Suppose a psychiatrist has to classify people as sick or well (hospitalized br not) on the basis of a psychological test. The test scores are normally distributed, with a = 8, and mean 00 = 100 if they are well or p, 120 if they are sick. The losses (regrets) of a wrong classification are obvious: if a healthy person is hospitalized, resources are wasted and the person himself may even be hurt by the treatment. Yet the other loss is even worse: if a sick person is not hospitalized, he may do damage, conceivably fatal. Suppose this second loss is considered roughly five times as serious. From past records it has been found that of the people taking the test, 60% are sick and 40% are healthy.
(a) (1) What should be the critical score above which the person is classified as sick? Then
(2) What is a? (Probability of type I error).
(3) What is fi? (Probability of type H error).
(b) (1) If a classical test is used, arbitrarily setting a = 5%, what then will be the critical score? Then
(2) What is /3?
(3) By how much has the average loss increased by using this less-than-optimal method?
(c) What would we have to assume the ratio of the two regrets to be in order to arrive at a Bayesian test having x = 5 %? Do you think it is reasonable?