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Labcratory has {ested trash bags and has found that no 30-gallon bogs Consider the trash 079" Probteat Suoporeeaat acande rcegih 0780 more. On tne basis of the...

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Labcratory has {ested trash bags and has found that no 30-gallon bogs Consider the trash 079" Probteat Suoporeeaat acande rcegih 0780 more. On tne basis of these rcsults; the producer of Inat are currenly on the markel have mcan orcaking strength pourds such bng on Ihc market if the ner trash bog' $ mean te ncw improved trash bag teels Sure Ihat #s 30-galon bag will bethe 5trongcsl 0t43 trash bag breaking strengtns In Table 1.10 Is preacing sirength can be shown {0 be at least 50 pound

labcratory has {ested trash bags and has found that no 30-gallon bogs Consider the trash 079" Probteat Suoporeeaat acande rcegih 0780 more. On tne basis of these rcsults; the producer of Inat are currenly on the markel have mcan orcaking strength pourds such bng on Ihc market if the ner trash bog' $ mean te ncw improved trash bag teels Sure Ihat #s 30-galon bag will bethe 5trongcsl 0t43 trash bag breaking strengtns In Table 1.10 Is preacing sirength can be shown {0 be at least 50 pounds Tre mean of thc samplc we Ict /denotc thc mcan tne brcaking strcngths of a"l possible (rash bags of Ihe new type and assume that & equals 50,560. 162: (0} Cnlculate 95 porcent and 99 pefcont confidencc intcrvals tox (Round your nnswers j0 3 declmal places } (0b porcant conutanka ilen= putrni condcncn Wletr 0a} Using thc 95 percent confidence interval can Ke be 95 percent confident rnat H is alleast 50 pounds? Csdoicetanhn



Answers

(a) identify the claim and state $H_{0}$ and $H_{a},(b)$ find the critical value and identify the rejection region, $(c)$ find the test statistic $F,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal. If convenient, use technology. The table shows the salaries of a sample of individuals from six large metropolitan areas. At $\alpha=0.05,$ can you conclude that the mean salary is different in at least one of the areas? (Adapted from U.S. Bureau of Economic Analysis) $$\begin{array}{|l|l|l|l|c|l|}\hline \text { Chicago } & \text { Dallas } & \text { Miami } & \text { Denver } & \text { San Diego } & \text { Seattle } \\\hline 43,581 & 36,524 & 49,357 & 37,790 & 48,370 & 57,678 \\37,731 & 33,709 & 53,207 & 38,970 & 45,470 & 48,043 \\46,831 & 40,209 & 40,557 & 42,990 & 43,920 & 45,943 \\53,031 & 51,704 & 52,357 & 46,290 & 54,670 & 52,543 \\52,551 & 40,909 & 44,907 & 49,565 & 41,770 & 57,418 \\42,131 & 53,259 & 48,757 & 40,390 & & \\& 47,269 & 53,557 & & & \\\hline\end{array}$$

All right. We want to test the hypothesis. New equals 64.5 versus the null hypothesis or alternative hypothesis does not equal 64.5 for population standard deviation 0.6 given the following data at alpha equals 0.1 confidence. This question is testing your knowledge of how to conduct a hypothesis test for population renew. When the standard deviation sigma is known. We see through the five steps listed here to solve first, we verify normality. So from the normal probability bar on the left, in the box, on the right, we conclude that the data is normal without outliers. Thus we can proceed to solve so and be we calculate that sample size and X five simple meat. This is N equals 22 X 5 64.7 Thus you see we can calculate our test statistic, this is zero equals experiments were not over segment about 10, which gives negative 38.53 for alpha equals 0.1 We continue the critical value in step D. For two tailed tests using a Z. Table where the area H 2.5 are critical value is plus or minus 2.58 Thus we can conclude in step E the following first. Let's rewrite our critical Z score correctly. Since our Xena is to the left of our negative 2.58 we can conclude that zero is in the critical region. So we reject H. Not because you know it's a critical region. We also address the fact that ALPHA equals 00.1 is more reasonable than Alpha equals 0.1 because there are severe punishments for making the type point ever in this problem I. E. There's a severe punishment for making an area generally. So we want to keep the 0.0.1 stringent criteria.

Following is the solution in number 14 at one way Innova test. Uh and this is about the mean sales prices for three cities. And the null hypothesis here is that the mean sale prices are the same for these certain houses. And then the alternative is that at least one of them is different. The second step is to find the critical value and you need three pieces of information to find the critical value. One is your alpha, Your significance level, that significance level. In this case, that's usually given to you is 10. They also needed the degrees of freedom for the numerator, which is the number of categories in this case, the number of cities minus one. So there were three cities that we looked at minus one is two, so degrees of freedom for the numerator is two degrees of freedom for the denominator is the total number of data values minus the number of categories. So in this case, if you counted up those data values, there were 31 -3 cities that we looked at. So 31 -3 is 28. So that's what we need. So from there you can use a table or you can use software. I'm gonna use software. So I wrote a program and I called it inverse. F. I'm not going to show you how to write this program. You can youtube it if you wish. But um it makes Makes it easier for me. So the area is the alpha value. So we'll put in .10 for that. Degrees of freedom from the numerator was too. And then degrees of freedom for the denominator was 28. And that's going to spit out my f. star my alpha value and my uh critical value which is about 2.503. Let's call it 2.503 is my F. stars 2.503. So anything greater than 2.503. We're going to reject h not anything less than 2.503. And we're gonna fail to reject the null. The next step is to find the f statistic and you can do that manually but especially with bigger data sets that can be really time consuming. So I went ahead and punch this into stat. Edit. And these are my data values. So these are the I think these are in thousands of dollars but these are the mean sale prices and if you go back to stat and then tests and the very last one in nova And you put in your columns just make sure you separate them. This is on the T. 84 by the way but make sure you separate them by commas otherwise it's not gonna read it right so nova for those three columns and that's gonna give us everything we need to. The f statistic is about 0.966 Let's go and write that down. So 0.9 66 which is somewhere over here. So that lands in the non rejection region. So that's actually gonna tell us why our fourth step which is the decision and we're going to fail to reject H not since the F statistic is less than the critical value. Now you can also use the P value method that's what this second piece of information is good for it. Now this other stuff doesn't really matter. Um You can just kind of ignore it because this is really what we need. We need the F statistic and we need the p value and the p values pretty large. It's about 0.39 And what you do is you explicitly compare the P value with your alpha value. So the p value in this case is greater than your alpha value. 0.39 is bigger than 0.10. And any time you're P values greater than alpha, you failed to reject H nine. If it's less than alpha then you you reject. And then the final step is to conclude this, you know, with actual words and bring it back to the question at hand. And so what we're going to say is that there is not enough evidence or there is not sufficient statistical evidence. So there's not sufficient evidence to suggest that the mean sales price prices of houses In the three cities are different. Okay that's the five step in Nova process.

Who can't move till question 1 75 Show part A were asked The weather, The American off the auto for inspect is less than one need a meter. So obviously the danger is going to be Yes, because all the data point we have all the merriment are less now. Want me demeanor? So we're confidence enough, Teoh, to say yes. For this question, all data points are less. Then one point narrow you a meter. Okay. And when will to move to part B? We're trying to find the 95% confidence interval. So we start with the simple sized and here it is equal to 36. And after the calculation, we will have the average off the sample to be karaoke wanks 254 So this is our sample size X bar. All right. And according Teoh, the question we have our sick of up, which is a center deviation, is going to be there a 0.1 all right. And our Alfa is equal to their oh, buying their own five. So the critical values that off Alfa over two is going to be 1.96 According Teoh Table for table for beat. Yeah. Okay. Now we're be able to calculate the maximum era. So e is equal to that off Alfa over two times Sigma over squared Rudolph and show we plugging numbers through it. We have 1.96 times there a 0.1 divided by a squared grove 36 we will have the maximum error to be there. Oh, point Narrow 33 How our read it again? Here he had a sequel to their applying their of 33 So our confidence interval is going to be equal to our ex far plus or minus maximum terror. So it is going to me they're open to 54 plus or minus. There are quite narrow 33 and we will get a interval. The lower bound off it is going to pee. Narrow point 2 to 1 in the upper bound is going to be there. Oh, bind! Teoh 87 So this is going to be our 95% confidence in trouble

Okay so the following is an a nova test uh for toothpaste and the mean cost per ounce For very good stain removal, good stain removal and fair stain removal. So the first step is to state your hypotheses and this is always the way it's gonna be. Is h not is that the means are equal and then that the alternative is at least one of them is different with just one of those means is different. The second step is you need to find the critical value. Now you can find the critical value either using a table or using a calculator. Now I've written a program in the calculator. Someone use a calculator but you can certainly use a table as well. I'm not going to go into detail on how to Create the program in the T. I. 84. You can search that online if you want. But the the uh there is a table for you to do this as well. So for a critical value F. Star we're going to call it uh you need the degrees of freedom for the numerator which in this case will be 10. And that's the number of columns we had. Very good, good and fair. So that's three columns minus the one gives us too. So that's the numerator. And then the degrees of freedom for the denominators 12. That's the total number of data values, which is 15 minus the number of columns, which is three. So that is enough and an alpha equals 0.5. So you also need to know your alpha. So alpha equals 0.5. So with these three pieces of information you should be able to find the F. Star in the book. But just to show you on the calculator. So I've written a program called inverse F. So this is going to give you my F. Star and the area is your alpha value. The degrees of freedom for the numerator was too and then for the Denominator was 12 and you should get about 3.89 or 3.885 will go to three decimal places. So that's your f. star. 3.885. So anything to the right or anything greater than 3.885 will reject the null hypothesis. So the next step is to find our critical value. Um Which is F. Now you can do that manually certainly but I'm going to go and use the calculator because it's so much easier. So I've actually already pre set this one up. You gotta step edit here your data values. So this was your very good column. The L one, the good column was L. Two and then the fair column was L. Three. Then if you go to stat and then air over two tests and it's your very last one in nova. And we go second L one comma second L two comma second L three. And then we close our parentheses and that gives us everything we need. So the F is about 4.8. We'll call it. So f is 4.8. Okay, Which is somewhere over here. And that means in step four we reject the null hypothesis, reject. H not now. You can also look at the P value. So this p value is 0.25 And remember that alpha value was point oh five. So um that's actually preferred to the P value whenever the p values less than alpha. That's whenever we reject the null hypothesis and the p value 0.2 is less than 0.5. But either way you reject you can use the traditional or the P value methods. So then the last step is where we just basically state what our conclusion is. So there is sufficient evidence. We'll just go and say there is sufficient evidence to suggest that at least one mean cost per ounce is different from the others.


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