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Researchers are comparing the average number of hours worked per week by employees at two different companies. Below are the results from two independent random samples. Assuming population standard deviations are unknown and unequal, calculate the $t$-score for the difference in means, but do not find a $P$-value or state a conclusion.

Company A: $n_1=25$; ${\bar{x}}_1=22.4$ hours; $s_1=3.2$ hours

Company B: $n_2=16$ ${\bar{x}}_2=21.1$ hours; $s_1=2.9$ hours

A researcher is comparing average number of hours spelt per night by college students who work part-time versus those who don't. From survey data, they calculate ${\bar{x}}_1=6.82$ hours and $x_{\bar{2}}=6.57$ hours with a margin of error of 0.41. Should they reject or fail to reject the claim that there is no difference in hours slept between the two groups?

Randomization with Commute Times Given the two samples of commute times (minutes) shown here, which of the following are randomizations of them?

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a. Boston: 10 10 60. New York: 5 20 25 30 45.

b. Boston: 10 10 60 20 25. New York: 5 30 45.

c. Boston: 5 10 25 25 60. New York: 5 30 30 60.

d. Boston: 10 10 60. New York: 5 20 25 30 45.

e. Boston: 10 10 10 10 10. New York: 60 60 60.

Finding Critical Values Assume that we have two treatments (A and B) that produce quantitative results, and we have only two observations for treatment A and two observations for treatment B. We cannot use the Wilcoxon signed-ranks test given in this section because both sample sizes do not exceed 10.

a. Complete the accompanying table by listing the five rows corresponding to the other five possible outcomes, and enter the corresponding rank sums for treatment A.

Comparing Two Means Treating the data as samples from larger populations, test the claim that there is a significant difference between the mean of presidents and the mean of popes.

Explain how to perform a two-sample z-test for the difference between two population means using independent samples with and known.