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

Testing a hypothesis about the mean $$\xi $$ of a population on the basis of a sample X₁, …, 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 σ2 < ∞ is the variance of the X’s), this suggests rejecting H when 1$$\frac{{\sqrt n \left| {\bar X - \left. {{\xi _0}} \right|} \right.}}{\sigma }$$ exceeds the appropriate critical value calculated from that distribution.