Chapter 10.5, Problem 61E

Let μ denote the mean lifetime (in hours) for a certain type of battery under controlled laboratory conditions. A test of $H_0:\mu =10$ versus $H_a:\mu <10$ will be based on a sample of size 36. Suppose that σ is known to be 0.6, so ${\sigma }_{\bar{x}}=0.1$. The appropriate test statistic is then

$z=\frac{\bar{x}-10}{0.1}$

1. a. What is α for the test procedure that rejects H₀ if $z\le -1.28$?
2. b. If the test procedure of Part (a) is used, calculate β when μ = 9.8, and interpret this error probability.
3. c. Without doing any calculation, explain how β when μ = 9.5 compares to β when μ = 9.8. Then check your assertion by calculating β when μ = 9.5.
4. d. What is the power of the test when μ = 9.8? when μ = 9.5?