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T-test on unpaired means with unknown variance

Do two samples come from populations with the same mean ?
$\displaystyle H_0:\mu_1-\mu_2$ $\displaystyle =$ 0 (6.5)
$\displaystyle H_1: \mu_1-\mu_2$ $\displaystyle \neq$ $\displaystyle 0 \nonumber$  

Two-sided test using a T test statistic based on the difference in sample means that has a Student's t distribution with $ n_1+n_2-2$ degrees of freedom
$\displaystyle T$ $\displaystyle =$ $\displaystyle \frac{\overline{X_1}-\overline{X_2}}{s_p/\sqrt{n}}\sim t_{n_1+n_2-2}$  

where $ \frac{1}{n}=\frac{1}{n_1}+\frac{1}{n_2}$ and $ s_p^2$ is the pooled estimate of variance
$\displaystyle s_p^2$ $\displaystyle =$ $\displaystyle \frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{(n_1+n_2-2)}$ (6.6)


David Stephenson 2005-09-30