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Use the accompanying partially completed two-way ANOVA summary table to complete parts through belowClick the icon t0 view the table.Reject the null hypothesis. The...

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Use the accompanying partially completed two-way ANOVA summary table to complete parts through belowClick the icon t0 view the table.Reject the null hypothesis. There is sufficient evidence to conclude that not all Factor_ means are equal: 0 D. It is inappropriate t0 draw conclusion from this test because the Factors and interacte) Using a = 0.05, are the Factor B means different?Identify the hypotheses to test for Factor B Choose the correct answer below:0A: Ho: HA = HB, Hq: KA #KB0 B. Ho: 431

Use the accompanying partially completed two-way ANOVA summary table to complete parts through below Click the icon t0 view the table. Reject the null hypothesis. There is sufficient evidence to conclude that not all Factor_ means are equal: 0 D. It is inappropriate t0 draw conclusion from this test because the Factors and interact e) Using a = 0.05, are the Factor B means different? Identify the hypotheses to test for Factor B Choose the correct answer below: 0A: Ho: HA = HB, Hq: KA #KB 0 B. Ho: 431 432, H1: Not all Factor B means are equa Ho: H81 #H82 H33, H1: KB1 VB2 VB3 0 D. Ho: 431 H3z H83, H1 Not all Factor means are equa Find the p-value for Factor B_ P-value = L (Round to three decimal places as needed ) Draw the appropriate conclusion for Factor B. Choose the correct answer below: Do not reject the null hypothesis. There is insufficient evidence to conclude that not all Factor B means are equal Do not reject the null hypothesis. There is sufficient evidence to conclude that the means differ: Reject the null hypothesis. There is sufficient evidence to conclude that not all Factor B means are equal It is inappropriate t0 draw conclusion from this test because the Factors and interact



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If the null hypothesis is rejected in a one-way ANOVA test of three or more means, then a Scheffé Test can be performed to find which means have a significant difference. In a Scheffé Test, the means are compared two at a time. For instance, with three means you would have the following comparisons: $\bar{x}_{1}$ versus $\bar{x}_{2}, \bar{x}_{1}$ versus $\bar{x}_{3},$ and $\bar{x}_{2}$ versus $\bar{x}_{3} .$ For each comparison, calculate $$\frac{\left(\bar{x}_{a}-\bar{x}_{b}\right)^{2}}{\frac{S S_{W}}{\Sigma\left(n_{i}-1\right)}\left(\frac{1}{n_{a}}+\frac{1}{n_{b}}\right)}$$ where $\bar{x}_{a}$ and $\bar{x}_{b}$ are the means being compared and $n_{a}$ and $n_{b}$ are the corresponding sample sizes. Calculate the critical value by multiplying the critical value of the one-way ANOVA test by $k-1 .$ Then compare the value that is calculated using the formula above with the critical value. The means have a significant difference when the value calculated using the formula above is greater than the critical value. Refer to the data in Exercise $8 .$ At $\alpha=0.01,$ perform a Scheffé Test to determine which means have a significant difference.

The following is a solution video to number 20. Using the chefs test to compare. I think they're vacuum cleaners. There's um bagged upright, bagless upright and then top canister for the means of I can't remember it's like cost or or weight or something. And um I have the data here. So here's the bag, upright, bag bagless upright in the top canister. And then I start finding everything that's needed for the formula. So I needed to find the sample means which is X. Bar. So ignore this stuff over here, we'll get to that minute. So I just found the average is so that's equals average and then A A two to a seven. B two to B. Seven and then see to see the seven. So those are the averages and right off the bat. The way that I think about this is I just look now it's not always the case but um the two means that are the farthest away from each other. I think that those are going to be the ones um that have a significant difference. Now, again, that's not always the case, but bagless, upright and top canister are pretty far apart, so that's kind of where I'm going to start actually now, it could be other ones, but that's actually the one that I'm going to start with now, I also found the sample variance because that's that s uh S I squared, uh part of the formula. And to do that, you find the sample standard deviation, there's probably a variance formula too, but I just do standard deviation squared. So I did the sample standard deviation, make sure it's dot s not dot P. If you're using excel a two day seven and that's squared, and then B two to B seven, that's squared, and then C two to C seven, that square. So these are the sample standard deviations, I'm sorry, sample variances. And then I have these in -1, so that's just the sample size for each category -1. So that's five and five and five. And then I found this S. S. W. And the way you do that, that's that's the formula for the n minus one is the summation of n minus one times the sample variance. So that's where I get this, A 16, that's five times 4.7 plus five times 3.6 plus five times 4.6. And that gives me the 62.16, 62 and 16. And then the summation of n minus one is just these added together. So five plus five plus five is 15. So those are all the pieces of the formula you need. And then the other thing that you need is the F. Star or the critical value. And in the book it says the way you do that is you find the F. Star from the problem at hand. And this was I think from number seven or something. Um And you can look back at data but it's 6.36 is the critical value. And you multiply by k minus one where K. Is the number of categories. Well there are three categories here, so three minus one is two. So I took the critical value in the original problem at the I think it was the five level of significance or something like that And you multiply by two. So anything larger than this. f. star this 12.72 we say that there is a significant difference, anything less than that critical value. We say there's not a significant difference. Okay, so like I said I started with the bagless upright and the top canister. Because those two means are the farthest apart. And I'm thinking that this is going to be a pretty significant difference. And it and it is because that um chef statistic is to 23.24 which is significantly larger than that 12.72. So the way I found that is I took the two means that I subtracted in my squared. Um So that's the B 10 minus the c 10 quantity squared. And then divided by Here that a 19 oversee 19. That's that SSW divided by the summation of uh -1. So that's the 62.16, repeating divided by the 15 and then times one over N plus one over N. Where in is this sample size for for each of those? And in this case it's the same number anyway. So it's 1/6 +16 Because there are six data values in each category. And then whenever you enter that then it's 23.24. And you can get this with the calculated to I just find excel to be easier. So for sure bagless upright and top canister. There is a significant difference in those two means. And then I went ahead and did the other ones. Um So bagless upright and then top canister and those do not have a significant here. So the bagged upright and top canister, you know it's it's kind of close but it's not quite uh significant. And then bagged upright and bagless upright. It's very very small difference actually. So bagless upright and top canister is the most significant difference due to that chefs statistic.

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.

In the following video, we're going to use chefs test uh to determine which of these means is different for very good, good and fair for toothpaste cost per per ounce or something. So they give you the formula of the book and Uh they involve x bar so the average, so here's the data set. Very good. Good and fair and I found the average here for the very good. It's .455 for the good, it's .606. And then for the fair it's .46. Now I have the technology to do this for me and that's nice. But if you are running out of time or something you can kind of look at these and what I would do is you're you're looking at which one is the most different which which means the most different. And it appears that this good is pretty far off the other ones. So I'm just going to make a wild guess. I mean I've already done it but I'm going to make a wild guess that Very good and good. They seem to be pretty far apart, so I think that's going to be the big difference. All right, So the next thing I did was I found the sample variance, because that's what you also need with that formula. And so that was S T D E V dot s. That's the sample variance of the first of the very good. Of the good and then of the fair, so it's the standard deviation and then squared. So I needed to find the variance squared and then I did these in minus ones. So in is the sample size for very good? Well, there were six minus one, is five, so that's where I get 54 and three. And then I found the S. S. W. The way you find ssw you just simply take the n minus one times the sample variance. That's why I found the sample variance. So it's five times 50.287 Plus. And it's the summation of all those plus four times .01 098 plus three Times .011467. And that gave me a uh within as 0.9267 And then I found the summation of n minus one. Which was that five plus four plus plus three. Okay, so that's basically everything that I need um to use the formula and then it also says for a critical value. Use the critical value found in an example five and that was 3.885 I won't show you how to do that again, but it's 0.33 point 885 And then times k minus one where K. Is the number of categories. So. Very good. Good. Fair. There are three categories. So I multiply that by two and I got 7.77 So if the um chefs statistic is larger than that critical value then that's the difference. That's the that's a significant difference in means if it's smaller than than it's not significant. So here we have these numbers and it is very good and good. That are the most different. So that's kind of what I thought it was going to be because that's larger than 7.77 Now how did I get that? Well I applied the formula so it's the two means subtracted and then squared. So it's the difference of the two means that's where I get the 8 10 minus B. 10 quantity squared divided by It's the SSW. Which is this a 19 Divided by the c. 19. So divided by that summation of N. I -1 and then times one over N plus one over N. So that's one over the number of very good which was six plus one over. Um the number of goods which was five. And that gave me this 8.055 and that's how I did this one and this one just different numbers. Um but you know, I won't go through that because it's essentially the same thing. But this is the statistic that is larger than that 7.77. So very good is um there is a significant difference between very good and good for the toothpaste problem.


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