Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

284 Some Elementary Statistical Inferences

which has, asn→∞, an approximateN(0,1) distribution (see Theorem 4.2.1).
Furthermore, as discussed in Example 5.3.6, the distribution ofY^2 is approximately
χ^2 (1). LetX 2 =n−X 1 and letp 2 =1−p 1 .LetQ 1 =Y^2 .ThenQ 1 may be
written as

Q 1 =

(X 1 −np 1 )^2

np 1 (1−p 1 )

=

(X 1 −np 1 )^2

np 1

+

(X 1 −np 1 )^2

n(1−p 1 )

=

(X 1 −np 1 )^2

np 1

+

(X 2 −np 2 )^2

np 2

(4.7.1)

because (X 1 −np 1 )^2 =(n−X 2 −n+np 2 )^2 =(X 2 −np 2 )^2. This result can be
generalized as follows.
LetX 1 ,X 2 ,...,Xk− 1 have a multinomial distribution with the parametersn
andp 1 ,...,pk− 1 , as in Section 3.1. LetXk =n−(X 1 +···+Xk− 1 )andlet
pk=1−(p 1 +···+pk− 1 ). DefineQk− 1 by

Qk− 1 =

∑k

i=1

(Xi−npi)^2

npi

.

It is proved in a more advanced course that, asn→∞,Qk− 1 has an approximate
χ^2 (k−1) distribution. Some writers caution the user of this approximation to be
certain thatnis large enough so that eachnpi,i=1, 2 ,...,k, is at least equal
to 5. In any case, it is important to realize thatQk− 1 does not have a chi-square
distribution, only an approximate chi-square distribution.
The random variableQk− 1 may serve as the basis of the tests of certain statis-
tical hypotheses which we now discuss. Let the sample spaceAof a random ex-
periment be the union of a finite numberkof mutually disjoint setsA 1 ,A 2 ,...,Ak.
Furthermore, letP(Ai)=pi,i=1, 2 ,...,k,wherepk =1−p 1 −···−pk− 1 ,
so thatpiis the probability that the outcome of the random experiment is an
element of the set Ai. The random experiment is to be repeatednindepen-
dent times andXirepresents the number of times the outcome is an element
of setAi.Thatis,X 1 ,X 2 ,...,Xk = n−X 1 − ··· −Xk− 1 are the frequen-
cies with which the outcome is, respectively, an element ofA 1 ,A 2 ,...,Ak.Then
the joint pmf ofX 1 ,X 2 ,...,Xk− 1 is the multinomial pmf with the parameters
n, p 1 ,...,pk− 1. Consider the simple hypothesis (concerning this multinomial pmf)
H 0 :p 1 =p 10 ,p 2 =p 20 ,...,pk− 1 =pk− 1 , 0 (pk=pk 0 =1−p 10 −···−pk− 1 , 0 ),
wherep 10 ,...,pk− 1 , 0 are specified numbers. It is desired to testH 0 against all
alternatives.
If the hypothesisH 0 is true, the random variable

Qk− 1 =

∑k

1

(Xi−npi 0 )^2

npi 0

has an approximate chi-square distribution withk−1 degrees of freedom. Since,

whenH 0 is true,npi 0 is the expected value ofXi, one would feel intuitively that

observed values ofQk− 1 should not be too large ifH 0 is true. Our test is then