Homework #3 Solution

Homework #3 Solution

8.1

(a)Matched pairs

(b)Independent samples

(c)Matched pairs

(d)Independent samples

8.2

(a)Experimental

(b)Observational

(c)Experimental

(d)Experimental

8.3

(a)Matched pairs

(b)Matched pairs

(c)Independent samples

(d)Independent samples

8.4

(a)Observational

(b)Observational

(c)Observational

(d)Experimental

8.5

(a)The clouds in each group are not matched to one another on some characteristic.

(b)Seeded cloud rainfalls tend to be larger than unseeded cloud rainfalls.

8.7

(a)The two different types of eyes are or belong to the same person.

(b)The pairs tend to lie pretty close to the 45 degree line through the origin. Eyes with glaucoma do not appear to have thicker corneas than unaffected eyes.

8.9

(a)The pooled SD is

Then a 95% CI for is given by

Since the CI does not contain 0, we reject and conclude that there is a significant difference between the two filters.

(b)Since

and

The degrees of freedom are

Then a 95% CI for is given by :

The results are similar, indicating that the assumption of equal variances is reasonable.

8.11

(a)Using , , , , , and , the pooled SD is

Then the test statistic is

Since , we reject and conclude that there are significant differences between the methods.

(b)Since

and

The degrees of freedom are

The test statistic is

Since , we do not reject at . This is the opposite conclusion s that obtained from assuming equal variances.

8.12

(a)One can see from the normal plots that the original data are highly skewed, while the log-transformed data appear reasonably normal. The t tests require that the data be normally distributed; therefore, we need to use the transformed data for the analysis.

(b)Using , , , , and , the pooled SD is

Then the test statistic is

Since , we reject and conclude that there are significant differences in average rainfall between the seeded and unseeded clouds.

(c)Since

and

The degrees of freedom are

The test statistic is

Since , we still reject . The conclusion is the same as that obtained from assuming equal variances, indicating that the assumption of equal variances is reasonable.

8.16

(a)Using and , the test statistic is

Since , do not reject at , and conclude that the average corneal thicknesses are unaffected by glaucoma.

(b)A 90% CI for is given by

8.18

The test statistic is

Since , reject and conclude that the new oven provides more even heating.

8.19

Using the log transformed data, and . Also, and . Then a 95% CI for is given by

Since this interval contains 1, we can recommend using the pooled variance t-test.

8.20

The test statistic is

Since , reject and conclude that the variances are unequal. Therefore, one should use a t-test with separate variances to test the equality of means.

ependent samples

8.21

The sample variances are and . Also, and . Then a 90% CI for is given by

Since this interval contains 1, we can recommend using the pooled variance t-test. But the results using the pooled variances and separate variances t differed, so it would be better to use separate variances t.

8.23

(a)The pooled SD is, using H=High fiber diet and L=Low fiber diet,

Then a 95% CI for is given by:

Treating the data as independent samples, the high and low fiber diets are not significantly different.

(b)Treating the samples as matched, a 95% CI for is given by

There is still no significant difference between high and low fiber diets, although the CI is much narrower.

(c)A 95% CI for , where B=Baseline diet, is given by

There is a significant difference in the mean cholesterol levels at baseline and under a high fiber diet.

(d)High/low fiber diets have a significant effect at of reducing the total cholesterol levels from baseline, but there is no significant difference between high and low fiber diets.

8.24

(a)

Then the ratio of the variances is

(b)If , then the independent samples variance is smaller. If , then the variances are the same. If , then the matched pairs variance is smaller. Because of the nature of the matching, we would expect for most cases. If one part of the pair has an above average response, we would expect the other part of the pair to also have an above average response, just shifted up or down by the treatment.

8.25

(a)The salary boxplots are right-skewed with numerous outliers. The (salary) boxplots are more symmetric with fewer outliers. The log transformed data would work better because the skewness and outliers inflate the variance and would make a difference in means harder to detect.

(b)The pooled SD is

Then the test statistic is

Since , reject at level , and conclude that the two groups are significantly different.

(c)The pooled SD is the average of the two sample variances, which since they are identical, is . Then the test statistic is

Since , reject at level , and conclude that the two groups are significantly different.

(d)The conclusions are the same for . However, note that the t-statistic is larger for the log-transformed data. Using log-transformed data yields a more significant result.

8.26

(a)The rejection rule is to reject if , or if

Then the power is

(b)The power is maximized when is minimized. So for a fixed sample size N, we want to minimize

With respect to . Do this by setting the derivative equal to 0, and solving for

And

Since , then

(c)To satisfy

We must have

Using and from above,

Then

And

(d)Using , , , , and , the required sample size is

Hence and

8.27

The rejection rule is to reject if

Then the power is

Where G is the cdf of the F distribution with and degree of freedom.

Using , , and ,