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Now, given that I have picked a fair coin, what is the probability that I will get heads twice? So let me write it this way. And this is just notation right here. So the probability of, I'll call it heads, heads, so I get two heads in a row, given that I have a fair coin. It looks like very fancy notation, but it's just like, look, if you knew for a fact that that coin you had is absolutely fair, that it has a 50% chance of coming up heads, what is the probability of getting two heads in a row? Well, then we can just say, well, that's just going to be 50%. So 50% times 50%, which is equal to 25%. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
So the probability of, I'll call it heads, heads, so I get two heads in a row, given that I have a fair coin. It looks like very fancy notation, but it's just like, look, if you knew for a fact that that coin you had is absolutely fair, that it has a 50% chance of coming up heads, what is the probability of getting two heads in a row? Well, then we can just say, well, that's just going to be 50%. So 50% times 50%, which is equal to 25%. Now, what is the probability that you, so if you want to know what is the probability that you picked a fair coin and you got two heads in a row? So given that you have a fair coin, it's a 25% chance that you have two heads in a row. But the probability of picking a fair coin and then given the fair coin getting two heads in a row will be the 5 8ths times the 25%. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
So 50% times 50%, which is equal to 25%. Now, what is the probability that you, so if you want to know what is the probability that you picked a fair coin and you got two heads in a row? So given that you have a fair coin, it's a 25% chance that you have two heads in a row. But the probability of picking a fair coin and then given the fair coin getting two heads in a row will be the 5 8ths times the 25%. So the probability, so this whole branch, I should maybe draw it this way, the probability of this whole series of events happening. So starting with you picking the fair coin and then getting two heads in a row will be, I'll write it this way, it will be 5 8ths, 5 over 8, times this right over here, times the 0.25. I want to make it very clear. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
But the probability of picking a fair coin and then given the fair coin getting two heads in a row will be the 5 8ths times the 25%. So the probability, so this whole branch, I should maybe draw it this way, the probability of this whole series of events happening. So starting with you picking the fair coin and then getting two heads in a row will be, I'll write it this way, it will be 5 8ths, 5 over 8, times this right over here, times the 0.25. I want to make it very clear. The 0.25 is the probability of getting two heads in a row given that you knew that you got a fair coin. But the probability of this whole series of events happening, you would have to multiply this times the probability that you actually got a fair coin. So another way of thinking about it is this is the probability that you got a fair coin and that you have two heads in a row. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
I want to make it very clear. The 0.25 is the probability of getting two heads in a row given that you knew that you got a fair coin. But the probability of this whole series of events happening, you would have to multiply this times the probability that you actually got a fair coin. So another way of thinking about it is this is the probability that you got a fair coin and that you have two heads in a row. Now let's do the same thing for the unfair coin. So the probability, I'll do that in the same green color, the probability that I get heads heads given that my coin is unfair. So if you were to somehow know that your coin is unfair, what is the probability of getting two heads in a row? | Dependent probability example Probability and Statistics Khan Academy.mp3 |
So another way of thinking about it is this is the probability that you got a fair coin and that you have two heads in a row. Now let's do the same thing for the unfair coin. So the probability, I'll do that in the same green color, the probability that I get heads heads given that my coin is unfair. So if you were to somehow know that your coin is unfair, what is the probability of getting two heads in a row? Well, in this unfair coin, it has a 60% chance of coming up heads. So it will be equal to 0.6 times 0.6, which is 0.36. So if you have an unfair coin, if you know for a fact that you have an unfair coin, if that is a given, you have a 36% chance of getting two heads in a row. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
So if you were to somehow know that your coin is unfair, what is the probability of getting two heads in a row? Well, in this unfair coin, it has a 60% chance of coming up heads. So it will be equal to 0.6 times 0.6, which is 0.36. So if you have an unfair coin, if you know for a fact that you have an unfair coin, if that is a given, you have a 36% chance of getting two heads in a row. Now, if you want to know the probability of this whole series of events, the probability that you picked an unfair coin and you get two heads in a row, so the probability of unfair and two heads in a row, given that you had that unfair coin, you would multiply this 3 8ths times the 0.36. So this will be equal to the 3 8ths times 0.36. And so let's get a calculator out and calculate these. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
So if you have an unfair coin, if you know for a fact that you have an unfair coin, if that is a given, you have a 36% chance of getting two heads in a row. Now, if you want to know the probability of this whole series of events, the probability that you picked an unfair coin and you get two heads in a row, so the probability of unfair and two heads in a row, given that you had that unfair coin, you would multiply this 3 8ths times the 0.36. So this will be equal to the 3 8ths times 0.36. And so let's get a calculator out and calculate these. So if I take 5 divided by 8 times 0.25, I get 15 point, or I'll just write it as a decimal, 0.15625. So this is equal to 0.15625. And then if I do the other part, so if I have 3 divided by 8 times 0.36, that gives me 0.135. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
And so let's get a calculator out and calculate these. So if I take 5 divided by 8 times 0.25, I get 15 point, or I'll just write it as a decimal, 0.15625. So this is equal to 0.15625. And then if I do the other part, so if I have 3 divided by 8 times 0.36, that gives me 0.135. So this is 0.135. So if someone were to ask you, what's the probability of picking the fair coin and then getting two heads in a row with that fair coin, you would get this number. If someone were to say, what's the probability of you pick the unfair coin and then get two heads in a row with that unfair coin, you would get this number. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
And then if I do the other part, so if I have 3 divided by 8 times 0.36, that gives me 0.135. So this is 0.135. So if someone were to ask you, what's the probability of picking the fair coin and then getting two heads in a row with that fair coin, you would get this number. If someone were to say, what's the probability of you pick the unfair coin and then get two heads in a row with that unfair coin, you would get this number. Now if someone were to say, either way, what's the probability of getting two heads in a row? Because that's what they're asking us here. What is the probability of getting two heads? | Dependent probability example Probability and Statistics Khan Academy.mp3 |
If someone were to say, what's the probability of you pick the unfair coin and then get two heads in a row with that unfair coin, you would get this number. Now if someone were to say, either way, what's the probability of getting two heads in a row? Because that's what they're asking us here. What is the probability of getting two heads? So we could get it through this method, by chance picking the fair coin, or through this method, by chance picking the unfair coin. So since we can do it either way, we can sum up the probabilities. Either of these events meet our constraints. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
What is the probability of getting two heads? So we could get it through this method, by chance picking the fair coin, or through this method, by chance picking the unfair coin. So since we can do it either way, we can sum up the probabilities. Either of these events meet our constraints. So we can just add these two things up. So let's do that. So we can add 0.135 plus 0.15625 gives us 0.29125. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
Either of these events meet our constraints. So we can just add these two things up. So let's do that. So we can add 0.135 plus 0.15625 gives us 0.29125. So 0.29125, that's when we add 0.15625 plus 0.135 will equal this. And if we want to write it as a percentage, you essentially just multiply this times 100 and add the percentage sign there. So this is equal to 29.125%. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
So we can add 0.135 plus 0.15625 gives us 0.29125. So 0.29125, that's when we add 0.15625 plus 0.135 will equal this. And if we want to write it as a percentage, you essentially just multiply this times 100 and add the percentage sign there. So this is equal to 29.125%. Or if we were to round to the nearest hundredths, then this would be the exact number, or we could say it's approximately 29.13%, depending on how much we need to round it. So we have a little less than a third chance of this happening. And the reason why, remember, if it was a fair coin, there would only be, if everything in the bag was a fair coin, there'd be a 25% chance of this happening. | Dependent probability example Probability and Statistics Khan Academy.mp3 |
We want to test the hypothesis that more than 30% of US households have internet access, with a significance level of 5%. We collect a sample of 150 households and find that 57 have access. So to do our hypothesis test, let's just establish our null hypothesis and our alternative hypothesis. So our null hypothesis is that the hypothesis is not correct. Our null hypothesis is that the proportion of US households that have internet access is less than or equal to 30%. And our alternative hypothesis is what our hypothesis actually is, is that the proportion is greater than 30%. We see it over here. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So our null hypothesis is that the hypothesis is not correct. Our null hypothesis is that the proportion of US households that have internet access is less than or equal to 30%. And our alternative hypothesis is what our hypothesis actually is, is that the proportion is greater than 30%. We see it over here. We want to test the hypothesis that more than 30% of US households have internet access. That's that right here. This is what we're testing. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
We see it over here. We want to test the hypothesis that more than 30% of US households have internet access. That's that right here. This is what we're testing. We're testing the alternative hypothesis. And the way we're going to do it is we're going to assume a p-value based on the null hypothesis. We're going to assume a proportion based on the null hypothesis for the population. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
This is what we're testing. We're testing the alternative hypothesis. And the way we're going to do it is we're going to assume a p-value based on the null hypothesis. We're going to assume a proportion based on the null hypothesis for the population. And given that assumption, what is the probability that 57 out of 150 of our samples actually have internet access? And if that probability is less than 5%, if it's less than our significance level, then we're going to reject the null hypothesis in favor of the alternative one. So let's think about this a little bit. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
We're going to assume a proportion based on the null hypothesis for the population. And given that assumption, what is the probability that 57 out of 150 of our samples actually have internet access? And if that probability is less than 5%, if it's less than our significance level, then we're going to reject the null hypothesis in favor of the alternative one. So let's think about this a little bit. So we're going to start off assuming the null hypothesis is true. And in that assumption, we're going to have to pick a population proportion or a population mean. We know that for Bernoulli distributions, they're the same thing. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So let's think about this a little bit. So we're going to start off assuming the null hypothesis is true. And in that assumption, we're going to have to pick a population proportion or a population mean. We know that for Bernoulli distributions, they're the same thing. And what I'm going to do is I'm going to pick a proportion so high so that it maximizes the probability of getting this over here. And we actually don't even know what that number is. And actually, so that we can think about it a little more intelligent, let's just find out what our sample portion even is. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
We know that for Bernoulli distributions, they're the same thing. And what I'm going to do is I'm going to pick a proportion so high so that it maximizes the probability of getting this over here. And we actually don't even know what that number is. And actually, so that we can think about it a little more intelligent, let's just find out what our sample portion even is. We had 57 people out of 150 having internet access. So 57 households out of 150. So our sample proportion is 0.38. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
And actually, so that we can think about it a little more intelligent, let's just find out what our sample portion even is. We had 57 people out of 150 having internet access. So 57 households out of 150. So our sample proportion is 0.38. So let me write that over here. Our sample proportion is equal to 0.38. So when we assume our null hypothesis to be true, we're going to assume a population proportion that maximizes the probability that we get this over here. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So our sample proportion is 0.38. So let me write that over here. Our sample proportion is equal to 0.38. So when we assume our null hypothesis to be true, we're going to assume a population proportion that maximizes the probability that we get this over here. So the highest population proportion that's within our null hypothesis that will maximize the probability of getting this is actually if we're right at 30%. So if we say our population proportion, we're going to assume this is true. This is our null hypothesis. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So when we assume our null hypothesis to be true, we're going to assume a population proportion that maximizes the probability that we get this over here. So the highest population proportion that's within our null hypothesis that will maximize the probability of getting this is actually if we're right at 30%. So if we say our population proportion, we're going to assume this is true. This is our null hypothesis. We're going to assume that it is 0.3 or 30%. And I want you to understand that. If we said 29% would have been in our null hypothesis. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
This is our null hypothesis. We're going to assume that it is 0.3 or 30%. And I want you to understand that. If we said 29% would have been in our null hypothesis. 28%, that would have been in our null hypothesis. But for 29% or 28%, the probability of getting this would have been even lower. So it wouldn't have been as strong of a test. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
If we said 29% would have been in our null hypothesis. 28%, that would have been in our null hypothesis. But for 29% or 28%, the probability of getting this would have been even lower. So it wouldn't have been as strong of a test. If we take the maximum proportion that still satisfies our null hypothesis, we're maximizing the probability that we get this. So if that number is still low, if it's still less than 5%, we can feel pretty good about the alternative hypothesis. So just to refresh ourselves, we're going to assume a population proportion of 0.3. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So it wouldn't have been as strong of a test. If we take the maximum proportion that still satisfies our null hypothesis, we're maximizing the probability that we get this. So if that number is still low, if it's still less than 5%, we can feel pretty good about the alternative hypothesis. So just to refresh ourselves, we're going to assume a population proportion of 0.3. And if we just think about the distribution, sometimes it's helpful to draw these things. So I will draw it. So this is what the population distribution looks like. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So just to refresh ourselves, we're going to assume a population proportion of 0.3. And if we just think about the distribution, sometimes it's helpful to draw these things. So I will draw it. So this is what the population distribution looks like. Based on our assumption, based on this assumption right over here. Our population distribution has point, or maybe I should write 30% have internet access. And I'll just signify that with a 1. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So this is what the population distribution looks like. Based on our assumption, based on this assumption right over here. Our population distribution has point, or maybe I should write 30% have internet access. And I'll just signify that with a 1. And then the rest don't have internet access. 70% do not have internet access. This is just a Bernoulli distribution. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
And I'll just signify that with a 1. And then the rest don't have internet access. 70% do not have internet access. This is just a Bernoulli distribution. We know that the mean over here is going to be the same thing as the proportion that has internet access. So the mean over here is going to be 0.3, same thing as 30%. This is the population mean. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
This is just a Bernoulli distribution. We know that the mean over here is going to be the same thing as the proportion that has internet access. So the mean over here is going to be 0.3, same thing as 30%. This is the population mean. And maybe I should write it this way. The mean assuming our null hypothesis is 0.3. And then the population standard deviation, let me write this over here in yellow. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
This is the population mean. And maybe I should write it this way. The mean assuming our null hypothesis is 0.3. And then the population standard deviation, let me write this over here in yellow. The population standard deviation, assuming our null hypothesis, and we've seen this when we first learned about Bernoulli distributions. It is going to be the square root of the percentage of the population that has internet access, so 0.3, times the proportion of the population that does not have internet access, times 0.7 right over here. So this would be the square root of 0.21. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
And then the population standard deviation, let me write this over here in yellow. The population standard deviation, assuming our null hypothesis, and we've seen this when we first learned about Bernoulli distributions. It is going to be the square root of the percentage of the population that has internet access, so 0.3, times the proportion of the population that does not have internet access, times 0.7 right over here. So this would be the square root of 0.21. And we could deal with this later using our calculator. Now, with that out of the way, we want to figure out the probability of getting a sample proportion that has of 0.38. So let's look at the distribution of sample proportions. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So this would be the square root of 0.21. And we could deal with this later using our calculator. Now, with that out of the way, we want to figure out the probability of getting a sample proportion that has of 0.38. So let's look at the distribution of sample proportions. So you could literally look at every combination of getting 150 households from this, and you would actually get a binomial distribution. And we've also seen this before. You would actually get a binomial distribution where you'd get a bunch of bars like that. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So let's look at the distribution of sample proportions. So you could literally look at every combination of getting 150 households from this, and you would actually get a binomial distribution. And we've also seen this before. You would actually get a binomial distribution where you'd get a bunch of bars like that. But if your n is suitably large, and in particular, and this is kind of the test for it, the test if n times p, and in this case, we're saying p is 30%, if n times p is greater than 5, and n times 1 minus p is greater than 5, you can assume that the distribution of the sample proportion is going to be normal. So if you looked at all of the different ways you could sample 150 households from this population, you'd get all of these bars. But since our n is pretty big, it's 150, and 150 times 0.3 is obviously greater than 5, 150 times 0.7 is also greater than 5. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
You would actually get a binomial distribution where you'd get a bunch of bars like that. But if your n is suitably large, and in particular, and this is kind of the test for it, the test if n times p, and in this case, we're saying p is 30%, if n times p is greater than 5, and n times 1 minus p is greater than 5, you can assume that the distribution of the sample proportion is going to be normal. So if you looked at all of the different ways you could sample 150 households from this population, you'd get all of these bars. But since our n is pretty big, it's 150, and 150 times 0.3 is obviously greater than 5, 150 times 0.7 is also greater than 5. You can approximate that with a normal distribution. So let me do that. So you can approximate it with a normal distribution. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
But since our n is pretty big, it's 150, and 150 times 0.3 is obviously greater than 5, 150 times 0.7 is also greater than 5. You can approximate that with a normal distribution. So let me do that. So you can approximate it with a normal distribution. So this is a normal distribution right over there. Now, the mean of the distribution of the proportion data that we're assuming is a normal distribution is going to be, and remember, we're working under the context that the null hypothesis is true. So this mean is going to be this mean right here. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So you can approximate it with a normal distribution. So this is a normal distribution right over there. Now, the mean of the distribution of the proportion data that we're assuming is a normal distribution is going to be, and remember, we're working under the context that the null hypothesis is true. So this mean is going to be this mean right here. So the mean of our sample proportions is going to be the same thing as our population mean. So this is going to be 0.3, same value as that. And the standard deviation, this comes straight from the central limit theorem. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So this mean is going to be this mean right here. So the mean of our sample proportions is going to be the same thing as our population mean. So this is going to be 0.3, same value as that. And the standard deviation, this comes straight from the central limit theorem. So the standard deviation of our sample proportions is going to be the square root. It's going to be our population standard deviation. The standard deviation we're assuming with our null hypothesis divided by the square root of the number of samples we have. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
And the standard deviation, this comes straight from the central limit theorem. So the standard deviation of our sample proportions is going to be the square root. It's going to be our population standard deviation. The standard deviation we're assuming with our null hypothesis divided by the square root of the number of samples we have. And in this case, we have 150 samples. It's going to be 150 samples, and we can calculate this. This value on top, we just figured out, is the square root of 0.21. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
The standard deviation we're assuming with our null hypothesis divided by the square root of the number of samples we have. And in this case, we have 150 samples. It's going to be 150 samples, and we can calculate this. This value on top, we just figured out, is the square root of 0.21. So this is the square root of 0.21 over the square root of 150, and I can get the calculator out to calculate this, so I'll just do it the way I wrote it. Square root of 0.21, and I'm going to divide that, so whatever answer is, I'm going to divide that by the square root of 150. So it's 0.037. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
This value on top, we just figured out, is the square root of 0.21. So this is the square root of 0.21 over the square root of 150, and I can get the calculator out to calculate this, so I'll just do it the way I wrote it. Square root of 0.21, and I'm going to divide that, so whatever answer is, I'm going to divide that by the square root of 150. So it's 0.037. So we figured out the standard deviation here of the distribution of our sample proportions is going to be, let me write this down, I'll scroll over to the right a little bit. It is 0.037. I think I'm falling off the screen a little bit. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So it's 0.037. So we figured out the standard deviation here of the distribution of our sample proportions is going to be, let me write this down, I'll scroll over to the right a little bit. It is 0.037. I think I'm falling off the screen a little bit. So we'll just say 0.037. Now, to figure out the probability of having a sample proportion of 0.38, we just have to figure out how many standard deviations that is away from our mean, or essentially calculate a z statistic for our sample, because a z statistic or a z score is really just how many standard deviations you are away from the mean, and then figure out whether the probability of getting that z statistic is more or less than 5%. So let's figure out how many standard deviations we are away from the mean. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
I think I'm falling off the screen a little bit. So we'll just say 0.037. Now, to figure out the probability of having a sample proportion of 0.38, we just have to figure out how many standard deviations that is away from our mean, or essentially calculate a z statistic for our sample, because a z statistic or a z score is really just how many standard deviations you are away from the mean, and then figure out whether the probability of getting that z statistic is more or less than 5%. So let's figure out how many standard deviations we are away from the mean. So just to remind ourselves, this sample proportion we got, we can view as just a sample from this distribution of all of the possible sample proportions. So how many standard deviations away from the mean is this? So if we take our sample proportion, subtract from that the mean of the distribution of sample proportions, and divide it by the standard deviation of the distribution of the sample proportions, we get 0.38 minus 0.3, all of that over this value, which we just figured out, was 0.037. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So let's figure out how many standard deviations we are away from the mean. So just to remind ourselves, this sample proportion we got, we can view as just a sample from this distribution of all of the possible sample proportions. So how many standard deviations away from the mean is this? So if we take our sample proportion, subtract from that the mean of the distribution of sample proportions, and divide it by the standard deviation of the distribution of the sample proportions, we get 0.38 minus 0.3, all of that over this value, which we just figured out, was 0.037. So what does that give us? The numerator over here is just 0.08. The denominator is 0.037. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So if we take our sample proportion, subtract from that the mean of the distribution of sample proportions, and divide it by the standard deviation of the distribution of the sample proportions, we get 0.38 minus 0.3, all of that over this value, which we just figured out, was 0.037. So what does that give us? The numerator over here is just 0.08. The denominator is 0.037. So let's figure this out. So our numerator is 0.08 divided by this last number right here, which is the 0.037. So second answer, and we get 2.14 standard deviations. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
The denominator is 0.037. So let's figure this out. So our numerator is 0.08 divided by this last number right here, which is the 0.037. So second answer, and we get 2.14 standard deviations. So this is equal to 2.14 standard deviations. Or we could say that our z statistic, we could call this our z score or our z statistic, the number of standard deviations we are away from our mean is 2.14. And to be exact, we're 2.14 standard deviations above the mean. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So second answer, and we get 2.14 standard deviations. So this is equal to 2.14 standard deviations. Or we could say that our z statistic, we could call this our z score or our z statistic, the number of standard deviations we are away from our mean is 2.14. And to be exact, we're 2.14 standard deviations above the mean. We're going to care about a one-tailed distribution. Now, is the probability of getting this more or less than 5%? If it's less than 5%, we're going to reject the null hypothesis in favor of our alternative. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
And to be exact, we're 2.14 standard deviations above the mean. We're going to care about a one-tailed distribution. Now, is the probability of getting this more or less than 5%? If it's less than 5%, we're going to reject the null hypothesis in favor of our alternative. So how do we think about that? Well, let's think about just a normalized normal distribution. Or maybe you could call it a z distribution if you want. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
If it's less than 5%, we're going to reject the null hypothesis in favor of our alternative. So how do we think about that? Well, let's think about just a normalized normal distribution. Or maybe you could call it a z distribution if you want. If you look at a normal distribution, a completely normalized normal distribution, its mean is at 0. And essentially, each of these values are essentially z scores, because if you're a value of 1, it literally means you're one standard deviation away from this mean over here. So we need to find a critical z value right over here. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
Or maybe you could call it a z distribution if you want. If you look at a normal distribution, a completely normalized normal distribution, its mean is at 0. And essentially, each of these values are essentially z scores, because if you're a value of 1, it literally means you're one standard deviation away from this mean over here. So we need to find a critical z value right over here. Let me call that a critical z. We could even say a critical z score or critical z value, so that the probability of getting a z value higher than that is 5%. So that this whole area right here is 5%. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So we need to find a critical z value right over here. Let me call that a critical z. We could even say a critical z score or critical z value, so that the probability of getting a z value higher than that is 5%. So that this whole area right here is 5%. And that's because that's what our significance level is. Anything that has a lower than 5% chance of occurring, for us, will be validation to reject our null hypothesis. Or another way of thinking about it, if that area is 5%, this whole area right over here is 95%. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So that this whole area right here is 5%. And that's because that's what our significance level is. Anything that has a lower than 5% chance of occurring, for us, will be validation to reject our null hypothesis. Or another way of thinking about it, if that area is 5%, this whole area right over here is 95%. And once again, this is a one-tailed test, because we only care about values greater than this. Z values greater than that will make us reject the null hypothesis. And to figure out this critical z value, we can literally just go to a z table. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
Or another way of thinking about it, if that area is 5%, this whole area right over here is 95%. And once again, this is a one-tailed test, because we only care about values greater than this. Z values greater than that will make us reject the null hypothesis. And to figure out this critical z value, we can literally just go to a z table. And we say, OK, the probability of getting a z value less than that is 95%. And that's exactly the number that this gives us, the cumulative probability of getting a value less than that. And so if we just scan this, we're looking for 95%. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
And to figure out this critical z value, we can literally just go to a z table. And we say, OK, the probability of getting a z value less than that is 95%. And that's exactly the number that this gives us, the cumulative probability of getting a value less than that. And so if we just scan this, we're looking for 95%. We have 0.9495, we have 0.9505. So I'll go with this, just to make sure we're a little bit closer. So this z value, and the z value here is 1.6, and the next digit is 5. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
And so if we just scan this, we're looking for 95%. We have 0.9495, we have 0.9505. So I'll go with this, just to make sure we're a little bit closer. So this z value, and the z value here is 1.6, and the next digit is 5. 1.65. So this critical z value is equal to 1.65. So the probability of getting a z value less than 1.65, or even in a completely normalized normal distribution, the probability of getting a value less than 1.65, or in any normal distribution, the probability of being less than 1.65 standard deviations away from the mean is going to be 95%. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So this z value, and the z value here is 1.6, and the next digit is 5. 1.65. So this critical z value is equal to 1.65. So the probability of getting a z value less than 1.65, or even in a completely normalized normal distribution, the probability of getting a value less than 1.65, or in any normal distribution, the probability of being less than 1.65 standard deviations away from the mean is going to be 95%. So that's our critical z value. Now, the z value, or the z statistic for our actual sample, is 2.14. Our actual z value we got is 2.14. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
So the probability of getting a z value less than 1.65, or even in a completely normalized normal distribution, the probability of getting a value less than 1.65, or in any normal distribution, the probability of being less than 1.65 standard deviations away from the mean is going to be 95%. So that's our critical z value. Now, the z value, or the z statistic for our actual sample, is 2.14. Our actual z value we got is 2.14. It's sitting all the way out here someplace. So the probability of getting that was definitely less than 5%. And actually, we could even say, what's the probability of getting that or something, or a more extreme result? | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
Our actual z value we got is 2.14. It's sitting all the way out here someplace. So the probability of getting that was definitely less than 5%. And actually, we could even say, what's the probability of getting that or something, or a more extreme result? And if you figured out this area, and you could actually figure it out by looking at a z table, you could figure out the p value of this result. But anyway, the whole exercise here is just to figure out if we're going to reject the null hypothesis with a significance level of 5%. We can. | Large sample proportion hypothesis testing Probability and Statistics Khan Academy.mp3 |
And so what we could do is we could set up some buckets of time studied and some buckets of percent correct, and then we could survey the students and or look at the data from the scores on the test. And then we can place students in these buckets. So what you see right over here, this is a two-way table, and you can also view this as a joint distribution along these two dimensions. So one way to read this is that 20 out of the 200 total students got between 60 and 79% on the test and studied between 21 and 40 minutes. So there's all sorts of interesting things that we could try to glean from this, but what we're going to focus on this video is two more types of distributions other than the joint distribution that we see in this data. One type is a marginal distribution. And a marginal distribution is just focusing on one of these dimensions. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
So one way to read this is that 20 out of the 200 total students got between 60 and 79% on the test and studied between 21 and 40 minutes. So there's all sorts of interesting things that we could try to glean from this, but what we're going to focus on this video is two more types of distributions other than the joint distribution that we see in this data. One type is a marginal distribution. And a marginal distribution is just focusing on one of these dimensions. And one way to think about it is you can determine it by looking at the margin. So for example, if you wanted to figure out the marginal distribution of the percent correct, what you could do is look at the total of these rows. So these counts right over here give you the marginal distribution of the percent correct. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
And a marginal distribution is just focusing on one of these dimensions. And one way to think about it is you can determine it by looking at the margin. So for example, if you wanted to figure out the marginal distribution of the percent correct, what you could do is look at the total of these rows. So these counts right over here give you the marginal distribution of the percent correct. 40 out of the 200 got between 80 and 100. 60 out of the 200 got between 60 and 79, so on and so forth. Now a marginal distribution could be represented as counts or as percentages. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
So these counts right over here give you the marginal distribution of the percent correct. 40 out of the 200 got between 80 and 100. 60 out of the 200 got between 60 and 79, so on and so forth. Now a marginal distribution could be represented as counts or as percentages. So if you represent it as percentages, you would divide each of these counts by the total, which is 200. So 40 over 200, that would be 20%. 60 out of 200, that'd be 30%. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
Now a marginal distribution could be represented as counts or as percentages. So if you represent it as percentages, you would divide each of these counts by the total, which is 200. So 40 over 200, that would be 20%. 60 out of 200, that'd be 30%. 70 out of 200, that would be 35%. 20 out of 200 is 10%. And 10 out of 200 is 5%. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
60 out of 200, that'd be 30%. 70 out of 200, that would be 35%. 20 out of 200 is 10%. And 10 out of 200 is 5%. So this right over here in terms of percentages gives you the marginal distribution of the percent correct based on these buckets. So you could say 10% got between a 20 and a 39. Now you could also think about marginal distributions the other way. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
And 10 out of 200 is 5%. So this right over here in terms of percentages gives you the marginal distribution of the percent correct based on these buckets. So you could say 10% got between a 20 and a 39. Now you could also think about marginal distributions the other way. You could think about the marginal distribution for the time studied in the class. And so then you would look at these counts right over here. You'd say a total of 14 students studied between zero and 20 minutes. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
Now you could also think about marginal distributions the other way. You could think about the marginal distribution for the time studied in the class. And so then you would look at these counts right over here. You'd say a total of 14 students studied between zero and 20 minutes. You're not thinking about the percent correct anymore. A total of 30 studied between 21 and 40 minutes. And likewise, you could write these as percentages. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
You'd say a total of 14 students studied between zero and 20 minutes. You're not thinking about the percent correct anymore. A total of 30 studied between 21 and 40 minutes. And likewise, you could write these as percentages. This would be 7%. This would be 15%. This would be 43%. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
And likewise, you could write these as percentages. This would be 7%. This would be 15%. This would be 43%. And this would be 35% right over there. Now another idea that you might sometimes see when people are trying to interpret a joint distribution like this or get more information or more realizations from it is to think about something known as a conditional distribution. Conditional distribution. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
This would be 43%. And this would be 35% right over there. Now another idea that you might sometimes see when people are trying to interpret a joint distribution like this or get more information or more realizations from it is to think about something known as a conditional distribution. Conditional distribution. And this is the distribution of one variable given something true about the other variable. So for example, an example of a conditional distribution would be the distribution, distribution of percent correct, correct, given that students, students, students study between, let's say, 41 and 60 minutes. Between 41 and 60 minutes. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
Conditional distribution. And this is the distribution of one variable given something true about the other variable. So for example, an example of a conditional distribution would be the distribution, distribution of percent correct, correct, given that students, students, students study between, let's say, 41 and 60 minutes. Between 41 and 60 minutes. Well, to think about that, you would first look at your condition. Okay, let's look at the students who have studied between 41 and 60 minutes. And so that would be this column right over here. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
Between 41 and 60 minutes. Well, to think about that, you would first look at your condition. Okay, let's look at the students who have studied between 41 and 60 minutes. And so that would be this column right over here. And then that column, the information in it, can give you your conditional distribution. Now an important thing to realize is a marginal distribution can be represented as counts for the various buckets or percentages, while the standard practice for conditional distribution is to think in terms of percentages. So the conditional distribution of the percent correct given that students study between 41 and 60 minutes, it would look something like this. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
And so that would be this column right over here. And then that column, the information in it, can give you your conditional distribution. Now an important thing to realize is a marginal distribution can be represented as counts for the various buckets or percentages, while the standard practice for conditional distribution is to think in terms of percentages. So the conditional distribution of the percent correct given that students study between 41 and 60 minutes, it would look something like this. Let me get a little bit more space. So if we set up the various categories, 80 to 100, 60 to 79, 40 to 59, continue it over here, 20 to 39, and zero to 19, what we'd wanna do is calculate the percentage that fall into each of these buckets, given that we're studying between 41 and 60 minutes. So this first one, 80 to 100, it would be 16 out of the 86 students. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
So the conditional distribution of the percent correct given that students study between 41 and 60 minutes, it would look something like this. Let me get a little bit more space. So if we set up the various categories, 80 to 100, 60 to 79, 40 to 59, continue it over here, 20 to 39, and zero to 19, what we'd wanna do is calculate the percentage that fall into each of these buckets, given that we're studying between 41 and 60 minutes. So this first one, 80 to 100, it would be 16 out of the 86 students. So we would write 16 out of 86, which is equal to, 16 divided by 86 is equal to, I'll just round to one decimal place, it's roughly 18.6%. 18.6, approximately equal to 18.6%. And then to get the full conditional distribution, we would keep doing that. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
We are told a restaurant owner installed a new automated drink machine. The machine is designed to dispense 530 milliliters of liquid on the medium-sized setting. The owner suspects that the machine may be dispensing too much in medium drinks. They decide to take a sample of 30 medium drinks to see if the average amount is significantly greater than 500 milliliters. What are appropriate hypotheses for their significance test? And they actually give us four choices here. I'll scroll down a little bit so that you can see all of the choices. | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
They decide to take a sample of 30 medium drinks to see if the average amount is significantly greater than 500 milliliters. What are appropriate hypotheses for their significance test? And they actually give us four choices here. I'll scroll down a little bit so that you can see all of the choices. So like always, pause this video and see if you can have a go at it. Okay, now let's do this together. So let's just remind ourselves what a null hypothesis is and what an alternative hypothesis is. | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
I'll scroll down a little bit so that you can see all of the choices. So like always, pause this video and see if you can have a go at it. Okay, now let's do this together. So let's just remind ourselves what a null hypothesis is and what an alternative hypothesis is. One way to view a null hypothesis, this is the hypothesis where things are happening as expected. Sometimes people will describe this as the no difference hypothesis. It'll often have a statement of equality where the population parameter is equal to the value where the value is what people were kind of assuming all along. | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
So let's just remind ourselves what a null hypothesis is and what an alternative hypothesis is. One way to view a null hypothesis, this is the hypothesis where things are happening as expected. Sometimes people will describe this as the no difference hypothesis. It'll often have a statement of equality where the population parameter is equal to the value where the value is what people were kind of assuming all along. The alternative hypothesis, this is a claim where if you have evidence to back up that claim, that would be new news. You are saying, hey, there's something interesting going on here. There is a difference. | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
It'll often have a statement of equality where the population parameter is equal to the value where the value is what people were kind of assuming all along. The alternative hypothesis, this is a claim where if you have evidence to back up that claim, that would be new news. You are saying, hey, there's something interesting going on here. There is a difference. And so in this context, the no difference, we would say the null hypothesis would be we would care about the population parameter, and here we care about the average amount of drink dispensed in the medium setting. So the population parameter there would be the mean, and that the mean would be equal to 530 milliliters because that's what the drink machine is supposed to do. And then the alternative hypothesis, this is what the owner fears, is that the mean actually might be larger than that, larger than 530 milliliters. | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
There is a difference. And so in this context, the no difference, we would say the null hypothesis would be we would care about the population parameter, and here we care about the average amount of drink dispensed in the medium setting. So the population parameter there would be the mean, and that the mean would be equal to 530 milliliters because that's what the drink machine is supposed to do. And then the alternative hypothesis, this is what the owner fears, is that the mean actually might be larger than that, larger than 530 milliliters. And so let's see, which of these choices is this? Well, these first two choices are talking about proportion, but it's really the average amount that we're talking about. We see it up here. | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
And then the alternative hypothesis, this is what the owner fears, is that the mean actually might be larger than that, larger than 530 milliliters. And so let's see, which of these choices is this? Well, these first two choices are talking about proportion, but it's really the average amount that we're talking about. We see it up here. They decided to take a sample of 30 medium drinks to see if the average amount, they're not talking about proportions here, they're talking about averages, and in this case, we're talking about estimating the population parameter, the population mean for how much drink is dispensed on that setting. And so this one is looking like this right over here. Only these two are even dealing with the mean. | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
We see it up here. They decided to take a sample of 30 medium drinks to see if the average amount, they're not talking about proportions here, they're talking about averages, and in this case, we're talking about estimating the population parameter, the population mean for how much drink is dispensed on that setting. And so this one is looking like this right over here. Only these two are even dealing with the mean. And the difference between this one and this one is this says the mean is greater than 530 milliliters, and that indeed is the owner's fear. And this over here, this alternative hypothesis is that it's dispensing, on average, less than 530 milliliters, but that's not what the owner is afraid of, and so that's not the kind of the news that we're trying to find some evidence for. So I would definitely pick choice C. Let's do another example. | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
Only these two are even dealing with the mean. And the difference between this one and this one is this says the mean is greater than 530 milliliters, and that indeed is the owner's fear. And this over here, this alternative hypothesis is that it's dispensing, on average, less than 530 milliliters, but that's not what the owner is afraid of, and so that's not the kind of the news that we're trying to find some evidence for. So I would definitely pick choice C. Let's do another example. The National Sleep Foundation recommends that teenagers aged 14 to 17 years old get at least eight hours of sleep per night for proper health and wellness. A statistics class at a large high school suspects that students at their school are getting less than eight hours of sleep on average. To test their theory, they randomly sample 42 of these students and ask them how many hours of sleep they get per night. | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
So I would definitely pick choice C. Let's do another example. The National Sleep Foundation recommends that teenagers aged 14 to 17 years old get at least eight hours of sleep per night for proper health and wellness. A statistics class at a large high school suspects that students at their school are getting less than eight hours of sleep on average. To test their theory, they randomly sample 42 of these students and ask them how many hours of sleep they get per night. The mean from this sample, the mean from the sample, is 7.5 hours. Here's their alternative hypothesis. The average amount of sleep students at their school get per night is, what is an appropriate ending to their alternative hypothesis? | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
To test their theory, they randomly sample 42 of these students and ask them how many hours of sleep they get per night. The mean from this sample, the mean from the sample, is 7.5 hours. Here's their alternative hypothesis. The average amount of sleep students at their school get per night is, what is an appropriate ending to their alternative hypothesis? So pause this video and see if you can think about that. So let's just first think about a good null hypothesis. So the null hypothesis is, hey, there's actually no news here that everything is what people were always assuming. | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
The average amount of sleep students at their school get per night is, what is an appropriate ending to their alternative hypothesis? So pause this video and see if you can think about that. So let's just first think about a good null hypothesis. So the null hypothesis is, hey, there's actually no news here that everything is what people were always assuming. And so the null hypothesis here is that, no, the students are getting at least eight hours of sleep per night. And so that would be that, and remember, we care about the population of students, and we care about the population of students at the school, and so we would say, well, the null hypothesis is that the parameter for the students at that school, the mean amount of sleep that they're getting, is indeed greater than or equal to eight hours. And a good clue for the alternative hypothesis is when you see something like this, where they say a statistics class at a large high school suspects, so they suspect that things might be different than what people have always been assuming or actually what's good for students. | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
So the null hypothesis is, hey, there's actually no news here that everything is what people were always assuming. And so the null hypothesis here is that, no, the students are getting at least eight hours of sleep per night. And so that would be that, and remember, we care about the population of students, and we care about the population of students at the school, and so we would say, well, the null hypothesis is that the parameter for the students at that school, the mean amount of sleep that they're getting, is indeed greater than or equal to eight hours. And a good clue for the alternative hypothesis is when you see something like this, where they say a statistics class at a large high school suspects, so they suspect that things might be different than what people have always been assuming or actually what's good for students. And so they suspect that students at their school are getting less than eight hours of sleep on average. And so they suspect that the population parameter, the population mean for their school is actually less than eight hours. And so if you wanted to write this out in words, the average amount of sleep students at their school get per night is less than eight hours. | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
And a good clue for the alternative hypothesis is when you see something like this, where they say a statistics class at a large high school suspects, so they suspect that things might be different than what people have always been assuming or actually what's good for students. And so they suspect that students at their school are getting less than eight hours of sleep on average. And so they suspect that the population parameter, the population mean for their school is actually less than eight hours. And so if you wanted to write this out in words, the average amount of sleep students at their school get per night is less than eight hours. Now, one thing to watch out for is, one, you wanna make sure you're getting the right parameter. Sometimes it's often a population mean, sometimes it's a population proportion. But the other thing that sometimes folks get stuck up on, but the other thing that sometimes confuses folks is, well, we are measuring, is that we are calculating a statistic from a sample. | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
And so if you wanted to write this out in words, the average amount of sleep students at their school get per night is less than eight hours. Now, one thing to watch out for is, one, you wanna make sure you're getting the right parameter. Sometimes it's often a population mean, sometimes it's a population proportion. But the other thing that sometimes folks get stuck up on, but the other thing that sometimes confuses folks is, well, we are measuring, is that we are calculating a statistic from a sample. Here we're calculating the sample mean, but the sample statistics are not what should be involved in your hypotheses. Your hypotheses are claims about your population that you care about. Here, the population is the students at the high school. | Examples of null and alternative hypotheses AP Statistics Khan Academy.mp3 |
So they all agree to put in their salaries into a computer, and so these are their salaries, they're measured in thousands, so one makes 35,000, 50,000, 50,000, 50,000, 56,000, two make 60,000, one make 75,000, and one makes 250,000, so she's doing very well for herself. And the computer spits out a bunch of parameters based on this data here. So it spits out two typical measures of central tendency. The mean is roughly 76.2, the computer would calculate it by adding up all of these numbers, these nine numbers, and then dividing by nine. And the median is 56. And median is quite easy to calculate, you just order the numbers and you take the middle number here, which is 56. Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure? | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
The mean is roughly 76.2, the computer would calculate it by adding up all of these numbers, these nine numbers, and then dividing by nine. And the median is 56. And median is quite easy to calculate, you just order the numbers and you take the middle number here, which is 56. Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure? All right, so let's think about this a little bit. I'm gonna plot it on a line here, I'm gonna plot my data so we get a better sense, so we just don't see them, so we just don't see things as numbers, but we see where those numbers sit relative to each other. So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure? All right, so let's think about this a little bit. I'm gonna plot it on a line here, I'm gonna plot my data so we get a better sense, so we just don't see them, so we just don't see things as numbers, but we see where those numbers sit relative to each other. So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough. I could draw this a little bit neater, but 60, 70, 80, 90, actually let me just clean this up a little bit more too, this one right over here would be a little bit closer. So this one, let me just put it right around here, so that's 40, and then this would be 30, 20, 10. Okay, that's pretty good. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough. I could draw this a little bit neater, but 60, 70, 80, 90, actually let me just clean this up a little bit more too, this one right over here would be a little bit closer. So this one, let me just put it right around here, so that's 40, and then this would be 30, 20, 10. Okay, that's pretty good. So let's plot this data. So one student makes 35,000, so that is right over there. Two make 50, or three make 50,000, so one, two, and three, I'll put it like that. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
Okay, that's pretty good. So let's plot this data. So one student makes 35,000, so that is right over there. Two make 50, or three make 50,000, so one, two, and three, I'll put it like that. One makes 56,000, which would put them right over here. One makes 60,000, or actually two make 60,000, so it's like that. One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
Two make 50, or three make 50,000, so one, two, and three, I'll put it like that. One makes 56,000, which would put them right over here. One makes 60,000, or actually two make 60,000, so it's like that. One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there. And then when we calculate the mean, as 76.2 is our measure of central tendency, 76.2 is right over there. So is this a good measure of central tendency? Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there. And then when we calculate the mean, as 76.2 is our measure of central tendency, 76.2 is right over there. So is this a good measure of central tendency? Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one. And the reason is is that you have this one, that our data is skewed significantly by this data point at $250,000. It is so far from the rest of the distribution, from the rest of the data, that it has skewed the mean. And this is something that you see in general. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one. And the reason is is that you have this one, that our data is skewed significantly by this data point at $250,000. It is so far from the rest of the distribution, from the rest of the data, that it has skewed the mean. And this is something that you see in general. If you have data that is skewed, and especially things like salary data, where someone might make, most people are making 50, 60, $70,000, but someone might make $2 million, and so that will skew the average, or skew the mean, I should say, when you add them all up and divide by the number of data points you have. In this case, especially when you have data points that would skew the mean, median is much more robust. The median at 56 sits right over here, which seems to be much more indicative for central tendency. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
And this is something that you see in general. If you have data that is skewed, and especially things like salary data, where someone might make, most people are making 50, 60, $70,000, but someone might make $2 million, and so that will skew the average, or skew the mean, I should say, when you add them all up and divide by the number of data points you have. In this case, especially when you have data points that would skew the mean, median is much more robust. The median at 56 sits right over here, which seems to be much more indicative for central tendency. And think about it. Even if you made this, instead of 250,000, if you made this 250,000,000, which would be $250 million, which is a ginormous amount of money to make, it would skew the mean incredibly, but it actually would not even change the median, because the median, it doesn't matter how high this number gets. This could be a trillion dollars. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
The median at 56 sits right over here, which seems to be much more indicative for central tendency. And think about it. Even if you made this, instead of 250,000, if you made this 250,000,000, which would be $250 million, which is a ginormous amount of money to make, it would skew the mean incredibly, but it actually would not even change the median, because the median, it doesn't matter how high this number gets. This could be a trillion dollars. This could be a quadrillion dollars. The median is going to stay the same. So the median is much more robust if you have a skewed data set. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
This could be a trillion dollars. This could be a quadrillion dollars. The median is going to stay the same. So the median is much more robust if you have a skewed data set. Mean makes a little bit more sense if you have a symmetric data set, or if you have things that are, where things are roughly above and below the mean, or things aren't skewed incredibly in one direction, especially by a handful of data points like we have right over here. So in this example, the median is a much better measure of central tendency. And so what about spread? | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
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