22.8 Ethnocentrism. A social psychologist reports, “In our sample, ethnocentrism was significantly higher (P < 0.05) among church attenders than among nonattenders.’’ Explain to someone who knows no statistics what this means.

22.9 Students’ earnings. The financial aid office of a university asks a sample of students about their employment and earnings. The report says, “For academic year earnings, a significant difference (P = 0.028) was found between the sexes, with men earning more on average than women. No difference (P = 0.576) was found between the earnings of black and white students.’’ Explain both of these conclusions, for the effects of sex and of race on mean earnings, in language understandable to someone who knows no statistics.

22.10 Diet and diabetes. Does eating more fiber reduce the blood cholesterol level of patients with diabetes? A randomized clinical trial compared normal and high-fiber diets. Here is part of the researchers’ conclusion: “The high-fiber diet reduced plasma total cholesterol concentrations by 6.7 percent (P = 0.02), triglyceride concentrations by 10.2 percent (P = 0.02), and very-low-density lipoprotein cholesterol concentrations by 12.5 percent (P = 0.01).’’ A doctor who knows no statistics says that a drop of 6.7% in cholesterol isn’t a lot—maybe it’s just an accident due to the chance assignment of patients to the two diets. Explain in simple language how “P = 0.02’’ answers this objection.

22.11 Diet and bowel cancer. It has long been thought that eating a healthier diet reduces the risk of bowel cancer. A large study cast doubt on this advice. The subjects were 2079 people who had polyps removed from their bowels in the past six months. Such polyps may lead to cancer. The subjects were randomly assigned to a low-fat, high-fiber diet or to a control group in which subjects ate their usual diets. Did polyps reoccur during the next four years?

22.12 Pigs and prestige in ancient China. It appears that pigs in Stone Age China were not just a source of food. Owning pigs was also a display of wealth. Evidence for this comes from examining burial sites. If the skulls of sacrificed pigs tend to appear along with expensive ornaments, that suggests that the pigs, like the ornaments, signal the wealth and prestige of the person buried. A study of burials from around 3500 B.C. concluded that “there are striking differences in grave goods between burials with pig skulls and burials without them. . . . A test indicates that the two samples of total artifacts are significantly different at the 0.01 level.’’ Explain clearly why “significantly different at the 0.01 level’’ gives good reason to think that there really is a systematic difference between burials that contain pig skulls and those that lack them.

22.13 Ancient Egypt. Settlements in Egypt before the time of the pharaohs are dated by measuring the presence of forms of carbon that decay over time. The first datings of settlements in the Nagada region used hair that had been excavated 60 years earlier. Now researchers have used newer methods and more recently excavated material. Do the dates differ? Here is the conclusion about one location: “There are two dates from Site KH6. Statistically, the two dates are not significantly different. They provide a weighted average corrected date of 3715 ± 90 B.C.’’ Explain to someone interested in ancient Egypt but not interested in statistics what “not significantly different’’ means.

22.14 What’s a gift worth? Do people value gifts from others more highly than they value the money it would take to buy the gift? We would like to think so because we hope that “the thought counts.’’ A survey of 209 adults asked them to list three recent gifts and then asked, “Aside from any sentimental value, if, without the giver ever knowing, you could receive an amount of money instead of the gift, what is the minimum amount of money that would make you equally happy?’’ It turned out that most people would need more money than the gift cost to be equally happy. The magic words “significant (P < 0.01)’’ appear in the report of this finding.

22.15 Attending church. A 2010 Gallup Poll found that 39% of American adults say they attended religious services last week. This is almost certainly not true.

22.16 Body temperature. We have all heard that 98.6 degrees Fahrenheit (or 37 degrees Celsius) is “normal body temperature.’’ In fact, there is evidence that most people have a slightly lower body temperature. You plan to measure the body temperature of a random sample of people very accurately. You hope to show that a majority have temperatures lower than 98.6 degrees.

22.17 Unemployment. The national unemployment rate in a recent month was 5.1%. You think the rate may be different in your city, so you plan a sample survey that will ask the same questions as the Current Population Survey. To see if the local rate differs significantly from 5.1%, what hypotheses will you test?

22.18 First-year students. A UCLA survey of college freshmen in the 2014–2015 academic year found that 19.4% of all first-year college students identify themselves as politically conservative. You wonder if this percentage is different at your school, but you have no idea whether it is higher or lower. You plan a sample survey of first-year students at your school. What hypotheses will you test to see if your school differs significantly from the UCLA survey result?

22.19 Do our athletes graduate? The National Collegiate Athletic Association (NCAA) requires colleges to report the graduation rates of their athletes. At one large university, 78% of all students who entered in 2004 graduated within six years. One hundred thirty-seven of the 190 students who entered with athletic scholarships graduated. Consider these 190 as a sample of all athletes who will be admitted under present policies. Is there evidence that the percentage of athletes who graduate is less than 78%?

22.20 Using the Internet. In 2013, 81.8% of first-year college students responding to a national survey said that they used the Internet frequently for research or homework. Administrators at a large state university believe that their current first-year students use the Internet more frequently than students in 2013 did. They find that 168 of an SRS of 200 of the university’s first-year students said that they used the Internet frequently for research or homework. Is the proportion of first-year students at this university who said that they used the Internet frequently for research or homework larger than the 2013 national value of 81.8%?

22.21 Vote for the best face? We often judge other people by their faces. It appears that some people judge candidates for elected office by their faces. Psychologists showed head-and-shoulders photos of the two main candidates in 32 races for the U.S. Senate to many subjects (dropping subjects who recognized one of the candidates) to see which candidate was rated “more competent’’ based on nothing but the photos. On election day, the candidates whose faces looked more competent won 22 of the 32 contests. If faces don’t influence voting, half of all races in the long run should be won by the candidate with the better face. Is there evidence that the proportion of times the candidate with the better face wins is more than 50%?

22.22 Do our athletes graduate? Is the result of Exercise 22.19 statistically significant at the 10% level? At the 5% level?

22.23 Using the Internet. Is the result of Exercise 22.20 statistically significant at the 5% level? At the 1% level?

22.24 Vote for the best face? Is the result of Exercise 22.21 statistically significant at the 5% level? At the 1% level?

22.25 Significant at what level? Explain in plain language why a result that is significant at the 1% level must always be significant at the 5% level. If a result is significant at the 5% level, what can you say about its significance at the 1% level?

22.26 Significance means what? Asked to explain the meaning of “statistically significant at the α = 0.05 level,’’ a student says: “This means that the probability that the null hypothesis is true is less than 0.05.’’ Is this explanation correct? Why or why not?

22.27 Finding a P-value by simulation. Is a new method of teaching reading to first-graders (Method B) more effective than the method now in use (Method A)? You design a matched pairs experiment to answer this question. You form 20 pairs of first-graders, with the two children in each pair carefully matched by IQ, socioeconomic status, and reading-readiness score. You assign at random one student from each pair to Method A. The other student in the pair is taught by Method B. At the end of first grade, all the children take a test to determine their reading skill. Assume that the higher the score on this test, the more proficient the student is at reading. Let p stand for the proportion of all possible matched pairs of children for which the child taught by Method B has the higher score. Your hypotheses are

The result of your experiment is that Method B gave the higher score in 12 of the 20 pairs, or $\widehat{p}=12/20=0.6$.

22.28 Finding a P-value by simulation. A classic experiment to detect extra-sensory perception (ESP) uses a shuffled deck of cards containing five suits (waves, stars, circles, squares, and crosses). As the experimenter turns over each card and concentrates on it, the subject guesses the suit of the card. A subject who lacks ESP has probability 1-in-5 of being right by luck on each guess. A subject who has ESP will be right more often. Julie is right in five of 10 tries. (Actual experiments use much longer series of guesses so that weak ESP can be spotted. No one has ever been right half the time in a long experiment!)

22.29 Using the Internet. Return to the study in Exercise 22.20, which found that 168 of 200 first-year students said that they used the Internet frequently for research or homework. Carry out the hypothesis test described in Exercise 22.20 and compute the P-value. How does your value compare with the value given in Exercise 22.16(d)?

22.30 Interpreting scatterplots. In 2014, the Pew Research Centers American Trends Panel sought to better understand what Americans know about science. It was observed that among a random selection of 3278 adults, 2065 adults could correctly interpret a scatterplot. Is this good evidence that more than 60% of Americans are able to correctly interpret scatterplots?

22.31 Side effects. An experiment on the side effects of pain relievers assigned arthritis patients to one of several over-the-counter pain medications. Of the 420 patients who took one brand of pain reliever, 21 suffered some “adverse symptom.’’

22.32 Do chemists have more girls? Some people think that chemists are more likely than other parents to have female children. (Perhaps chemists are exposed to something in their laboratories that affects the sex of their children.) The Washington State Department of Health lists the parents’ occupations on birth certificates. Between 1980 and 1990, 555 children were born to fathers who were chemists. Of these births, 273 were girls. During this period, 48.8% of all births in Washington State were girls. Is there evidence, at a significance level of 0.05, that the proportion of girls born to chemists is higher than the state proportion?

22.33 Speeding. It often appears that most drivers on the road are driving faster than the posted speed limit. Situations differ, of course, but here is one set of data. Researchers studied the behavior of drivers on a rural interstate highway in Maryland where the speed limit was 55 miles per hour. They measured speed with an electronic device hidden in the pavement and, to eliminate large trucks, considered only vehicles less than 20 feet long. They found that 5690 out of 12,931 vehicles were exceeding the speed limit. Is this good evidence, at a significance level of 0.05, that (at least in this location) fewer than half of all drivers are speeding?

22.34 Student attitudes. The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures students’ study habits and attitudes toward school. Scores range from 0 to 200. The mean score for U.S. college students is about 115, and the standard deviation is about 30. A teacher suspects that older students have better attitudes toward school. She gives the SSHA to 25 students who are at least 30 years old. Assume that scores in the population of older students are Normally distributed with standard deviation σ = 30. The teacher wants to test the hypotheses

H₀: μ = 115

H_a: μ > 115

22.35 Mice in a maze. Experiments on learning in animals sometimes measure how long it takes mice to find their way through a maze. The mean time is 19 seconds for one particular maze. A researcher thinks that a loud noise will cause the mice to complete the maze faster. She measures how long each of several mice takes to find its way through a maze with a noise as stimulus. What are the null hypothesis H₀ and alternative hypothesis H_a?

22.36 Response time. Last year, your company’s service technicians took an average of 2.5 hours to respond to trouble calls from business customers who had purchased service contracts. Do this year’s data show a significantly different average response time? What null and alternative hypotheses should you test to answer this question?

22.37 Testing a random number generator. Our statistical software has a “random number generator’’ that is supposed to produce numbers scattered at random between 0 and 1. If this is true, the numbers generated come from a population with μ = 0.5. A command to generate 100 random numbers gives outcomes with $\overline{x}$ = 0.536 and s = 0.312. Is this good evidence that the mean of all numbers produced by this software is not 0.5?

22.38 Will they charge more? A bank wonders whether omitting the annual credit card fee for customers who charge at least $3000 in a year will increase the amount charged on its credit cards. The bank makes this offer to an SRS of 400 of its credit card customers. It then compares how much these customers charge this year with the amount that they charged last year. The mean increase in the sample is $246, and the standard deviation is $112. Is there significant evidence at the 1% level that the mean amount charged increases under the no-fee offer? State H₀ and H_a and carry out a significance test. Use significance level 0.01.

22.39 Bad weather, bad tip? People tend to be more generous after receiving good news. Are they less generous after receiving bad news? The average tip left by adult Americans is 20%. Give 20 patrons of a restaurant a message on their bill warning them that tomorrow’s weather will be bad and record the tip percentage they leave. Here are the tips as a percentage of the total bill:

18.0 19.1 19.2 18.8 18.4 19.0

18.5 16.1 16.8 18.2 14.0 17.0

13.6 17.5 20.0 20.2 18.8 18.0

23.2 19.4

Suppose that tip percentages are Normal with σ = 2, and assume that the patrons in this study are a random sample of all patrons of this restaurant. Is there good evidence that the mean tip percentage for all patrons of this restaurant is less than 20 when they receive a message warning them that tomorrow’s weather will be bad? State H₀ and H_a and carry out a significance test. Use significance level 0.05.

22.40 Why should the significance level matter? On June 15, 2005, an article by Lawrence K. Altman appeared in the New York Times. The title of the article was “Studies Rebut Earlier Report on Pledges of Virginity.” The article began by stating the following: “Challenging earlier findings, two studies from the Heritage Foundation reported yesterday that young people who took virginity pledges had lower rates of acquiring sexually transmitted diseases and engaged in fewer risky sexual behaviors.” The new findings were based on the same national survey used by earlier studies and conducted by the Department of Health and Human Services. But the authors of the new study used different methods of statistical analysis from those in an earlier one that was widely publicized, making direct comparisons difficult. One particular criticism of the new study was that the result of a statistical test at a 0.10 level of significance was reported when journals generally use a lower level of 0.05. Why might this be a concern?