In this paper an approach is presented how to test fuzzily formulated hypotheses with crisp-data. The quantities α and β, the probabilities of the errors of type I and of type II, are suitably generalized and the concept of a best test is introduced. Within the framework of a one-parameter exponential distribution family the search for a best test is considerably reduced. Furthermore, it is shown under very weak conditions that α and β can simultaneously be diminished by increasing the sample size even in the case of testing H0 against the omnibus alternative H1: not H0, a result completely different from the case of crisp sets H0 and H1: not H0.
