© 2010 Pearson Prentice Hall. All rights reserved The Chi-Square Test of Homogeneity

12-2 In a chi-square test for homogeneity of proportions, we test whether different populations have the same proportion of individuals with some characteristic.

12-3 The procedures for performing a test of homogeneity are identical to those for a test of independence.

12-4 The following question was asked of a random sample of individuals in 1992, 2002, and 2008: “Would you tell me if you feel being a teacher is an occupation of very great prestige?” The results of the survey are presented below: Test the claim that the proportion of individuals that feel being a teacher is an occupation of very great prestige is the same for each year at the  = 0.01 level of significance. Source: The Harris Poll Parallel Example 5: A Test for Homogeneity of Proportions Yes No

12-5 Step 1: The null hypothesis is a statement of “no difference” so the proportions for each year who feel that being a teacher is an occupation of very great prestige are equal. We state the hypotheses as follows: H 0 : p 1 = p 2 = p 3 H 1 : At least one of the proportions is different from the others. Step 2: The level of significance is  =0.01. Solution

12-6 Step 3: (a) The expected frequencies are found by multiplying the appropriate row and column totals and then dividing by the total sample size. They are given in parentheses in the table below, along with the observed frequencies. Solution Yes 418 ( ) 479 ( ) 525 ( ) No 602 ( ) 541 ( ) 485 ( )

12-7 Step 3: (b)Since none of the expected frequencies are less than 5, the requirements are satisfied. (c)The test statistic is Solution

12-8 Step 4: There are r = 2 rows and c =3 columns so we find the P-value using (2-1)(3-1) = 2 degrees of freedom. The P-value is the area under the chi-square distribution with 2 degrees of freedom to the right of which is approximately 0. Solution: P-Value Approach

12-9 Step 5: Since the P-value is less than the level of significance  = 0.01, we reject the null hypothesis. Solution: P-Value Approach

12-10 Step 6: There is sufficient evidence at the  = 0.01 level of significance to say the proportion of individuals who believe that teaching is a very prestigious career is different for at least one of the three years. Solution Note: As with the other Chi-square tests, if the null hypothesis is rejected, we must explain what appears to be happening between the populations.