Two Normal Means, Unknown Equal Variance

Test statistic

$$t=\frac{\bar{x}-\bar{y}}{\sqrt{s_p^2(\frac{1}{n_x}+\frac{1}{n_y})}}$$

should follow $t_{n-2}$ distribution under $H_0$, where $n=n_x+n_y$ and the pool sample variance

$$s_p^2=\frac{(n_x-1)s_x^2 + (n_y-1)s_y^2}{n_x+n_y-2}=\frac{\sum_{i=1}^{n_x} (x_i-\bar{x})^2 + \sum_{j=1}^{n_y} (y_j-\bar{y})^2 }{n_x+n_y-2}$$

Decision Rule: reject $H_0$ if $t\gt t_{n-2,\alpha}$ in (i), $t\lt t_{n-2,\alpha}$ in (ii), $|t|\gt t_{n-2,\alpha/2}$ in (iii)