Mathematical Methods for Physics and Engineering - Matematica.NET

31.7 HYPOTHESIS TESTINGWe now turn to Fisher’s F-test. Let us suppose that two independent samplesof sizes N 1 and N 2 are drawn from Gaussian distributions with means andvariances µ 1 ,σ1 2 and µ 2,σ2 2 respectively, and we wish to distinguish between thetwo hypothesesH 0 : σ 2 1 = σ 2 2 and H 1 : σ 2 1 ≠ σ 2 2.In this case, the generalised likelihood ratio is found to beλ = (N [1 + N 2 ) (N1+N2)/2 F(N1 − 1)/(N 2 − 1) ] N 1/2N N1/21N N2/2 [2 1+F(N1 − 1)/(N 2 − 1) ] ,(N 1+N 2)/2where F is given by the variance ratioF = N 1s 2 1 /(N 1 − 1)N 2 s 2 2 /(N 2 − 1) ≡ u2v 2 (31.123)and s 1 and s 2 are the standard deviations of the two samples. On plotting λ as afunction of F, it is apparent that the rejection region λ