Corrections for Nonnormality in the Use of the Two-Sample $t$- and $F$-Test at High Significance Levels

Abstract

Correct factors to the probabilities that the two-sample $t$ and $F$ statistics shall exceed fixed positive values $t_0$ and $F_0$ either numerically or arithmetically and to the probabilities that $t$ shall be exceeded by fixed negative values $t_0$ have been derived geometrically. The derivations, when the population has a normal density function, produce a novel method of obtaining the usual distributions of $t$ and of $F$. The correct factors permit the use of existing tables and the results are asymptotically correct for numerically large values of the test statistics. There is some indication that the correction factors are better for small sample sizes than large ones. Useful bounds on the errors committed by using the asymptotic corrections at usual significance levels have not been obtained and are a subject for future investigation. For difficult cases a method of approximate evaluation of the correction factors is provided.