Idea Transcript
Hypothesis Tests for 1 sample Proportions 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 0 : pˆ 0.96 H A : pˆ 0.96
SRS, 0.96(200) > 10 188 0.94 ; SD pˆ 200 0.96 0.94 z 1.18 0.017
0.94 0.06 200
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 0 : p 0.9 H A : p 0.9
SRS, 750 > 10 657 0.876 ; SD pˆ 750 0.876 0.94 z 2 0.012
0.88 0.12 750
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 Pvalue? 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 Pvalue? d) Explain carefully what the P-value means in this context. e) What's your conclusion? f) Do environmental chemicals cause autism? 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 hypothesis and state your conclusion. Be sure the appropriate assumptions and conditions are satisfied before you proceed. 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 before you proceed. 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 hypothesis and state your conclusion. Be sure the appropriate assumptions and conditions are satisfied before you proceed.
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.
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 before you proceed.
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.
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 before you proceed.
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.
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 hypothesis and state your conclusion. Be sure the appropriate assumptions and conditions are satisfied before you proceed.
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 at least 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.
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.
Hypothesis Tests for 1 sample Proportions Answers 7. 1) Use p in hypotheses, not pˆ . 2) The question was about failing to meet the goal, so HA should be p< 0.96. 3) Did not check 0.04(200) = 8.
.96 .04
0.014 200 0.876 0.9 5) z is incorrect; should be z 1.43 0.014
4) 188/200 = 0.94: SD( pˆ ) =
6) P = P(z < -1.43) = 0.076 7) Since p-value is greater than alpha (0.05), we will fail to reject that at least 96% of customers succeed programming their VCRs. 8. 1) Use p in hypotheses, not pˆ . 2) The question asks, "not accurate," so HA should be two sided: p 0.9. 3) The correct conditions are SRS, (0.9)(750) > 10, and (0.10)(750) > 10.
.9 .1
0.011 750 0.876 0.9 2.18 5) 2 is incorrect; should be z 0.011
4) p = 657/750 = 0.876; SD( pˆ ) =
6) P = 2P(z < -2.18) = 0.029 7) Since p-value is less than alpha (0.05), we will reject that the proportion of adults who drink milk here is 90%. 9. a) H0: p = 0.30; HA: 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) Since p-value is greater than alpha (0.05), we will fail to reject that the dowser's chance of finding water is the same as normal drilling. 10. a) H0: p = 0.05; HA: p > 0.05 b) SRS (not clear from information provided), 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) Since p-value is less than alpha (0.05), we will reject that the rate of autism is 5%. f) We do not know that chemicals cause autism, only that the rate is higher now than in the past. 13. H0: p = 0.20; HA: 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. H0: p = 0.02; HA: 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. H0: p = 0.03; HA: 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. Since p-value is less than alpha (0.05), we will reject that the rate of twins born to teenage girls at this hospital is 3%. It is not clear whether this can be generalized to all teenagers. 16. H0:p = 0.50; HA: 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. Since p-value is less than alpha (0.05), we will reject that the home team does not have an advantage; they win more than 50% of games at home. 17. H0: p = 0.25; HA: 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. Since p-value is greater than alpha (0.05), we will fail to reject that 25% of current readers would subscribe; the company should not go ahead with the WebZine on the basis of these data. 18. H0: p = 0.92; HA: 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. Since p-value is less than alpha (0.05), we will reject that these seeds have lost viability; their germination rate is 92%.
19. H0: p = 0.40; HA: 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. Since p-value is greater than alpha (0.05), we will fail to reject that the proportion of women executives is 40% women in the company in general. 20. H0: p = 0.19; HA: p < 0.19. p = 0.125. z = -1.41; Pvalue = 0.0793. Since p-value is greater than alpha (0.05), we will fail to reject that Hispanics are represented in the jury pool is 19% proportion in the population in general. 21. H0: p = 0.109; HA: p > 0.109. p = 0.118; z = 1.198; P-value = 0.115. Since p-value is greater than alpha (0.05), we will fail to reject that the dropout rate is 10.9%. 22. H0: p = 0.15; HA: p > 0.15. p = 0.25; z = 2.80; Pvalue = 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). Since p-value is less than alpha (0.05), we will reject that the proportion of trees damaged by acid rain in the Hopkins forest is 15%. 23. H0: p = 0.90; HA: p < 0.90. p = 0.844; z = -2.05; Pvalue = 0.0201. Since p-value is less than alpha (0.05), we will reject that the actual rate at which passengers with lost luggage are reunited with it within 24 hours is the 90% claimed by the airline. 24. H0: p = 0.40; HA: p > 0.40. p = 0.431; z = 1.29; Pvalue = 0.0977. Since p-value is greater than alpha (0.05), we will fail to reject that 40% of the public recognizes the brand; I would not recommend they continue to advertise during Super Bowls on the basis of these data.