Chapter 9

As the United States has struggled with the growing obesity of its citizens, diets have become big business. Among the many competing regimens for those seeking weight reduction are the Atkins and Zone diets. In a comparison of these two diets for one-year weight loss, a study (64) found that seventy- seven subjects on the Atkins diet had an average weight loss of \(\bar{x}=-4.7 \mathrm{~kg}\) and a sample standard deviation of \(s_{X}=7.05 \mathrm{~kg}\). Similar figures for the seventy-nine people on the Zone diet were \(\bar{y}=-1.6 \mathrm{~kg}\) and \(s_{Y}=5.36 \mathrm{~kg}\). Is the greater reduction with the Atkins diet statistically significant? Test for \(\alpha=0.05\).

A medical researcher believes that women typically have lower serum cholesterol than men. To test this hypothesis, he took a sample of four hundred seventy-six men between the ages of nineteen and forty-four and found their mean serum cholesterol to be \(189.0 \mathrm{mg} / \mathrm{dl}\) with a sample standard deviation of \(34.2\). A group of five hundred ninety-two women in the same age range averaged \(177.2 \mathrm{mg} / \mathrm{dl}\) and had a sample standard deviation of \(33.3\). Is the lower average for the women statistically significant? Set \(\alpha=0.05\). Assume the variances are equal.

The University of Missouri-St. Louis gave a validation test to entering students who had taken calculus in high school. The group of ninety-three students receiving no college credit had a mean score of \(4.17\) on the validation test with a sample standard deviation of \(3.70\). For the twenty- eight students who received credit from a high school dual-enrollment class, the mean score was \(4.61\) with a sample standard deviation of \(4.28\). Is there a significant difference in these means at the \(\alpha=0.01\) level? Assume the variances are equal.

Ring Lardner was one of this country's most popular writers during the \(1920 \mathrm{~s}\) and \(1930 \mathrm{~s}\). He was also a chronic alcoholic who died prematurely at the age of forty-eight. The following table lists the life spans of some of Lardner's contemporaries (39). Those in the sample on the left were all problem drinkers; they died, on the average, at age sixty-five. The twelve (sober) writers on the right tended to live a full ten years longer. Can it be argued that an increase of that magnitude is statistically significant? Test an appropriate null hypothesis against a one-sided \(H_{1}\). Use the \(0.05\) level of significance. (Note: The pooled sample standard deviation for these two samples is 13.9.)

A company markets two brands of latex paint regular and a more expensive brand that claims to dry an hour faster. A consumer magazine decides to test this claim by painting ten panels with each product. The average drying time of the regular brand is \(2.1\) hours with a sample standard deviation of 12 minutes. The fast-drying version has an average of \(1.6\) hours with a sample standard deviation of 16 minutes. Test the null hypothesis that the more expensive brand dries an hour quicker. Use a onesided \(H_{1}\). Let \(\alpha=0.05\).

(a) Suppose \(H_{0}: \mu_{X}=\mu_{Y}\) is to be tested against \(H_{1}: \mu_{X} \neq \mu_{Y}\). The two sample sizes are 6 and 11. If \(s_{p}=\) \(15.3\), what is the smallest value for \(|\bar{x}-\bar{y}|\) that will result in \(H_{0}\) being rejected at the \(\alpha=0.01\) level of significance? (b) What is the smallest value for \(\bar{x}-\bar{y}\) that will lead to the rejection of \(H_{0}: \mu_{X}=\mu_{Y}\) in favor of \(H_{1}: \mu_{X}>\mu_{Y}\) if \(\alpha=0.05, s_{P}=214.9, n=13\), and \(m=8\) ?

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?

An executive has two routes that she can take to and from work each day. The first is by interstate; the second requires driving through town. On the average it takes her 33 minutes to get to work by the interstate and 35 minutes by going through town. The standard deviations for the two routes are 6 and 5 minutes, respectively. Assume the distributions of the times for the two routes are approximately normally distributed. (a) What is the probability that on a given day, driving through town would be the quicker of her choices? (b) What is the probability that driving through town for an entire week (ten trips) would yield a lower average time than taking the interstate for the entire week?

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 \(\sigma^{2}\).

Let \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{m}\) be independent random samples drawn from normal distributions with means \(\mu_{X}\) and \(\mu_{Y}\), respectively, and with the same known variance \(\sigma^{2}\). Use the generalized likelihood ratio criterion to derive a test procedure for choosing between \(H_{0}: \mu_{X}=\mu_{Y}\) and \(H_{1}: \mu_{X} \neq \mu_{Y}\).

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.

Let \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{m}\) be independent random samples from normal distributions with means \(\mu_{X}\) and \(\mu_{Y}\) and standard deviations \(\sigma_{X}\) and \(\sigma_{Y}\), respectively, where \(\mu_{X}\) and \(\mu_{Y}\) are known. Derive the GLRT for \(H_{0}: \sigma_{X}^{2}=\sigma_{Y}^{2}\) versus \(H_{1}: \sigma_{X}^{2}>\sigma_{Y}^{2}\).

The phenomenon of handedness has been extensively studied in human populations. The percentages of adults who are right-handed, left-handed, and ambidextrous are well documented. What is not so well known is that a similar phenomenon is present in lower animals. Dogs, for example, can be either right-pawed or left-pawed. Suppose that in a random sample of two hundred beagles, it is found that fifty-five are left-pawed and that in a random sample of two hundred collies, forty are left-pawed. Can we conclude that the difference in the two sample proportions of left-pawed dogs is statistically significant for \(\alpha=0.05 ?\)

Water witching, the practice of using the movements of a forked twig to locate underground water (or minerals), dates back over four hundred years. Its first detailed description appears in Agricola's De re Metallica, published in 1556. That water witching works remains a belief widely held among rural people in Europe and throughout the Americas. [In 1960 the number of "active" water witches in the United States was estimated to be more than 20,000 (205).] Reliable evidence supporting or refuting water witching is hard to find. Personal accounts of isolated successes or failures tend to be strongly biased by the attitude of the observer. Of all the wells dug in Fence Lake, New Mexico, twenty-nine "witched" wells and thirty-two "nonwitched" wells were sunk. Of the "witched"' wells, twenty-four were successful. For the "nonwitched"' wells, there were twentyseven successes. What would you conclude?

In some criminal cases, the judge and the defendant's lawyer will enter into a plea bargain, where the accused pleads guilty to a lesser charge. The proportion of time this happens is called the mitigation rate. A Florida Corrections Department study showed that Escambia County had the state's fourth highest rate, \(61.7 \%\) (1033 out of 1675 cases). Concerned that the guilty were not getting appropriate sentences, the state attorney put in new policies to limit the number of plea bargains. A followup study (143) showed that the mitigation rate dropped to \(52.1 \%\) (344 out of 660 cases). Is it fair to conclude that the drop was due to the new policies, or can the decline be written off to chance? Test at the \(\alpha=0.01\) level.

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 \(\alpha=0.05\).

The kittiwake is a seagull whose mating behavior is basically monogamous. Normally, the birds separate for several months after the completion of one breeding season and reunite at the beginning of the next. Whether or not the birds actually do reunite, though, may be affected by the success of their "relationship" the season before. A total of seven hundred sixty-nine kittiwake pairbonds were studied (33) over the course of two breeding seasons; of those seven hundred sixty-nine, some six hundred nine successfully bred during the first season; the remaining one hundred sixty were unsuccessful. The following season, one hundred seventy-five of the previously successful pair-bonds "divorced," as did one hundred of the one hundred sixty whose prior relationship left something to be desired. Can we conclude that the difference in the two divorce rates \((29 \%\) and \(63 \%)\) is statistically significant?

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?

Suppose that \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{m}\) are independent random samples from normal distributions with means \(\mu_{X}\) and \(\mu_{Y}\) and known standard deviations \(\sigma_{X}\) and \(\sigma_{Y}\), respectively. Derive a \(100(1-\alpha) \%\) confidence interval for \(\mu_{X}-\mu_{Y}\).

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?