A Note on Uniformly Most Powerful Two-Sided Tests

One of the standard examples (generally given in a first-year course on inference) of a testing situation in which there exists no uniformly most powerful test is that of the two-sided test of a normal mean.A fact, however, that is rarely mentioned about this example (also applicable to many other two-sided testing problems) is that the usual two-sided test is actually uniformly most powerful within the class of tests with critical regions that are symmetric about the null hypothesis (which is often the only class worth considering for the two-sided problem, in practice).Whereas this fact may be regarded as a straightforward application of the general theory of uniformly most powerful invariant tests; this particular point is not emphasized, nor is the fact generally mentioned in nonmathematical discussions.In this article, a simple, direct proof of this property of two-sided tests is given, thus offering the less theoretically oriented statistician a greater sense of security in the appropriateness of a test that theoreticians generally label as being nonoptimal.