1.NEGATIVES. 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. After the political ad campaign 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. 2. 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 results we observed, so it’s reasonable to conclude that the die is fair. d. There’s a 3% chance that a loaded die could randomly produce results we observed, so it’s reasonable to conclude that the die is loaded 3. 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. 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. 4. 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. 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. 5. 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? How many mistakes are in the work below? H 0 : ^p =0.9 Use p in hypotheses, not H A : ^p <0.9 The question asks, “not accurate”, so SRS, 750 > 10 ^p should be two sided: p≠ 0.9 The correct conditions are SRS, (0.9)(750) > 10 and (0.1)(750) > 10 √ ( 0.88 ) (0.12) 657 =0.876 ; SD ( ^p ) =0.012 ; 750 750 z= HA 0.876−0.94 =-2 0.012 P=P ( z>−2 ) =0.977 √ ( 0.9 ) (0.10) 657 ^ P= =0.876 ; SD ( ^p ) =0.011 ; 750 750 z is incorrect and should be; z= P=2 P ( z <−2.18 )=0.029 0.876−0.9 =-2.18 0.011 Z= 6. One day an inspector randomly checks 60 washers and finds that 5 of them have scratches or dents. Does this provide evidence that the company has reached its goal? Using your confidence interval. 9.6% b. A random survey of 350 customers found only 10 complaints. A recent study examined 384 children and found that 46 of them showed signs of some form of autism.20)(22) < 10 DO NOT PROCEED with a test. Data from a large city hospital found only 7 sets of twins were born to 469 teenage girls.0000000002 (almost 0). (13)POLLUTION.05)(384) >10 and (0.) H A : p>0. H A : p<0. (0. Check the necessary assumptions. so the company has met its goal.95)(384) > 10 Perform the mechanics of the test. not an SRS. Does that suggest that mothers under age 20 may be less likely to have twins? Test an appropriate hypothesis and state your conclusion. There is only a 2.03)(469) >10 and (0. H A : p>0. test an appropriate hypothesis and state your conclusion. A company with a fleet of 150 cars found that the emissions system of 7 out of 22 they tested failed to meet pollution control guidelines. the chance of observing at least 46 autistic children in a sample of 384 is 0.28 What is the P-value? P = 0.03 ^p=0. (10) AUTISM. < 10% of all children. so observations are independent. H 0 : p=0.There is more than a 97% chance that the stated percentage is correct for this region. Sometimes in handling them the appliances get damaged. c. a. An appliance manufacturer stockpiles washers and dryers in a large warehouse for shipment to retail stores.2 . so we believe the proportion of adults who drink milk here is different from the claimed 90% 6.2 SRS(not clear from information provided). z = -1. SRS.025 and conclude the proportion of dissatisfied customers is less than 5%. H A : p>0. H 0 : p=0. In the 1980 it was generally believed that autism affected about 5% of the nation’s children. One mother having twins willnot impact another. d.0000000002 Explain carefully what the P-value means in this context.91.05 Write appropriate hypotheses. (0. Do environmental chemicals cause autism? We do not know that chemicals cause autism. 7. (12) SATISFACTION. the mothers at this hospital may not be representative of all teenagers. we will reject H 0 : p=0. Create a 955 confidence interval for the true level of dissatisfaction among customers. (0. only that the rate is higher now than in the past. In 2001 a national vital statistics report indicated that about 3% of all births produced twins. (14)SCRATCH AND DENT.05 . . Some doctors suspect that young mothers have fewer multiple births. A company hopes to improve customer satisfaction. Is this strong evidence that the level of autism has increased? We consider a P-value of around 5% to represent strong evidence.02 . setting as a goal no more than 5% negative comments. (0. H A : p<0.9% chance of observing a ^p as far from 0. These data show the rate of autism is now more than 5% e.02)(60) < 10 DO NOT PROCEED with a test. Be sure the appropriate assumptions and conditions are satisfied before you proceed. P-value = . H 0 : p=0. Be sure the appropriate assumptions and conditions are satisfied before you proceed.05 Since 5% is not in the H 0 at α = 0. (15)TWINS. Be sure the appropriate assumptions and conditions are satisfied before you proceed.02 H 0 : p=0. the company must sell those machines at drastically reduced prices.9 by sampling variation. Is this strong evidence that more than 20% of the fleet might be out of compliance? Test an appropriate hypothesis and state your conclusion. Even though the damage may be minor.05 . sample is less than 10% of all births. H 0 . 10.015 .03 . 95% confidence interval. Some people believed that the increase in the number of chemicals in the environment has led to an increase in the incidence of autism. b. However. less than 10% of all washers made by the company. we are 95% confident that the proportion of dissatisfied customers is between 1. SRS(not clear from information provided). Based on these data. If the autism rate has not increased. 22 is more than 10% of the population of 150. a. What’s your conclusion? Reject f. The company goal is to keep the level of damaged machines below 2%. 8.1% and 4. Is this strong evidence that the warehouse is failing to meet the company goal? Test an appropriate hypothesis and state your conclusion.97)(469) > 10. Would he have made the same decision at α =. With a Pvalue of 0. The P-value is < 0. P-value = 0.. 14.9 ^p=0. Is this strong evidence of a home field advantage in professional football? ? Test an appropriate hypothesis and state your conclusion. which should be representative of all seasons. 90% is recovered and delivered to its owner within 24 hours. (23)ALPHA. He made this decision based on a test using α =. The magazine plans to go ahead only if it’s convinced that more than 25% of current readers would subscribe. LOST LUGGAGE. (0. H 0 . which isn’t necessarily the case. . It is not clear whether this can be generalized to all teenagers. there is a 17% chance of seeing an observed difference this large or larger by natural sampling variability.0101. With a P-value this low.25 z=1. data are all results for one season. It further claims that on those occasions when luggage is lost.24 . Sample size is less than 10% of all games. (0.01. reject H 0 . Have harsher penalties and ad campaigns increase seatbelt use among drivers and passengers? Observations of commuter traffic failed to find evidence of a significant change comp0ared with three years ago. H 0 : p=0.05. . so games are independent. SRS.5 ..5 Results of one game should not impact another. The magazine contacts a simple random sample of 500 current subscribers. If there is no difference in effectiveness. so it’s less than 0. Be sure the appropriate assumptions and conditions are satisfied before you proceed. What should the company do? Test an appropriate hypothesis and state your conclusion. 11. reject H 0 . A magazine is considering the launch of an online edition.1: yes. Explain what the study’s P-value of 0. H 0 : p=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%.05. Explain what the P-value means in this context. and 137 of those surveyed expressed interest.05.7% by natural sampling variability. the P=value must be below 0. α = . Does this cast doubt on the airline’s claim? Explain. H A : p>0. These data show strong evidence that the home team does have an advantage. 13. z=2. During the 2000 season.844 z=−2. Be sure the appropriate assumptions and conditions are satisfied before you proceed.9 . 12. (22)ANOTHER P-VALUE. (17) WEBZINE. With a P-value this low. they win more than 50% of games at home. sample size is less than 10% of potential subscribers.17 means in this context. (16)FOOTBALL.75)(500) >10 H A : p>0. the company should not go ahead with the WebZi8ne on the basis of these data. The P-value is high so do not reject H 0 . An airline’s public relations department says that the airline rarely loses passengers’ luggage. With a P=value this low. he concludes the new treatment is more effective. These data do not show evidence that more than 25% of current readers would subscribe. P-value = 0.25 . (0.01? Explain. 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.10? How about α =.0201. 15.0278. A medical researcher has tested a new treatment for poison ivy against the traditional ointment. A researcher developing scanners to search for hidden weapons at airports has concluded that a new device is significantly better than the current scanner.01. the home team won 138 of the 240 regular season National Football League games.0. But to reject H 0 at α = .25)(500) >10 and P-value = 01076.5)(24) >10 .32 . (21)P-VALUE.10. the chance of seeing an observed difference this large or larger is 4. There is strong evidence that the actual rate at which passengers with last luggage are reunited with it within 24 hours is less than the 90% claimed by this airline. If harsher penalties and ad campaigns have made no difference in seatbelt use. reject H A : p<0. H 0 : p=0.047. This is a cluster sample. and finds that 181 of them know that this company manufactures printers. TV ADS.4 . P-value = 0. A random survey of medical records at many schools across the country found that among more than 13.29 . with 1.000 children a year vaccinated. We conclude that p is below 0. SEEDS.16..10: they reach the same decision only if the P-value is > 0.39.05.01. reject z=−1. a pollster contacts 420 randomly chosen adults.899) Most likely. but the center is concerned that the seeds may not germinate at the same rate a year later. What do you think? Test an appropriate hypothesis and state your conclusion.889. Would you recommend that company continue to advertise during Super Bowls? Explain. H 0 : p=0.4% vaccinated by natural sampling variation if 90% have really been vaccinated.08)(200 > 10 . At α = . (0. A garden center wants to store leftover packets of vegetable seeds for sale the following spring. They based this conclusion on a test using α =. The company hopes that name recognition will be worth the high cost of the ads. P-value = 0. a decrease from 90% to 89. H 0 : p=0. Be sure the appropriate assumptions and conditions are satisfied before you proceed..011. Although the packet claims a germination rate of 92%. it’s not surprising given that only 40% of all their employees are women. (19) WOMAN EXECUTIVES. Would they have made the same decision at α =. A company is criticized because only 13 of 43 people in executive-level positions are women. The day after the game.431 z=1. 20. Because the P-value is high. reject H 0 .01? Explain.9% would not be considered important. We only know that it’s > 0. It decides to gamble by running commercials during the Super Bowl.10. a. we These data do not show that at least 40% of the public recognizes the brand. H 0 : p=0.31. but is it important? Comment.0955 The P-value is high so we fail to H 0 . Is this evidence that the seeds have lost viability during a year in storage? Test an appropriate hypothesis and state your conclusion.1% chance of seeing a sample proportion as low as 89. z=−3.000.4% had been vaccinated. We may view this cluster as representative of all year-old seeds.92 Seeds in a single packet may not be independent of each other.0977. 18.05.60)(43) >10. H A : p<0. A statistician would reject the 90% hypothesis with a P-value of P=0. There is strong evidence that the seeds have lost viability. 0. 19. With a P-value this low. There is only a 1. b. (24)ALPHA AGAIN. even 0. P-value = 0. Be sure the appropriate assumptions and conditions are satisfied before you proceed.1% represents about 1000 kids—so this may very well be important. fail to reject H0 H A : p>0. only 171 of 200 test sedds sprout.4 ^p=0.01 they reach the same decision. Explain what the P-Value means in this context. On the other hand.000children only 89. . The company explains that although this proportion is lower than it might wish.40)(43) >10 and (0. The manager finds a packet of last year’s green bean seeds and plants them as a test. H A : p<0. The result is statistically significant. A start-up company is about to market a new computer printer. in which case the sample is less than 10% of all seeds. But we know that its > 0. I would not recommend they continue to advertise during the Super Bowl on the basis of these data.4 . but a 95% confidence interval would show that the true proportion is between (0. Environmentalists concerned about the impact of high-frequency radio transmissions on birds found that there was no evidence of higher mortality rate among hatchling in nests near cell towers. (25)SIGNIFICANT? Public health officials believe that 90% of children have been vaccinated against measles. 17. (0. their germination rate is less than 92%. So at > 0. These data do not show evidence that the proportion of women executives is less than the 40% women in the company in general.92 .92)(200) >10 and (0. The goal of the company is that at least 40% of the public recognize its brand name and associate it with computer equipment. .10? How about α =.0004.4 Data are for all executives in this company and may not be able to be generalized to all companies.9.
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