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A SRS of size n is drawn from a population. Each individual in the
sample is classified according to two categorical variables. The
probabilities for the row classification are 
and the
probabilities for the column classification are 
.
The null hypothesis is the the row and column classifications are
independent; that is, there is no relationship between the row and
column classifications. Letting 
denote the probability of an
observation being classified in row i and in column j, the null
hypothesis is
The alternative hypothesis is the the row and column classifications are dependent; that is, the row and column classifications are related in some way. We write this alternative as
The second model is a natural extension of the comparison of two
proportions we studied in Section 9.2. That is, the c populations
are independently sampled and the number of possible outcomes in each
population is r where 
.