The following is a solution video to number 22 with the chef test with comparing different regions of the country, Northeast, Midwest, south and West with the well being index. And here's your data set that was given in the book and there are four categories, which means we have actually a few more um scenarios to look at. So we need to look at each pair and I just wrote them all out. I exhausted them. Also, you ignore these numbers for now I'll show you how to get these in a second, but Northeast and midwest, we're gonna compare those North east to south, will compare those north east to west, will compare those and then midwest, Well, I didn't do Midwest. Northeast has already did that one. You can't do midwest with itself. So I did Midwest and south and then Midwest and west. And then already did south and Northeast up here already in the south and midwest right here and you can't do south and south. So then I did south and west and then that's all them exhausted. So we're looking at six pairs here. Um, and we use the formula in the book. So the first thing I did, as I found the sample means for all four categories and the way you do that, you just do equals average. And then it's just the data set. So Northeast would be a two to a seven. Midwest would be to to be nine. South is the largest sample size C2 to c. 12 and then the West is D two to D 10. So you find the X. Bars and I like to look at these Because it gives me a good information, kind of a good idea going into it. What I think the difference would be if there is in fact a difference and I see northeast is 67.4, And then west is 67.46. So those are all, you know, pretty much the same. So I'm thinking that those three probably don't have a significant difference. However, the south, I mean, that's A full 2-2 and a half points lower. So I'm thinking if anything South Will be the different one, no, I don't know for sure. It may not, none of them might not be different. I don't know. But if there is a difference, it's probably gonna be south now again, I don't know that for sure, but it's looking that way. The next thing I found is the sample variance, because that's what the stand the formula calls for us, that S one squared, or S I squared, and the way that I did that. Now, there's other ways you can do this, but I just did the sample standard deviation, make sure it's dot s, it's s t d e v dot s not dot p. Because this is a sample, not population of each data set, and then I squared it to get that sample variance. So that's what I did on all these. Okay, so the standard deviation for the sample quantity squared, and those are the sample variances. The next thing I did was I found, I just actually wrote down the n minus one, so that's the sample size for each category, minus one. So there are six data values here, so minus one is five and 7, 10 and eight. So I just took each um sample size and attractive one, and then I found this S S w the way you find that is, you sort of combine these last two things, it's the summation of In -1, the sample size -1 times its corresponding sample variance. So that's why I did the summation, or a 20 times a. 17 plus B 20 times B 17 plus C 20 times C. 17 Plus d. 20 times d. 17. And that's my ssw about 54.7. And then I went and did summation of in minus one. That's just that five plus seven plus 10 plus eight, because that's also part of the formula. So that's all the information I need for the formula. The only other thing I need is this F. Star and that's the critical value. And the way you find that is you take your original critical value that you found on. I think it was like number 11, whatever this problem was, And that was 2.276. And then you multiply by the number of categories -1. So you multiply in this case by three because there are four categories, so 4 -1 is three. So anything larger than this. F Star. This critical value means there is a significant relationship or significant difference between sample means anything smaller than that. F Star means that there's not much of a difference between the means. So let's look at what we did here. So this is the formula in Excel form. So I compared northeast Midwest first and I got .05. So there is not a significant difference between northeast and Midwest, which I assumed because those are so close to the same thing, so that's just sort of verified it now, How did I get that? Well, this is the two sample means a 14 -714 quantity squared. That's the numerator of that formula in your book. The denominator Is this SSW. So that's where I get this. 8:24 Divided by c. The summation of N -1 times one over N plus one over N. With the ends being the corresponding sample sizes. So remember this is Northeast. So there are six data values there and Midwest and their eight data values there Plug that in and you get the .05. So that is not a significant difference because it's not bigger than that. 6.8-8 and I do that six more times. But the only thing I change, so look at this compared to this, not much has changed. One. The meanest changed because I'm now I'm comparing Northeast which is the a column With the south which is the c. column. So I changed that to c. 14. And then also my sample size changed. 1/6 hasn't changed because it's still northeast. But the south has a larger sample size, it's 11, so it's 1/11 and that is 12.13 So look at that that is almost double what that F. Star is. So there is a significant difference between the north east and the south with their well being index. Okay, back to so look at this formula compared to this one again, almost the exact same thing. North east and west. The only difference is the um see change to A. D. So it's D. 14 because that's my sample mean there and then the other thing that changed was the sample size. So Northeast is 1/6. But the west there are nine data values so it's 1/9 and that is a very very small should face statistics. So north east and west, You know and you can kind of verify that these X bars they're they're almost idea I mean there are only 2.03 off. Right, so we're very small test statistic there. So there is no difference between north east and west. Same thing with Midwest and south. There is a pretty significant difference there. It's again almost double the F statistic. Everything has stayed the same. I'm just comparing midwest with the south so be 14 minus c 14 quantity squared Divided by the Ssw divided by N -1, summation of N -1 Times one over the Midwest sample size. There are eight data values plus one over the south sample size. That there are 11 of those and you get 12.5. Mhm. Okay. Midwest and west. Same idea. Just different slightly different numbers. You should get a test value of .10 about and then Southwest and West. The most significant difference there of 15.9, which is significantly larger than the F. Star. So it appears that the original assumption of South being much different with them. Um Well being index is different because look at this North, east, south, midwest, South and West and South. All three of those Are significantly different, whereas the other three Are almost identical at zero. So that is the shape test for the well being index for different regions in the country.