9.1 - Two Independent Proportions

Two independent proportions tests are used to compare the proportions in two unrelated groups. In StatKey these were known as "Difference in Proportions" tests. 

Given that \(n_1 p_1 \ge 10\), \(n_1(1-p_1) \ge 10\), \(n_2 p_2 \ge 10\), and \(n_2(1-p_2) \ge 10\), where the subscript 1 represents one groups and the subscript 2 represents the second group, the sampling distribution will be approximately normal with a standard deviation (i.e., standard error) of \( \sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}\).  Given that the population proportions are not known, they are estimated using the sample proportions. In other words, if there are at least 10 "successes" and at least 10 "failures" in both groups, then the sampling distribution for the difference in proportions will be approximately normal.

If the assumption above is met, the normal approximation method is typically preferred. The normal approximation method uses the z distribution to approximate the sampling distribution, similar to the procedures we used in Lesson 7.

When this assumption is not met, Fisher's exact method, bootstrapping, or randomization methods may be used.