Power of a two-sided test

Question

A two-sided test of H0: \mu = 0 against H1: \mu 
e 0 uses n = 64 observations, known \sigma, and level \alpha = 0.05. Compute the power to detect a true mean of \mu = 0.3\sigma.

Accepted Answer

Rejection region. With known \sigma and a two-sided \alpha = 0.05, we reject when Z 1.96, where Z = \bar X / (\sigma/\sqrt n). Distribution under the alternative. When \mu = 0.3\sigma, the statistic Z is normal with variance 1 and mean (the noncentrality) \frac{\mu}{\sigma/\sqrt n} = 0.3 	imes \sqrt{64} = 0.3 	imes 8 = 2.4. Power. Add up the chance of landing in each rejection tail: 	ext{Power} = P(Z 1.96) + P(Z < -1.96) = \Phi(2.4 - 1.96) + \Phi(-2.4 - 1.96). The first term is \Phi(0.44) \approx 0.67; the second is \Phi(-4.36) \approx 0, negligible because a positive effect almost never t…

Power of a two-sided test

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