Mathematical Methods for Physics and Engineering - Matematica.NET

31.7 HYPOTHESIS TESTINGIn the last equality, we rewrote the expression in matrix notation by defining thecolumn vector f with elements f i = f(x i ; a). The value χ 2 (â) at this minimum canbe used as a statistic to test the null hypothesis H 0 , as follows. The N quantitiesy i − f(x i ; a) are Gaussian distributed. However, provided the function f(x j ; a) islinear in the parameters a, the equations (31.98) that determine the least-squaresestimate â constitute a set of M linear constraints on these N quantities. Thus,as discussed in subsection 30.15.2, the sampling distribution of the quantity χ 2 (â)will be a chi-squared distribution with N − M degrees of freedom (d.o.f), which hasthe expectation value and varianceE[χ 2 (â)] = N − M and V [χ 2 (â)] = 2(N − M).Thus we would expect the value of χ 2 (â) to lie typically in the range (N − M) ±√ 2(N − M). A value lying outside this range may suggest that the assumed modelfor the data is incorrect. A very small value of χ 2 (â) is usually an indication thatthe model has too many free parameters and has ‘over-fitted’ the data. Morecommonly, the assumed model is simply incorrect, and this usually results in avalue of χ 2 (â) that is larger than expected.One can choose to perform either a one-tailed or a two-tailed test on thevalue of χ 2 (â). It is usual, for a given significance level α, to define the one-tailedrejection region to be χ 2 (â) >k,wheretheconstantk satisfies∫ ∞kP (χ 2 n ) dχ2 n = α (31.127)and P (χ 2 n) is the PDF of the chi-squared distribution with n = N − M degrees offreedom (see subsection 30.9.4).◮An experiment produces the following data sample pairs (x i ,y i ):x i : 1.85 2.72 2.81 3.06 3.42 3.76 4.31 4.47 4.64 4.99y i : 2.26 3.10 3.80 4.11 4.74 4.31 5.24 4.03 5.69 6.57where the x i -values are known exactly but each y i -value is measured only to an accuracyof σ =0.5. At the one-tailed 5% significance level, test the null hypothesis H 0 that theunderlying model for the data is a straight line y = mx + c.These data are the same as those investigated in section 31.6 and plotted in figure 31.9. Asshown previously, the least squares estimates of the slope m and intercept c are given byˆm =1.11 and ĉ =0.4. (31.128)Since the error on each y i -value is drawn independently from a Gaussian distribution withstandard deviation σ, we haveN∑[ ] 2χ 2 yi − f(x i ; a)N∑ [ yi − mx i − c] 2(a) ==. (31.129)σσi=1i=1Inserting the values (31.128) into (31.129), we obtain χ 2 ( ˆm, ĉ) =11.5. In our case, thenumber of data points is N = 10 and the number of fitted parameters is M = 2. Thus, the1297