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?
𝐻0 : 𝑝̂ = 0.9
Use p in hypotheses, not 𝑝̂
𝐻𝐴 : 𝑝̂ < 0.9
The question asks, “not accurate”, so 𝐻𝐴 should be two sided: 𝑝 ≠ 0.9
SRS, 750 > 10
657
750
The correct conditions are SRS, (0.9)(750) > 10 and (0.1)(750) > 10
= 0.876; 𝑆𝐷(𝑝̂ )√
𝑧=
0.876−0.94
0.012
=-2
(0.88)(0.12)
750
(0.9)(0.10)
657
= 0.012; 𝑃̂ =
= 0.876; 𝑆𝐷(𝑝̂ )√
= 0.011;
750
z is incorrect and should be; 𝑧 =
750
0.876−0.9
0.011
=-2.18
𝑃 = 𝑃(𝑧 > −2) = 0.977 𝑃 = 2𝑃(𝑧 < −2.18) = 0.029
There is more than a 97% chance that the stated percentage is correct for this region.
There is only a 2.9% chance of observing a 𝑝̂ as far from 0.9 by sampling variation, so we believe the proportion of adults
who drink milk here is different from the claimed 90%
6.
(10) AUTISM. In the 1980 it was generally believed that autism affected about 5% of the nation’s children. Some people
believed 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. 𝐻0 : 𝑝 = 0.05;
𝐻𝐴 : 𝑝 > 0.05
b. Check the necessary assumptions. SRS(not clear from information provided), < 10% of all children;
(0.05)(384) >10 and (0.95)(384) > 10
c. Perform the mechanics of the test. Z= 6.28 What is the P-value? P = 0.0000000002
d. Explain carefully what the P-value means in this context. If the autism rate has not increased, the chance of
observing at least 46 autistic children in a sample of 384 is 0.0000000002 (almost 0).
e. What’s your conclusion? Reject 𝐻0 . These data show the rate of autism is now more than 5%
f. Do environmental chemicals cause autism? We do not know that chemicals cause autism, only that the rate is higher
now than in the past.
7.
(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 complaints.
a. Create a 955 confidence interval for the true level of dissatisfaction among customers. Based on these data, we are
95% confident that the proportion of dissatisfied customers is between 1.1% and 4.6%
b. Does this provide evidence that the company has reached its goal? Using your confidence interval, test an
appropriate hypothesis and state your conclusion. 𝐻0 : 𝑝 = 0.05;
𝐻𝐴 : 𝑝 < 0.05 Since 5% is not in the 95%
confidence interval, we will reject 𝐻0 at α = 0.025 and conclude the proportion of dissatisfied customers is less
than 5%, so the company has met its goal.
8.
(13)POLLUTION. 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. 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. 𝐻0 : 𝑝 = 0.2;
𝐻𝐴 : 𝑝 > 0.2 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.
9.
(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. . 𝐻0 : 𝑝 = 0.02;
𝐻𝐴 : 𝑝 > 0.02
SRS; less than 10% of all washers made by the company; (0.02)(60) < 10 DO NOT PROCEED with a test.
10. (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.
𝐻0 : 𝑝 = 0.03;
𝐻𝐴 : 𝑝 < 0.03 𝑝̂ = 0.015 ; One mother having twins willnot 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) >10 and (0.97)(469) > 10; z = -1.91; P-value = 0.0278. With a P=value
this low, reject 𝐻0 . . 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.
11. (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.
𝐻0 : 𝑝 = 0.5;
𝐻𝐴 : 𝑝 > 0.5 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 size is less than 10% of all games; (0.5)(24) >10 ;
𝑧 = 2.32; P-value = 0.0101. With a P-value this low, reject 𝐻0 .. These data show strong evidence that the home team does
have an advantage; they win more than 50% of games at home.
12. (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.
𝐻0 : 𝑝 = 0.25;
𝐻𝐴 : 𝑝 > 0.25 SRS; sample size is less than 10% of potential subscribers; (0.25)(500) >10 and
(0.75)(500) >10𝑧 = 1.24; P-value = 01076. The P-value is high so do not reject 𝐻0 .. These data do not show evidence that
more than 25% of current readers would subscribe; the company should not go ahead with the WebZi8ne on the basis of
these data.
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.
𝐻0 : 𝑝 = 0.9;
𝐻𝐴 : 𝑝 < 0.9 𝑝̂ = 0.844 𝑧 = −2.05; P-value = 0.0201. With a P-value this low, reject 𝐻0 . 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.
13. (21)P-VALUE. A medical researcher has tested a new treatment for poison ivy against the traditional ointment. With a Pvalue of 0.047, he concludes the new treatment is more effective. Explain what the P-value means in this context. If there is
no difference in effectiveness, the chance of seeing an observed difference this large or larger is 4.7% by natural sampling
variability.
14. (22)ANOTHER P-VALUE. 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. Explain
what the study’s P-value of 0.17 means in this context. If harsher penalties and ad campaigns have made no difference in
seatbelt use, there is a 17% chance of seeing an observed difference this large or larger by natural sampling variability.
15. (23)ALPHA. A researcher developing scanners to search for hidden weapons at airports has concluded that a new device is
significantly better than the current scanner. He made this decision based on a test using α =.05. Would he have made the
same decision at α =.10? How about α =.01? Explain. α = .1: yes. The P-value is < 0.05, so it’s less than 0.10. But to reject
𝐻0 at α = .01, the P=value must be below 0.01, which isn’t necessarily the case.
16. (24)ALPHA AGAIN. 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. They based this conclusion on
a test using α =.05. Would they have made the same decision at α =.10? How about α =.01? Explain. At α = .10: they reach
the same decision only if the P-value is > 0.10. We only know that it’s > 0.05. But we know that its > 0.01. So at > 0.01 they
reach the same decision.
17. (25)SIGNIFICANT? Public health officials believe that 90% of children have been vaccinated against measles. A random
survey of medical records at many schools across the country found that among more than 13,000children only 89.4% had
been vaccinated. A statistician would reject the 90% hypothesis with a P-value of P=0.011.
a.
b.
Explain what the P-Value means in this context. There is only a 1.1% chance of seeing a sample proportion as low
as 89.4% vaccinated by natural sampling variation if 90% have really been vaccinated.
The result is statistically significant, but is it important? Comment. We conclude that p is below 0.9, but a 95%
confidence interval would show that the true proportion is between (0.889, 0.899) Most likely, a decrease from 90%
to 89.9% would not be considered important. On the other hand, with 1,000,000 children a year vaccinated, even
0.1% represents about 1000 kids—so this may very well be important.
18. 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 company continue to advertise during Super Bowls? Explain.
𝐻0 : 𝑝 = 0.4;
𝐻𝐴 : 𝑝 > 0.4 𝑝̂ = 0.431 𝑧 = 1.29; P-value = 0.0977. Because the P-value is high, we fail to reject 𝐻0
These data do not show that at least 40% of the public recognizes the brand; I would not recommend they continue to
advertise during the Super Bowl on the basis of these data.
19. 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 sedds
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.
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.
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 and (0.08)(200 > 10 ; 𝑧 = −3.39; P-value = 0.0004. With a P-value this low, reject 𝐻0 .. There is strong
evidence that the seeds have lost viability; their germination rate is less than 92%.
.
20. (19) WOMAN 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.
𝐻0 : 𝑝 = 0.4;
𝐻𝐴 : 𝑝 < 0.4 Data are for all executives in this company and may not be able to be generalized to all
companies; (0.40)(43) >10 and (0.60)(43) >10; 𝑧 = −1.31; P-value = 0.0955 The P-value is high so we fail to reject 𝐻0 ..
These data do not show evidence that the proportion of women executives is less than the 40% women in the company in
general.