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3 |
The alternative is that it's less than 18. So to test that null hypothesis, Miriam takes a sample. Sample size is equal to seven. From that, she would calculate her sample mean and her sample standard deviation. And from that, she would calculate this t-statistic. The way she would do that, or if they didn't tell us ahead of time what that was, they would say, okay, well we would say the t-statistic is equal to her sample mean minus the assumed mean from the null hypothesis, that's what we have over here, divided by, and this is a mouthful, our approximation of the standard error of the mean. And the way we get that approximation, we take our sample standard deviation and divide it by the square root of our sample size. | Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3 |
From that, she would calculate her sample mean and her sample standard deviation. And from that, she would calculate this t-statistic. The way she would do that, or if they didn't tell us ahead of time what that was, they would say, okay, well we would say the t-statistic is equal to her sample mean minus the assumed mean from the null hypothesis, that's what we have over here, divided by, and this is a mouthful, our approximation of the standard error of the mean. And the way we get that approximation, we take our sample standard deviation and divide it by the square root of our sample size. Well, they've calculated this ahead of time for us. This is equal to negative 1.9. And so if we think about a t-distribution, I'll try to hand-draw a rough t-distribution really fast, and if this is the mean of the t-distribution, what we are curious about, because our alternative hypothesis is that the mean is less than 18, so what we care about is, well, what is the probability of getting a t-value that is more than 1.9 below the mean? | Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3 |
And the way we get that approximation, we take our sample standard deviation and divide it by the square root of our sample size. Well, they've calculated this ahead of time for us. This is equal to negative 1.9. And so if we think about a t-distribution, I'll try to hand-draw a rough t-distribution really fast, and if this is the mean of the t-distribution, what we are curious about, because our alternative hypothesis is that the mean is less than 18, so what we care about is, well, what is the probability of getting a t-value that is more than 1.9 below the mean? So this right over here, negative 1.9. So it's this area right there. And so I'm gonna do this with a TI-84, at least an emulator of a TI-84, and all we have to do is we would go to second distribution and then I would use the t-cumulative distribution function. | Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3 |
And so if we think about a t-distribution, I'll try to hand-draw a rough t-distribution really fast, and if this is the mean of the t-distribution, what we are curious about, because our alternative hypothesis is that the mean is less than 18, so what we care about is, well, what is the probability of getting a t-value that is more than 1.9 below the mean? So this right over here, negative 1.9. So it's this area right there. And so I'm gonna do this with a TI-84, at least an emulator of a TI-84, and all we have to do is we would go to second distribution and then I would use the t-cumulative distribution function. So let's go there, that's the number six right there, click Enter. And so my lower bound, yeah, I essentially want it to be negative infinity, and so we can just call that negative infinity, it's an approximation of negative infinity, very, very low number. Our upper bound would be negative 1.9, negative 1.9. | Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3 |
And so I'm gonna do this with a TI-84, at least an emulator of a TI-84, and all we have to do is we would go to second distribution and then I would use the t-cumulative distribution function. So let's go there, that's the number six right there, click Enter. And so my lower bound, yeah, I essentially want it to be negative infinity, and so we can just call that negative infinity, it's an approximation of negative infinity, very, very low number. Our upper bound would be negative 1.9, negative 1.9. And then our degrees of freedom, that's our sample size minus one. Our sample size is seven, so our degrees of freedom would be six, and so there we have it. And then so this would be, our p-value would be approximately 0.053. | Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3 |
Our upper bound would be negative 1.9, negative 1.9. And then our degrees of freedom, that's our sample size minus one. Our sample size is seven, so our degrees of freedom would be six, and so there we have it. And then so this would be, our p-value would be approximately 0.053. So our p-value would be approximately 0.053. And then what Miriam would do is, would compare this p-value to her preset significance level, to alpha. If this is below alpha, then she would reject her null hypothesis, which would suggest the alternative. | Using TI calculator for P-value from t statistic AP Statistics Khan Academy.mp3 |
And this is in the context of significance testing. So just as a little bit of review, in order to do a significance test, we first come up with a null and an alternative hypothesis. And we'll do this on some population in question. These will say some hypotheses about a true parameter for this population. And the null hypothesis tends to be kind of what was always assumed or the status quo, while the alternative hypothesis, hey, there's news here, there's something alternative here. And to test it, and we're really testing the null hypothesis, we're gonna decide whether we want to reject or fail to reject the null hypothesis, we take a sample. We take a sample from this population. | Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3 |
These will say some hypotheses about a true parameter for this population. And the null hypothesis tends to be kind of what was always assumed or the status quo, while the alternative hypothesis, hey, there's news here, there's something alternative here. And to test it, and we're really testing the null hypothesis, we're gonna decide whether we want to reject or fail to reject the null hypothesis, we take a sample. We take a sample from this population. Using that sample, we calculate a statistic, we calculate a statistic that's trying to estimate the parameter in question. And then using that statistic, we try to come up with the probability of getting that statistic, the probability of getting that statistic that we just calculated from that sample of a certain size, given, if we were to assume that our null hypothesis, if our null hypothesis is true. And if this probability, which is often known as a p-value, is below some threshold that we set ahead of time, which is known as the significance level, then we reject the null hypothesis. | Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3 |
We take a sample from this population. Using that sample, we calculate a statistic, we calculate a statistic that's trying to estimate the parameter in question. And then using that statistic, we try to come up with the probability of getting that statistic, the probability of getting that statistic that we just calculated from that sample of a certain size, given, if we were to assume that our null hypothesis, if our null hypothesis is true. And if this probability, which is often known as a p-value, is below some threshold that we set ahead of time, which is known as the significance level, then we reject the null hypothesis. Let me write this down. So this right over here, this is our p-value. This should be all we review, we introduce it in other videos. | Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3 |
And if this probability, which is often known as a p-value, is below some threshold that we set ahead of time, which is known as the significance level, then we reject the null hypothesis. Let me write this down. So this right over here, this is our p-value. This should be all we review, we introduce it in other videos. We have seen in other videos if our p-value is less than our significance level, then we reject, reject our null hypothesis. And if our p-value is greater than or equal to our significance level, alpha, then we fail to reject, fail to reject our null hypothesis. And when we reject our null hypothesis, some people say that might suggest the alternative hypothesis. | Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3 |
This should be all we review, we introduce it in other videos. We have seen in other videos if our p-value is less than our significance level, then we reject, reject our null hypothesis. And if our p-value is greater than or equal to our significance level, alpha, then we fail to reject, fail to reject our null hypothesis. And when we reject our null hypothesis, some people say that might suggest the alternative hypothesis. And the reason why this makes sense is if the probability of getting this statistic from a sample of a certain size, if we assume that the null hypothesis is true, is reasonably low, if it's below a threshold, maybe this threshold is 5%, if the probability of that happening was less than 5%, then hey, maybe it's reasonable to reject it. But we might be wrong in either of these scenarios, and that's where these errors come into play. Let's make a grid to make this clear. | Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3 |
And when we reject our null hypothesis, some people say that might suggest the alternative hypothesis. And the reason why this makes sense is if the probability of getting this statistic from a sample of a certain size, if we assume that the null hypothesis is true, is reasonably low, if it's below a threshold, maybe this threshold is 5%, if the probability of that happening was less than 5%, then hey, maybe it's reasonable to reject it. But we might be wrong in either of these scenarios, and that's where these errors come into play. Let's make a grid to make this clear. So there's the reality. Let me put reality up here. So the reality is there's two possible scenarios in reality. | Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3 |
Let's make a grid to make this clear. So there's the reality. Let me put reality up here. So the reality is there's two possible scenarios in reality. One is is that the null hypothesis is true, and the other is that the null hypothesis is false. And then based on our significance test, there's two things that we might do. We might reject the null hypothesis, or we might fail to reject the null hypothesis. | Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3 |
So the reality is there's two possible scenarios in reality. One is is that the null hypothesis is true, and the other is that the null hypothesis is false. And then based on our significance test, there's two things that we might do. We might reject the null hypothesis, or we might fail to reject the null hypothesis. And so let's put a little grid here to think about the different combinations, the different scenarios here. So in a scenario where the null hypothesis is true, but we reject it, that feels like an error. We shouldn't reject something that is true, and that indeed is a type one error. | Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3 |
We might reject the null hypothesis, or we might fail to reject the null hypothesis. And so let's put a little grid here to think about the different combinations, the different scenarios here. So in a scenario where the null hypothesis is true, but we reject it, that feels like an error. We shouldn't reject something that is true, and that indeed is a type one error. Type one error. You shouldn't reject the null hypothesis if it was true. You should reject the null hypothesis if it was true. | Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3 |
We shouldn't reject something that is true, and that indeed is a type one error. Type one error. You shouldn't reject the null hypothesis if it was true. You should reject the null hypothesis if it was true. And you could even figure out what is the probability of getting a type one error? Well, that's gonna be your significance level. Because if your null hypothesis is true, let's say that your significance level is 5%. | Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3 |
You should reject the null hypothesis if it was true. And you could even figure out what is the probability of getting a type one error? Well, that's gonna be your significance level. Because if your null hypothesis is true, let's say that your significance level is 5%. Well, 5% of the time, even if your null hypothesis is true, you're going to get a statistic that's going to make you reject the null hypothesis. So one way to think about the probability of a type one error is your significance level. Now, if your null hypothesis is true and you fail to reject it, well, that's good. | Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3 |
Because if your null hypothesis is true, let's say that your significance level is 5%. Well, 5% of the time, even if your null hypothesis is true, you're going to get a statistic that's going to make you reject the null hypothesis. So one way to think about the probability of a type one error is your significance level. Now, if your null hypothesis is true and you fail to reject it, well, that's good. This, we could write this as, this is a correct, correct conclusion. The good thing just happened to happen this time. Now, if your null hypothesis is false and you reject it, that's also good. | Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3 |
Now, if your null hypothesis is true and you fail to reject it, well, that's good. This, we could write this as, this is a correct, correct conclusion. The good thing just happened to happen this time. Now, if your null hypothesis is false and you reject it, that's also good. That is the correct, correct conclusion. But if your null hypothesis is false and you fail to reject it, well, then that is a type two error. That is a type two error. | Introduction to Type I and Type II errors AP Statistics Khan Academy.mp3 |
So let's go back to an example that we've seen before. We're rolling a fair six-sided die. And so there's six possibilities. We could get a one, a two, a three, a four, a five, or a six. Now let's say we ask ourselves, what is the probability of rolling, of rolling a number that is less than or equal to, less than or equal to two? What is this going to be? Well, there are six equally likely possibilities. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
We could get a one, a two, a three, a four, a five, or a six. Now let's say we ask ourselves, what is the probability of rolling, of rolling a number that is less than or equal to, less than or equal to two? What is this going to be? Well, there are six equally likely possibilities. And rolling less than or equal to two, well, that means I'm either rolling a one or a two. So two, one, two, of the six equally likely possibilities meet my constraints. So there is a 2 6th probability of rolling a number less than or equal to two. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
Well, there are six equally likely possibilities. And rolling less than or equal to two, well, that means I'm either rolling a one or a two. So two, one, two, of the six equally likely possibilities meet my constraints. So there is a 2 6th probability of rolling a number less than or equal to two. Or I could just rewrite that as an equivalent fraction. I could say there's a 1 3rd probability. I could go either way. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
So there is a 2 6th probability of rolling a number less than or equal to two. Or I could just rewrite that as an equivalent fraction. I could say there's a 1 3rd probability. I could go either way. Now let's ask ourselves another question. What is the probability of rolling a number greater than or equal, greater than or equal to three? Well, once again, there are six equally likely possibilities. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
I could go either way. Now let's ask ourselves another question. What is the probability of rolling a number greater than or equal, greater than or equal to three? Well, once again, there are six equally likely possibilities. And how many of them involve rolling greater than or equal to three? Let's see, one, two, three, four. These possibilities right over here. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
Well, once again, there are six equally likely possibilities. And how many of them involve rolling greater than or equal to three? Let's see, one, two, three, four. These possibilities right over here. Roll a three, a four, a five, or a six. So four out of the six equally likely possibilities. Or I could rewrite this as an equivalent fraction as 2 3rds. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
These possibilities right over here. Roll a three, a four, a five, or a six. So four out of the six equally likely possibilities. Or I could rewrite this as an equivalent fraction as 2 3rds. So what's more likely? Rolling a number that's less than or equal to two or rolling a number that's greater than or equal to three? Well, you can see right over here. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
Or I could rewrite this as an equivalent fraction as 2 3rds. So what's more likely? Rolling a number that's less than or equal to two or rolling a number that's greater than or equal to three? Well, you can see right over here. The probability of rolling greater than or equal to three is 2 3rds, while the probability of rolling less than or equal to two is only 1 3rd. This number is greater. So this has a greater probability. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
Well, you can see right over here. The probability of rolling greater than or equal to three is 2 3rds, while the probability of rolling less than or equal to two is only 1 3rd. This number is greater. So this has a greater probability. Or another way of thinking about it, rolling greater than or equal to three is more likely than rolling less than or equal to two. In fact, not only is it more likely, you see that 2 3rds is twice 1 3rd. This right over here is twice as likely. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
So this has a greater probability. Or another way of thinking about it, rolling greater than or equal to three is more likely than rolling less than or equal to two. In fact, not only is it more likely, you see that 2 3rds is twice 1 3rd. This right over here is twice as likely. You're twice as likely to roll a number greater than or equal to three than you are to roll a number less than or equal to two. And you can even see right over here that you have twice as many possibilities of the six equally likely ones, four versus two. Four versus two here. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
This right over here is twice as likely. You're twice as likely to roll a number greater than or equal to three than you are to roll a number less than or equal to two. And you can even see right over here that you have twice as many possibilities of the six equally likely ones, four versus two. Four versus two here. And so you say, okay, I get it, Sal. You know, if the probability is a larger number, the event is more likely. It makes sense. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
Four versus two here. And so you say, okay, I get it, Sal. You know, if the probability is a larger number, the event is more likely. It makes sense. And in this case, it's twice, the number is twice as large, so it's twice as likely. But what's the range of possible probabilities? How low can a probability get? | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
It makes sense. And in this case, it's twice, the number is twice as large, so it's twice as likely. But what's the range of possible probabilities? How low can a probability get? And how high can a probability get? So let's think about the first question. So how low, how low can a probability go? | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
How low can a probability get? And how high can a probability get? So let's think about the first question. So how low, how low can a probability go? How low? So what's the lowest probability that you can imagine for anything? Well, let's give ourselves, let's give ourselves a little bit of a experiment. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
So how low, how low can a probability go? How low? So what's the lowest probability that you can imagine for anything? Well, let's give ourselves, let's give ourselves a little bit of a experiment. Let's ask ourselves the probability of rolling, of rolling a seven. Well, once, and pause the video and try to figure it out on your own. Well, there are six equally likely possibilities. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
Well, let's give ourselves, let's give ourselves a little bit of a experiment. Let's ask ourselves the probability of rolling, of rolling a seven. Well, once, and pause the video and try to figure it out on your own. Well, there are six equally likely possibilities. And how many of them involve rolling a seven? Well, none of them. It's impossible to roll a seven. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
Well, there are six equally likely possibilities. And how many of them involve rolling a seven? Well, none of them. It's impossible to roll a seven. So none of the six. So we could say this probability is zero. And if you see a probability of zero, someone says the probability of that thing happening is zero. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
It's impossible to roll a seven. So none of the six. So we could say this probability is zero. And if you see a probability of zero, someone says the probability of that thing happening is zero. That means it's impossible. That means in no, in no world can that happen if it's exactly zero. So this right here, the probability is zero. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
And if you see a probability of zero, someone says the probability of that thing happening is zero. That means it's impossible. That means in no, in no world can that happen if it's exactly zero. So this right here, the probability is zero. That means it is impossible. It is impossible. Now how high can a probability get? | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
So this right here, the probability is zero. That means it is impossible. It is impossible. Now how high can a probability get? So how, how high can a probability get? Well, let's think about, let's say probability of rolling, rolling any number, any number from one to six. Well, I have six equally, I have six, I have six equally likely possibilities. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
Now how high can a probability get? So how, how high can a probability get? Well, let's think about, let's say probability of rolling, rolling any number, any number from one to six. Well, I have six equally, I have six, I have six equally likely possibilities. And any one of those six meets this constraint. I would have rolled a number, any number from one to six, including one and six. So there's six equally likely possibilities. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
Well, I have six equally, I have six, I have six equally likely possibilities. And any one of those six meets this constraint. I would have rolled a number, any number from one to six, including one and six. So there's six equally likely possibilities. And so the probability is one. And so if someone says the probability is zero, it's impossible. And if someone says the probability is one, that means it's definitely going to happen. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
So there's six equally likely possibilities. And so the probability is one. And so if someone says the probability is zero, it's impossible. And if someone says the probability is one, that means it's definitely going to happen. Going, going, going, it's definitely going to happen. So the maximum probability for anything is one. The minimum probability is zero. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
And if someone says the probability is one, that means it's definitely going to happen. Going, going, going, it's definitely going to happen. So the maximum probability for anything is one. The minimum probability is zero. You don't have negative probabilities, and you don't have probabilities greater than one. And you might be thinking, wait, wait, you know, I've seen things that, you know, they look like larger numbers than one. And you're probably thinking of seeing this as a percentage. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
The minimum probability is zero. You don't have negative probabilities, and you don't have probabilities greater than one. And you might be thinking, wait, wait, you know, I've seen things that, you know, they look like larger numbers than one. And you're probably thinking of seeing this as a percentage. So one as a percentage, you could also write this as 100%. This right over here as a percentage is 100%. But 100% is the same thing as one. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
And you're probably thinking of seeing this as a percentage. So one as a percentage, you could also write this as 100%. This right over here as a percentage is 100%. But 100% is the same thing as one. You can't have a probability at 110%. 110% would be the same thing as 1.1. Now this is really interesting, because you'd often see someone say, hey, you know, something for sure is going to happen, or something is impossible. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
But 100% is the same thing as one. You can't have a probability at 110%. 110% would be the same thing as 1.1. Now this is really interesting, because you'd often see someone say, hey, you know, something for sure is going to happen, or something is impossible. But even a lot of the things that we think for sure are going to happen, there's some probability, or some chance that they don't happen. So for example, you might hear someone say, well, what's the probability that the sun will rise tomorrow? Well, you might say it's going to happen for sure. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
Now this is really interesting, because you'd often see someone say, hey, you know, something for sure is going to happen, or something is impossible. But even a lot of the things that we think for sure are going to happen, there's some probability, or some chance that they don't happen. So for example, you might hear someone say, well, what's the probability that the sun will rise tomorrow? Well, you might say it's going to happen for sure. But you gotta remember, you know, some type of weird cosmological event might occur, some kind of strange, huge, planet-sized object in space might come and knock the Earth out of its rotation. Who knows what could happen. All of these have a very low, a very low likelihood, very, very, very, very, very low likelihood. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
Well, you might say it's going to happen for sure. But you gotta remember, you know, some type of weird cosmological event might occur, some kind of strange, huge, planet-sized object in space might come and knock the Earth out of its rotation. Who knows what could happen. All of these have a very low, a very low likelihood, very, very, very, very, very low likelihood. But it's hard to say it's exactly one. If I'd said the probability that the sun will rise, sun will rise tomorrow, tomorrow, instead of saying one, I would probably say it's 0.999, and I would throw a lot of nines over here. I wouldn't say it's 0.9 repeating forever, and actually there's an interesting proof that 0.9 repeating forever is actually the same thing as one, which is a little counterintuitive. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
All of these have a very low, a very low likelihood, very, very, very, very, very low likelihood. But it's hard to say it's exactly one. If I'd said the probability that the sun will rise, sun will rise tomorrow, tomorrow, instead of saying one, I would probably say it's 0.999, and I would throw a lot of nines over here. I wouldn't say it's 0.9 repeating forever, and actually there's an interesting proof that 0.9 repeating forever is actually the same thing as one, which is a little counterintuitive. But I would say there's a very high probability, but even something that's such a high probability, it's going to be close to one, but I won't say it's exactly one because there could be some kind of quasar that blasts us with gamma rays, or who knows what might happen. But it's a very, very high probability. Same thing, you know, the probability, the probability here, probability that my pet, my pet gopher, my pet gopher could compose, could write the next great novel, writes, writes a novel, and actually not just a novel, a great novel. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
I wouldn't say it's 0.9 repeating forever, and actually there's an interesting proof that 0.9 repeating forever is actually the same thing as one, which is a little counterintuitive. But I would say there's a very high probability, but even something that's such a high probability, it's going to be close to one, but I won't say it's exactly one because there could be some kind of quasar that blasts us with gamma rays, or who knows what might happen. But it's a very, very high probability. Same thing, you know, the probability, the probability here, probability that my pet, my pet gopher, my pet gopher could compose, could write the next great novel, writes, writes a novel, and actually not just a novel, a great novel. Just a novel wouldn't be that impressive for a gopher, but writes a great novel, well, once again, you know, this gopher sitting there typing at a keyboard, it would seem somewhat random, but there is some probability that it actually does it. There's some chance it does it, so I would put this at a very low one. I would say it's exactly zero. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
Same thing, you know, the probability, the probability here, probability that my pet, my pet gopher, my pet gopher could compose, could write the next great novel, writes, writes a novel, and actually not just a novel, a great novel. Just a novel wouldn't be that impressive for a gopher, but writes a great novel, well, once again, you know, this gopher sitting there typing at a keyboard, it would seem somewhat random, but there is some probability that it actually does it. There's some chance it does it, so I would put this at a very low one. I would say it's exactly zero. If we had an infinite number of gophers doing this forever, who knows? Maybe one of them might write that great novel. In fact, if we had an infinite number doing it forever, eventually, a lot of people would say, at some point you would. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
I would say it's exactly zero. If we had an infinite number of gophers doing this forever, who knows? Maybe one of them might write that great novel. In fact, if we had an infinite number doing it forever, eventually, a lot of people would say, at some point you would. But just one gopher trying to write a novel, what's the probability they write a great novel? Well, I would say it's pretty close to zero. I would throw a lot of zeros. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
In fact, if we had an infinite number doing it forever, eventually, a lot of people would say, at some point you would. But just one gopher trying to write a novel, what's the probability they write a great novel? Well, I would say it's pretty close to zero. I would throw a lot of zeros. I would throw a lot of zeros here, and at some point, you might have something like this. So once again, not absolutely impossible, but pretty close to, pretty, pretty close to impossible. And so, big takeaways, higher probability, more likely. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
I would throw a lot of zeros. I would throw a lot of zeros here, and at some point, you might have something like this. So once again, not absolutely impossible, but pretty close to, pretty, pretty close to impossible. And so, big takeaways, higher probability, more likely. The lowest probability you can get to, zero. Highest probability is one. If your probability is more, if, you know, when you talk about coin flipping, if you say the probability of heads for a fair coin, you say, well, that's 1.5, that means it's equally likely to happen or not happen. | Intuitive sense of probabilities Statistics and probability 7th grade Khan Academy.mp3 |
In the next few videos, I'm going to embark on something that will just result in a formula that's pretty straightforward to apply. And in most statistics classes, you'll just see that end product. But I actually want to show how to get there. But I just want to warn you right now. It's going to be a lot of hairy math, most of it hairy algebra, and then we're actually going to have to do a little bit of calculus near the end. We're going to have to do a few partial derivatives. So if any of that sounds daunting or sounds like something that will discourage you in some way, you don't have to watch it. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
But I just want to warn you right now. It's going to be a lot of hairy math, most of it hairy algebra, and then we're actually going to have to do a little bit of calculus near the end. We're going to have to do a few partial derivatives. So if any of that sounds daunting or sounds like something that will discourage you in some way, you don't have to watch it. You could skip to the end and just get the formula that we're going to derive. But I at least find it pretty satisfying to actually derive it. So what we're going to think about here is, let's say we have n points on a coordinate plane. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
So if any of that sounds daunting or sounds like something that will discourage you in some way, you don't have to watch it. You could skip to the end and just get the formula that we're going to derive. But I at least find it pretty satisfying to actually derive it. So what we're going to think about here is, let's say we have n points on a coordinate plane. And they all don't have to be in the first quadrant. But just for simplicity or visualization, I'll draw them all in the first quadrant. So let's say I have this point right over here. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
So what we're going to think about here is, let's say we have n points on a coordinate plane. And they all don't have to be in the first quadrant. But just for simplicity or visualization, I'll draw them all in the first quadrant. So let's say I have this point right over here. Let me do them in different colors. Let's say I have this point right over here. And that coordinate is x1, y1. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
So let's say I have this point right over here. Let me do them in different colors. Let's say I have this point right over here. And that coordinate is x1, y1. And then let's say I have another point over here. I have, let me do that in a different color. Say I have another point over here. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
And that coordinate is x1, y1. And then let's say I have another point over here. I have, let me do that in a different color. Say I have another point over here. The coordinates there are x2, y2. And then I could keep adding points and I could keep drawing them. We'd just have a ton of points there and there and there. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
Say I have another point over here. The coordinates there are x2, y2. And then I could keep adding points and I could keep drawing them. We'd just have a ton of points there and there and there. And we go all the way to the nth point. All the way to the actual nth point. Maybe it's over here. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
We'd just have a ton of points there and there and there. And we go all the way to the nth point. All the way to the actual nth point. Maybe it's over here. The nth point is over here. And we're just going to call that xn, yn. So we have n points here. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
Maybe it's over here. The nth point is over here. And we're just going to call that xn, yn. So we have n points here. I haven't drawn all of the actual points. But what I want to do is find a line that minimizes the squared distances to these different points. So let's think about it. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
So we have n points here. I haven't drawn all of the actual points. But what I want to do is find a line that minimizes the squared distances to these different points. So let's think about it. Let's visualize that line for a second. So there's going to be some line. And I'm going to try to draw a line that kind of approximates what these points are doing. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
So let's think about it. Let's visualize that line for a second. So there's going to be some line. And I'm going to try to draw a line that kind of approximates what these points are doing. So let me draw this line here. So maybe the line might look something like this. I'm going to try my best to approximate it. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
And I'm going to try to draw a line that kind of approximates what these points are doing. So let me draw this line here. So maybe the line might look something like this. I'm going to try my best to approximate it. So maybe it looks something like that. Actually, let me draw it a little bit different. Maybe it looks something like that. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
I'm going to try my best to approximate it. So maybe it looks something like that. Actually, let me draw it a little bit different. Maybe it looks something like that. I don't even know what it looks like right now. What we want to do is minimize the squared error from each of these points to the line. So let's think about what that means. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
Maybe it looks something like that. I don't even know what it looks like right now. What we want to do is minimize the squared error from each of these points to the line. So let's think about what that means. So if the equation of this line right here is y is equal to mx plus b. And this just comes straight out of algebra 1. This is the slope of the line. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
So let's think about what that means. So if the equation of this line right here is y is equal to mx plus b. And this just comes straight out of algebra 1. This is the slope of the line. And this is the y-intercept. This is actually the point 0b right here. What I want to do is I want to find, and that's what the topic of the next few videos are going to be, I want to find an m and a b. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
This is the slope of the line. And this is the y-intercept. This is actually the point 0b right here. What I want to do is I want to find, and that's what the topic of the next few videos are going to be, I want to find an m and a b. So I want to find these two things that define this line so that it minimizes the squared error. So let me define what the error even is. So for each of these points, the error between it and the line is the vertical distance. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
What I want to do is I want to find, and that's what the topic of the next few videos are going to be, I want to find an m and a b. So I want to find these two things that define this line so that it minimizes the squared error. So let me define what the error even is. So for each of these points, the error between it and the line is the vertical distance. So this right here we can call error 1. And then this right here would be error 2. It would be the vertical distance between that point and the line. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
So for each of these points, the error between it and the line is the vertical distance. So this right here we can call error 1. And then this right here would be error 2. It would be the vertical distance between that point and the line. Or you could think of it the y value of this point and the y value of the line. And you just keep going all the way to the end point between the y value of this point and the y value of the line. So this error right here, error 1, if you think about it, it is this value right here, this y value. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
It would be the vertical distance between that point and the line. Or you could think of it the y value of this point and the y value of the line. And you just keep going all the way to the end point between the y value of this point and the y value of the line. So this error right here, error 1, if you think about it, it is this value right here, this y value. It's equal to y1 minus this y value. Well, what's this y value going to be? Well, over here we have x is equal to x1. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
So this error right here, error 1, if you think about it, it is this value right here, this y value. It's equal to y1 minus this y value. Well, what's this y value going to be? Well, over here we have x is equal to x1. And this point is the point mx1 plus b. You take x1 into this equation of the line, and you're going to get this point right over here. So that's literally going to be equal to mx1 plus b. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
Well, over here we have x is equal to x1. And this point is the point mx1 plus b. You take x1 into this equation of the line, and you're going to get this point right over here. So that's literally going to be equal to mx1 plus b. That's that first error. We can keep doing it with all of the points. This error right over here is going to be y2 minus mx2 plus b, so y2. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
So that's literally going to be equal to mx1 plus b. That's that first error. We can keep doing it with all of the points. This error right over here is going to be y2 minus mx2 plus b, so y2. And then this right here, this point right here, is mx2 plus b, the value when you take x2 into this line. And then we keep going all the way to our nth point. This error right here is going to be yn minus mxn plus b. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
This error right over here is going to be y2 minus mx2 plus b, so y2. And then this right here, this point right here, is mx2 plus b, the value when you take x2 into this line. And then we keep going all the way to our nth point. This error right here is going to be yn minus mxn plus b. Now, what we want to do, so if we wanted to just take the straight up sum of the errors, we could just sum these things up. But what we want to do is minimize the square of the error between each of these points, each of these end points in the line. So let me define, I'll do this in a new color. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
This error right here is going to be yn minus mxn plus b. Now, what we want to do, so if we wanted to just take the straight up sum of the errors, we could just sum these things up. But what we want to do is minimize the square of the error between each of these points, each of these end points in the line. So let me define, I'll do this in a new color. Let me define the squared error against this line as being equal to the sum of these squared errors. So this error right here, or error 1 we could call it, is y1 minus mx1 plus b. And we're going to square it. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
So let me define, I'll do this in a new color. Let me define the squared error against this line as being equal to the sum of these squared errors. So this error right here, or error 1 we could call it, is y1 minus mx1 plus b. And we're going to square it. So this is the error 1 squared. And then we're going to go to error 2 squared. Error 2 squared is y2 minus mx2 plus b. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
And we're going to square it. So this is the error 1 squared. And then we're going to go to error 2 squared. Error 2 squared is y2 minus mx2 plus b. And then we're going to square that error. We're squaring this error. And then we keep going. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
Error 2 squared is y2 minus mx2 plus b. And then we're going to square that error. We're squaring this error. And then we keep going. We're going to keep going. We're going to go n spaces, or n points I should say. We keep going all the way to this nth error. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
And then we keep going. We're going to keep going. We're going to go n spaces, or n points I should say. We keep going all the way to this nth error. The nth error is going to be yn minus mxn plus b. And then we're going to square it. And then we are going to square it. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
We keep going all the way to this nth error. The nth error is going to be yn minus mxn plus b. And then we're going to square it. And then we are going to square it. So this is the squared error of a line. And I want to find, and what we're going to do over the next few videos, is I want to find the m and b that minimizes this value. That minimizes the squared error of this line right here. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
And then we are going to square it. So this is the squared error of a line. And I want to find, and what we're going to do over the next few videos, is I want to find the m and b that minimizes this value. That minimizes the squared error of this line right here. So if you view this as the best metric for how good a fit a line is, we're going to try to find the best fitting line for these points. And I'll continue in the next video, because I find that with these very hairy math problems, it's good to kind of just deliver one concept at a time. And it also minimizes my probability of making a mistake. | Squared error of regression line Regression Probability and Statistics Khan Academy.mp3 |
What are the odds of making 10 free throws in a row? Here's my good friend Sal with the answer. That's a great question LeBron and I think the answer might surprise you. So I looked up your career free throw percentage and you're right at around 75% which is a little bit higher than my free throw percentage. And one way to interpret that, if we have a million LeBron James, as you can imagine any large number of LeBron James is taking a free throw. So let's say that this line represents all of the LeBron James that take that first free throw. Let's call that free throw number one. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
So I looked up your career free throw percentage and you're right at around 75% which is a little bit higher than my free throw percentage. And one way to interpret that, if we have a million LeBron James, as you can imagine any large number of LeBron James is taking a free throw. So let's say that this line represents all of the LeBron James that take that first free throw. Let's call that free throw number one. We would expect on average that 75% of them would make that first free throw. So let me draw 75%. So this is about halfway. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
Let's call that free throw number one. We would expect on average that 75% of them would make that first free throw. So let me draw 75%. So this is about halfway. This would be 25. This would get us to 75. So we would expect 75% of them would make that first free throw. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
So this is about halfway. This would be 25. This would get us to 75. So we would expect 75% of them would make that first free throw. And then the other 25% we would expect on average would miss that first free throw. Now what we care about are the ones that keep making the free throws. We want 10 in a row. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
So we would expect 75% of them would make that first free throw. And then the other 25% we would expect on average would miss that first free throw. Now what we care about are the ones that keep making the free throws. We want 10 in a row. So let's just focus on the 75% that made the first one. Some of these 25% might make some free throws going forward but we don't care about them anymore. They're kind of out of the game. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
We want 10 in a row. So let's just focus on the 75% that made the first one. Some of these 25% might make some free throws going forward but we don't care about them anymore. They're kind of out of the game. So let's go to free throw number two. Free throw number two. What percentage of the folks who made of the LeBron Jameses that made that first free throw, what percentage would we expect to make the second one? | Free throwing probability Probability and Statistics Khan Academy.mp3 |
They're kind of out of the game. So let's go to free throw number two. Free throw number two. What percentage of the folks who made of the LeBron Jameses that made that first free throw, what percentage would we expect to make the second one? And we're going to assume that whether or not you made the first one has no bearing on the probability of you making the second. This continues to be the probability of a LeBron James making a given free throw. So we would expect 75% of these LeBron Jameses to also make the second one. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
What percentage of the folks who made of the LeBron Jameses that made that first free throw, what percentage would we expect to make the second one? And we're going to assume that whether or not you made the first one has no bearing on the probability of you making the second. This continues to be the probability of a LeBron James making a given free throw. So we would expect 75% of these LeBron Jameses to also make the second one. So we're going to take 75% of 75%. So this is about half of that 75%. This would be a quarter. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
So we would expect 75% of these LeBron Jameses to also make the second one. So we're going to take 75% of 75%. So this is about half of that 75%. This would be a quarter. This would be 3 4ths, which is exactly 75%. So right over here. So this represents of the ones that made the first one, how many also made the second one. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
This would be a quarter. This would be 3 4ths, which is exactly 75%. So right over here. So this represents of the ones that made the first one, how many also made the second one. So you could say the percentage of the LeBron Jameses that we would expect on average to make the first two free throws. So this length right over here is 75% of this 75% right over there. And I think you might begin to see a pattern emerging. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
So this represents of the ones that made the first one, how many also made the second one. So you could say the percentage of the LeBron Jameses that we would expect on average to make the first two free throws. So this length right over here is 75% of this 75% right over there. And I think you might begin to see a pattern emerging. Let's go to the third free throw. Free throw number three. So what percentage of these folks are going to make the third one? | Free throwing probability Probability and Statistics Khan Academy.mp3 |
And I think you might begin to see a pattern emerging. Let's go to the third free throw. Free throw number three. So what percentage of these folks are going to make the third one? Well 75% of them are going to make the third one. So 75% are going to make the third one. So what is this going to be? | Free throwing probability Probability and Statistics Khan Academy.mp3 |
So what percentage of these folks are going to make the third one? Well 75% of them are going to make the third one. So 75% are going to make the third one. So what is this going to be? This is going to be 75% of this number, of this length, which is 75% of 75%. And if you were to go all the way to free throw number 10, and I think you see the pattern here. If we were to go all the way to free throw number 10, so I'm just skipping a bunch, we're going to get some very, very, very small fraction that have made all 10. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
So what is this going to be? This is going to be 75% of this number, of this length, which is 75% of 75%. And if you were to go all the way to free throw number 10, and I think you see the pattern here. If we were to go all the way to free throw number 10, so I'm just skipping a bunch, we're going to get some very, very, very small fraction that have made all 10. It's essentially going to be 75% times 75% times 75%, 10 times. 75% being multiplied repeatedly 10 times. So this is going to be what we're left off with, is going to be 75% times 75%. | Free throwing probability Probability and Statistics Khan Academy.mp3 |
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