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Studying for a test? Prepare with these 2 lessons on Inference for categorical data (chi-square tests).
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I'm thinking about buying a restaurant, so I go and ask the current owner, what is the distribution of the number of customers you get each day? And he says, oh, I've already figure that out. And he gives me this distribution over here, which essentially says 10% of his customers come in on Monday, 10% on Tuesday, 15% on Wednesday, so forth, and so on. They're closed on Sunday. So this is 100% of the customers for a week. If you add that up, you get 100%. I obviously am a little bit suspicious, so I decide to see how good this distribution that he's describing actually fits observed data. So I actually observe the number of customers, when they come in during the week, and this is what I get from my observed data. So to figure out whether I want to accept or reject his hypothesis right here, I'm going to do a little bit of a hypothesis test. So I'll make the null hypothesis that the owner's distribution-- so that's this thing right here-- is correct. And then the alternative hypothesis is going to be that it is not correct, that it is not a correct distribution, that I should not feel reasonably OK relying on this. It's not the correct-- I should reject the owner's distribution. And I want to do this with a significance level of 5%. Or another way of thinking about it, I'm going to calculate a statistic based on this data right here. And it's going to be chi-square statistic. Or another way to view it is it that statistic that I'm going to calculate has approximately a chi-square distribution. And given that it does have a chi-square distribution with a certain number of degrees of freedom and we're going to calculate that, what I want to see is the probability of getting this result, or getting a result like this or a result more extreme less than 5%. If the probability of getting a result like this or something less likely than this is less than 5%, then I'm going to reject the null hypothesis, which is essentially just rejecting the owner's distribution. If I don't get that, if I say, hey, the probability of getting a chi-square statistic that is this extreme or more is greater than my alpha, than my significance level, then I'm not going to reject it. I'm going to say, well, I have no reason to really assume that he's lying. So let's do that. So to calculate the chi-square statistic, what I'm going to do is-- so here we're assuming the owner's distribution is correct. So assuming the owner's distribution was correct, what would have been the expected observed? So we have expected percentage here, but what would have been the expected observed? So let me write this right here. Expected. I'll add another row, Expected. So we would have expected 10% of the total customers in that week to come in on Monday, 10% of the total customers of that week to come in on Tuesday, 15% to come in on Wednesday. Now to figure out what the actual number is, we need to figure out the total number of customers. So let's add up these numbers right here. So we have-- I'll get the calculator out. So we have 30 plus 14 plus 34 plus 45 plus 57 plus 20. So there's a total of 200 customers who came into the restaurant that week. So let me write this down. So this is equal to-- so I wrote the total over here. Ignore this right here. I had 200 customers come in for the week. So what was the expected number on Monday? Well, on Monday, we would have expected 10% of the 200 to come in. So this would have been 20 customers, 10% times 200. On Tuesday, another 10%. So we would have expected 20 customers. Wednesday, 15% of 200, that's 30 customers. On Thursday, we would have expected 20% of 200 customers, so that would have been 40 customers. Then on Friday, 30%, that would have been 60 customers. And then on Friday 15% again. 15% of 200 would have been 30 customers. So if this distribution is correct, this is the actual number that I would have expected. Now to calculate chi-square statistic, we essentially just take-- let me just show it to you, and instead of writing chi, I'm going to write capital X squared. Sometimes someone will write the actual Greek letter chi here. But I'll write the x squared here. And let me write it this way. This is our chi-square statistic, but I'm going to write it with a capital X instead of a chi because this is going to have approximately a chi-squared distribution. I can't assume that it's exactly, so this is where we're dealing with approximations right here. But it's fairly straightforward to calculate. For each of the days, we take the difference between the observed and expected. So it's going to be 30 minus 20-- I'll do the first one color coded-- squared divided by the expected. So we're essentially taking the square of almost you could kind of do the error between what we observed and expected or the difference between what we observed and expect, and we're kind of normalizing it by the expected right over here. But we want to take the sum of all of these. So I'll just do all of those in yellow. So plus 14 minus 20 squared over 20 plus 34 minus 30 squared over 30 plus-- I'll continue over here-- 45 minus 40 squared over 40 plus 57 minus 60 squared over 60, and then finally, plus 20 minus 30 squared over 30. I just took the observed minus the expected squared over the expected. I took the sum of it, and this is what gives us our chi-square statistic. Now let's just calculate what this number is going to be. So this is going to be equal to-- I'll do it over here so you don't run out of space. So we'll do this a new color. We'll do it in orange. This is going to be equal to 30 minus 20 is 10 squared, which is 100 divided by 20, which is 5. I might not be able to do all of them in my head like this. Plus, actually, let me just write it this way just so you can see what I'm doing. This right here is 100 over 20 plus 14 minus 20 is negative 6 squared is positive 36. So plus 36 over 20. Plus 34 minus 30 is 4, squared is 16. So plus 16 over 30. Plus 45 minus 40 is 5 squared is 25. So plus 25 over 40. Plus the difference here is 3 squared is 9, so it's 9 over 60. Plus we have a difference of 10 squared is plus 100 over 30. And this is equal to-- and I'll just get the calculator out for this-- this is equal to, we have 100 divided by 20 plus 36 divided by 20 plus 16 divided by 30 plus 25 divided by 40 plus 9 divided by 60 plus 100 divided by 30 gives us 11.44. So let me write that down. So this right here is going to be 11.44. This is my chi-square statistic, or we could call it a big capital X squared. Sometimes you'll have it written as a chi-square, but this statistic is going to have approximately a chi-square distribution. Anyway, with that said, let's figure out, if we assume that it has roughly a chi-square distribution, what is the probability of getting a result this extreme or at least this extreme, I guess is another way of thinking about it. Or another way of saying, is this a more extreme result than the critical chi-square value that there's a 5% chance of getting a result that extreme? So let's do it that way. Let's figure out the critical chi-square value. And if this is more extreme than that, then we will reject our null hypothesis. So let's figure out our critical chi-square values. So we have an alpha of 5%. And actually the other thing we have to figure out is the degrees of freedom. The degrees of freedom, we're taking one, two, three, four, five, six sums, so you might be tempted to say the degrees of freedom are six. But one thing to realize is that if you had all of this information over here, you could actually figure out this last piece of information, so you actually have five degrees of freedom. When you have just kind of n data points like this, and you're measuring kind of the observed versus expected, your degrees of freedom are going to be n minus 1, because you could figure out that nth data point just based on everything else that you have, all of the other information. So our degrees of freedom here are going to be 5. It's n minus 1. So our significance level is 5%. And our degrees of freedom is also going to be equal to 5. So let's look at our chi-square distribution. We have a degree of freedom of 5. We have a significance level of 5%. And so the critical chi-square value is 11.07. So let's go with this chart. So we have a chi-squared distribution with a degree of freedom of 5. So that's this distribution over here in magenta. And we care about a critical value of 11.07. So this is right here. Oh, you actually even can't see it on this. So if I were to keep drawing this magenta thing all the way over here, if the magenta line just kept going, over here, you'd have 8. Over here you'd have 10. Over here, you'd have 12. 11.07 is maybe some place right over there. So what it's saying is the probability of getting a result at least as extreme as 11.07 is 5%. So we could write it even here. Our critical chi-square value is equal to-- we just saw-- 11.07. Let me look at the chart again. 11.07. The result we got for our statistic is even less likely than that. The probability is less than our significance level. So then we are going to reject. So the probability of getting that is-- let me put it this way-- 11.44 is more extreme than our critical chi-square level. So it's very unlikely that this distribution is true. So we will reject what he's telling us. We will reject this distribution. It's not a good fit based on this significance level.