21.9 A student survey. Tonya wants to estimate what proportion of the students in her dormitory like the dorm food. She interviews an SRS of 50 of the 175 students living in the dormitory. She finds that 14 think the dorm food is good.

21.10 Fire the coach? A college president says, “99% of the alumni support my firing of Coach Boggs.’’ You contact an SRS of 200 of the college’s 15,000 living alumni and find that 66 of them support firing the coach.

21.11 Are teachers engaged in their work? Results from a Gallup survey conducted in 2013 and 2014 reveal that 30% of Kindergarten through Grade 12 school teachers report feeling engaged in their work. The report from this random sample of 6711 teachers stated that, with 95% confidence, the margin of sampling error was ±1.0%. Explain to someone who knows no statistics what the phrase “95% confidence’’ means in this report.

21.12 Gun control. A December 2014 Pew Research Center poll asked a sample of 1507 adults whether they thought it was important to control gun ownership. A total of 693 of the poll respondents were in support of controls on gun ownership. Although the samples in national polls are not SRSs, they are similar enough that our method gives approximately correct confidence intervals.

21.13 Using public libraries. The Pew Research Center Libraries 2015 Survey examined the attitudes and behaviors of a nationally representative sample of 2004 people ages 16 and older who were living in the United States. One survey question asked whether the survey respondent had visited, in person, a library or a bookmobile in the last year. A total of 922 of those surveyed answered Yes to this question. Although the samples in national polls are not SRSs, they are similar enough that our method gives approximately correct confidence intervals.

21.14 Computer crime. Adults are spending more and more time on the Internet, and the number experiencing computer- or Internet-based crime is rising. A 2010 Gallup Poll of 1025 adults, aged 18 and over, found that 113 of those in the sample said that they or a member of their household were victims of computer or Internet crime on their home computer in the past year. Although the samples in national polls are not SRSs, they are similar enough that our method gives approximately correct confidence intervals.

21.15 Gun control. In Exercise 21.12, you constructed a 95% confidence interval based on a random sample of n = 1507 adults. How large a sample would be needed to get a margin of error half as large as the one in Exercise 21.12? You may find it helpful to refer to the discussion surrounding Example 5 in Chapter 3 (page 47).

21.16 The effect of sample size. A December 2014 CBS News/New York Times poll found that 52% of its sample thought that basic medical care in the United States was affordable. Give a 95% confidence interval for the proportion of all adults who feel this way, assuming that the result comes from a sample of size

21.17 Random digits. We know that the proportion of 0s among a large set of random digits is p = 0.1 because all 10 possible digits are equally probable. The entries in a table of random digits are a random sample from the population of all random digits. To get an SRS of 200 random digits, look at the first digit in each of the 200 five-digit groups in lines 101 to 125 of Table A in the back of the book. How many of these 200 digits are 0s? Give a 95% confidence interval for the proportion of 0s in the population from which these digits are a random sample. Does your interval cover the true parameter value, p = 0.1?

21.18 Tossing a thumbtack. If you toss a thumbtack on a hard surface, what is the probability that it will land point up? Estimate this probability p by tossing a thumbtack 100 times. The 100 tosses are an SRS of size 100 from the population of all tosses. The proportion of these 100 tosses that land point up is the sample proportion . Use the result of your tosses to give a 95% confidence interval for p. Write a brief explanation of your findings for someone who knows no statistics but wonders how often a thumbtack will land point up.

21.19 Don’t forget the basics. The Behavioral Risk Factor Surveillance System survey found that 792 individuals in its 2010 random sample of 6911 college graduates in California said that they had engaged in binge drinking in the past year. We can use this finding to calculate confidence intervals for the proportion of all college graduates in California who engaged in binge drinking in the past year. This sample survey may have bias that our confidence intervals do not take into account. Why is some bias likely to be present? Does the sample proportion 11.5% probably overestimate or underestimate the true population proportion?

21.20 Count Buffon’s coin. The eighteenth-century French naturalist Count Buffon tossed a coin 4040 times. He got 2048 heads. Give a 95% confidence interval for the probability that Buffon’s coin lands heads up. Are you confident that this probability is not 1/2? Why?

21.21 Share the wealth. The New York Times conducted a nationwide poll of 1650 randomly selected American adults. Of these, 1089 felt that money and wealth in this country should be more evenly distributed among more people. We can consider the sample to be an SRS.

21.22 Harley motorcycles. In 2013, it was reported that 55% of the new motorcycles that were registered in the United States were Harley-Davidson motorcycles. You plan to interview an SRS of 600 new motorcycle owners.

21.23 Do you jog? Suppose that 10% of all adults jog. An opinion poll asks an SRS of 400 adults if they jog.

21.24 The quick method. The quick method of Chapter 3 (page 46) uses as a rough recipe for a 95% confidence interval for a population proportion. The margin of error from the quick method is a bit larger than needed. It differs most from the more accurate method of this chapter when is close to 0 or 1. An SRS of 500 motorcycle registrations finds that 68 of the motorcycles are Harley-Davidsons. Give a 95% confidence interval for the proportion of all motorcycles that are Harleys by the quick method and then by the method of this chapter. How much larger is the quick-method margin of error?

21.25 68% confidence. We used the 95 part of the 68–95–99.7 rule to give a recipe for a 95% confidence interval for a population proportion p.

21.26 Simulating confidence intervals. In Exercise 21.25, you found the recipe for a 68% confidence interval for a population proportion p. Suppose that (unknown to anyone) 60% of Americans actively tried to avoid drinking regular soda or pop in 2015.

21.27 Gun control. Exercise 21.12 reports a Pew Research Center poll in which 693 of a random sample of 1507 adults were in support of controls on gun ownership. Use Table 21.1 to give a 90% confidence interval for the proportion of all adults who feel this way. How does your interval compare with the 95% confidence interval from Exercise 21.12?

21.28 Using public libraries. Exercise 21.13 reports a Pew Research Center Libraries 2015 Survey that found that 922 in a random sample of 2004 American adults said that they had visited, in person, a library or bookmobile in the last year. Use Table 21.1 to give a 99% confidence interval for the proportion of all American adults who have done this. How does your interval compare with the 95% confidence interval of Exercise 21.13?

21.29 Organic Food. A 2014 Consumer Reports National Research Center survey on food labeling found that 49% of a random sample of 1004 American adults report looking for information on food labels about whether the food they are purchasing is organic. Use this survey result and Table 21.1 to give 70%, 80%, 90%, and 99% confidence intervals for the proportion of all adults who feel this way. What do your results show about the effect of changing the confidence level?

21.30 Unhappy HMO patients. How likely are patients who file complaints with a health maintenance organization (HMO) to leave the HMO? In one year, 639 of the more than 400,000 members of a large New England HMO filed complaints. Fifty-four of the complainers left the HMO voluntarily. (That is, they were not forced to leave by a move or a job change.) Consider this year’s complainers as an SRS of all patients who will complain in the future. Give a 90% confidence interval for the proportion of complainers who voluntarily leave the HMO.

21.31 Estimating unemployment. The Bureau of Labor Statistics (BLS) uses 90% confidence in presenting unemployment results from the monthly Current Population Survey (CPS). The September 2015 survey reported that of the 156,715 individuals surveyed in the civilian labor force, 148,800 were employed and 7915 were unemployed. The CPS is not an SRS, but for the purposes of this exercise, we will act as though the BLS took an SRS of 156,715 people. Give a 90% confidence interval for the proportion of those surveyed who were unemployed. (Note: Example 3 in Chapter 8 on page 166 explains how unemployment is measured.)

21.32 Safe margin of error. The margin of error is 0 when is 0 or 1 and is largest when is 1/2. To see this, calculate for = 0, 0.1, 0.2, . . . , 0.9, and 1. Plot your results vertically against the values of horizontally. Draw a curve through the points. You have made a graph of . Does the graph reach its highest point when ? You see that taking gives a margin of error that is always at least as large as needed.

21.33 The idea of a sampling distribution. Figure 21.1 (page 496) shows the idea of the sampling distribution of a sample proportion in picture form. Draw a similar picture that shows the idea of the sampling distribution of a sample mean .

21.34 IQ test scores. Here are the IQ test scores of 31 seventh-grade girls in a Midwest school district:

114   100   104   89   102   91   114   114   103   105

108   130   120   132   111   128   118   119   86   72

111   103   74   112   107   103   98   96   112   112   93

21.35 Averages versus individuals. Scores on the ACT college entrance examination vary Normally with mean μ = 18 and standard deviation σ = 6. The range of reported scores is 1 to 36.

21.36 Blood pressure. A randomized comparative experiment studied the effect of diet on blood pressure. Researchers divided 54 healthy white males at random into two groups. One group received a calcium supplement, and the other group received a placebo. At the beginning of the study, the researchers measured many variables on the subjects. The average seated systolic blood pressure of the 27 members of the placebo group was reported to be with a standard deviation of .

21.37 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 mean and s = 0.312. Give a 90% confidence interval for the mean of all numbers produced by the software.

21.38 Will they charge more? A bank wonders whether omitting the annual credit card fee for customers who charge at least $2500 in a year will increase the amount charged on its credit cards. The bank makes this offer to an SRS of 200 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 $346, and the standard deviation is $112. Give a 99% confidence interval for the mean amount charges would have increased if this benefit had been extended to all such customers.

21.39 A sampling distribution. Exercise 21.37 concerns the mean of the random numbers generated by a computer program. The mean is supposed to be 0.5 because the numbers are supposed to be spread at random between 0 and 1. We asked the software to generate samples of 100 random numbers repeatedly. Here are the sample means for 50 samples of size 100:

0.532 0.450 0.481 0.508 0.510 0.530 0.499 0.461 0.543 0.490

0.497 0.552 0.473 0.425 0.449 0.507 0.472 0.438 0.527 0.536

0.492 0.484 0.498 0.536 0.492 0.483 0.529 0.490 0.548 0.439

0.473 0.516 0.534 0.540 0.525 0.540 0.464 0.507 0.483 0.436

0.497 0.493 0.458 0.527 0.458 0.510 0.498 0.480 0.479 0.499

The sampling distribution of is the distribution of the means from all possible samples. We actually have the means from 50 samples. Make a histogram of these 50 observations. Does the distribution appear to be roughly Normal, as the central limit theorem says will happen for large enough samples?

21.40 Will they charge more? In Exercise 21.38, you carried out the calculations for a confidence interval based on a bank’s experiment in changing the rules for its credit cards. You ought to ask some questions about this study.

21.41 A sampling distribution, continued. Exercise 21.39 presents 50 sample means from 50 random samples of size 100. Using a calculator, find the mean and standard error of these 50 values. Then answer these questions.

21.42 Plus four confidence intervals for a proportion. The large-sample confidence interval for a sample proportion p is easy to calculate. It is also easy to understand because it rests directly on the approximately Normal distribution of . Unfortunately, confidence levels from this interval can be inaccurate, particularly with smaller sample sizes where there are only a few successes or a few failures. The actual confidence level is usually less than the confidence level you asked for in choosing the critical value z*. That’s bad. What is worse, accuracy does not consistently get better as the sample size increases. There are “lucky’’ and “unlucky’’ combinations of the sample size and the true population proportion p.

Fortunately, there is a simple modification that is almost magically effective in improving the accuracy of the confidence interval. We call it the “plus four’’ method because all you need to do is add four imaginary observations, two successes and two failures. With the added observations, the plus four estimate of p is

The formula for the confidence interval is exactly as before, with the new sample size and number of successes. To practice using the plus four confidence interval, consider the following problem. Cocaine users commonly snort the powder up the nose through a rolled-up paper currency bill. Spain has a high rate of cocaine use, so it’s not surprising that euro paper currency in Spain often shows traces of cocaine. Researchers collected 20 euro bills in each of several Spanish cities. In Madrid, 17 out of 20 bore traces of cocaine. The researchers note that we can’t tell whether the bills had been used to snort cocaine or had been contaminated in currency-sorting machines. Use the plus four confidence interval method to estimate the proportion of all euro bills in Madrid that have traces of cocaine.