Confidence Intervals for Two Means: Unequal and Unknown Population Variance

Construct a Welch t-confidence interval for the difference between two population means when the population variances are unknown and unequal. Covers the conservative degrees of freedom choice, the Welch-Satterthwaite degrees of freedom formula, and full applications in context.

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Tutorial

Introduction

When the population variances $\sigma_1^2$ and $\sigma_2^2$ are unknown, we replace them with the sample variances $s_1^2$ and $s_2^2$ and use a $t$ -critical value rather than a $z$ -critical value. The resulting interval is called the Welch $t$ -interval.

Given independent samples from two populations, a $(1-\alpha)100\%$ confidence interval for $\mu_1 - \mu_2$ is

(\bar{x}_1 - \bar{x}_2) \pm t^*_{\alpha/2,\, df}\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}.

When the variances are unknown and unequal, this statistic does not follow a $t$ -distribution exactly. A simple, conservative choice of degrees of freedom is

df = \min(n_1 - 1,\, n_2 - 1).

This choice gives a wider interval than strictly necessary but requires no extra computation. For example, with $n_1 = 8$ and $n_2 = 12$ , we would use $df = \min(7, 11) = 7$ .