Two-Sample Tests and Intervals

Every interval in the previous two lessons answered a one-sample question: what is $\mu$, or $\sigma^2$, for a single population? A desk more often asks a comparative question — is Strategy A's mean daily return actually different from Strategy B's, or is the apparent gap just sampling noise? Is Manager A's realised volatility genuinely higher than Manager B's? This note extends the pivoting machinery of Pivotal Quantities and Exact Confidence Intervals to two samples, in four standard configurations built in a deliberate order: the pooled two-sample $t$-test, valid when the two populations share a common variance; Welch's $t$-test, the fix when they don't; the two-sample $F$-test, which is exactly how a desk decides which of the previous two to trust; and the paired $t$-test, which reduces a two-sample problem back to a one-sample problem by differencing, and can be dramatically more powerful when the two series are correlated — as two strategies' daily P&L, both driven by the same market, typically are.

Learning objectives. By the end of this note you should be able to: