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 $t$?

Determine the critical value for the following test: , $\alpha =0.10$, $n_1=12$, $n_2=9$. Assume random, independent samples, normal populations, and equal variances.

A researcher is comparing the average recovery times for two medications. Test the claim that the mean recovery time for medication A is less than that for medication B at the $\alpha =0.05$ significance level. Assume independent random samples and equal population variances. Sample statistics: ${\bar{x}}_A=4.2$ days, $s_A=1.1$, $n_A=15$; ${\bar{x}}_B=5.0$ days, $s_B=0.9$, $n_B=12$.

Suppose we are comparing the average monthly grocery expenditures (in dollars) for households in three different regions. The following table presents the sample data for each region, where each sample has the same size.

Compute the test statistic $F$ for testing whether the mean monthly grocery expenditure differs between the two regions. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal. Use a significance level of $\alpha =0.10$.

A researcher wishes to use the $z$-test to compare the means of two groups. Which requirement is not necessary for the $z$-test?

A researcher compares the mean weekly hours spent on exercise for residents of two towns. A random sample from Town A has size $n_1=18$, mean ${\bar{x}}_1=12.4$ hours, and standard deviation $s_1=3.1$. A random sample from Town B has size $n_2=22$, mean ${\bar{x}}_2=9.7$ hours, and standard deviation $s_2=2.8$. Assume both populations are approximately normal with equal variances. Construct a $90\%$ confidence interval for ${\mu }_1-{\mu }_2$.

A researcher wants to test if the mean score of group A is greater than that of group B at the $\alpha =0.05$ significance level. The populations are normal, independent, and have known standard deviations. Population statistics: ${\sigma }_1=25$, ${\sigma }_2=20$. Sample statistics: ${\bar{x}}_1=82$, $n_1=64$; ${\bar{x}}_2=78$, $n_2=49$. What is the result of the hypothesis test?