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Choosing the Optimum Number of Classes in the Chi-Square Test for Arbitrary Power Levels

Harrison, Richard H.
Publication Year:
Report Number:
ETS Research Report
Document Type:
Page Count:
Subject/Key Words:
Data Collection, Goodness of Fit, Statistical Analysis


When a set of observations X sub 1, X sub 2, ..., X sub N are fitted to a hypothesized statistical distribution, the chi-square test is sometimes used to determine wh the fit is acceptable. When the postulated distribution is continuous and completely specified, A. Wald and H. B. Mann suggest a method for determining the optimum number of classes k into which the N observations should be divided. They prove that for large enough N the power p of the chi-square test against a fixed alternative is always one-half or greater, provided that k is determined by a formula involving N , a distance Delta between the distribution the data and a significance level Alpha. In this report we give more general formulas relating N, k, Delta and Alpha when the power must exceed some arbitra constant Beta. To introduce the above techniques, artifical data is fitted to a hypothesized gamma distribution. Tables and graphs for an optimum number of classes k sub N as function of N are provided for various power and alpha level Further suggestions are given for applying the method to real observations. (38pp.)

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