Testing for Differences between Two Proportions

TESTING FOR DIFFERENCES BETWEEN TWO PROPORTIONS

The foregoing chi-square tests also may be considered tests of proportion and may be used as an alternative to the binomial test of proportions (Chapter 10). Tests for differences among groups are based on whether or not the proportions are equal. So a test of independence between gender and smoking is the same as testing that the proportion of male smokers equals the proportion of female smokers. The binomial test is called an exact test of significance, whereas the chi-square test is an approximate test of the comparison of two or more proportions. The chi-square test statistic under the null hypothesis has an approximate chi-square distribution based on as-ymptotic theory, but the exact probability distribution is not a chi-square. Hence, the significance level based on the table of the chi-square distribution is only an ap-proximation to the true significance level. On the other hand, the binomial distribu-tion is the exact probability of the test statistic and so an exact significance level can be found by referring to the appropriate binomial distribution under the null hypothesis.