Abstract
In the bivariate normal, n=2 case, when testing H0:μx=μy=0,σ2 x=σ2 y=1, ρ=0 vs. H1:μx=μy=0,σ2 x=σ2 y=1, 0<ρ<1, it is shown that the median p-values given by the locally most powerful test and the distantly most powerful test are both beaten everywhere by the median of a third test.