Type II error with a strong bias, small sample

Question

You flip a coin n = 64 times to test H0: p = 0.5 against H1: p 0.5. The rule rejects H0 if the number of heads is 40 or more. Suppose the coin is strongly biased with true p = 0.65. What is the probability of a type II error (missing the bias)?

Accepted Answer

Model the count under H1. With p = 0.65 and n = 64: mean = 64 	imes 0.65 = 41.6, and 	ext{sd} = \sqrt{64 	imes 0.65 	imes 0.35} = \sqrt{14.56} \approx 3.82. Apply the normal approximation with continuity correction. "Fail to reject" is X \le 39, boundary 39.5: \beta = P(X \le 39.5 \mid p = 0.65) = \Phi\!\left(\frac{39.5 - 41.6}{3.82}ight) = \Phi(-0.55) \approx 0.291. So even a strong 15-point bias is missed about 29% of the time when you only flip 64 times; power is about 0.71. A large true effect helps, but a small sample keeps the noise wide enough that misses remain common. (For refe…

Type II error with a strong bias, small sample

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