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

## TL;DR

This paper explores the foundational principles of the t-test and F-test, revealing their underlying algebraic and geometric structures, and discusses implications for hypothesis testing and residual diagnostics.

## Contribution

It demonstrates how simple algebraic manipulations align the t-test with the recommended null hypothesis approach and extends these insights to Gaussian linear models.

## Key findings

- Algebraic manipulations reveal equivalence of t-test procedures
- Geometric intuition clarifies test statistic interpretation
- Application impacts residual diagnostics in practice

## Abstract

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.