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,
*p*_{1}and*p*_{2} - The two sample sizes,
*n*_{1}and*n*_{2} - The two numbers with the certain characteristic,
*x*_{1}and*x*_{2} - The two sample proportions,
*p̂*_{1}=*x*_{1 }/*n*_{1}and*p̂*_{2}=*x*_{2 }/*n*_{2}

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, *n*_{1} and *n*_{2}, must satisfy the two basic requirements:

*n*_{1}*p*_{1}(1 –*p*_{1}) ≥ 10 and*n*_{2}*p*_{2}(1 –*p*_{2}) ≥ 10*n*_{1}≤ 0.05(*N*_{1}) and*n*_{2}≤ 0.05(*N*_{2}), 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.