Show that the test of Problem 6.10(i) reduces to

(i) [x(n) − x(1)]/S < c for normal versus uniform;

(ii) [ ¯x − x(1)]/S < c for normal versus exponential;

(iii) [ ¯x − x(1)]/[x(n) − x(1)] < c for uniform versus exponential.

(Uthoff, 1970.)

Note. When testing for normality, one is typically not interested in distinguishing the normal from some other given shape but would like to know more generally whether the data are or are not consonant with a normal distribution. This is a special case of the problem of testing for goodness of fit, which is briefly discussed at the end of Section 6.13 and forms the topic of Chapter 16; also, see the many references in the notes to Chapter 16.