Hypothesis Tests for 1 sample Proportions 1. Hypotheses. Write the null and alternative hypotheses you would use to test each of the following situations. a) A governor is concerned about his "negatives" the percentage of state residents who express disapproval of his job performance. His political committee pays for a series of TV ads, hoping that they can keep the negatives below 30%. They will use follow-up polling to assess the ads' effectiveness. b) Is a coin fair? c) Only about 20% of people who try to quit smoking succeed. Sellers of a motivational tape claim that listening to the recorded messages can help people quit. 2. More hypotheses. Write the null and alternative hypotheses you would use to test each of the following situations. a) In the 1950s only about 40% of high school graduates went on to college. Has the percentage changed? b) 20% of cars of a certain model have needed costly transmission work after being driven between 50,000 and 100,000 miles. The manufacturer hopes that redesign of a transmission component has solved this problem. c) We field test a new flavor soft drink, planning to market it only if we are sure that at least 60% of the people like the flavor. 3. Negatives. After the political ad campaign described in Exercise la, pollsters check the governor's negatives. They test the hypothesis that the ads produced no change against the alternative that the negatives are now below 30%, and find a P-value of 0.22. Which conclusion is appropriate? a) There's a 22% chance that the ads worked. b) There's a 78% chance that the ads worked. c) There's a 22% chance that the poll they conducted is correct. d) There's a 22% chance that poll results could be just natural sampling variation rather than a real change in public opinion. 4. Dice. The seller of a loaded die claims that it will favor the outcome 6. We don't believe that claim, and roll the die 200 times to test an appropriate hypothesis. Our P- value turns out to be 0.03. Which conclusion is appropriate? a) There's a 3% chance that the die is fair. b) There's a 97% chance that the die is fair. c) There's a 3% chance that a loaded die could randomly produce the results we observed, so it's reasonable to conclude that the die is fair. d) There's a 3% chance that a fair die could randomly produce the results we observed, so it's reasonable to conclude that the die is loaded. 5. Relief. A company's old antacid formula provided relief for 70% of the people who used it. The company tests a new formula to see if it is better, and gets a P-value of 0.27. Is it reasonable to conclude that the new formula and the old one are equally effective? Explain. 6. Cars. A survey investigating whether the proportion of today's high school seniors who own their own cars is higher than it was a decade ago finds a P- value of 0.017. Is it reasonable to conclude that more high schoolers have cars? Explain. 7. Blinking timers. Many people have trouble programming their VCRs, so a company has developed what it hopes will be easier instructions. The goal is to have at least 96% of customers succeed. The company tests the new system on 200 people, of whom 188 were successful. Is this strong evidence that the new system fails to meet the company's goal? A student's test of this hypothesis is shown below. How many mistakes can you find? H : p 0.96 0 ˆ H : ˆ A p 0.96 SRS, 0.96(200) > 10 188 0.94 200 ; 0.94 0.06 SD pˆ 0.017 200 0.96 0.94 z 1.18 0.017 P = P(z > 1.18) = 0.12 There is strong evidence that the new system does not work. 8. Got milk? In November 2001, The Ag Globe Trotter newsletter reported that 90% of adults drink milk. A regional farmers' organization planning a new marketing campaign across its multi-county" area polls a random sample of 750 adults living there. In this sample, 657 people said that they drink milk. Do these responses provide strong evidence that the 90% figure is not accurate for this region? Correct the mistakes you find in a student's attempt to test an appropriate hypothesis. H : p 0.9 0 HA : p 0.9 SRS, 750 > 10 657 0.876 750 ; 0.88 0.12 SD pˆ 0.012 750 0.876 0.94 z 2 0.012 P = P(z > -2) = 0.977 There is more than a 97% chance that the stated percentage is correct for this region.
9. Dowsing. In a rural area only about 30% of the wells that are drilled find adequate water at a depth of 100 feet or less. A local man claims to be able to find water by "dowsing" using a forked stick to indicate where the well should be drilled. You check with 80 of his customers and find that 27 have wells less than 100 feet deep. What do you conclude about his claim? (We consider a P-value of around 5% to represent strong evidence.) a) Write appropriate hypotheses. b) Check the necessary assumptions. c) Perform the mechanics of the test. What is the P- value? d) Explain carefully what the P-value means in this context. e) What's your conclusion? 10. Autism. In the 1980s it was generally believed that autism affected about 5% of the nation's children. Some people believe that the increase in the number of chemicals in the environment has led to an increase in the incidence of autism. A recent study examined 384 children and found that 46 of them showed signs of some form of autism. Is this strong evidence that the level of autism has increased? (We consider a P-value of around 5% to represent strong evidence.) a) Write appropriate hypotheses. b) Check the necessary assumptions. c) Perform the mechanics of the test. What is the P- value? d) Explain carefully what the P-value means in this context. e) What's your conclusion? f) Do environmental chemicals cause autism? 11. Smoking. National data in the 1960s showed that about 44% of the adult population had never smoked cigarettes. In 1995 a national health survey interviewed a random sample of 881 adults and found that 52% had never been smokers. a) Create a 95% confidence interval for the proportion of adults (in 1995) who had never been smokers. b) Does this provide evidence of a change in behavior among Americans? Using your confidence interval, test an appropriate hypothesis and state your conclusion. 12. Satisfaction. A company hopes to improve customer satisfaction, setting as a goal no more than 5% negative comments. A random survey of 350 customers found only 10 with complaints. a) Create a 95% confidence interval for the true level of dissatisfaction among customers. b) Does this provide evidence that the company has reached its goal? Using your confidence interval, test an appropriate hypothesis and state your conclusion. 13. Pollution. A company with a fleet of 150 cars found that the emissions systems of 7 out of the 22 they tested failed to meet pollution control guidelines. Is this strong evidence that more than 20% of the fleet might be out of compliance? Test an appropriate 14. Scratch and dent. An appliance manufacturer stockpiles washers and dryers in a large warehouse for shipment to retail stores. Sometimes in handling them the appliances get damaged. Even though the damage may be minor, the company must sell those machines at drastically reduced prices. The company goal is to keep the level of damaged machines below 2%. One day an inspector randomly checks 60 washers and finds that 5 of them have scratches or dents. Is this strong evidence that the warehouse is failing to meet the company goal? Test an appropriate hypothesis and state your conclusion. Be sure the appropriate assumptions and conditions are satisfied before you proceed. 15. Twins. Some doctors suspect that young mothers have fewer multiple births. In 2001 a national vital statistics report indicated that about 3% of all births produced twins. Data from a large city hospital found only 7 sets of twins were born to 469 teenage girls. Does that suggest that mothers under age 20 may be less likely to have twins? Test an appropriate hypothesis and state your conclusion. Be sure the appropriate assumptions and conditions are satisfied 16. Football. During the 2000 season, the home team won 138 of the 240 regular season National Football League games. Is this strong evidence of a home field advantage in professional football? Test an appropriate 17. WebZine. A magazine is considering the launch of an online edition. The magazine plans to go ahead only if it's convinced that more than 25% of current readers would subscribe. The magazine contacts a simple random sample of 500 current subscribers, and 137 of those surveyed expressed interest. What should the company do? Test an appropriate hypothesis and state your conclusion. Be sure the appropriate assumptions and conditions are satisfied
18. Seeds. A garden center wants to store leftover packets of vegetable seeds for sale the following spring, but the center is concerned that the seeds may not germinate at the same rate a year later. The manager finds a packet of last year's green bean seeds and plants them as a test. Although the packet claims a germination rate of 92%, only 171 of 200 test seeds sprout. Is this evidence that the seeds have lost viability during a year in storage? Test an appropriate hypothesis and state your conclusion. Be sure the appropriate assumptions and conditions are satisfied 19. Women executives. A company is criticized because only 13 of 43 people in executive-level positions are women. The company explains that although this proportion is lower than it might wish, it's not surprising given that only 40% of all their employees are women. What do you think? Test an appropriate 20. Jury. Census data for a certain county shows that 19% of the adult residents are Hispanic. Suppose 72 people are called for jury duty, and only 9 of them are Hispanic. Does this call into question the fairness of the jury selection system? Explain. 21. Dropouts. Some people are concerned that new tougher standards and high stakes tests adopted in many states may drive up the high school dropout rate. The National Center for Education Statistics reported that the high school dropout rate for the year 2000 was 10.9%. One school district, whose dropout rate has always been very close to the national average, reports that 210 of their 1782 students dropped out last year. Is their experience evidence that the dropout rate may be increasing? Explain. 22. Acid rain. A study of the effects of acid rain on trees in the Hopkins Forest shows that of 100 trees sampled, 25 of them exhibited some sort of damage from acid rain. This rate seemed to be higher than the 15% quoted in a recent Environmetrics article on the average proportion of damaged trees in the Northeast. Does the sample suggest that trees in the Hopkins Forest are more susceptible than the rest of the region? Comment, and write up your own conclusions based on an appropriate confidence interval as well as a hypothesis test. Include any assumptions you made about the data. 23. Lost luggage. An airline's public relations department says that the airline rarely loses Passengers' luggage. It further claims that on those occasions when luggage is lost, 90% is recovered and delivered to its owner within 24 hours. A consumer group who surveyed a large number of air travelers found that only 103 of 122 people who lost luggage on that airline were reunited with the missing items by the next day. Does this cast doubt on the airline's claim? Explain. 24. TV ads. A start-up company is about to market a new computer printer. It decides to gamble by running commercials during the Super Bowl. The company hopes that name recognition will be worth the high cost of the ads. The goal of the company is that greater than 40% of the public recognize its brand name and associate it with computer equipment. The day after the game, a pollster contacts 420 randomly chosen adults, and finds that 181 of them know that this company manufactures printers. Would you recommend that the company continue to advertise during Super Bowls? Explain.
Hypothesis Tests for 1 sample Proportions Answers 1. a) H 0 : p = 0.30; H A : p < 0.30 b) H 0 : p = 0.50; H A : p 0.50 c) H 0 : p = 0.20; H A : p > 0.20 2. a) H 0 : p = 0.40; H A : p 0.40 b) H 0 : p = 0.20; H A : p < 0.20 c) H 0 : p = 0.60; H A : p < 0.60 3. Statement d is correct. 4. Statement d is correct. 5. No, we can say only that there is a 27% chance of seeing the observed effectiveness just from natural sampling variation. There is no evidence that the new formula is more effective, but we can't conclude that they are equally effective. 6. Yes. If there is no difference, there is only a 1.7% chance of seeing such a high sample proportion just from natural sampling variation. 7. 1) Use p in hypotheses, not ˆp. 2) The question was about failing to meet the goal, so H A should be p< 0.96. 3) Did not check 0.04(200) = 8 or N > 10(200).96.04 4) 188/200 = 0.94: SD( p ) = 0.014 200 0.94 0.96 5) z is incorrect; should be z 1.43 0.014 6) P = P(z < -1.43) = 0.076 7) There is only weak evidence that the new system does not work. 8. 1) Use p in hypotheses, not ˆp. 2) The question asks, "not accurate," so H A should be two sided: p 0.9. 3) The correct conditions are SRS, (0.9)(750) > 10, and (0.10)(750) > 10..9.1 4) p = 657/750 = 0.876; SD( p ) = 0.011 750 0.876 0.9 5) 2 is incorrect; should be z 2.18 0.011 6) P = 2P(z < -2.18) = 0.029 7) There is only a 2.9% chance of observing a ˆp as far from 0.90 by sampling variation, so we believe the proportion of adults who drink milk here is different from the claimed 90%. 9. a) H 0 : p = 0.30; H A : p > 0.30 b) Possibly an SRS; we don't know if the sample is less than 10% of his customers but could be viewed as less than 10% of all possible customers; (0.3)(80) > 10 and (0.7)(80) > 10. Wells are independent only if customers don't have farms on the same underground springs. c) z = 0.73; P-value = 0.232 d) If his dowsing is no different from standard methods, there is more than a 23% chance of seeing results as good as those of the dowsers, or better, by natural sampling variation. e) These data provide no evidence that the dowser's chance of finding water is any better than normal drilling. 10. a) H 0 : p = 0.05; H A : p > 0.05 b) SRS (not clear from information provided), <10% of all children, (0.05)(384) > 10, and (0.95)(384) > 10. c) z = 6.28, P = 2 X 10-10 d) If the autism rate has not increased, the chance of observing at least 46 autistic children in a sample of 384 is 2 x 10-10 (almost 0). e) Reject H 0. These data show the rate of autism is now more than 5%. f) We do not know that chemicals cause autism, only that the rate is higher now than in the past. 11. a) Based on these data, we are 95% confident the proportion of adults in 1995 who had never smoked cigarettes is between 48.7% and 55.3%. b) H 0 : p = 0.44; H A : p 0.44. Since 44% is not in the confidence interval, we will reject Hg and conclude the proportion of adults in 1995 who had never smoked was more than in the 1960s. 12. a) Based on these data, we are 95% confident the proportion of dissatisfied customers is between 1.1% and 4.6%. b) H 0 : p = 0.05; H A : p < 0.05. Since 5% is not in the 95% confidence interval, we will reject H 0 at = 0.025 and conclude the proportion of dissatisfied customers is less than 5%, so the company has met its goal. 13. H 0 : p = 0.20; H A : p > 0.20. SRS (not clear from information provided); 22 is more than 10% of the population of 150; (0.20)(22) < 10. Do not proceed with a test. 14. H 0 : p = 0.02; H A : p > 0.02. SRS; less than 10% of all washers and dryers made by the company; (0.02)(60) < 10. Do not proceed with a test. 15. H 0 : p = 0.03; H A : p < 0.03. p = 0.015. One mother having twins will not impact another, so observations are independent; not an SRS; sample is less than 10% of all births. However, the mothers at this hospital may not be representative of all teenagers; (0.03)(469) = 14.07 > 10; (0.97)(469) > 10. z = -1.91; P-value = 0.0278. With a P-value this low, reject H 0. These data show evidence that the rate of twins born to teenage girls at this hospital is less than the national rate of 3%. It is not clear whether this can be generalized to all teenagers. 16. H 0 :p = 0.50; H A : p > 0.50. Results of one game should not impact another, so games are independent; data are all results for one season,
which should be representative of all seasons; sample is less than 10% of all games; (0.50)(240) > 10; (0.50)(240) > 10. z = 2.32; P-value = 0.0101. With a P-value this low, reject H 0. These data show strong evidence that the home team does have an advantage; they win more than 50% of games at home. 17. H 0 : p = 0.25; H A : p > 0.25. SRS; sample is less than 10% of all potential subscribers; (0.25)(500) > 10; (0.75)(500) > 10. z = 1.24; P-value = 0.1076. The P- value is high, so do not reject H 0. These data do not show that more than 25% of current readers would subscribe; the company should not go ahead with the WebZine on the basis of these data. 18. H 0 : p = 0.92; H A : p < 0.92. Seeds in a single packet may not be independent of each other. This is a cluster sample of all seeds in the packet. We may view this cluster as representative of all year-old seeds, in which case the sample is less than 10% of all seeds; (0.92)(200) > 10; (0.08)(200) > 10. z = - 3.39; P-value = 0.0004. Because the P-value is very low, we reject H 0. There is strong evidence that these seeds have lost viability; their germination rate is less than 92%. 19. H 0 : p = 0.40; H A : p < 0.40. Data are for all executives in this company and may not be able to be generalized to all companies; (0.40)(43) > 10; (0.60)(43) > 10. z = -1.31; P-value = 0.0955. Because the P-value is high, we fail to reject H 0. These data do not show that the proportion of women executives is less than the 40% women in the company in general. 20. H 0 : p = 0.19; H A : p < 0.19. ˆp = 0.125. z = -1.41; P- value = 0.0793. Because the P-value is high, we fail to reject H 0. These data do not show convincing evidence that Hispanics are represented in the jury pool at less than their 19% proportion in the population in general, but indicates that it is below what was expected. 21. H 0 : p = 0.109; H A : p > 0.109. ˆp = 0.118; z = 1.198; P-value = 0.115. Because the P-value is high, we fail to reject H 0. These data do not show the dropout rate has increased from 10.9%. 22. H 0 : p = 0.15; H A : p > 0.15. ˆp = 0.25; z = 2.80; P- value = 0.0026. The 95% confidence interval is (0.165, 0.335). We must assume the trees sampled are a SRS of the trees in the area and are less than 10% of all trees in the forest. The results are generalizable only to the Hopkins forest (or nearby if the forest is viewed as representative). Because the P-value is so low, we reject H 0. There is strong evidence that the proportion of trees damaged by acid rain in the Hopkins forest is higher than the 15% average for the Northeast. 23. H 0 : p = 0.90; H A : p < 0.90. ˆp = 0.844; z = -2.05; P- value = 0.0201. Because the P-value is so low, we reject H 0. There is strong evidence that the actual rate at which passengers with lost luggage are reunited with it within 24 hours is less than the 90% claimed by the airline. 24. H 0 : p = 0.40; H A : p > 0.40. ˆp = 0.431; z = 1.29; P- value = 0.0977. Because the P-value is high, we fail to reject H 0. These data do not show that at least 40% of the public recognizes the brand; I would not recommend they continue to advertise during Super Bowls on the basis of these data.