Mathematical Methods for Physics and Engineering - Matematica.NET

STATISTICShowever, it is difficult to determine a and b for a given significance level α, soaslightly different rejection region, which we now describe, is usually adopted.The sampling distribution P (u|H 0 ) may be found straightforwardly from thesampling distribution of s given in (31.35). Let us first determine P (s 2 |H 0 )bydemanding thatfrom which we findP (s 2 |H 0 )= P (s|H 0)2sP (s|H 0 ) ds = P (s 2 |H 0 ) d(s 2 ),( ) (N−1)/2 N (s 2 ) (N−3)/2 ( )=2σ02 Γ ( 12 (N − 1)) exp − Ns22σ02 . (31.122)Thus, the sampling distribution of u = Ns 2 /σ0 2 is given byP (u|H 0 )=12 (N−1)/2 Γ ( 12 (N − 1))u(N−3)/2 exp ( − 1 2 u) .We note, in passing, that the distribution of u is precisely that of an (N − 1) thorderchi-squared variable (see subsection 30.9.4), i.e. u ∼ χ 2 N−1 . Although it doesnot give quite the best test, one then takes the rejection region to be0