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We discuss the role that the null hypothesis should play in the constructionof a test statistic used to make a decision about that hypothesis. To constructthe test statistic for a point null hypothesis about a binomial proportion, acommon recommendation is to act as if the null hypothesis is true. We arguethat, on the surface, the one-sample t-test of a point null hypothesis about aGaussian population mean does not appear to follow the recommendation. We showhow simple algebraic manipulations of the usual t-statistic lead to anequivalent test procedure consistent with the recommendation. We providegeometric intuition regarding this equivalence and we consider extensions totesting nested hypotheses in Gaussian linear models. We discuss an applicationto graphical residual diagnostics where the form of the test statistic makes apractical difference. By examining the formulation of the test statistic frommultiple perspectives in this familiar example, we provide simple, concreteillustrations of some important issues that can guide the formulation ofeffective solutions to more complex statistical problems.

Rediscovering a little known fact about the t-test and the F-test: Algebraic, Geometric, Distributional and Graphical Considerations