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Statistics / Probability and Statistics 4 / Chapter 9.8 / Problem 4

Probability and Statistics | 4th Edition | ISBN: 9780321500465 | Authors: Morris H. DeGroot, Mark J. Schervish

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Chapter 9, Problem 4

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QUESTION:

Suppose that the proportion p of defective items in a large manufactured lot is unknown, and it is desired to test the following simple hypotheses: H0: p = 0.3, H1: p = 0.4. Suppose that the prior probability that p = 0.3 is 1/4, and the prior probability that p = 0.4 is 3/4; also suppose that the loss from choosing an incorrect decision is 1 unit, and the loss from choosing a correct decision is 0. Suppose that a random sample of n items is selected from the lot. Show that the Bayes test procedure is to reject H0 if and only if the proportion of defective items in the sample is greater than log 7 6 + 1 n log 1 3 log 14 9

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QUESTION:

Suppose that the proportion p of defective items in a large manufactured lot is unknown, and it is desired to test the following simple hypotheses: H0: p = 0.3, H1: p = 0.4. Suppose that the prior probability that p = 0.3 is 1/4, and the prior probability that p = 0.4 is 3/4; also suppose that the loss from choosing an incorrect decision is 1 unit, and the loss from choosing a correct decision is 0. Suppose that a random sample of n items is selected from the lot. Show that the Bayes test procedure is to reject H0 if and only if the proportion of defective items in the sample is greater than log 7 6 + 1 n log 1 3 log 14 9

ANSWER:

Step 1 of 2

Here the losses of making an incorrect decision are the same, that is and are the same and finite. Moreover, the prior probabilities are and .

The number of defectives follows where is the sample size.

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