91Ó°ÊÓ

Suppose that \(H_{0}: \mu_{X}=\mu_{Y}\) is being tested against \(H_{1}\) : \(\mu_{X} \neq \mu_{Y}\), where \(\sigma_{X}^{2}\) and \(\sigma_{Y}^{2}\) are known to be \(17.6\) and \(22.9\), respectively. If \(n=10, m=20, \bar{x}=81.6\), and \(\bar{y}=79.9\), what \(P\)-value would be associated with the observed \(Z\) ratio?

If \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{m}\) are independent random samples from normal distributions with the same \(\sigma^{2}\), prove that their pooled sample variance, \(S_{p}^{2}\), is an unbiased estimator for \(\sigma^{2}\).

In a study designed to investigate the effects of a strong magnetic field on the early development of mice (7), ten cages, each containing three 30 -day- old albino female mice, were subjected for a period of 12 days to a magnetic field having an average strength of \(80 \mathrm{Oe} / \mathrm{cm}\). Thirty other mice, housed in ten similar cages, were not put in the magnetic field and served as controls. Listed in the table are the weight gains, in grams, for each of the twenty sets of mice. Test whether the variances of the two sets of weight gains are significantly different. Let \(\alpha=0.05\). For the mice in the magnetic field, \(s_{X}=5.67\); for the other mice, \(s_{Y}=3.18\).

Crosstown busing to compensate for de facto segregation was begun on a fairly large scale in Nashville during the \(1960 \mathrm{~s}\). Progress was made, but critics argued that too many racial imbalances were left unaddressed. Among the data cited in the early 1970 s are the following figures, showing the percentages of African-American students enrolled in a random sample of eighteen public schools (176). Nine of the schools were located in predominantly African-American neighborhoods; the other nine, in predominantly white neighborhoods. Which version of the two-sample \(t\) test, Theorem \(9.2 .2\) or the Behrens-Fisher approximation given in Theorem \(9.2 .3\), would be more appropriate for deciding whether the difference between \(35.9 \%\) and \(19.7 \%\) is statistically significant? Justify your answer.

Suppose \(H_{0}: \quad p_{X}=p_{Y}\) is being tested against \(H_{1}: p_{X} \neq p_{Y}\) on the basis of two independent sets of one hundred Bernoulli trials. If \(x\), the number of successes in the first set, is sixty and \(y\), the number of successes in the second set, is forty-eight, what \(P\)-value would be associated with the data?

A total of 8605 students are enrolled full-time at State University this semester, 4134 of whom are women. Of the 6001 students who live on campus, 2915 are women. Can it be argued that the difference in the proportion of men and women living on campus is statistically significant? Carry out an appropriate analysis. Let \(\alpha=0.05\).

A utility infielder for a National League club batted . 260 last season in three hundred trips to the plate. This year he hit \(.250\) in two hundred at- bats. The owners are trying to cut his pay for next year on the grounds that his output has deteriorated. The player argues, though, that his performances the last two seasons have not been significantly different, so his salary should not be reduced. Who is right?

Flonase is a nasal spray for diminishing nasal allergic symptoms. In clinical trials for side effects, 782 sufferers from allergic rhinitis were given a daily dose of \(200 \mathrm{mcg}\) of Flonase. Of this group, 126 reported headaches. A group of 758 subjects were given a placebo, and 111 of them reported headaches. Find a \(95 \%\) confidence interval for the difference in proportion of headaches for the two groups. Does the confidence interval suggest a statistically significant difference in the frequency of headaches for Flonase users?