Power of a two-sided reaction-time test

Question

A lab tests whether a treatment changes the mean reaction time from the known baseline of 100 ms: H0: \mu = 100 against the two-sided H1: \mu 
eq 100. Reaction times have a known standard deviation of \sigma = 20 ms. They collect n = 25 subjects and test at two-sided \alpha = 0.05. Suppose the true mean is \mu = 110 ms. What is the power of this test?

Accepted Answer

Standard error. \sigma/\sqrt{n} = 20/\sqrt{25} = 20/5 = 4 ms. Effect in SE units. The true mean is (110 - 100)/4 = 2.5 standard errors above the null. Power for a two-sided test. A two-sided test rejects if the sample mean is far in either tail, using the cutoff z{\alpha/2} = 1.96 on each side. Power adds the chance of landing past each cutoff: 	ext{power} = \Phi(	ext{effect} - z{\alpha/2}) + \Phi(-	ext{effect} - z{\alpha/2}). = \Phi(2.5 - 1.96) + \Phi(-2.5 - 1.96) = \Phi(0.54) + \Phi(-4.46) \approx 0.705 + 0.000 \approx 0.71. The lower-tail piece is negligible because a true mean well abo…

Power of a two-sided reaction-time test

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