The Formula for \(z\)-test comparing 2 means:

\[z=\frac{\Big(\overline{x}_1-\overline{x}_2\Big)-\Big(\mu_1-\mu_2\Big)}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}\]

The Formula for the test statistic for two means when \(\sigma\) is unknown:

\[t=\frac{\Big(\overline{x}_1-\overline{x}_2\Big)-\Big(\mu_1-\mu_2\Big)}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\]

where the degrees of freedom are equal to the smaller of \(n_1-1\) and \(n_2-1\).