Consider a two-way contingency table with three rows and three columns. Suppose that, for i = 1, 2, 3 and j = 1, 2, 3, the probability pij that an individual selected at random from a given population will be classified in the ith row and the jth column of Table 10.20.
Table 10.20 Data for Exercise 7
a. Show that the rows and columns of this table are independent by verifying that the values pij satisfy the null hypothesis H0 in Eq. (10.3.3).
b. Generate a random sample of 300 observations from the given population using a uniform pseudo-random number generator. Select 300 pseudo-random numbers between 0 and 1 and proceed as follows: Since p11 = 0.15, classify a pseudo-random number x in the first cell if x < 0.15. Since p11 + p12 = 0.24, classify a pseudo-random number x in the second cell if 0.15 ≤ x < 0.24. Continue in this way for all nine cells. For example, since the sum of all probabilities except p33 is 0.92, a pseudo-random number x will be classified in the lower right cell of the table if x ≥ 0.92.
c. Consider the 3 × 3 table of observed values Nij generated in part (b). Pretend that the probabilities pij were unknown, and test the hypotheses (10.3.3).
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