Chapter 5.12, Problem 17E

Refer to Exercise 5.16. Graph the power curve for rejecting H₀: μ ≥ 26 for the following values of μ: 20, 21, 22, 23, 24, 25, and 26.

1. a. Describe the change in the power as the value of μ decreases from μ₀ = 26.
2. b. Suppose the value of n remains at 50 but α is decreased to α = .01. Without recalculating the values of the power, superimpose on the graph for α = .05 and n = 50 the power curve for α = .01 and n = 50.
3. c. Suppose the value of n is decreased to 35 but α is kept at α = .05. Without recalculating the values of the power, superimpose on the graph for α = .05 and n = 50 the power curve for α = .05 and n = 35.

5.16 A study is designed to test the hypotheses H₀: μ ≥ 26 versus H_a: μ < 26. A random sample of 50 units was selected from a specified population, and the measurements were summarized to $\bar{y}$ = 25.9 and s = 7.6.

1. a. With α = .05, is there substantial evidence that the population mean is less than 26?
2. b. Calculate the probability of making a Type II error if the actual value of the population mean is at most 24.
3. c. If the sample size is doubled to 100, what is the probability of making a Type II error if the actual value of the population mean is at most 24?