Creating a Confidence Interval for the Difference of Two Means

The goal of many surveys and studies is to compare two populations, such as men versus women, low versus high income families, and Republicans versus Democrats. When the characteristic being compared is numerical (for example, height, weight, or income), the object of interest is the amount of difference in the means (averages) for the two populations.

For example, you may want to compare the difference in average age of Republicans versus Democrats, or the difference in average incomes of men versus women. You estimate the difference between two population means, µ₁ – µ₂, by taking a sample from each population (say, sample 1 and sample 2) and using the difference of the two sample means ₁ - ₂, plus or minus a margin of error. The result is a confidence interval for the difference of two population means, µ₁ – µ₂. The formula for the CI is different depending on certain conditions, as seen in the following sections; I call them Case 1 and Case 2.