Miss rate for a sensor with wider spread

Question

A sensor reads a value X. In the normal state (H0), X \sim 	ext{Normal}(10, 2). In the fault state (H1), X \sim 	ext{Normal}(16, 2). The rule declares "fault" whenever X 13. What is the probability of a type II error (missing a real fault)?

Accepted Answer

A type II error means a fault is present but X \le 13. Compute \beta. Under H1, X \sim 	ext{Normal}(16, 2), so standardize using \sigma = 2: \beta = P(X \le 13 \mid H1) = \Phi\!\left(\frac{13 - 16}{2}ight) = \Phi(-1.5) \approx 0.067. So the rule misses a real fault about 6.7% of the time; power is about 0.93. For context. The cutoff sits exactly halfway between the two means (13 is midway between 10 and 16), each 1.5 standard deviations away. That symmetry makes the two error rates equal: \alpha = P(X 13 \mid H0) = 1 - \Phi\!\left(\frac{13 - 10}{2}ight) = 1 - \Phi(1.5) \approx 0.067. Tak…

Miss rate for a sensor with wider spread

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