A psychologist is studying the effect of background music on concentration levels. One group of students took a test with classical music playing in the background, while another group took the same test with no music. The scores were analyzed for variability. The results are as follows:

Classical Music: $n=28$, $\bar{x}=75.2$, $s=8.1$

No Music: $n=30$, $\bar{x}=78.5$, $s=5.6$

At the $0.05$ significance level, test the claim that the variances of test scores are the same for both groups. Assume that the samples are from normally distributed populations.

A company wants to compare the proportion of employees who use public transportation to commute in two of its branches. In Branch A, $80$ out of $200$ employees use public transportation. In Branch B, $110$ out of $220$ employees use public transportation. At the $0.05$ significance level, test whether the proportions are the same in both branches.

A nutritionist is comparing the average daily calorie intake of two groups: vegetarians and non-vegetarians. A sample of $15$ vegetarians has a mean intake of $2,200$ calories with a standard deviation of $180$ calories, a sample of $20$ non-vegetarians has a mean intake of $2,500$ calories with a standard deviation of $210$ calories. At the $0.05$ significance level, what is the value of the test statistic for testing whether there is a significant difference in mean daily calorie intake between the vegetarian and non-vegetarian groups?

A confidence interval for the difference in means is calculated as (1, 5). What does this suggest about the difference in population means?

A nutritionist wants to determine if a new diet plan affects cholesterol levels. She measures the cholesterol levels (in $\text{mg/dL}$) of $8$ patients before and after following the diet for three months. The data are as follows:

Using the Wilcoxon signed-ranks test at a $0.05$ significance level, test the claim that the matched pairs have differences that come from a population with a median equal to zero. Assume that the population of differences has a distribution that is approximately symmetric.

In a Wilcoxon Signed-Ranks Test, suppose you have $n$ nonzero paired differences and the sum of the negative ranks is $t$. What is the expression for the sum of the positive ranks in terms of $n$ and $t$?