Type II error of a defect-rate test

Question

A factory tests whether its defect rate has risen above the target of $10\%$ : $H_0: p = 0.10$ against $H_1: p > 0.10$ . It inspects $n = 400$ items and rejects $H_0$ if the number of defects is $52$ or more. Suppose the true defect rate has risen to $p = 0.15$ .

What is the probability of a type II error (failing to flag the higher rate)?

Accepted Answer

Model the count under H1. With p = 0.15 and n = 400: mean = 400 	imes 0.15 = 60, and 	ext{sd} = \sqrt{400 	imes 0.15 	imes 0.85} = \sqrt{51} \approx 7.14. Apply the normal approximation with continuity correction. "Fail to reject" is X \le 51, boundary 51.5: \beta = P(X \le 51.5 \mid p = 0.15) = \Phi\!\left(\frac{51.5 - 60}{7.14}ight) = \Phi(-1.19) \approx 0.117. So a true jump to a 15% defect rate is missed only about 12% of the time; power is about 0.88. The alternative's mean (60 defects) sits well above the cutoff of 52, so the higher rate usually produces enough defects to trigger …

Type II error of a defect-rate test

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