SummaryThe use of unilateral and bilateral critical regions in the testing of hypotheses.This paper endeavours to explain in simple terms the principles of the Neyman‐Pearson theory.Let H0be the hypothesis to be tested. Then the observations availuble for testing H0are first condensed into a single statistic, x, the distribution of which can be evaluated when H0is true. Out of the possible range of values of this statistic a critical region is selected, and H0is rejected when x falls in this region, and not rejected when x falls outside. This critical region is chosen so that(A). the probability of rejecting H0when true has a prescribed upper limit a (or preferably is equal to a);(B). the probability of rejecting H0is higher when an alternative hypothesis H1, is true than when H0itself is true; and(C). if possible, the probability of rejecting H0is a maximum when any hypothesis h out of a set of alternative hypotheses is true.When the set of alternative hypotheses is specified by a single parameter θ, H0corresponding to θ=θ0, these requirements will, under conditions of a general nature, lead to the use of unilateral or of bilateral tail‐errors according to the range of values of θ taken into consideration. If it may be assumed that either θ=θ0or θ=θθ0, the critical regions must be unilateral, but if θ can be both greater or smaller than θ0, they have to be bilateral.The arguments are illustrated by a