4025_ch08_p398-443.pdf

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I n f e r e n c e s f r o m Tw o S a m p l e s
Using Technology
STATDISK
Select Analysis from the main menu bar, then
number of trials for Sample 1, in cell C1 enter the number of suc- select Hypothesis Testing, then Proportion-Two Samples. Enter
cesses for Sample 2, and in cell D1 enter the number of trials for the required items in the dialog box. Confidence interval limits Sample 2. Click on DDXL. Select Hypothesis Tests and Summ
are included with the hypothesis test results.
2 Var Prop Test or select Confidence Intervals and Summ 2
Var Prop Interval. In the dialog box, click on the four pencil
Minitab can now handle summary statistics for icons and enter A1, B1, C1, and D1 in the four input boxes. Click two samples. Select Stat from the main menu bar, then select
OK. Proceed to complete the new dialog box.
Basic Statistics, then 2 Proportions. Click on the button for
Summarize data. Click on the Options bar. Enter the desired
TI-83 Plus
The TI-83 Plus calculator can be used for hy- confidence level, enter the claimed value of p 2 pothesis tests and confidence intervals. Press STAT and select
format for the alternative hypothesis, and click on the box to use TESTS. Then choose the option of 2-PropZTest (for a hypothe-
the pooled estimate of p for the test. Click OK twice.
sis test) or 2-PropZInt (for a confidence interval). When testing
hypotheses, the TI-83 Plus calculator will display a P-value in-
You must use the Data Desk XL add-in, which stead of critical values, so the P-value method of testing hypothe- is a supplement to this book. First make these entries: In cell A1 enter the number of successes for Sample 1, in cell B1 enter the 8-2 Basic Skills and Concepts
Finding Number of Successes. In Exercises 1– 4, find the number of successes x sug-
gested by the given statement.
1. From the Arizona Department of Weights and Measures: Among 37 inspections at
2. From the New York Times: Among 240 vinyl gloves subjected to stress tests, 63%
3. From Sociological Methods and Research: When 294 central-city residents were sur-
4. From a Time>CNN survey: 24% of 205 single women said that they “definitely want
Calculations for Testing Claims. In Exercises 5 and 6, assume that you plan to use a sig-
nificance level of a 5 0.05 to test the claim that p 5
p2. Use the given sample sizes and numbers of successes to find (a) the pooled estimate p, (b) the z test statistic, (c) the criti-cal z values, and (d) the P-value. 5. Workers
6. Low Activity
7. E-Mail and Privacy A survey of 436 workers showed that 192 of them said that it was
seriously unethical to monitor employee e-mail. When 121 senior-level bosses weresurveyed, 40 said that it was seriously unethical to monitor employee e-mail (based Copyright 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley Copyright 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley 4025_CH08_p398-443 01/02/04 12:34 PM Page 409 I n f e r e n c e s A b o u t Tw o P r o p o r t i o n s
on data from a Gallup poll). Use a 0.05 significance level to test the claim that forthose saying that monitoring e-mail is seriously unethical, the proportion of employ-ees is greater than the proportion of bosses.
8. E-Mail and Privacy Refer to the sample data given in Exercise 7 and construct a 90%
confidence interval estimate of the difference between the two population propor-tions. Is there a substantial gap between the employees and bosses? 9. Exercise and Coronary Heart Disease In a study of women and coronary heart dis-
ease, the following sample results were obtained: Among 10,239 women with a lowlevel of physical activity (less than 200 kcal>wk), there were 101 cases of coronaryheart disease. Among 9877 women with physical activity measured between 200 and600 kcal>wk, there were 56 cases of coronary heart disease (based on data from“Physical Activity and Coronary Heart Disease in Women” by Lee, Rexrode, et al.,Journal of the American Medical Association, Vol. 285, No. 11). Construct a 90%confidence interval estimate for the difference between the two proportions. Does thedifference appear to be substantial? Does it appear that physical activity correspondsto a lower rate of coronary heart disease? 10. Exercise and Coronary Heart Disease Refer to the sample data in Exercise 9 and use a
0.05 significance level to test the claim that the rate of coronary heart disease is higherfor women with the lower levels of physical activity. What does the conclusion suggest? 11. Instant Replay in Football In the 2000 football season, 247 plays were reviewed by
officials using instant video replays, and 83 of them resulted in reversal of the originalcall. In the 2001 football season, 258 plays were reviewed and 89 of them were re-versed (based on data from “Referees Turn to Video Aid More Often” by RichardSandomir, New York Times). Is there a significant difference in the two reversal rates?Does it appear that the reversal rate was the same in both years? 12. Effectiveness of Smoking Bans The Joint Commission on Accreditation of Health-
care Organizations mandated that hospitals ban smoking by 1994. In a study of theeffects of this ban, subjects who smoke were randomly selected from two differentpopulations. Among 843 smoking employees of hospitals with the smoking ban, 56quit smoking one year after the ban. Among 703 smoking employees from work-places without a smoking ban, 27 quit smoking a year after the ban (based on datafrom “Hospital Smoking Bans and Employee Smoking Behavior” by Longo, Brown-son, et al., Journal of the American Medical Association, Vol. 275, No. 16). Is there asignificant difference between the two proportions at a 0.05 significance level? Isthere a significant difference between the two proportions at a 0.01 significance level?Does it appear that the ban had an effect on the smoking quit rate? 13. Testing Effectiveness of Vaccine In a USA Today article about an experimental nasal
spray vaccine for children, the following statement was presented: “In a trial involv-ing 1602 children only 14 (1%) of the 1070 who received the vaccine developed theflu, compared with 95 (18%) of the 532 who got a placebo.” The article also referredto a study claiming that the experimental nasal spray “cuts children’s chances of get-ting the flu.” Is there sufficient sample evidence to support the stated claim? 14. Color Blindness in Men and Women In a study of red>green color blindness, 500
men and 2100 women are randomly selected and tested. Among the men, 45 havered>green color blindness. Among the women, 6 have red>green color blindness(based on data from USA Today).
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a. Is there sufficient evidence to support the claim that men have a higher rate of red>
green color blindness than women? Use a 0.01 significance level.
b. Construct the 98% confidence interval for the difference between the color blind-
ness rates of men and women. Does there appear to be a substantial difference? c. Why would the sample size for women be so much larger than the sample size
15. Seat Belts and Hospital Time A study was made of 413 children who were hospital-
ized as a result of motor vehicle crashes. Among 290 children who were not usingseat belts, 50 were injured severely. Among 123 children using seat belts, 16 were in-jured severely (based on data from “Morbidity Among Pediatric Motor Vehicle CrashVictims: The Effectiveness of Seat Belts,” by Osberg and Di Scala, American Journalof Public Health, Vol. 82, No. 3). Is there sufficient sample evidence to conclude, atthe 0.05 significance level, that the rate of severe injuries is lower for children wear-ing seat belts? Based on these results, what action should be taken? 16. Drinking and Crime Karl Pearson, who developed many important concepts in statis-
tics, collected crime data in 1909. Of those convicted of arson, 50 were drinkers and43 abstained. Of those convicted of fraud, 63 were drinkers and 144 abstained. Use a0.01 significance level to test the claim that the proportion of drinkers among con-victed arsonists is greater than the proportion of drinkers among those convicted offraud. Does it seem reasonable that drinking might have had an effect on the type ofcrime? Why? 17. Interpreting a Computer Display A U.S. Department of Justice report (NCJ-156831)
included the claim that “in spouse murder cases, wife defendants were less likely tobe convicted than husband defendants.” Sample data consisted of 277 convictionsamong 318 husband defendants, and 155 convictions among 222 wife defendants.
Test the stated claim and identify one possible explanation for the result. The Minitabresults are shown here.
0.871069
TI-83 Plus
0.698198
Estimate for p(1) 2 p(2): 0.172871
95% lower bound for p(1) 2 p(2): 0.113511
Test for p(1) 2 p(2) 5 0 (vs . 0): Z 5 4.94 P-value 5 0.000
18. Effectiveness of Salk Vaccine for Polio In initial tests of the Salk vaccine, 33 of
200,000 vaccinated children later developed polio. Of 200,000 children vaccinatedwith a placebo, 115 later developed polio. The TI-83 Plus calculator display is shownhere. At the 0.01 significance level, test the claim that the Salk vaccine is effective inlowering the polio rate. Does it appear that the vaccine is effective? 19. Failed Inspections When conducting tests of auto parts stores, the Arizona Depart-
ment of Weights and Measures conducted 100 inspections of Autozone stores andfound that 63% of those inspections failed. Among 37 inspections at NAPA AutoParts stores, 81% failed. Use a 0.05 significance level to determine whether there is asignificant difference between those two rates of failures. Does it appear that eitherstore is a better choice for consumers? Copyright 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley 4025_CH08_p398-443 01/02/04 12:34 PM Page 411 I n f e r e n c e s A b o u t Tw o P r o p o r t i o n s
20. Airline Load Factor In a recent year, Southwest Airlines had 3,131,727 aircraft seats
available on all of its flights, and 2,181,604 of them were occupied by passengers.
America West had 2,091,859 seats available, and 1,448,255 of them were occupied.
The percentage of seats occupied is called the load factor, so these results show that
the load factor is 69.7% (rounded) for Southwest Airlines and 69.2% (rounded) for
America West. (The data are from the U.S. Department of Transportation.) Answer
the following by assuming that the results are from randomly selected samples.
a. Test the claim that both airlines have the same load factor.
b. Given that 69.7% and 69.2% appear to be so obviously close, how do you explain
c. Generalize the key point of this example by completing the following statement:
“If two sample sizes are extremely large, even seemingly small differences in sam-ple proportions . . . ” 21. Attitudes Toward Marriage In a Time>CNN survey, 24% of 205 single women said
that they “definitely want to get married.” In the same survey, 27% of 260 single mengave that same response. Construct a 99% confidence interval estimate of the differ-ence between the proportions of single women and single men who definitely want toget married. Is there a gender gap on this issue? 22. Attitudes Toward Marriage Refer to the same sample data in Exercise 21 and use a
0.01 significance level to test the claim that there is a difference between the propor-tion of men and the proportion of women who definitely want to get married. Doesthere appear to be a difference? 23. Violent Crime and Age Group The newly appointed head of the state mental health
agency claims that a smaller proportion of the crimes committed by persons youngerthan 21 years of age are violent crimes (when compared to the crimes committed bypersons 21 years of age or older). Of 2750 randomly selected arrests of criminalsyounger than 21 years of age, 4.25% involve violent crimes. Of 2200 randomly se-lected arrests of criminals 21 years of age or older, 4.55% involve violent crimes(based on data from the Uniform Crime Reports). Construct a 95% confidence inter-val for the difference between the two proportions of violent crimes. Does the confi-dence interval indicate that there isn’t a significant difference between the two ratesof violent crimes? 24. Testing Laboratory Gloves The New York Times ran an article about a study in which
Professor Denise Korniewicz and other Johns Hopkins researchers subjected labora-tory gloves to stress. Among 240 vinyl gloves, 63% leaked viruses. Among 240 latexgloves, 7% leaked viruses. At the 0.005 significance level, test the claim that vinylgloves have a larger virus leak rate than latex gloves.
25. Written Survey and Computer Survey In a study of 1700 teens aged 15–19, half were
given written surveys and half were given surveys using an anonymous computer pro-
gram. Among those given the written surveys, 7.9% say that they carried a gun within
the last 30 days. Among those given the computer surveys, 12.4% say that they car-
ried a gun within the last 30 days (based on data from the Urban Institute).
a. The sample percentages of 7.9% and 12.4% are obviously not equal, but is the dif-
b. Construct a 99% confidence interval estimate of the difference between the two
population percentages, and interpret the result.
26. Adverse Drug Reactions The drug Viagra has become quite well known, and it has
had a substantial economic impact on its producer, Pfizer Pharmaceuticals. In prelim- Copyright 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley 4025_CH08_p398-443 01/02/04 12:34 PM Page 412 C H A P T E R 8
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inary tests for adverse reactions, it was found that when 734 men were treated with
Viagra, 16% of them experienced headaches. (There’s some real irony there.) Among
725 men in a placebo group, 4% experienced headaches (based on data from Pfizer
Pharmaceuticals).
a. Using a 0.01 significance level, is there sufficient evidence to support the claim
that among those men who take Viagra, headaches occur at a rate that is greaterthan the rate for those who do not take Viagra? b. Construct a 99% confidence interval estimate of the difference between the rate of
headaches among Viagra users and the headache rate for those who are given aplacebo. What does the confidence interval suggest about the two rates? 27. Poll Refusal Rate Professional pollsters are becoming concerned about the growing
rate of refusals among potential survey subjects. In analyzing the problem, there is aneed to know if the refusal rate is universal or if there is a difference between the ratesfor central-city residents and those not living in central cities. Specifically, it was foundthat when 294 central-city residents were surveyed, 28.9% refused to respond. A sur-vey of 1015 residents not living in a central city resulted in a 17.1% refusal rate (basedon data from “I Hear You Knocking But You Can’t Come In,” by Fitzgerald and Fuller,Sociological Methods and Research, Vol. 11, No. 1). At the 0.01 significance level, testthe claim that the central-city refusal rate is the same as the refusal rate in other areas.
28. Home Field Advantage When games were sampled from throughout a season, it was
found that the home team won 127 of 198 professional basketball games, and thehome team won 57 of 99 professional football games (based on data from “PredictingProfessional Sports Game Outcomes from Intermediate Game Scores,” by Cooper etal., Chance, Vol. 5, No. 3–4). Construct a 95% confidence interval for the differencebetween the proportions of home wins. Does there appear to be a significant differ-ence between the proportions of home wins? What do you conclude about the homefield advantage? 29. Alcohol and Tobacco in Children’s Movies Test the claim that the proportion of 25 of
50 randomly selected children’s movies showing some use of alcohol is significantlyless than the sample proportion of 28 of 50 other such movies showing some use oftobacco. Do the results apply to Data Set 7? 30. Health Survey Refer to Data Set 1 in Appendix B and use the sample data to test the
claim that the proportion of men over the age of 30 is equal to the proportion ofwomen over the age of 30.
8-2 Beyond the Basics
31. Interpreting Overlap of Confidence Intervals In the article “On Judging the Signifi-
cance of Differences by Examining the Overlap Between Confidence Intervals” by
Schenker and Gentleman (The American Statistician, Vol. 55, No. 3), the authors con-
sider sample data in this statement: “Independent simple random samples, each of
size 200, have been drawn, and 112 people in the first sample have the attribute,
whereas 88 people in the second sample have the attribute.”
a. Use the methods of this section to construct a 95% confidence interval estimate
p2. What does the result suggest about the equality of p1 b. Use the methods of Section 6-2 to construct individual 95% confidence interval es-
timates for each of the two population proportions. After comparing the overlap Copyright 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley 4025_CH08_p398-443 01/02/04 12:34 PM Page 413 I n f e r e n c e s A b o u t Tw o M e a n s : I n d e p e n d e n t S a m p l e s
between the two confidence intervals, what do you conclude about the equality ofp1 and p2? c. Use a 0.05 significance level to test the claim that the two population proportions
d. Based on the preceding results, what should you conclude about equality of p1 and
p2? Which of the three preceding methods is least effective in testing for equalityof p1 and p2? 32. Equivalence of Hypothesis Test and Confidence Interval Two different simple ran-
dom samples are drawn from two different populations. The first sample consists of20 people with 10 having a common attribute. The second sample consists of 2000people with 1404 of them having the same common attribute. Compare the resultsfrom a hypothesis test of p 5 p2 (with a 0.05 significance level) and a 95% confi- 33. Same Proportions with Larger Samples This section used the sample data in Table
p2 and to construct a confidence interval estimate of p2. How are the results affected if the sample data in Table 8-1 are modified so 1 becomes 240 2000 instead of 24 200, and p2 becomes 1470 14,000 instead of 147>1400? Note that both sample proportions remain the same, but the sample sizesare larger. Is there now sufficient evidence to support the claim that the proportion ofblack drivers stopped by the police is greater than the proportion of white drivers whoare stopped? 34. Testing for Constant Difference To test the null hypothesis that the difference between
two population proportions is equal to a nonzero constant c, use the test statistic pˆ1s1 2 pˆ1d 1 pˆ2s1 2 pˆ2d As long as n1 and n2 are both large, the sampling distribution of the test statistic z willbe approximately the standard normal distribution. Refer to Exercise 26 and use a0.05 significance level to test the claim that the headache rate of Viagra users is 10percentage points more than the percentage for those who are given a placebo.
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