On Proving That Xbar and S^2 are Independent

The need to establish the independence of the sample mean and the sample variance in sampling from a normal population arises early in a course in statistics.For the result is an essential ingredient in the derivation of the Student-t distribution for statistical inference.Often this need arises before the tools, notably multivariate methods, for a rigorous proof are available.Occasionally one will find attempts to derive this result using only bivariate assumptions.A recent article in this journal, as well as some current textbooks, offer such a proof.In all cases there are serious questions about the validity of the proofs.