Testing Claims about Proportions. In

Determining Sample Size The sample size needed to estimate the difference between two population proportions to within a margin of error E with a confidence level of \(1-\alpha\) can be found by using the following expression: \(\mathrm {E=z\ \alpha}\ /\text{ 2 p 1 q 1 n 1+p 2 q 2 n 2}\) Replace n 1 and n 2 by n in the formula above (assuming that both samples have the same size) and replace each of p 1 , q 1 , p 2 , and q 2 by 0.5 (because their values are not known). Solving for n results in this expression: \(\mathrm {n=z\ \alpha}\ /\text{ 2 2 2 E 2}\) Use this expression to find the size of each sample if you want to estimate the difference between the proportions of adult men and women who are college graduates. Assume that you want 90% confidence that your error is no more than 0.02. ________________ Equation Transcription: Text Transcription: 1-alpha E = z alpha / 2 p 1 q 1 n 1 + p 2 q 2 n 2 n = z alpha / 2 2 2 E 2

Problem 1BSC Verifying Requirements In the largest clinical trial ever conducted, 401,974 children were randomly assigned to two groups. The treatment group consisted of 201,229 children given the Salk vaccine for polio, and the other 200,745 children were given a placebo. Among those in the treatment group, 33 developed polio, and among those in the placebo group, 115 developed polio. If we want to use the methods of this section to test the claim that the rate of polio is less for children given the Salk vaccine, are the requirements for a hypothesis test satisfied? Explain.

Problem 3BSC Hypotheses and Conclusions Refer to the hypothesis test described in Exercise. a. Identify the null hypothesis and the alternative hypothesis. ________________ b. If the P-value for the test is reported as “less than 0.001,” what should we conclude about the original claim? Exercise Verifying Requirements In the largest clinical trial ever conducted, 401,974 children were randomly assigned to two groups. The treatment group consisted of 201,229 children given the Salk vaccine for polio, and the other 200,745 children were given a placebo. Among those in the treatment group, 33 developed polio, and among those in the placebo group, 115 developed polio. If we want to use the methods of this section to test the claim that the rate of polio is less for children given the Salk vaccine, are the requirements for a hypothesis test satisfied? Explain.

Notation For the sample data given in Exercise 1, consider the Salk vaccine treatment group to be the first sample. Identify the values of n 1 , p ^ 1 , q ^ 1 , n 2 , p ^ 2 , q ^ 2 , \(\mathrm {p^-}\), and \(\mathrm {q^-}\). Round all values so that they have six significant digits. ________________ Equation Transcription: Text Transcription: p^- q^-

Interpreting Displays. In Exercises 5 and 6, use the results from the given displays. Flu Vaccine A USA Today article about an experimental nasal spray vaccine for children stated, “In a trial involving 1602 children only 14 (1%) of the 1070 who received the vaccine developed the flu, compared with 95 (18%) of the 532 who got a placebo.” The accompanying TI-­83/84 Plus calculator display results from a test of the claim that the vaccine had no effect. Identify the test statistic and P-­value. What should you conclude about the claim? TI­-83/84 PLUS

E­mail Privacy Among 436 workers surveyed in a Gallup poll, 192 said that it was seriously unethical to monitor employee e-­mail. Among 121 senior­-level managers, 40 said that it was seriously unethical to monitor employee e-­mail. Consider the claim that for those saying that monitoring e­-mail is seriously unethical, the proportion of workers is the same as the proportion of managers. Identify the test statistic and P-­value. If using a 0.01 significance level, what should you conclude about the claim? MINITAB

Problem 7BSC Testing Claims about Proportions. In Exercises, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim. Dreaming in Black and White A study was conducted to determine the proportion of people who dream in black and white instead of color. Among 306 people over the age of 55, 68 dream in black and white, and among 298 people under the age of 25, 13 dream in black and white (based on data from “Do We Dream in Color?” by Eva Murzyn, Consciousness and Cognition, Vol. 17, No. 4). We want to use a 0.01 significance level to test the claim that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25. a. Test the claim using a hypothesis test. ________________ b. Test the claim by constructing an appropriate confidence interval. ________________ c. An explanation given for the results is that those over the age of 55 grew up exposed to media that was mostly displayed in black and white. Can the results from parts (a) and (b) be used to verify that explanation?

Problem 4BSC Using Confidence Intervals a. Assume that we want to use a 0.05 significance level to test the claim that p1 < p2. If we want to test that claim by using a confidence interval, what confidence level should we use? ________________ b. If we test the claim in part (a) using the sample data in Exercise, we get this confidence interval: –0.000508 < p1– p2 < ?0.000309. What does this confidence interval suggest about the claim? ________________ c. In general, when dealing with inferences for two population proportions, which two of the following are equivalent: confidence interval method; P-value method; critical value method? Exercise Verifying Requirements In the largest clinical trial ever conducted, 401,974 children were randomly assigned to two groups. The treatment group consisted of 201,229 children given the Salk vaccine for polio, and the other 200,745 children were given a placebo. Among those in the treatment group, 33 developed polio, and among those in the placebo group, 115 developed polio. If we want to use the methods of this section to test the claim that the rate of polio is less for children given the Salk vaccine, are the requirements for a hypothesis test satisfied? Explain.

Problem 8BSC Testing Claims about Proportions. In Exercises, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim. Ginkgo for Dementia The herb ginkgo biloba is commonly used as a treatment to prevent dementia. In a study of the effectiveness of this treatment, 1545 elderly subjects were given ginkgo and 1524 elderly subjects were given a placebo. Among those in the ginkgo treatment group, 246 later developed dementia, and among those in the placebo group, 277 later developed dementia (based on data from “Ginkgo Biloba for Prevention of Dementia,” by DeKosky et al., Journal of the American Medical Association, Vol. 300, No. 19). We want to use a 0.01 significance level to test the claim that ginkgo is effective in preventing dementia. a. Test the claim using a hypothesis test. ________________ b. Test the claim by constructing an appropriate confidence interval. ________________ c. Based on the results, is ginkgo effective in preventing dementia?

Problem 10BSC Testing Claims about Proportions. In Exercises, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim. Cardiac Arrest at Day and Night A study investigated survival rates for in-hospital patients who suffered cardiac arrest. Among 58,593 patients who had cardiac arrest during the day, 11,604 survived and were discharged. Among 28,155 patients who suffered cardiac arrest at night, 4139 survived and were discharged (based on data from “Survival from In-Hospital Cardiac Arrest During Nights and Weekends,” by Peberdy et al., Journal of the American Medical Association, Vol. 299, No. 7). We want to use a 0.01 significance level to test the claim that the survival rates are the same for day and night. a. Test the claim using a hypothesis test. ________________ b. Test the claim by constructing an appropriate confidence interval. ________________ c. Based on the results, does it appear that for in-hospital patients who suffer cardiac arrest, the survival rate is the same for day and night?

Problem 12BSC Testing Claims about Proportions. In Exercises, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim. Bednets to Reduce Malaria In a randomized controlled trial in Kenya, insecticide-treated bednets were tested as a way to reduce malaria. Among 343 infants using bednets, 15 developed malaria. Among 294 infants not using bednets, 27 developed malaria (based on data from “Sustainability of Reductions in Malaria Transmission and Infant Mortality in Western Kenya with Use of Insecticide-Treated Bednets,” by Lindblade et A., Journal of the American Medical Association, Vol. 291, No. 21). We want to use a 0.01 significance level to test the claim that the incidence of malaria is lower for infants using bednets. a. Test the claim using a hypothesis test. ________________ b. Test the claim by constructing an appropriate confidence interval. ________________ c. Based on the results, do the bednets appear to be effective?

Problem 11BSC Testing Claims about Proportions. In Exercises, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim. Is Echinacea Effective for Colds? Rhinoviruses typically cause common colds. In a test of the effectiveness of echinacea, 40 of the 45 subjects treated with echinacea developed rhinovirus infections. In a placebo group, 88 of the 103 subjects developed rhinovirus infections (based on data from “An Evaluation of Echinacea Angustifolia in Experimental Rhinovirus Infections,” by Turner et al., New England Journal of Medicine, Vol. 353, No. 4). We want to use a 0.05 significance level to test the claim that echinacea has an effect on rhinovirus infections. a. Test the claim using a hypothesis test. ________________ b. Test the claim by constructing an appropriate confidence interval. ________________ c. Based on the results, does echinacea appear to have any effect on the infection rate?

Problem 9BSC Testing Claims about Proportions. In Exercises, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim. Are Seat Belts Effective? A simple random sample of front-seat occupants involved in car crashes is obtained. Among 2823 occupants not wearing seat belts, 31 were killed. Among 7765 occupants wearing seat belts, 16 were killed (based on data from “Who Wants Airbags?” by Meyer and Finney, Chance, Vol. 18, No. 2). We want to use a 0.05 significance level to test the claim that seat belts are effective in reducing fatalities. a. Test the claim using a hypothesis test. ________________ b. Test the claim by constructing an appropriate confidence interval. ________________ c. What does the result suggest about the effectiveness of seat belts?

Problem 14BSC Testing Claims about Proportions. In Exercises, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim. Police Gunfire In a study of police gunfire reports during a recent year, it was found that among 540 shots fired by New York City police, 182 hit their targets; and among 283 shots fired by Los Angeles police, 77 hit their targets (based on data from the New York Times). We want to use a 0.05 significance level to test the claim that New York City police and Los Angeles police have the same proportion of hits. a. Test the claim using a hypothesis test. ________________ b. Test the claim by constructing an appropriate confidence interval. ________________ c. Based on the results, does there appear to be a difference between the hit rates of New York City police and Los Angeles police?

Problem 13BSC Testing Claims about Proportions. In Exercises, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim. Tennis Challenges Since the Hawk-Eye instant replay system for tennis was introduced at the U.S. Open in 2006, men challenged 1412 referee calls, with the result that 421 of the calls were overturned. Women challenged 759 referee calls, and 220 of the calls were overturned. We want to use a 0.05 significance level to test the claim that men and women have equal success in challenging calls. a. Test the claim using a hypothesis test. ________________ b. Test the claim by constructing an appropriate confidence interval. ________________ c. Based on the results, does it appear that men and women have equal success in challenging calls?

Problem 15BSC Testing Claims about Proportions. In Exercises, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim. Headache Treatment In a study of treatments for very painful “cluster” headaches, 150 patients were treated with oxygen and 148 other patients were given a placebo consisting of ordinary air. Among the 150 patients in the oxygen treatment group, 116 were free from headaches 15 minutes after treatment. Among the 148 patients given the placebo, 29 were free from headaches 15 minutes after treatment (based on data from “High-Flow Oxygen for Treatment of Cluster Headache,” by Cohen, Burns, and Goadsby, Journal of the American Medical Association, Vol. 302, No. 22). We want to use a 0.01 significance level to test the claim that the oxygen treatment is effective. a. Test the claim using a hypothesis test. ________________ b. Test the claim by constructing an appropriate confidence interval. ________________ c. Based on the results, is the oxygen treatment effective?

Problem 17BSC Testing Claims about Proportions. In Exercises, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim. Lefties In a random sample of males, it was found that 23 write with their left hands and 217 do not. In a random sample of females, it was found that 65 write with their left hands and 455 do not (based on data from “The Left-Handed: Their Sinister History,” by Elaine Fowler Costas, Education Resources Information Center, Paper 399519). We want to use a 0.01 significance level to test the claim that the rate of left-handedness among males is less than that among females. a. Test the claim using a hypothesis test. ________________ b. Test the claim by constructing an appropriate confidence interval. ________________ c. Based on the results, is the rate of left-handedness among males less than the rate of left-handedness among females?

Testing Claims About Proportions. In Exercises 7–18, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-­value or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim. Spending Large and Small Bills In the same study cited in Example 1, another trial was conducted with 75 women in China given a 100­-Yuan bill, while another 75 women in China were given 100 Yuan in the form of smaller bills (a 50­-Yuan bill plus two 20­-Yuan bills plus two 5­-Yuan bills). Among those given the single bill, 60 spent some or all of the money. Among those given the smaller bills, 68 spent some or all of the money. We want to use a 0.05 significance level to test the claim that when given a single large bill, a smaller proportion of women in China spend some or all of the money when compared to the proportion of women in China given the same amount in smaller bills. a. Test the claim using a hypothesis test. b. Test the claim by constructing an appropriate confidence interval. c. If the significance level is changed to 0.01, does the conclusion change?

Problem 20BB Equivalence of Hypothesis Test and Confidence Interval Two different simple random samples are drawn from two different populations. The first sample consists of 20 people with 10 having a common attribute. The second sample consists of 2000 people with 1404 of them having the same common attribute. Compare the results from a hypothesis test of p1 = p2(with a 0.05 significance level) and a 95% confidence interval estimate of p1? p2.

Problem 19BB Interpreting Overlap of Confidence Intervals In the article “On Judging the Significance of Differences by Examining the Overlap Between Confidence Intervals,” by Schenker and Gentleman (American Statistician, Vol. 55, No. 3), the authors consider 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 of the difference p1– p2. What does the result suggest about the equality of p1 and p2? ________________ b. Use the methods of Section 7-2 to construct individual 95% confidence interval estimates for each of the two population proportions. After comparing the overlap between the two confidence intervals, what do you conclude about the equality of p1 and p2? ________________ c. Use a 0.05 significance level to test the claim that the two population proportions are equal. What do you conclude? ________________ d. Based on the preceding results, what should you conclude about the equality of p1 and p2? Which of the three preceding methods is least effective in testing for the equality of p1 and p2?