Now we look at how you deal with tests of two means when the population standard deviation \(\sigma\) is NOT known.

For this method to valid we also need to assume that our samples are independent, as well as the sample variance DO NOT EQUAL.

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 \(n_2-1\).