Suppose in the disk drive factory of example 88 we can

Question

Suppose in the disk drive factory of Example 8.8, we can observe K, the number of failed devices out of n devices tested. As in the example, let H_i denote the hypothesis that the failure rate is q_i.

(a) Assuming q_0 1, what is the ML hypothesis test based on an observation of K?

(b) What are the conditional probabilities of error P_FA = P[A₁|H₀] and PMISS = P[A₀|H₁]? Calculate these probabilities for n = 500, q₀ = 10^-4, q₁ = 10^-2.

(c) Compare this test to that considered in Example 8.8. Which test is more reliable? Which test is easier to implement?

Example 8.8

At a computer disk drive factory, the manufacturing failure rate is the probability that a randomly chosen new drive fails the first time it is powered up. Normally the production of drives is very reliable, with a failure rate q₀ = 10^-4. However, from time to time there is a production problem that causes the failure rate to jump to q₁ = 10^-1. Let H_i denote the hypothesis that the failure rate is q_i. Every morning, an inspector chooses drives at random from the previous day's production and tests them. If a failure occurs too soon, the company stops production and checks the critical part of the process. Production problems occur at random once every ten days, so that P[H₁] = 0.1 = 1 - P[H₀]. Based on N, the number of drives tested up to and including the first failure, design a MAP hypothesis test. Calculate the conditional error probabilities P_FA and P_MISS and the total error probability P_ERR.