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The paper deals with the problem of the interpretation of the results of statistical tests in terms of the theory of possibility. The well known concept in statistics of the observed test size p (also known as p-value and significance) has been given a new possibilistic interpretation and generalised for the case of imprecisely defined statistical hypotheses and vague statistical data. The proposed approach allows a practitioner to evaluate the test results using intuitive concepts of possibility, necessity or indifference.