In the goodness-of-fit test procedure of Section 4.5, instead of the estimated critical value \(c_{\alpha}\), give a formula for a an estimated \(p\)-value.

Data from Section 4.5

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For a test of univariate normality, we apply the statistic (4.2). Suppose that 11,..., In is a random sample and the null hypothesis is that the sampled distribution is normal with mean and variance 2. Then it can be derived that for any fixed x, under Ho, Ex-X=2(x-)F(x) +20 f(x) (x ), - (4.4) where F and f denote the cdf and density of the hypothesized N(,0) dis- tribution, and 20 E|X X'|= The energy goodness-of-fit statistic for testing the simple hypothesis H : X ~ N(u, ) is therefore nn (X; , ) = n n i=1 - {2(xi) F(xi)+20 f(xi) (xi )} 20 - n2 n i,j=1 - (4.5)