Introduction to Probability and Statistics for Engineers and Scientists

496 Chapter 11:Goodness of Fit Tests and Categorical Data Analysis

The different members of the population will be assumed to be independent. Also, let

pi=P{X=i}=

∑s

j= 1

Pij, i=1,...,r

and

qj=P{Y=j}=

∑r

i= 1

Pij, j=1,...,s

That is, pi is the probability that an arbitrary member of the population will have
X-characteristici, andqjis the probability it will haveY-characteristicj.
We are interested in testing the hypothesis that a population member’s X- and
Y-characteristics are independent. That is, we are interested in testing

H 0 :Pij=piqj, for all i=1,...,r

j=1,...,s

against the alternative

H 1 :Pij=piqj, for some i,ji=1,...,r

j=1,...,s

To test this hypothesis, suppose thatnmembers of the population have been sampled, with
the result thatNijof them have simultaneously hadX-characteristiciandY-characteristic
j,i=1,...,r,j=1,...,s.
Since the quantitiespi,i=1,...,r, andqj,j=1,...,sare not specified by the null
hypothesis, they must first be estimated. Now since

Ni=

∑s

j= 1

Nij, i=1,...,r

represents the number of the sampled population members that haveX-characteristici,
a natural (in fact, the maximum likelihood) estimator ofpiis

pˆi=

Ni

n

, i=1,...,r

Similarly, letting

Mj=

∑r

i= 1

Nij, j=1,...,s