The Modified Bayesian-Frequentist test of Berger, Brown and Wolpert (1994), is considered here in the context of normal hypothesis testing. We focus attention on the testing of a precise null hypothesis versus a composite alternative, either the one-sided or the two-sided type. We study the properties of the corresponding modified Bayesian-Frequentist test and in particular the large-sample behavior of its {\it no decision region} under two different classes of prior p.d.fs; the shifted conjugate class and a domain-restricted noninformative class. It is shown that under these prior classes, the size of the no-decision region of the test is rather small, compared to the relevant sample size. A lower bound on the conditional probability of the type I error is also provided. Postscript File (547kB)