Inference for 2 Proportions Mean and Standard Deviation

When comparing proportions from two populations we use a modification of the Z interval and Z test just described for one proportion. The formula for the difference of two proportions is simply the difference of the two proportions. Seeing the formula makes this easier to understand. i.e., so,

These are the most complicated formulas in this course, but once you think about them you will see that they are quite simply derived using the Rule for Variances. Our standard deviation formulas work only for independent random variables. Recall, for X 1 and X 2, the variance of the difference is:

For proportions the variance is: So, for the difference between X 1 and X 2 : or

This leads us to the formula for a confidence interval for the difference of two proportions:

The corresponding formula for the Z test for the difference of 2 proportions has one more twist to it: Since our null hypothesis is H0:H0:we expect the proportions to be the same. Since we don’t know what p 1 and p 2 are, we use our sample data to estimate. As we expect p 1 and p 2 to be the same, we pool the data together and calculate a combined (or pooled) where c is the count of “successes” of each sample

The standard error formula now using the pooled sample proportion is For example, if

The formula for the test statistic Z is: or, as we usually write it,

THE END