What is the power to detect a shift to 108?

Question

A quality engineer tests H0: \mu = 100 against H1: \mu 100 with a known \sigma = 15, a sample of n = 25, at the one-sided 5% level. Suppose the process has actually shifted to a true mean of \mu = 108. Find the power of the test (the probability it correctly rejects H0) and the Type II error rate \beta.

Accepted Answer

Power is the chance of rejecting H0 when the alternative is true. First find the rejection rule, then evaluate its probability under the true mean. Standard error: 	ext{SE} = \sigma/\sqrt{n} = 15/\sqrt{25} = 15/5 = 3. The one-sided test rejects when \bar x exceeds the critical value: \bar x{	ext{crit}} = 100 + z{0.05}\,	ext{SE} = 100 + 1.645(3) = 104.935. Now evaluate the probability of landing in that rejection region when the true mean is 108: 	ext{Power} = P(\bar x 104.935 \mid \mu = 108) = P\!\left(Z \frac{104.935 - 108}{3}ight) = P(Z -1.022). 	ext{Power} = \Phi(1.022) \approx 0.85…

What is the power to detect a shift to 108?

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