A Graphical Determination of Sample Size for Wilks' Tolerance Limits

Abstract

To determine the smallest sample size for which the minimum and the maximum of a sample are the $100 \beta %$ distribution-free tolerance limits at the probability level $\epsilon$, one has to solve the equation $N\beta^{N-1} - (N - 1)\beta^N = 1 - \epsilon$ given by S. S. Wilks [1]. A direct numerical solution of (1) by trial requires rather laborious tabulations. An approximate formula for the solution has been indicated by H. Scheffe and J. W. Tukey [2], however an analytic proof for this approximation does not seem to be available. The present note describes a graph which makes it possible to solve (1) with sufficient accuracy for all practically useful values of $\beta$ and $\epsilon$.