6.124 Telemarketing wages. An advertisement in the student newspaper asks you to consider working for a telemarketing company. The ad states, “Earn between $500 and $1000 per week.” Do you think that the ad is describing a confidence interval? Explain your answer.

6.125 Exercise and statistics exams. A study examined whether light exercise performed an hour before the final exam in statistics affects how students perform on the exam. The P-value was given as 0.13.

6.126 Roulette. A roulette wheel has 18 red slots among its 38 slots. You observe many spins and record the number of times that red occurs. Now you want to use these data to test whether the probability of a red has the value that is correct for a fair roulette wheel. State the hypotheses H₀ and H_a that you will test.

6.127 Food selection by children in school cafeterias. A group of researchers examined whether children’s food selection in a school cafeteria met the standards set by the School Meals Initiative. They measured food selection and food intake of 2049 fourth- through sixth-grade students in 33 schools over a three-day period using digital photography. The following table summarizes some of the food intake measurements.³¹

Given the large sample sizes, we can assume that the sample standard deviations are the population standard deviations.

6.128 Coverage percent of 95% confidence interval. For this exercise, you will use the Confidence Interval applet. Set the confidence level at 95% and click the “Sample” button 10 times to simulate 10 confidence intervals. Record the percent hit. Simulate another 10 intervals by clicking another 10 times (do not click the “Reset” button). Record the percent hit for your 20 intervals. Repeat the process of simulating 10 additional intervals and recording the results until you have a total of 200 intervals. Create a time plot of your results and write a summary of what you have found.

6.129 Coverage percent of 90% confidence interval. Refer to the previous exercise. Do the simulations and report the results for 90% confidence.

6.130 Effect of sample size on significance. You are testing the null hypothesis that μ = 0 versus the alternative μ > 0 using α = 0.05. Assume that σ = 16. Suppose that and n = 10. Calculate the test statistic and its P-value. Repeat assuming the same value of but with n = 20. Do the same for sample sizes of 30, 40, and 50. Plot the values of the test statistic versus the sample size. Do the same for the P-values. Summarize what this demonstration shows about the effect of the sample size on significance testing.

6.131 Survey response and margin of error. Suppose that a business conducts a marketing survey. As is often done, the survey is conducted by telephone. As it turns out, the business was only able to elicit responses from less than 10% of the randomly chosen customers. The low response rate is attributable to many factors, including caller ID screening. Undaunted, the marketing manager was pleased with the sample results because the margin of error was quite small, and thus the manager felt that the business had a good sense of the customers’ perceptions on various issues. Do you think the small margin of error is a good measure of the accuracy of the survey’s results? Explain.

6.132 Reporting margins of error. A U.S. News & World Report article from July 17, 2014, reported Commerce Department estimates of changes in the construction industry:

Construction fell 9.3 percent last month to a seasonally adjusted annual rate of 893,000 homes.

If we turn to the original Commerce Department report (released the same day), it states:

Privately owned housing starts in June were at a seasonally adjusted annual rate of 893,000. This is 9.3 percent (10.3%) below the revised May estimate of 985,000.

The Commerce Department has no evidence that privately owned housing starts rose or fell in June from the previous month.

6.133 Blood phosphorus level in dialysis patients. Patients with chronic kidney failure may be treated by dialysis, in which a machine removes toxic wastes from the blood, a function normally performed by the kidneys. Kidney failure and dialysis can cause other changes, such as retention of phosphorus, that must be corrected by changes in diet. A study of the nutrition of dialysis patients measured the level of phosphorus in the blood of several patients on six occasions. Here are the data for one patient (in milligrams of phosphorus per deciliter of blood):³²

5.4 5.2 4.5 4.9 5.7 6.3

The measurements are separated in time and can be considered an SRS of the patient’s blood phosphorus level. Assume that this level varies Normally with σ = 0.9 mg/dl.

6.134 Cellulose content in alfalfa hay. An agronomist examines the cellulose content of a variety of alfalfa hay. Suppose that the cellulose content in the population has standard deviation σ = 8 milligrams per gram (mg/g). A sample of 15 cuttings has mean cellulose content mg/g.

6.135 Odor threshold of future wine experts. Many food products contain small quantities of substances that would give an undesirable taste or smell if they are present in large amounts. An example is the “off-odors” caused by sulfur compounds in wine. Oenologists (wine experts) have determined the odor threshold, the lowest concentration of a compound that the human nose can detect. For example, the odor threshold for dimethyl sulfide (DMS) is given in the oenology literature as 25 micrograms per liter of wine (μg/l). Untrained noses may be less sensitive, however. Here are the DMS odor thresholds for 10 beginning students of oenology:

31 31 43 36 23 34 32 30 20 24

Assume (this is not realistic) that the standard deviation of the odor threshold for untrained noses is known to be σ = 7 μg/l.

6.136 Where do you buy? Consumers can purchase nonprescription medications at food stores, mass merchandise stores such as Target and Walmart, or pharmacies. About 45% of consumers make such purchases at pharmacies. What accounts for the popularity of pharmacies, which often charge higher prices?

A study examined consumers’ perceptions of overall performance of the three types of stores, using a long questionnaire that asked about such things as “neat and attractive store,” “knowledgeable staff,” and “assistance in choosing among various types of nonprescription medication.” A performance score was based on 27 such questions. The subjects were 201 people chosen at random from the Indianapolis telephone directory. Here are the means and standard deviations of the performance scores for the sample:³³

We do not know the population standard deviations, but a sample standard deviation s from so large a sample is usually close to σ. Use s in place of the unknown σ in this exercise.

6.137 CEO pay. A study of the pay of corporate chief executive officers (CEOs) examined the increase in cash compensation of the CEOs of 104 companies, adjusted for inflation, in a recent year. The mean increase in real compensation was %, and the standard deviation of the increases was s = 55 %. Is this good evidence that the mean real compensation μ of all CEOs increased that year? The hypotheses are

H₀: μ = 0 (no increase)

H_a: μ > 0 (an increase)

Because the sample size is large, the sample s is close to the population σ, so take σ = 55 %.

6.138 Meaning of “statistically significant.” When asked to explain the meaning of “statistically significant at the α = 0.01 level,” a student says, “This means there is only probability 0.01 that the null hypothesis is true.” Is this an essentially correct explanation of statistical significance? Explain your answer.

6.139 More on the meaning of “statistically significant.” Another student, when asked why statistical significance appears so often in research reports, 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.

6.140 Increasing the power. Refer to Example 6.17 (page 375). Suppose the Deely Laboratory wants to make sure the power is at least 80% for μ = 15.30 ppm. It cannot reduce σ, so the options are changing the significance level α and/or sample size n.

6.141 Simulation study of the confidence interval. Use a computer to generate n = 15 observations from a Normal distribution with mean 20 and standard deviation 5: N(20, 5). Find the 95% confidence interval for μ. Repeat this process 100 times and then count the number of times that the confidence interval includes the value μ = 20. Explain your results.

6.142 Simulation study of a test of significance. Use a computer to generate n = 15 observations from a Normal distribution with mean 20 and standard deviation 5: N(20, 5). Test the null hypothesis that μ = 20 using a two-sided significance test. Repeat this process 100 times and then count the number of times that you reject H₀. Explain your results.

6.143 Another simulation study of a test of significance. Use the same procedure for generating data as in the previous exercise. Now test the null hypothesis that μ = 24. Explain your results.

6.144 Simulation study of power. Refer to the previous two exercises. What is the power of detecting a difference of four units (H₀: μ = 24 versus μ = 20) in this setting? Compare this power with the proportion of times you rejected H₀ in the previous exercise. Explain your results.

6.145 Find published studies with confidence intervals. Search the Internet or some journals that report research in your field and find two reports that provide an estimate with a margin of error or a confidence interval. For each report,