Again consider the automobile parts supplier situation. Remember that a problem-solving team will be assigned to rectify the process producing the cylindrical engine parts if the null hypothesis H0: μ = 3 is rejected in favor of Ha: μ = 3. In this exercise we calculate probabilities of various Type II errors in the context of this situation.
a. Suppose that the parts supplier's hypothesis test is based on a sample of n = 100 diameters and that σ equals .023. If the parts supplier sets α = .05, calculate the probability of a Type II error for each of the following alternative values of μ: 2.990. 2.995, 3.005, 3.010.
b. If we want both the probabilities of making a Type II error when μ equals 2.995 and when μ equals 3.005 to be very small, is the parts supplier's hypothesis test adequate? Explain why or why not. If not. and if we wish to maintain the value of a at .05, what must be done?
c. Plot the power of the test versus the alternative values of μ in part a. What happens to the power of the test as the alternative value of μ moves away from 3?