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 .

**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.

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