Statistical Inference: A Short Course

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

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 at each of its points of discontinuity ...