## Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 20, Number 2 (1949), 313-316.

### A Graphical Determination of Sample Size for Wilks' Tolerance Limits

Z. W. Birnbaum and H. S. Zuckerman

#### 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 $\epsilon$.

#### Article information

**Source**

Ann. Math. Statist., Volume 20, Number 2 (1949), 313-316.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177730044

**Digital Object Identifier**

doi:10.1214/aoms/1177730044

**Mathematical Reviews number (MathSciNet)**

MR30176

**Zentralblatt MATH identifier**

0041.26101

**JSTOR**

links.jstor.org

#### Citation

Birnbaum, Z. W.; Zuckerman, H. S. A Graphical Determination of Sample Size for Wilks' Tolerance Limits. Ann. Math. Statist. 20 (1949), no. 2, 313--316. doi:10.1214/aoms/1177730044. https://projecteuclid.org/euclid.aoms/1177730044