A proposal of fuzzy test for statistical hypothesis

Taking up the typical test problem of statistical hypotheses, the paper studies fuzzy statistical hypotheses, their test method and its significance. Concretely, the null hypothesis H₀ that the population mean μ is roughly μ₀(μ=μ¯₀ ) is tested by use of sample data x_i(i=1,...,n) which is independently obtained from normal population, under the alternative one H₁ that is not roughly μ₀(μ≠μ¯₀). Their fuzzy hypotheses bring the possible rejection region and the necessary one corresponding to the conventional method. New aspects of this study are as follows: (1) this method arrives at a soft decision by referring to the grade of the possibility or the necessity that the sample value experimentally obtained belong to the above possible rejection region or the necessary one. On the other hand, the conventional one gives the strict and mechanical decision whether the hypothesis is acceptable or not. (2) The conventional method has the defect that the power of the test heavily decreases at μ=μ₁ close to μ₀ The proposed one can improve the defect thanks to the fuzzy hypotheses