 Hi, this is Dr. Don and this is a continuation of that problem 5.2.13 about sampling distributions and variance and checking to see if a sample statistic is an unbiased estimate of the population parameter. To do that, we've got to find of course the population parameters and then the sample statistic that we're going to compare those. The sample statistic is in our case, because we've got a distribution, a sampling distribution, then the sample statistic we've got to compare is the mean of that sampling distribution and I'll show you how we find that. And then we compare that mean, in this case we're looking at the variance, it would be s squared, is it equal to sigma squared, the population parameter, and if they're equal then sigma, excuse me, s squared is an unbiased estimator of sigma squared. So how are we going to do that? We're going to go back and click on the icon again and bring up this table. This time we want the population itself, which is in the small table there, and we just click open in Excel and we open that again the way we did it before. Here's the spreadsheet that contains the information we have about the population. Let's start by getting expected values. The expected value of x in each of these cases is just equal to the value of x times its probability. And so that's the expected value of x and I'll just drag that down so we've got our e of x's and we will sum those to get mu, the mean of those. So I'm going to use the sum, sum those up, hit enter, and this is our mu. This problem is unfortunately one of those where we are going to have to use the formulas that are given in the text and in the show me an example. In this particular part the variance sigma squared of the population is equal to the summation of the probabilities of x times x minus the mean, we just found mu, that value squared. So let's do that. Let's start out here. I'm going to in this column call it x minus mu squared. That's what we're going to find in that column. So we go with equal, that's x minus, whoops, minus our value of mu. Now we want to be able to copy this formula down so I'm going to hit F4 to turn the references for that particular cell to absolute. And before I raise this to a power I need to put in some parentheses because we want the whole thing raised and then I'm going to put that formula, parentheses A minus C7 locked down to make it absolute, raise to power. And so that gives me my x minus mu 3.24 and I'll just drag that down to get all those values. The next thing we need to do is to get the, multiply that times the probability and that's pretty straightforward. I'm just going to hit equal, probability times my x minus mu squared hit enter and that gives me a value that I can drag down to get all of my, that part of the equation. Now we need to sum those up. So we just hit equal SUM, sum, select that range hit enter. So our variance is equal to 1.96, put an apostrophe equal SIGMA squared. That's our value of the population variance sigma squared. Now back to the first table. We need to get the expected values of our sampling variances in order to add those up to get the mean of those expected values which would give us our S square, our sampling variance. So I'm going to start here in this cell and just take the variance which is zero times its probability that we calculated, 0.22. Now I could copy those down more carefully but I'm going to do it a little faster there because all these zeros don't affect us anyway. We need to get the SUM, get that and select that range. So you can see our S square equals 1.96 and I'm pretty sure that's also the value we got here. So because the sampling statistic S square is equal to the population parameter sigma squared, we can say that S square is an unbiased estimator of sigma squared. The last two parts we need to find the sampling distribution of the standard deviation S which is similar to what we did for the variance of the sample and we also need to determine if S is an unbiased estimator of sigma. So let's jump back into our tables. When I have this first, this second spreadsheet up, I'm going to go ahead and get sigma. Remember sigma is just the square root of the variance. So S, Q, or T of the variance, 1.4 equal. So that's our population sigma 1.4. Now we need to check to see if S is equal to 1.4 to see if it's an unbiased estimator. As I just did for the population, S is just equal to the square root S, Q, or T, select that, of the variance, if I go over here to the variance there, that's zero. So that's my S and I'm going to do the quick and dirty thing again, just drag that down to save a little bit of time. So our sampling distribution of the standard deviation is just equal to zero with a probability of .22, .707 with a probability of .32, 1.414 with a probability of .22, 2.121 with a probability of .16, 2.828 with a probability of .08, which is the answer they wanted over there. Now we need to get the expected values of these standard deviations, and again in the same way, that's just equal to S times the probability associated with it, which is the probability of the variance, which is zero, and now I can drag those down. And again, because I've got those, oops, I've got one more to go there, because I've got those zeroes in there, I can ignore those. And again, the mean of the expected values, well I'll go down here in this cell and start entering sum, and then I'll sum up that range again, because the zeroes don't hurt us, hit enter. Now our value of S is 1.10, and that does not equal sigma of 1.4, therefore S is not an unbiased estimator of sigma. This is a biased estimator of sigma, so I hope this helps.