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Statistical Inference: A Short Course by Michael J. Panik

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Appendix 10.C Lilliefors Goodness-of-Fit Test for Normality

In Appendix 7.B, we developed the concept of a normal probability plot as a graphical device for determining if a sample data set could be treated “as if” it was extracted from a normal population. Let us now consider an alternative and more formal “test for normality.” Specifically, we shall test for the goodness-of-fit of a set of sample observations to a normal distribution with unknown mean and standard deviation—the Lilliefors Test. To set the stage for the development of this procedure, let us first review the concept of a cumulative distribution function (CDF) and then develop the notion of a sample or empirical cumulative distribution function (ECDF).

We previously defined the CDF of a discrete random variable X (Section 5.1) as

img

where f is a probability mass function. The ECDF, denoted as S(X), is formed by ordering the set of sample values from the smallest to largest (we form the set of order statistics X(1), X(2), . . . , X(n)) and then plotting the cumulative relative frequencies. Clearly, S(X) is a discrete random variable that exhibits the proportion of the sample values that are less than or equal to X. It plots as a step function (Fig. 10.C.1)—it jumps or increases by at least img at each of its points of discontinuity ...

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