Mathematical Methods for Physics and Engineering - Matematica.NET

31.7 HYPOTHESIS TESTING◮Ten independent sample values x i , i =1, 2,...,10, are drawn at random from a Gaussiandistribution with standard deviation σ =1. The sample values are as follows:2.22 2.56 1.07 0.24 0.18 0.95 0.73 −0.79 2.09 1.81Test the null hypothesis H 0 : µ =0at the 10% significance level.We must test the (simple) null hypothesis H 0 : µ = 0 against the (composite) alternativehypothesis H 1 : µ ≠ 0. Thus, the subspace S is the single point µ =0,whereasA is theentire µ-axis. The likelihood function is1L(x; µ) =(2π) exp [ ∑− 1 N/2 2 i (x i − µ) 2] ,which has its global maximum at µ = ¯x. The test statistic t is then given byt(x) = L(x;0)L(x; ¯x) = exp [ ∑ ]− 1 2 i x2 iexp [ − 1 2∑i (x i − ¯x) 2] =exp( − 1 N¯x2) .2It is in fact more convenient to consider the test statisticv = −2lnt = N¯x 2 .Since −2lnt is a monotonically decreasing function of t, the rejection region now becomesv>v crit ,where∫ ∞P (v|H 0 ) dv = α, (31.111)v critα being the significance level of the test. Thus it only remains to determine the samplingdistribution P (v|H 0 ). Under the null hypothesis H 0 ,weexpect¯x to be Gaussian distributed,with mean zero and variance 1/N. Thus, from subsection 30.9.4, v will follow a chi-squareddistribution of order 1. Substituting the appropriate form for P (v|H 0 ) in (31.111) and settingα =0.1, we find by numerical integration (or from table 31.2) that v crit = N¯x 2 crit =2.71.Since N = 10, the rejection region on ¯x at the 10% significance level is thus¯x 0.52.As noted before, for this sample ¯x =1.11, and so we may reject the null hypothesisH 0 : µ = 0 at the 10% significance level. ◭The above example illustrates the general situation that if the maximumlikelihoodestimates â of the parameters fall in or near the subspace S then thesample will be considered consistent with H 0 and the value of t will be nearunity. If â is distant from S then the sample will not be in accord with H 0 andordinarily t will have a small (positive) value.It is clear that in order to prescribe the rejection region for t, orforarelatedstatistic u or v, it is necessary to know the sampling distribution P (t|H 0 ). If H 0is simple then one can in principle determine P (t|H 0 ), although this may provedifficult in practice. Moreover, if H 0 is composite, then it may not be possibleto obtain P (t|H 0 ), even in principle. Nevertheless, a useful approximate form forP (t|H 0 ) exists in the large-sample limit. Consider the null hypothesisand the a 0 iH 0 :(a 1 = a 0 1,a 2 = a 0 2,...,a R = a 0 R),where R ≤ Mare fixed numbers. (In fact, we may fix the values of any subset1283