23.8 A television poll. A television news program conducts a call-in poll about the salaries of business executives. Of the 2372 callers, 1921 believe that business executives are vastly overpaid and that their pay should be substantially reduced. The station, following recommended practice, makes a confidence statement: “81% of the Channel 13 Pulse Poll sample believe that business executives are vastly overpaid and that their pay should be substantially reduced. We can be 95% confident that the true proportion of citizens who believe that business executives are vastly overpaid and that their pay should be substantially reduced is within 1.6% of the sample result.” The confidence interval calculation is correct, but the conclusion is not justified. Why not?

23.9 Soccer salaries. Paris Saint-Germain, a French professional soccer team, led the 2015 ESPN/Sporting Intelligence Global Salary Survey with the largest payroll for all professional sports teams. A soccer fan looks up the salaries for all of the players on the 2015 Paris Saint-Germain team. Because the fan took a statistics course, the fan uses all of these salaries to get a 95% confidence interval for the mean salary for all 2015 Paris Saint-Germain players. This makes no sense. Why not?

23.10 How do we feel? A Gallup Poll taken in late October 2011 found that 58% of the American adults surveyed, reflecting on the day before they were surveyed, said they experienced a lot of happiness and enjoyment without a lot of stress and worry. The poll had a margin of sampling error of ±3 percentage points at 95% confidence. A news commentator at the time said the poll is surprising, given the current economic climate, in that it supports the notion that the majority of American adults are experiencing a lot of happiness and enjoyment without a lot of stress and worry. Why do you think the news commentator said this?

23.11 Legalizing marijuana. An October 2015 Gallup Poll found that 58% of the American adults surveyed support the legalization of marijuana in the United States. The poll had a margin of sampling error of ±4 percentage points at 95% confidence.

23.12 How far do rich parents take us? How much education children get is strongly associated with the wealth and social status of their parents. In social science jargon, this is “socioeconomic status,’’ or SES. But the SES of parents has little influence on whether children who have graduated from college go on to yet more education. One study looked at whether college graduates took the graduate admissions tests for business, law, and other graduate programs. The effects of the parents’ SES on taking the LSAT test for law school were “both statistically insignificant and small.’’

23.13 Searching for ESP. A researcher looking for evidence of extrasensory perception (ESP) tests 200 subjects. Only one of these subjects does significantly better (P < 0.01) than random guessing.

23.14 Are the drugs really effective? A March 29, 2012, article in the Columbus Dispatch reported that a former researcher at a major pharmaceutical company found that many basic studies on the effectiveness of new cancer drugs appeared to be unreliable. Among the studies the former researcher reviewed was one that had been published in a reputable journal. In this published study, a cancer drug was reported as having a statistically significant positive effect on treating cancer. For purposes of this problem, assume that statistically significant means significant at level 0.05.

23.15 Comparing bottle designs. A company compares two designs for bottles of an energy drink by placing bottles with both designs on the shelves of several markets in a large city. Checkout scanner data on more than 10,000 bottles bought show that more shoppers bought Design A than Design B. The difference is statistically significant (P = 0.018). Can we conclude that consumers strongly prefer Design A? Explain your answer.

23.16 Color blindness in Africa. An anthropologist suspects that color blindness is less common in societies that live by hunting and gathering than in settled agricultural societies. He tests a number of adults in two populations in Africa, one of each type. The proportion of color-blind people is significantly lower (P < 0.05) in the hunter-gatherer population. What additional information would you want to help you decide whether you accept the claim about color blindness?

23.17 Blood types in Southeast Asia. One way to assess whether two human groups should be considered separate populations is to compare their distributions of blood types. An anthropologist finds significantly different (P = 0.01) proportions of the main human blood types (A, B, AB, O) in different tribes in central Malaysia. What other information would you want before you agree that these tribes are separate populations?

23.18 Why we seek significance. Asked why statistical significance appears so often in research reports, a student says, “Because saying that results are significant tells us that they cannot easily be explained by chance variation alone.’’ Do you think that this statement is essentially correct? Explain your answer.

23.19 What is significance good for? Which of the following questions does a test of significance answer?

23.20 What distinguishes those who have schizophrenia? Psychologists once measured 77 variables on a sample of people who had schizophrenia and a sample of people who did not have schizophrenia. They compared the two samples using 77 separate significance tests. Two of these tests were significant at the 5% level. Suppose that there is, in fact, no difference in any of the 77 variables between people who do and do not have schizophrenia in the adult population. That is, all 77 null hypotheses are true.

23.21 Why are larger samples better? Statisticians prefer large samples. Describe briefly the effect of increasing the size of a sample (or the number of subjects in an experiment) on each of the following.

23.22 Is this convincing? You are planning to test a vaccine for a virus that now has no vaccine. Because the disease is usually not serious, you will expose 100 volunteers to the virus. After some time, you will record whether or not each volunteer has been infected.

23.23 In the courtroom. A criminal trial can be thought of as a decision problem, the two possible decisions being “guilty” and “not guilty.’’ Moreover, in a criminal trial there is a null hypothesis in the sense of an assertion that we will continue to hold until we have strong evidence against it. Criminal trials are, therefore, similar to hypothesis testing.

23.24 Acceptance sampling. You are a consumer of potatoes in an acceptance sampling situation. Your acceptance sampling plan has probability 0.01 of passing a truckload of potatoes that does not meet quality standards. You might think that the truckloads that pass are almost all good. Alas, it is not so.

23.25 Do our athletes graduate? Return to the study in Exercise 22.19 (page 540), which found that 137 of 190 athletes admitted to a large university graduated within six years. The proportion of athletes who graduated was significantly lower (P = 0.033) than the 78% graduation rate for all students. It may be more informative to give a 95% confidence interval for the graduation rate of athletes. Do this.

23.26 Using the Internet. Return to the study in Exercise 22.20 (page 541), which found that 168 of an SRS of 200 entering students at a large state university said that they used the Internet frequently for research or homework. This differed significantly (P = 0.242) from the 81.8% of all first-year students in the country who claim this.

23.27 Holiday spending. An October 2015 Gallup Poll asked a random sample of 1015 American adults “Roughly how much money do you think you personally will spend on Christmas gifts this year?” The mean holiday spending estimate in the sample was . We will treat these data as an SRS from a Normally distributed population with standard deviation .

23.28 Is it significant? Over several years and many thousands of students, 85% of the high school students in a large city have passed the competency test that is one of the requirements for a diploma. Now reformers claim that a new mathematics curriculum will increase the percentage who pass. A random sample of 1000 students follow the new curriculum. The school board wants to see an improvement that is statistically significant at the 5% level before it will adopt the new program for all students. If p is the proportion of all students who would pass the exam if they followed the new curriculum, we must test

H₀: p = 0.85

H_a: p > 0.85

23.29 We like confidence intervals. The previous exercise compared significance tests about the proportion p of all students who would pass a competency test, based on data showing that either 868 or 869 of an SRS of 1000 students passed. Give the 95% confidence interval for p in both cases. The intervals make clear how uncertain we are about the true value of p and how little difference there is between the two sample outcomes.