Statistical properties of the fuzzy p-value

We have considered different approaches for the calculation of the p-value for fuzzy statistical tests. We have considered several approaches known from literature, and a new one, based on Lehmann's concept of the interval statistical hypothesis. For the particular problem of testing hypotheses about the mean in the normal distribution with known standard deviation, and a certain type of fuzziness (both in data and tested hypotheses), we have found approximate probability distributions of the respective defuzzified p-values. These distributions let us evaluate the compatibility of the observed data with the assumed hypothetical model.