# Chi-Square Tests

Edited by Paul Ducham

CHI-SQUARE TEST

Multinomial probabilities Sometimes we collect count data in order to study how the counts are distributed among several categories or cells. As an example, we might study consumer preferences for four different brands of a product. To do this, we select a random sample of consumers, and we ask each survey participant to indicate a brand preference. We then count the number of consumers who prefer each of the four brands. Here we have four categories (brands), and we study the distribution of the counts in each category in order to see which brands are preferred.

We often use categorical data to carry out a statistical inference. For instance, suppose that a major wholesaler in Cleveland, Ohio, carries four different brands of microwave ovens. Historically, consumer behavior in Cleveland has resulted in the market shares shown in Table 12.1. The wholesaler plans to begin doing business in a new territory—Milwaukee, Wisconsin. To study whether its policies for stocking the four brands of ovens in Cleveland can also be used in Milwaukee, the wholesaler compares consumer preferences for the four ovens in Milwaukee with the historical market shares observed in Cleveland. A random sample of 400 consumers in Milwaukee gives the preferences shown in Table 12.2.

To compare consumer preferences in Cleveland and Milwaukee, we must consider a multinomial experiment. This is similar to the binomial experiment. However, a binomial experiment concerns count data that can be classified into two categories, while a multinomial experiment concerns count data that are classified into more than two categories. Specifically, the assumptions for the multinomial experiment are as follows: The Multinomial Experiment 1 We perform an experiment in which we carry out n identical trials and in which there are k possible outcomes on each trial. 2 The probabilities of the k outcomes are denoted p1, p2, . . . , pk where p1