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

Abstract

Testing a hypothesis about the mean ξξ\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 X¯X¯{\bar X} is considered to be the natural estimate of ξξ\xi , the hypothesis H:ξ=ξ0H:ξ=ξ0H:\xi = {\xi _0} should be rejected when X¯X¯{\bar X} differs sufficiently from ξ0ξ0{\xi _0}. Furthermore, since for large n the distribution of n−−√(X¯−ξ0)n(X¯−ξ0)\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 n−−√∣∣X¯−ξ0|σn|X¯−ξ0|σ\frac{{\sqrt n \left| {\bar X - \left. {{\xi _0}} \right|} \right.}}{\sigma } (1) exceeds the appropriate critical value calculated from that distribution.