Confidence Intervals for Two Means: Equal and Unknown Population Variance

Suppose we have two independent random samples drawn from normal populations with means $\mu_1$ and $\mu_2$. When we assume both populations share the same (but unknown) variance $\sigma^2$, we combine the two sample variances into a single estimate of $\sigma^2$ called the pooled sample variance:

$$s_p^2 = \dfrac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}.$$

Using this pooled estimate, a $100(1-\alpha)\%$ confidence interval for $\mu_1 - \mu_2$ is

$$(\bar{x}_1 - \bar{x}_2) \;\pm\; t_{\alpha/2,\, n_1+n_2-2} \cdot s_p \sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}.$$

The $t$-distribution has $n_1 + n_2 - 2$ degrees of freedom — the sum of the degrees of freedom of the two sample variances.

Note: Unlike the unequal-variances procedure, this interval uses a single standard deviation $s_p$ in place of two separate $s_1, s_2$, and the degrees of freedom is exactly $n_1+n_2-2$ (no Welch–Satterthwaite approximation needed).