In this lesson, we'll continue our investigation of hypothesis testing. In this case, we'll focus our attention on a hypothesis test for the difference in two population means \(\mu_1-\mu_2\) for two situations:

• a hypothesis test based on the \(t\)-distribution, known as the pooled two-sample \(t\)-test, for \(\mu_1-\mu_2\) when the (unknown) population variances \(\sigma^2_X\) and \(\sigma^2_Y\) are equal
• a hypothesis test based on the \(t\)-distribution, known as Welch's \(t\)-test, for \(\mu_1-\mu_2\) when the (unknown) population variances \(\sigma^2_X\) and \(\sigma^2_Y\) are not equal

Of course, because population variances are generally not known, there is no way of being 100% sure that the population variances are equal or not equal. In order to be able to determine, therefore, which of the two hypothesis tests we should use, we'll need to make some assumptions about the equality of the variances based on our previous knowledge of the populations we're studying.