Two-Sample t-Interval for a Difference of Means

When you want to compare two groups, like commute times for drivers versus cyclists, you care about one number: how far apart their true average commutes really are. Building two separate intervals and eyeballing whether they overlap answers a fuzzier question than the one you actually want answered. A two-sample t-interval targets mu1 - mu2 directly, so the margin of error reflects the uncertainty in the difference itself, not two separate uncertainties you're trying to compare by eye.

The standard error for a difference of two independent means is √((s1² / n1) + (s2² / n2)). Each sample contributes its own variance divided by its own sample size, and because the samples are independent, those two pieces of variance add under the square root rather than cancel or average. The interval is then (x-bar1 - x-bar2) plus or minus t-star times that standard error, where t-star comes from a t-distribution using a conservative degrees of freedom approach (often the smaller of n1 - 1 and n2 - 1, or technology's more exact value).

This topic is the natural next step after one-sample t-intervals: same t-machinery, same logic of estimate plus or minus margin of error, just applied to a difference instead of a single mean. It sets up two-sample t-tests, which ask whether that difference could plausibly be zero. Keep the independence of the two samples straight in your head, since dependent or paired data needs a completely different method built on the differences within pairs, not between separate groups.