The Fool's Type IIa Error.
AbstractThe thesis of the present paper is that there exists a sampling subspace in the testing of hypotheses where a Type II error is made for completely illogical reasons from the standpoint of probability. The hypothesized distribution, A, contains an α error rate for a rejection level, while the alternate distribution, B, contains a (usually larger) β error rate. The unit normal Z values for α and β are coincident in the sampling space, but the actual intersection of A and B is closer to the mean of A, so that there are more B’s than A’s in the sampling subspace between the intersection and the (Zα, Zβ) point. It is in this sampling subspace that the Fool’s Type IIa error is made; i.e., the tested hypothesis is not rejected, but there are obviously more B’s than A’s. The thesis of the present paper is that, from the standpoint of maximum likelihood and union intersection, only a fool would make that kind of error. Consideration is also given to the case of unequal probabilities of populations or condi...