Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

Suppose that two classes of students take the same mathematics examination and the

following percentage marks are obtained:

Class1:6662345577805560694750

Class2:64907656817270

Assuming that the two sets of examinations marksare drawn from Gaussian distributions,

test the hypothesisH 0 :σ^21 =σ^22 at the5%significance level.

The variances of the two samples ares^21 =(12.8)^2 ands^22 =(10.3)^2 and the sample sizes
areN 1 =11andN 2 = 7. Thus, we have

u^2 =

N 1 s^21

N 1 − 1

= 180.2andv^2 =

N 2 s^22

N 2 − 1

= 123. 8 ,

where we have takenu^2 to be the larger value. Thus,F=u^2 /v^2 =1.46 to two decimal
places. Since the first sample contains elevenvalues and the secondcontains seven values,
we taken 1 =10andn 2 = 6. Consulting table 31.4, wesee that, at the 5% significance
level,Fcrit=4.06. Since our value lies comfortably below this, we conclude that there is
no statistical evidence for rejecting the hypothesis that the two samples were drawn from
Gaussian distributions with a common variance.

It is also common to define the variablez=^12 lnF, the distribution of which

can be found straightfowardly from (31.126). This is a useful change of variable

since it can be shown that, for large values ofn 1 andn 2 , the variablezis

distributed approximately as a Gaussian with mean^12 (n− 21 −n− 11 ) and variance
1
2 (n

− 1

2 +n

− 1

1 ).

31.7.7 Goodness of fit in least-squares problems

We conclude our discussion of hypothesis testing with an example of a goodness-

of-fit test. In section 31.6, we discussed the use of the method of least squares in

estimating the best-fit values of a set of parametersain a given modely=f(x;a)

for a data set (xi,yi),i=1, 2 ,...,N. We have not addressed, however, the question

of whether the best-fit modely=f(x;aˆ) does, in fact, provide a good fit to the

data. In other words, we have not considered thus far how to verify that the

functional formfof our assumed model is indeed correct. In the language of

hypothesis testing, we wish to distinguish between the two hypotheses

H 0 : model is correct and H 1 : model is incorrect.

Given the vague nature of the alternative hypothesisH 1 , we clearly cannot use

the generalised likelihood-ratio test. Nevertheless, it is still possible to test the

null hypothesisH 0 at a given significance levelα.

The least-squares estimates of the parametersaˆ 1 ,aˆ 2 ,...,aˆM, as discussed in

section 31.6, are those values that minimise the quantity

χ^2 (a)=

∑N

i,j=1

[yi−f(xi;a)](N−^1 )ij[yj−f(xj;a)] = (y−f)TN−^1 (y−f).