Sampling Distribution for a Difference of Means

When you take two independent samples, each sample mean wobbles around its own population mean with its own spread. Subtracting one from the other combines both sources of wobble into a single new random quantity, x1-bar minus x2-bar. Its sampling distribution has a mean equal to mu1 minus mu2, because averages subtract the way you'd expect. But its spread is not a simple subtraction. Since the two samples are chosen independently, the variances of the two sampling distributions add together, and only then do you take a square root to get the standard error of the difference. That's why the gap between two sample means can land positive in one study and negative in another: both x1-bar and x2-bar bounce around their true means by chance, so sometimes group 1 happens to sample high, sometimes group 2 does.

This machinery sets up every two-sample t-procedure that follows. Once you know the center and spread of x1-bar minus x2-bar, you can standardize an observed difference into a t-statistic, check it against a t-distribution with a conservative degrees-of-freedom estimate, and build confidence intervals or run significance tests for comparing two population means. The population standard deviations are almost never known, so sample standard deviations s1 and s2 estimate them, which is exactly why t replaces z here, just as it did for one-sample mean inference.