A Graphical Determination of Sample Size for Wilks' Tolerance Limits

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 ε, one has to solve the equation NβN-1- (N - 1)βN= 1 - ε given by S. S. Wilks [1]. A direct numerical solution of (1) by trial requir...

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Bibliographic Details
Published in:The Annals of mathematical statistics 1949-06, Vol.20 (2), p.313-316
Main Authors: Birnbaum, Z. W., Zuckerman, H. S.
Format: Article
Language:English
Subjects:
Geometric lines
Iterative solutions
Sample size
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Summary: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 ε, one has to solve the equation NβN-1- (N - 1)βN= 1 - ε 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 β and ε.
ISSN:0003-4851
2168-8990
DOI:10.1214/aoms/1177730044