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DiseusS Ihe nalure of the dala. In particular; the dala based on an observalional study controlled experinut. Shoull the "malclcu pairs" 4 mnucnendeIIL sp...

Question

DiseusS Ihe nalure of the dala. In particular; the dala based on an observalional study controlled experinut. Shoull the "malclcu pairs" 4 mnucnendeIIL sples' melhods be #pplicd:Sdup appropriale hypotheses. Should the altemative be one [wo-sided? Why?Calculate Ihe test stalislie and stale Ihe conclusion reject not rcicchInc null hypothesis using 0.05. Do nol ASSume that the Iwo populations bCug comparcd have the same standard deviationd) Find the approximate -luctor nanoldowminter

DiseusS Ihe nalure of the dala. In particular; the dala based on an observalional study controlled experinut. Shoull the "malclcu pairs" 4 mnucnendeIIL sples' melhods be #pplicd: Sdup appropriale hypotheses. Should the altemative be one [wo-sided? Why? Calculate Ihe test stalislie and stale Ihe conclusion reject not rcicchInc null hypothesis using 0.05. Do nol ASSume that the Iwo populations bCug comparcd have the same standard deviation d) Find the approximate -luctor nanol dowm interval) Calculale conlidence interval for the diference [n body fat for the (Wo grups. Interpret the result: Write brief conclusion plain English; based VOut findings above;



Answers

The null hypothesis is $H_{0}: \mu_{1}=\mu_{2}$ and the alternative hypothesis is as specified. We have provided data from a simple random paired sample from the two populations under consideration. In each case, use the paired Wilcoxon signed-rank test to perform the required hypothesis test at the $10 \%$ significance level. (Note: These problems were presented as Exercises $10.135-10.140$ in Section $10.5,$ where they were to be solved by using the paired $t$ -test. $$H_{\mathrm{u}}: \mu_{1}<\mu_{2}$$

So we are going to be performing left tail test hypothesis analysis using confidence intervals. The tail test is this at whatever significance level you're choosing? Yeah. And we are going to be rejecting the null hypothesis in favor of the alternative. If and only if the confidence interval has an upper bound that is negative. So for 10.71 are given the confidence or significance level is your .05. Mm. And we find that the difference between the two means is going to be negative 3.14. Really? Cool, don't you think? And the critical value at 5° of freedom and half of 5% is going to be 2.571 times the square roots of the standard deviation square for the first one and the second one. So that's going to be 1.2, 2 squared plus 4.59 Squared divided by the respective sample sizes. Mm Yeah. And what we get is negative 3.14 Plus or -4.89. This will clearly get us a zero in this confidence interval and the upper interval will also be positive. Therefore we failed to reject At the significance level of .05. Now on to the next part, 10.72 are given significance level is going to be 1%, so alpha is equal 2.01 and 2° of freedom, which we find out by calculating from before It was 25 uh huh. When we go look in the back of the book, we're gonna look for 05 at 25° of freedom. Which is going to be 2.787. So we'll go ahead and write that in for a confidence interval Plus or -2.787 sq roots of its standard deviation 5.49 squared divided by its sample size plus 94 Squared divided by its sample size. Mhm. The difference between the two means is going to be 67.9 -66.81, Which is 1.9 A little strange that we're doing a left tailed test on that. Mhm. Yeah. And we get 1.09 plus or minus 6.6, which clearly will not have a negative in the upper bound there, so we again failed to reject At the alpha equals .1 significance level. Yeah.

Oh it's given to us that the alternative hypothesis is going to be oh I'm using my razor put away that shapes board. The alternative hypothesis is given that the anyone is less than Youtube which tells us that we are using a left tail test. So this side right here chop and next thing we can do is determine the differences. Um so it's gonna be 7 -13 Which is going to be 36 -6. And then four and 9 has a difference of negative five 10 and six difference of 40 and two difference of a negative 2 2019 difference of one -1 -5. That's a difference of six or pulled up. Yeah It's a difference of six. But is it positive or negative? Well yeah. Yeah. Yeah it's a difference of six and 12 -10 difference of two. So now that we have a lot of values right there we're gonna number them off from greatest to least greatest. The smallest least greatest The largest. So this is obviously our biggest or smallest value for my second smallest third smallest 4th 5th 6th and 7th R w not is found by calculating the positive values are just adding up these ones right here right there right here and right here I'm actually a little confusing but so we're just gonna add six plus four plus seven Plus 5 and we get a 22 rank and then now we need to find out where this green line is. Yeah so Help you do that is go to the back of the book Table five. We have a sample of seven And our significance level is given to be at .05. Yeah. Oh wait .1. Okay I got reviews of the last one So yeah 10% significance level. This time we're doing a one tailed test. So we are doing .1 Sample size of seven And are significant value is 22. Yeah, 22 is on 22. Therefore we fail to reject the null hypothesis. Okay. Yeah. Okay. The Means of the two samples are equal.

Okay. So we got another beautiful problem right here you one is less than Youtube's or alternative hypothesis, which means that we are looking at a left tail test and we have seven samples down there With the significance level of .1. We have to use this special formula to get to our critical value. And what it is is one w is one minus a So .1. So we're gonna get a nine. But that's just going to be our critical value for the left tail test And a sample size of seven times 8 divided by two minus the critical value that it would be if it were a retail test. So we're gonna go out and look that up in the back of the book there. Um 7.122. So I jumped the gun there with writing the 22 there. And what we do here is 56 divided by two minus 22 Is going to be equal to six. So that dropping black point is six right there. And then what we do is take the difference between the two samples. So we have here Is I want to see a problem with 68. Go find that table one, difference of seven and 13 is negative. six difference between four is 9, 49 is negative. Five difference between 10 and six is four, five and one. So it's five, it's gonna be one and then negative one and five. The difference but there is going to be negative six. Yeah. Oh and then I lost it and then the last 1 12 minus 10 to So we just add up our results here, at least the positive ones we add up and what we get is four plus one plus two, which is going to be seven, seven is greater than six. Lands us in the do not reject songs. So we failed to reject the null hypothesis at the alpha equals 60.1 significance level

All right. So, we got an alternative hypothesis of new one is greater than Youtube, which means that we have a right tail test as shown here. And that critical value for the right tail test is going to be for our eight samples At a 10% significance level. Which is going to get us a critical value of 28. Yeah. And we're going to compare that to the differences. So we do uh the difference over there with the two tests or two columns that we have. So we do 7 -3, which is 44 -5 -1 and so on. We go down the list and we get all these numbers here. The next part to find out W zero or W not, we just add up four plus one is five plus three and we get 13 which is less than 28 meaning that we landed. That do not reject region. Therefore, we failed to reject the null hypothesis at the Point or .1 Significance Level.


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