Type II error when the bias is small

Question

You flip a coin n = 200 times to test H0: p = 0.5 against H1: p 0.5. The rule rejects H0 if the number of heads is 112 or more. Suppose the coin is actually biased with true p = 0.55. What is the probability of a type II error (missing the real bias)?

Accepted Answer

Model the count under H1. With p = 0.55 and n = 200: mean = 200 	imes 0.55 = 110, and 	ext{sd} = \sqrt{200 	imes 0.55 	imes 0.45} = \sqrt{49.5} \approx 7.04. Apply the normal approximation with continuity correction. "Fail to reject" is X \le 111, boundary 111.5: \beta = P(X \le 111.5 \mid p = 0.55) = \Phi\!\left(\frac{111.5 - 110}{7.04}ight) = \Phi(0.21) \approx 0.584. So the test misses this small bias about 58% of the time; its power is only about 0.42. The rejection cutoff of 112 sits barely above the alternative's own mean of 110, so the biased coin often does not produce enough he…

Type II error when the bias is small

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