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8.9 Redefining success: In Example 8.5 (page 491) , we performed a sig nificance test t0 compare your product with your competitors Success defined as the outcome ...

Question

8.9 Redefining success: In Example 8.5 (page 491) , we performed a sig nificance test t0 compare your product with your competitors Success defined as the outcome where your product provided was better protection; Now; take the viewpoint of your competitor where success is defined to be the outcome where your competitors product provides better protection In other words, n remains the same, but X is now 7. (a) Perform the two-sided significance test and report the results: How do these compar

8.9 Redefining success: In Example 8.5 (page 491) , we performed a sig nificance test t0 compare your product with your competitors Success defined as the outcome where your product provided was better protection; Now; take the viewpoint of your competitor where success is defined to be the outcome where your competitors product provides better protection In other words, n remains the same, but X is now 7. (a) Perform the two-sided significance test and report the results: How do these compare with what we found in Example 8.5? (b) Find the 95% confidence interval for this setting, and compare it with the interval calculated when success is defined as the outcome where your product provides better protection.



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The test-retest method is one way of establishing the reliability of a test. The test is administered, and then, at a later date, the same test is readministered to the same individuals. The correlation coefficient is computed between the two sets of scores. The following test scores were obtained in a test-retest situation. $$\begin{array}{lllllllllll} \text { First Score } & 75 & 87 & 60 & 75 & 98 & 80 & 68 & 84 & 47 & 72 \\ \hline \text { Second Score } & 72 & 90 & 52 & 75 & 94 & 78 & 72 & 80 & 53 & 70 \end{array}$$ Find $r$ and set a $95 \%$ confidence interval for $\rho$.

Okay, so we're doing a 90% confidence interval which means a significant level of 10% and the other ones at 98%. So we have a 2% significance level for the second one. Our values is calculated from problem for 83. Page 483 on 10.142 are going to be as follows. In red here, Our degrees of freedom is one less than the sample size, which is nine. And in table four we will be able to find out what the critical value is. So in nine degrees of freedom. And at the 10% significance level we find that it is 1.833 Which gives us an upper bound, sorry, a lower bound of 1.17 And a lower bound of 3.49. This is going to be at the 10% significance level. So this right here is our 90% confidence interval Now, as for our 98% confidence interval. The only thing that's going to change this t critical value, which is 2.8-1, And then that gives us a lower bound of 0.544 for And an upper bound of 4.116. And again, this is that the 2% significance level, so this is our 98% confidence interval.

We are re examining the tread measuring methods this time, we're doing in 95% confidence interval, which means that our significance level is at 5%. As previously calculated, the values are as follows. Notice that a sample sizes 11, which means that our degrees freedom is 10. Uh huh. We're going to take half of the significance level. So we are looking in the back of the book for T of 0.25 for our critical value. And that number turns out to be in table four 2.2 to 8. And here is going to be the basis for our confidence interval calculating this out, we find that the lower bound is 1.59 mm and the upper bound is a 5.92 So this is our 95% confidence interval. We are 95% confident the true difference of means lies in here.

Okay. Okay. So we're doing confidence intervals. It's this formula right here we're doing it at the 5% and the 1% significance level. So for tea at the 5% significance level we have that it is a critical value of 2.145 Mhm. This is going to turn out to get us. And points of 0.28 and 41 832 at the alpha equals 5% significance level. Now of course we can repeat this for the other significance level of point A. One. And we see that the critical value adds T point oh five is 2.977 Yeah. So there, yeah. Mhm. So that gets us a lower bound of negative 8079 And our upper bound is going to be 49 939 This is our 99% confidence interval and there we go.

In this question, we're comparing essentially the size of different confidence intervals and where they go negative and positive in the first question were given that a 95% confidence interval goes from a negative number two, a positive number. Well, if we change our confidence level to 99%,, that's gonna make this interval be wider. Which will mean at our low end we're going to get more negative in our high end, we're going to get more positive so the interval will stay negatives and positives With a 90% confidence interval. Now then the interval is going to get narrower, it's gonna shrink up a little bit, which means yes, at the high end it's still going to stay positive. But at the low end we don't know whether that if it gets narrower, the negative might go into zero or into the positives by shortening up that low end. So we aren't sure that it will stay negative, it might turn positive, get the hot at both end points. And then the second question says, Okay, well if our 94 95% confidence interval is all positive, what's going to happen if we change to a 99% confidence interval? Well, again, this will make our interval become wider, which means that the high end obviously it's going to get more positive, but at the low end, depending on what that low end point is, if we go even lower, farther to the left, along the number line that could end up being negative. So we have a question mark there, it might be negative or it might be positive, might be zero at that low end. So we can't come to a firm conclusion in terms of the the sign of this low number For a 90% confidence interval again, this is gonna go narrower. Well, if our interval started out positive deposited, it's going to stay positive to positive. If we just shrink it up a little bit


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