Confidence Intervals for Two Means: Known and Unequal Population Variances

Suppose we have two independent samples drawn from populations with known variances $\sigma_1^2$ and $\sigma_2^2.$ Let $\bar{x}_1$ and $\bar{x}_2$ be the sample means based on sample sizes $n_1$ and $n_2,$ respectively.

A $100(1-\alpha)\%$ confidence interval for $\mu_1 - \mu_2$ is

$$(\bar{x}_1 - \bar{x}_2) \pm z_{\alpha/2}\sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}},$$

where $z_{\alpha/2}$ is the critical value from the standard normal distribution. For a $95\%$ confidence interval, $z_{0.025} = 1.96.$

For example, suppose $\bar{x}_1 = 50,$ $\bar{x}_2 = 47,$ $\sigma_1^2 = 16,$ $\sigma_2^2 = 9,$ and $n_1 = n_2 = 100.$ The standard error is

$$\sqrt{\dfrac{16}{100} + \dfrac{9}{100}} = \sqrt{0.25} = 0.5,$$

and the $95\%$ confidence interval is

$$(50 - 47) \pm 1.96(0.5) = 3 \pm 0.98 = (2.02,\, 3.98).$$