Consider the problem of testing the hypothesisH0: θ ∈W0against the alternativeH1: θ ∈W1when a certain subsetE0⊂W1is given. This subsetE0is such that the risk of acceptingH0will be significant economically only if θ ∈E0. For this problem a test obtained by generalising the equal probability test of Arrow (1960) is suggested. Applications of the test procedure to the problems of combining several independent tests and of comparing several populations with a standard are discussed.