How much power do you lose going from 5% to 1%?

Question

A one-sided test uses H0: \mu = 0 against H1: \mu 0 with a known \sigma = 10, a sample of n = 25. The true mean is \mu = 6. At \alpha = 0.05 the power is 0.91. Recompute the power if the significance level is tightened to \alpha = 0.01, and comment on the trade-off.

Accepted Answer

Standard error: 	ext{SE} = 10/\sqrt{25} = 10/5 = 2. At \alpha = 0.01 the one-sided critical value is z{0.01} = 2.326, so the test rejects when \bar x 0 + 2.326(2) = 4.652. Evaluate the power under the true mean \mu = 6: 	ext{Power} = P(\bar x 4.652 \mid \mu = 6) = P\!\left(Z \frac{4.652 - 6}{2}ight) = P(Z -0.674) = \Phi(0.674) \approx 0.75. Compared with power 0.91 at \alpha = 0.05 (whose cutoff was 1.645 	imes 2 = 3.29), tightening to 1% pushes the cutoff out to 4.652 and drops power from about 0.91 to about 0.75. Conclusion: cutting the false-positive rate from 5% to 1% raised \beta fr…

How much power do you lose going from 5% to 1%?

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