Body Mass Index. Refer to Exercise 12.93

SPEAKER WOOFER DRIVER MANUFACTURING

12.99 Body Mass Index. Refer to Exercise 12.93

a. Determine and interpret a 90% conﬁdence interval for the difference between the percentages of adults in the two degree categories who have an above healthy weight.

b. Repeat part (a) for an 80% conﬁdence interval.

In each of Exercises12.100–12.102, use the technology of your choice to conduct the required analyses.

12.100 Hormone Therapy and Dementia. An issue of Sci- ence News (Vol. 163, No. 22, pp. 341–342) reported that the Women’s Health Initiativecast doubts on the beneﬁt of hormone- replacement therapy. Researchers randomly divided 4532 healthy women over the age of 65 years into two groups. One group, consisting of 2229 women, received hormone-replacement therapy; the other group, consisting of 2303 women, received placebo. Over 5 years, 40 of the women receiving the hormone- replacement therapy were diagnosed with dementia, compared with 21 of those getting placebo.

a. At the 5% signiﬁcance level, do the data provide sufﬁcient evidence to conclude that healthy women over 65 years old who take hormone-replacement therapy are at greater risk for dementia than those who do not?

b. Determine and interpret a 95% conﬁdence interval for the difference in dementia risk rates for healthy women over 65 years old who take hormone-replacement therapy and those who do not.

12.101 Women in the Labor Force. TheOrganization for Eco- nomic Cooperation and Development(OECD) summarizes data on labor-force participation rates inOECD in Figures. Indepen- dent simple random samples were taken of 300 U.S. women and 250 Canadian women. Of the U.S. women, 215 were found to be in the labor force; of the Canadian women, 186 were found to be in the labor force.

a. At the 5% signiﬁcance level, do the data suggest that there is a difference between the labor-force participation rates of U.S.

and Canadian women?

b. Find and interpret a 95% conﬁdence interval for the difference between the labor-force participation rates of U.S. and Cana- dian women.

12.102 Neutropenia. Neutropenia is an abnormally low number of neutrophils (a type of white blood cell) in the blood.

Chemotherapy often reduces the number of neutrophils to a level that makes patients susceptible to fever and infections.

G. Bucaneve et al. published a study of such cancer patients in the paper “Levofloxacin to Prevent Bacterial Infection in Pa- tients With Cancer and Neutropenia” (New England Journal of Medicine, Vol. 353, No. 10, pp. 977–987). For the study, 375 patients were randomly assigned to receive a daily dose of levofloxacin, and 363 were given placebo. In the group receiving levofloxacin, fever was present in 243 patients for the duration of neutropenia, whereas fever was experienced by 308 patients in the placebo group.

a. At the 1% significance level, do the data provide sufficient evidence to conclude that levofloxacin is effective in reducing the occurrence of fever in such patients?

b. Find a 99% conﬁdence level for the difference in the proportions of such cancer patients who would experience fever for the duration of neutropenia.

Extending the Concepts and Skills

12.103 Eating Out Vegetarian. In this exercise, apply Formu- las 12.3 and 12.4 on page 564 to the study on ordering vegetarian considered in Examples 12.8–12.10.

a. Obtain the margin of error for the estimate of the difference between the proportions of men and women who sometimes order veg by taking half the length of the conﬁdence interval found in Example 12.10 on page 568. Interpret your answer in words.

b. Obtain the margin of error for the estimate of the difference between the proportions of men and women who sometimes order veg by applying Formula 12.3.

c. Without making a guess for the observed values of the sample proportions, ﬁnd the common sample size that will ensure a margin of error of at most 0.01 for a 90% conﬁdence interval.

d. Find a 90% conﬁdence interval forp1−p2if, for samples of the size determined in part (c), 38.3% of the men and 43.7% of the women sometimes order veg.

e. Determine the margin of error for the estimate in part (d), and compare it to the required margin of error speciﬁed in part (c).

f. Repeat parts (c)–(e) if you can reasonably presume that at most 41% of the men sampled and at most 49% of the women sampled will be people who sometimes order veg.

g. Compare the results obtained in parts (c)–(e) to those obtained in part (f ).

In each of Exercises12.104–12.109, we have given the numbers of successes and the sample sizes for simple random samples for independent random samples from two populations. In each case, a. use the two-proportions plus-four z-interval procedure as dis- cussed on page 569 to ﬁnd the required conﬁdence interval for the difference between the two population proportions.

b. compare your result with the corresponding confidence inter- val found in parts (d) of Exercises 12.82–12.87, if finding such a confidence interval was appropriate.

12.104 x1 =10,n1=20, x2=18, n2=30; 80% conﬁdence interval

12.105 x1 =18,n1=40, x2=30, n2=40; 80% conﬁdence interval

12.106 x1 =14, n1 =20, x2=8, n2=20; 90% conﬁdence interval

12.107 x1 =15,n1=20, x2=18, n2=30; 90% conﬁdence interval

12.108 x1 =18,n1=30, x2=10, n2=20; 95% conﬁdence interval

12.109 x1 =30,n1=80, x2=15, n2=20; 95% conﬁdence interval

In each of Exercises 12.110–12.113, use the two-proportions plus-four z-interval procedure as discussed on page 569 to ﬁnd the required conﬁdence interval. Interpret your results.

12.110 The Afghan War. TwoUSA TODAY/Galluppolls of 979 U.S. adults each, one in November 2001 and the other in March 2009, asked “Did the United States make a mistake in sending military forces to Afghanistan?” The numbers of af- ﬁrmative responses in the two polls were 90 and 418, respectively. Determine a 95% conﬁdence interval for the difference between the percentages of all U.S. adults who, during the two time

periods, thought sending military forces to Afghanistan was a mistake.

12.111 Unemployment Rates. TheOrganization for Economic Cooperation and Development(OECD) conducts studies on unemployment rates by country and publishes its ﬁndings in the documentMain Economic Indicators.Independent random samples of 100 and 75 people in the civilian labor forces of Finland and Denmark, respectively, revealed 7 and 3 unemployed, respectively, Find a 95% conﬁdence interval for the difference between the unemployment rates in Finland and Denmark.

12.112 Federal Gas Tax. TheQuinnipiac University Pollcon- ducts nationwide surveys as a public service and for research. In one poll, participants were asked whether they thought eliminating the federal gas tax for the summer months is a good idea. The following problems are based on the results of that poll.

a. Of 611 Republicans, 275 thought it a good idea, and, of 872 Democrats, 366 thought it a good idea. Obtain a 90% con- ﬁdence interval for the difference between the proportions of Republicans and Democrats who think that eliminating the federal gas tax for the summer months is a good idea.

b. Of 907 women, 417 thought it a good idea, and, of 838 men, 310 thought it a good idea. Obtain a 90% conﬁdence interval for the difference between the percentages of women and men who think that eliminating the federal gas tax for the summer months is a good idea.

12.113 Blockers and Cancer. AWall Street Journalarticle, ti- tled “Hypertension Drug Linked to Cancer,” reported on a study of several types of high-blood-pressure drugs and links to cancer. For one type, called calcium-channel blockers, 27 of 202 elderly patients taking the drug developed cancer. For another type, called beta-blockers, 28 of 424 other elderly patients developed cancer. Find a 90% conﬁdence interval for the difference between the cancer rates of elderly people taking calcium-channel blockers and those taking beta-blockers.Note:The results of this study were challenged and questioned by several sources that claimed, for example, that the study was ﬂawed and that several other studies have suggested that calcium-channel blockers are safe.

CHAPTER IN REVIEW

You Should Be Able to

1. use and understand the formulas in this chapter.

2. ﬁnd a large-sample conﬁdence interval for a population proportion.

3. compute the margin of error for the estimate of a population proportion.

4. understand the relationship between the sample size, conﬁ- dence level, and margin of error for a conﬁdence interval for a population proportion.

5. determine the sample size required for a speciﬁed conﬁdence level and margin of error for the estimate of a population proportion.

6. perform a large-sample hypothesis test for a population proportion.

7. perform large-sample inferences (hypothesis tests and conﬁ- dence intervals) to compare two population proportions.

8. understand the relationship between the sample sizes, conﬁ- dence level, and margin of error for a conﬁdence interval for the difference between two population proportions.

9. determine the sample sizes required for a speciﬁed conﬁ- dence level and margin of error for the estimate of the difference between two population proportions.

576 CHAPTER 12 Inferences for Population Proportions

Key Terms

margin of error,549, 568 number of failures,546 number of successes,546

one-proportion plus-fourz-interval procedure,551

one-proportionz-interval procedure,548

one-proportionz-test,558

pooled sample proportion (pˆp),565 population proportion (p),546 sample proportion (p),ˆ 546 sampling distribution of the

difference between two sample proportions,564

sampling distribution of the sample proportion,547

two-proportions plus-fourz-interval procedure,569

two-proportionsz-interval procedure,567

two-proportionsz-test,565

REVIEW PROBLEMS

Understanding the Concepts and Skills

1. Medical Marijuana? AHarris Pollwas conducted to estimate the proportion of Americans who feel that marijuana should be legalized for medicinal use in patients with cancer and other painful and terminal diseases. Identify the

a. speciﬁed attribute. b. population.

c. population proportion.

d. According to the poll, 80% of the 83,957 respondents said that marijuana should be legalized for medicinal use. Is the proportion 0.80 (80%) a sample proportion or a population proportion? Explain your answer.

2. Why is a sample proportion generally used to estimate a population proportion instead of obtaining the population proportion directly?

3. Explain what each phrase means in the context of inferences for a population proportion.

a. Number of successes b. Number of failures 4. Fill in the blanks.

a. The mean of all possible sample proportions is equal to

the .

b. For large samples, the possible sample proportions have approximately a distribution.

c. A rule of thumb for using a normal distribution to approxi- mate the distribution of all possible sample proportions is that

both and are or greater.

5. What does the margin of error for the estimate of a population proportion tell you?

6. Holiday Blues. A poll was conducted byOpinion Research Corporationto estimate the proportions of men and women who get the “holiday blues.” Identify the

a. speciﬁed attribute. b. two populations.

c. two population proportions.

d. two sample proportions.

e. According to the poll, 34% of men and 44% of women get the “holiday blues.” Are the proportions 0.34 and 0.44 sample proportions or population proportions? Explain your answer.

7. Suppose that you are using independent samples to compare two population proportions. Fill in the blanks.

a. The mean of all possible differences between the two sample proportions equals the .

b. For large samples, the possible differences between the two sample proportions have approximately a distribution.

8. Smallpox Vaccine. ABCNEWS.compublished the results of a poll that asked U.S. adults whether they would get a smallpox shot if it were available. Sampling, data collection, and tabulation were done byTNS Intersearchof Horsham, Pennsylvania. When the risk of the vaccine was described in detail, 4 in 10 of those surveyed said they would take the smallpox shot. According to the article, “the results have a three-point margin of error” (for a 0.95 conﬁdence level). Use the information provided to obtain a 95% conﬁdence interval for the percentage of all U.S. adults who would take a smallpox shot, knowing the risk of the vaccine.

9. Suppose that you want to ﬁnd a 95% conﬁdence interval based on independent samples for the difference between two population proportions and that you want a margin of error of at most 0.01.

a. Without making an educated guess for the observed sample proportions, ﬁnd the required common sample size.

b. Suppose that, from past experience, you are quite sure that the two sample proportions will be 0.75 or greater. What common sample size should you use?

10. Getting a Job. TheNational Association of Colleges and Employerssponsors theGraduating Student and Alumni Survey.

Part of the survey gauges student optimism in landing a job after graduation. According to one year’s survey results, published in American Demographics, among the 1218 respondents, 733 said that they expected difficulty finding a job. Use these data to obtain and interpret a 95% confidence interval for the proportion of students who expect difficulty finding a job.

11. Getting a Job. Refer to Problem 10.

a. Find the margin of error for the estimate ofp.

b. Obtain a sample size that will ensure a margin of error of at most 0.02 for a 95% conﬁdence interval without making a guess for the observed value of p.ˆ

c. Find a 95% confidence interval for pif, for a sample of the size determined in part (b), 58.7% of those surveyed say that they expect difficulty finding a job.

d. Determine the margin of error for the estimate in part (c), and compare it to the required margin of error speciﬁed in part (b).

e. Repeat parts (b)–(d) if you can reasonably presume that the percentage of those surveyed who say that they expect difﬁ- culty ﬁnding a job will be at least 56%.

f. Compare the results obtained in parts (b)–(d) with those obtained in part (e).

12. Justice in the Courts? In an issue ofParade Magazine, the editors reported on a national survey on law and order. One ques- tion asked of the 2512 U.S. adults who took part was whether they believed that juries “almost always” convict the guilty and free the innocent. Only 578 said that they did. At the 5% significance level, do the data provide sufﬁcient evidence to conclude that less than one in four Americans believe that juries “almost always” convict the guilty and free the innocent?

13. Height and Breast Cancer. In the article “Height and Weight at Various Ages and Risk of Breast Cancer” (Annals of Epidemiology, Vol. 2, pp. 597–609), L. Brinton and C. Swanson discussed the relationship between height and breast cancer. The study, sponsored by theNational Cancer Institute, took 5 years and involved more than 1500 women with breast cancer and 2000 women without breast cancer; it revealed a trend between height and breast cancer: “. . . taller women have a 50 to 80 per- cent greater risk of getting breast cancer than women who are closer to 5 feet tall.” Christine Swanson, a nutritionist who was involved with the study, added, “. . . height may be associated with the culprit, . . . but no one really knows” the exact relationship between height and the risk of breast cancer.

a. Classify this study as either an observational study or a de- signed experiment. Explain your answer.

b. Interpret the statement made by Christine Swanson in light of your answer to part (a).

14. Views on the Economy. State and local governments often poll their constituents about their views on the economy. In two polls taken approximately 1 year apart,O’Neil Associatesasked 600 Maricopa County, Arizona, residents whether they thought the state’s economy would improve over the next 2 years. In the first poll, 48% said “yes”; in the second poll, 60% said “yes.” At the 1% significance level, do the data provide sufficient evidence to conclude that the percentage of Maricopa County residents who thought the state’s economy would improve over the next 2 years was less during the time of the first poll than during the time of the second?

15. Views on the Economy. Refer to Problem 14.

a. Determine a 98% conﬁdence interval for the difference, p1−p2, between the proportions of Maricopa County residents who thought that the state’s economy would improve over the next 2 years during the time of the ﬁrst poll and during the time of the second poll.

b. Interpret your answer from part (a).

16. Views on the Economy. Refer to Problems 14 and 15.

a. Take half the length of the conﬁdence interval found in Prob- lem 15(a) to obtain the margin of error for the estimate of the difference between the two population proportions. Interpret your result in words.

b. Solve part (a) by applying Formula 12.3 on page 564.

c. Obtain the common sample size that will ensure a margin of error of at most 0.03 for a 98% conﬁdence interval without making a guess for the observed values of the sample proportions.

d. Find a 98% conﬁdence interval for p1−p2 if, for samples of the size determined in part (c), the sample proportions are 0.475 and 0.603, respectively.

e. Determine the margin of error for the estimate in part (d) and compare it to the required margin of error speciﬁed in part (c).

17. Bulletproof Vests. In theNew York Timesarticle “A Com- mon Police Vest Fails the Bulletproof Test,” E. Lichtblau reported on aU.S. Department of Justicestudy of 103 bulletproof vests containing a ﬁber known as Zylon. In ballistics tests, only 4 of these vests produced acceptable safety outcomes (and resulted in immediate changes in federal safety guidelines). Find a 95% con- ﬁdence interval for the proportion of all such vests that would produce acceptable safety outcomes by using the

a. one-proportionz-interval procedure.

b. one-proportion plus-fourz-interval procedure.

c. Explain the large discrepancy between the two methods.

d. Which conﬁdence interval would you use? Explain your answer.

In each of Problems18–21, use the technology of your choice to conduct the required analyses.

18. March Madness. The NCAA Men’s Division I Basketball Championship is held each spring and features 65 college basketball teams. This 20-day tournament is colloquially known as

“March Madness.” AHarris Pollasked 2435 randomly selected U.S. adults whether they would participate in an office pool for March Madness; 317 said they would. Use these data to find and interpret a 95% confidence interval for the percentage of U.S. adults who would participate in an office pool for March Madness.

19. Abstinence and AIDS. In aHarris Pollof 1961 randomly selected U.S. adults, 1137 said that they do not believe that abstinence programs are effective in reducing or preventing AIDS. At the 5% signiﬁcance level, do the data provide sufﬁcient evidence to conclude that a majority of all U.S. adults feel that way?

20. Bug Buster. N. Hill et al. conducted a clinical study to compare the standard treatment for head lice infestation with the Bug Buster kit, which involves using a ﬁne-toothed comb on thor- oughly wet hair four times at 4-day intervals. The researchers published their ﬁndings in the paper “Single Blind, Ran- domised, Comparative Study of the Bug Buster Kit and over the Counter Pediculicide Treatments against Head Lice in the United Kingdom” (British Medical Journal, (Vol. 331, pp. 384–387).

For the study, 56 patients were randomly assigned to use the Bug Buster kit and 70 were assigned to use the standard treatment. Thirty-two patients in the Bug Buster kit group were cured, whereas nine of those in the standard treatment group were cured.

a. At the 5% signiﬁcance level, do these data provide sufﬁcient evidence to conclude that a difference exists in the cure rates of the two types of treatment?

b. Determine a 95% conﬁdence interval for the difference in cure rates for the two types of treatment.

21. Finasteride and Prostate Cancer. In the article “The Influ- ence of Finasteride on the Development of Prostate Cancer” (New England Journal of Medicine, Vol. 349, No. 3, pp. 215–224), I. Thompson et al. reported the results of a major study to ex- amine the effect of finasteride in reducing the risk of prostate cancer. The study, known as the Prostate Cancer Prevention Trial (PCPT), was sponsored by theU.S. Public Health Service and theNational Cancer Institute. In the PCPT trial, 18,882 men 55 years old or older with normal physical exams and prostate- specific antigen (PSA) levels of 3.0 nanograms per milliliter or lower were randomly assigned to receive 5 milligrams of finasteride daily or placebo. At 7 years, of the 9060 men included in the final analysis, 4368 had taken finasteride and 4692 had received placebo. For those who took finasteride, 803 cases of

Body Mass Index. Refer to Exercise 12.93

The critical value for a two-tailed test

Inferences for Two Population Means, Using Paired Samples