Introduction to Probability and Statistics for Engineers and Scientists

11.2Goodness of Fit Tests When All Parameters are Specified 485

becoming exact asnapproaches infinity) a chi-square distribution withk−1 degrees of
freedom. Hence, fornlarge,ccan be taken to equalχα^2 ,k− 1 ; and so the approximateα
level test is

reject H 0 if T≥χα^2 ,k− 1

accept H 0 otherwise

If the observed value ofTisT=t, then the preceding test is equivalent to rejectingH 0
if the significance levelαis at least as large as thep-value given by

p-value=PH 0 {T≥t}

≈P{χk^2 − 1 ≥t}

whereχk^2 − 1 is a chi-square random variable withk−1 degrees of freedom.
An accepted rule of thumb as to how largenneed be for the foregoing to be a good
approximation is that it should be large enough so thatnpi≥1 for eachi,i=1,...,k,
and also at least 80 percent of the valuesnpishould exceed 5.

REMARKS

(a)A computationally simpler formula forTcan be obtained by expanding the square
in Equation 11.2.1 and using the results that

∑

ipi=1 and

∑
iXi=n(why is this
true?):

T=

∑k

i= 1

Xi^2 − 2 npiXi+n^2 p^2 i

npi

(11.2.2)

=

∑

i

Xi^2 /npi− 2

∑

i

Xi+n

∑

i

pi

=

∑

i

Xi^2 /npi−n

(b)The intuitive reason whyT, which depends on thekvaluesX 1 ,...,Xk, has onlyk− 1
degrees of freedom is that 1 degree of freedom is lost because of the linear relationship∑

iXi=n.
(c)Whereas the proof that, asymptotically,Thas a chi-square distribution is advanced, it
can be easily shown whenk=2. In this case, sinceX 1 +X 2 =n, andp 1 +p 2 =1, we