We discuss the role that the null hypothesis should play in the construction
of a test statistic used to make a decision about that hypothesis. To construct
the test statistic for a point null hypothesis about a binomial proportion, a
common recommendation is to act as if the null hypothesis is true. We argue
that, on the surface, the one-sample t-test of a point null hypothesis about a
Gaussian population mean does not appear to follow the recommendation. We show
how simple algebraic manipulations of the usual t-statistic lead to an
equivalent test procedure consistent with the recommendation. We provide
geometric intuition regarding this equivalence and we consider extensions to
testing nested hypotheses in Gaussian linear models. We discuss an application
to graphical residual diagnostics where the form of the test statistic makes a
practical difference. By examining the formulation of the test statistic from
multiple perspectives in this familiar example, we provide simple, concrete
illustrations of some important issues that can guide the formulation of
effective solutions to more complex statistical problems.Comment: 22 pages, 5 figure