Introduction to Student (1908) The Probable Error of a Mean

Abstract

Testing a hypothesis about the mean $\xi$ of a population on the basis of a sample X ₁, …, X _n from that population was treated throughout the 19th century by a large-sample approach that goes back to Laplace. If the sample mean ${\bar X}$ is considered to be the natural estimate of $\xi$, the hypothesis $H:\xi = {\xi _0}$ should be rejected when ${\bar X}$ differs sufficiently from ${\xi _0}$. Furthermore, since for large n the distribution of $\sqrt n \left( {\bar X} \right. - \left. {{\xi _0}} \right)$ / σ is approximately standard normal under H (where σ ² < ∞ is the variance of the X’s), this suggests rejecting H when

$$\frac{{\sqrt n \left| {\bar X - \left. {{\xi _0}} \right|} \right.}}{\sigma }$$

(1)

exceeds the appropriate critical value calculated from that distribution.