## 2. Inference Methods for Two Population Proportions

At this time, it would probably be helpful to refer back to sub-competencies 8 and 10, where you learned about confidence intervals and hypothesis testing for population proportions. The inference formulas for comparing two population proportions are going to seem more complicated, but they are fundamentally the same and will look remarkably similar!

First, we need some notation and assumptions. We have two values for each variable, one for each of the two samples:

• The two hypothesized proportions, p1 and p2
• The two sample sizes, n1 and n2
• The two numbers with the certain characteristic, x1 and x2
• The two sample proportions, 1 = x1 / n1 and 2 = x2 / n2

The biggest difference, again, is with the standard deviation of the difference in sample proportions. Whereas with one sample

with two independent samples the sample standard deviation for the difference in two proportions is:

Recall that when dealing with two proportions, our sample sizes, n1 and n2, must satisfy the two basic requirements:

• n1p1 (1 – p1) ≥ 10 and n2p2 (1 – p2) ≥ 10
• n1 ≤ 0.05(N1) and n2 ≤ 0.05(N2), which state that the sample sizes be no larger than 5% of the population size. This condition ensures that our samples remain independent.

These requirements must be satisfied in order for our results from both hypothesis testing and confidence intervals to be valid.