6.110 Make a recommendation. Your manager has asked you to review a research proposal that includes a section on sample size justification. A careful reading of this section indicates that the power is 18% for detecting an effect that would be considered important. Write a short report for your manager explaining what this means and make a recommendation on whether or not this study should be run.

6.111 Explain power and sample size. Two studies are identical in all respects except for the sample sizes. Consider the power versus a particular sample size. Will the study with the larger sample size have more power or less power than the one with the smaller sample size? Explain your answer in terms that could be understood by someone with very little knowledge of statistics.

6.112 Power for a different alternative. The power for a two-sided test of the null hypothesis μ = 0 versus the alternative μ = 6 is 0.83. What is the power versus the alternative μ = −6? Explain your answer.

6.113 More on the power for a different alternative. A one-sided test of the null hypothesis μ = 20 versus the alternative μ = 30 has power equal to 0.73. Will the power for the alternative μ = 35 be higher or lower than 0.73? Draw a picture and use this to explain your answer.

6.114 Effect of changing the alternative μ on power. The Statistical Power applet illustrates the power calculation similar to that in Figure 6.16 (page 393). Open the applet and keep the default settings for the null (μ = 0) and the alternative (μ > 0) hypotheses, the sample size (n = 10), the standard deviation (σ = 1), and the significance level (α = 0.05). In the “alt μ =” box, enter the value 1. What is the power? Repeat for alternative μ equal to 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9. Make a table giving μ and the power. What do you conclude?

6.115 Other changes and the effect on power. Refer to the previous exercise. For each of the following changes, explain what happens to the power for each alternative μ in the table.

6.116 Power of the random north–south distribution of trees test. In Exercise 6.70 (page 382), you performed a two-sided significance test of the null hypothesis that the average north–south location of the longleaf pine trees sampled in the Wade Tract was μ = 100. There were 584 trees in the sample and the standard deviation was assumed to be 58. The sample mean in that analysis was . Use the Statistical Power applet to compute the power for the alternative μ = 99 using a two-sided test at the 5% level of significance.

6.117 Power of the random east–west distribution of trees test. Refer to the previous exercise. Note that in the east–west direction, the average location was 113.8. Use the Statistical Power applet to find the power for the alternative μ = 110.

6.118 Planning another test to compare consumption. Example 6.15 (page 372) gives a test of a hypothesis about the mean consumption of sugar-sweetened beverages at your university based on a sample of size n = 100. The hypotheses are

H₀: μ = 286
H_a: μ ≠ 286

While the result was not statistically significant, it did provide some evidence that the mean was smaller than 286. Thus, you plan to recruit another sample of students from your university, but this time use a one-sided alternative. You were thinking of surveying n = 100 students but now wonder if this sample size gives adequate power to detect a decrease of 15 calories per day to μ = 271.

6.119 Planning the dining court survey. Exercise 6.38 (page 364) describes a survey to assess whether a newly designed dining court is viewed more favorably than the old design. The organizers are considering randomly surveying n = 100 student patrons but would like some statistical advice. The hypotheses are

H₀: μ = 4

H_a: μ > 4

and they’ve decided they want adequate power to detect a mean of at least 4.25.

6.120 Choose the appropriate distribution. You must decide which of two discrete distributions a random variable X has. We will call the distributions p₀ and p₁. Here are the probabilities they assign to the values x of X:

You have a single observation on X and wish to test

H₀: p₀ is correct

H_a: p₁ is correct

One possible decision procedure is to reject H₀ only if X ≤ 1.

6.121 Power of the mean SATM score test. Example 6.16 (page 374) gives a test of a hypothesis about the SATM scores of California high school students based on an SRS of 500 students. The hypotheses are

H₀: μ = 485
H_a: μ > 485

Assume that the population standard deviation is σ = 100. The test rejects H₀ at the 1% level of significance when z ≥ 2.326, where

Is this test sufficiently sensitive to usually detect an increase of 14 points in the population mean SATM score? Answer this question by calculating the power of the test to detect the alternative μ = 499.

6.122 More on choosing the appropriate distribution. Refer to Exercise 6.120. Suppose that instead of a single observation X, you obtained two observations and use the decision rule to reject when .

6.123 Computer-assisted career guidance systems. A wide variety of computer-assisted career guidance systems have been developed over the last decade. These programs use factors such as student interests, aptitude, skills, personality, and family history to recommend a career path. For simplicity, suppose that a program recommends a high school graduate either to go to college or to join the workforce.