 Hi everyone, it's MJ and we're looking at question 3 of the September 2017 paper. And this question, as we're going to see, it is with regards to sampling and the distributions of the sample variance. So the question says, let x1 to xn be a random sample of independent random variables from the normal distribution. It's on the shape of the sampling distribution of the sample variance s squared with respect to sample size n. Now this question, it is only with two marks, but a lot of people will struggle to get the second mark. Because I mean, the first mark, it's basically regurgitating what you've learned from the notes, which is as follows. That the sample variance is distributed accordingly to this as the chi-square distribution with n minus 1 degrees of freedom. What we now need to do, so you're probably going to get one mark for saying that. The tricky part comes with, what we want to do is comment on the shape. We want to talk about the shape with respect to the sample size n. So how does the shape and n, how are these two related? What we need to say is for when n is small, what is the shape going to be, and for when n is large, what is the shape going to be? And in order to answer this question, in order to get your marks, is you need to actually know a little bit more about the chi-square distribution, which is something that we've looked at in an earlier course. What we know is that when n is small, we're going to see that this chi-square distribution is heavily skewed to the right. Remember, we're going to get a shape almost that looks like that. Oh, I did not draw that very nicely, like something like that. But when we know that n is large, we know that chi-squared with k degrees of freedom is approximately distributed normally with k and 2k, when k is large. Which means that when n is large, because we know what the shape of the normal distribution is, we know it is going to therefore be symmetrical. And bam, that's where we're going to be picking up the rest of our marks for this part of the question here. So this question, not a lot of maths. In fact, there was no maths at all. Instead, it was recalling the theory side. So make sure you do learn the stuff. It is very difficult to just work out this in the exam. This is stuff that you should be learning, and it is just a little bit of memory recall. So make sure you learn the little quirks. I mean, they could ask you a much harder exam question where it is mathematical, and you need to know this result in order to continue. So they have been quite nice. If you haven't learned, it is only two marks that you're losing out on. But you want to get as many marks as you can get. If we look at part two now, which is determine the variance of s squared based on its sampling distribution, well, then what we're basically going to be doing is we are going to be looking at the variance of this thing here, n minus 1 s squared divided by sigma squared over here. Now, what we know or where this will take us is the following. We are going to get 2n minus 1. Now, how does that work out? Well, we know from over here that the variance of the chi-square distribution is this 2k, so it's these two degrees of freedom. So how are we getting this result? So we are going to go slowly just so that everyone is on the same page. So this, we're taking the variance of the side over here. And if we're taking the variance of the chi-square, so we know that the variance of a chi-square with n minus 1 is equal to 2n minus 1. OK, that is how we got our answer. Now, what we're going to do is we're going to solve this equation or we want to get the s on its own. So we want to write this in the sense of that equals to something, something, so we want to get these terms on the other side. So to do that, what we're going to be doing is we're going to see that the variance of s squared, OK, we're going to be taking this part out. And when we take out a constant out, we're going to be squaring it. We are going to see the following. 4n minus 1 squared, because remember, we're taking it out, so we're squaring it, and 2n minus 1. OK, is everyone happy with where we got that? Remember, we took this sigma squared here, which then becomes sigma 4. And then because we're putting it on this side of the equation, it goes from being the denominator now to the numerator. Same with this n minus 1. We're taking it out, which makes it n minus 1 squared. And then because we're taking it on the other side of the equation, because it was the numerator, it now becomes the denominator. OK, everyone happy with that. And then what we're going to see is we're going to cancel out some of these terms. And we're going to see that the variance of s squared is equal to 2 or n minus 1. OK, and there we go. Easy mocks. So the tricky part with this question was to first write out your equation, which was the variance of n minus 1 s squared divided by sigma squared is equal to the variance of sigma squared n minus 1. Then the whole idea here was to take n minus 1 squared. I'm just repeating this in case it didn't make enough sense. s equal to 2 n minus 1. And then we have variance s squared is equal to sigma 4 to n minus 1 divided by n minus 1 squared, which means variance s squared is equal to 2 sigma 4 divided by n minus 1. OK, so just a neat way of writing it out, then trying to go all over the place. But yeah, there we go. I mean, again, that is something straight from the core reading that is straight from the notes. This, I think the examiner will be very disappointed if you don't get these full marks. It does show that you maybe didn't study. But yeah, you should be getting these full marks. And I think that's what they do. The beginning, this is still question number three. They're warming you up. We're going to see later on as the questions increase. They do get a little bit more tricky. Anyway, if you've got any questions, please feel free to put them in the comments section below, and I'll get back to you. Thanks guys for watching. Cheers.