How big a sample for 90% power?

Question

You are planning a one-sided test of H0: \mu = 0 against H1: \mu 0 at level \alpha = 0.025, with known \sigma. You want 90\% power to detect an effect of size \delta = 0.4\sigma. How many observations do you need?

Accepted Answer

Power 90\% means the standardized effect must sit far enough above the critical value that a standard normal exceeds it 90\% of the time. Writing z\beta for the 90\% point (z{0.10} = 1.28) and z\alpha for the critical value (z{0.025} = 1.96), the requirement \frac{\delta\sqrt n}{\sigma} - z\alpha \ge z\beta rearranges to the standard sample-size formula n \ge \left(\frac{(z\alpha + z\beta)\,\sigma}{\delta}\right)^2 = \frac{(z\alpha + z\beta)^2}{(\delta/\sigma)^2}. Plugging in \delta/\sigma = 0.4: n \ge \frac{(1.96 + 1.28)^2}{0.4^2} = \frac{3.24^2}{0.16} = \frac{10.50}{0.16} \approx 65.6. Roun…

How big a sample for 90% power?

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