Two Normal Means, Unknown Unequal Variances

We test

$$t=\frac{\bar x-\bar y}{\sqrt{\frac{s_x^2}{n_x} + \frac{s_y^2}{n_y}}}$$

whose null distribution can be approximated by $t_v$ distribution, where

$$v=\frac{(\frac{s_x^2}{n_x} + \frac{s_y^2}{n_y})^2}{(\frac{s_x^2}{n_x})^2/(n_x-1) + (\frac{s_y^2}{n_y})^2/(n_y-1)}$$

Decision Rule: reject $H_0$ if $t\gt t_{v,\alpha}$ in (i)