Statistical Inference: A Short Course

Appendix 12.A Assessing Normality (Appendix 7.B Continued)

Appendix 7.B addressed the issue of determining if a simple random sample taken from an unknown population is normally distributed. The approach offered therein for assessing the normality of a data set was the normal probability plot of the ordered observed sample data values against their normal scores in the sample or their “expected” Z-scores (see Table 7.B.2 and Fig. 7.B.2). As Fig. 7.B.2 reveals, we do obtain a fairly good approximation to a straight line (the case of perfect normality). Is getting a “fairly good” approximation to a straight line obtained by simply “eyeballing” the data points (X_j, ) (columns 3 and 4, respectively, in Table 7.B.2) a satisfactory way to proceed? Can we devise a more sophisticated approach to determining if a linear equation actually obtains? And if one does obtain, can we test its statistical significance? To answer these questions, one need only look to the developments in this chapter. The task of assessing normality can be addressed via the application of regression analysis.

Let us take the X_j and data values and regress Z_p on X so as to obtain the equation of the line that best fits the data points in Fig. 7.B.2. Then, once we estimate the least squares regression line, we can test ...