We have data {(xi,yi)}\{ (x_i, y_i) \}, where xi−yi∼N(μx−μy,σd2)x_i-y_i\sim\mathcal{N}(\mu_x-\mu_y, \sigma_d^2), but xi,yix_i, y_i itself need not be normally distributed, and μx,μy,σd\mu_x,\mu_y,\sigma_d are unknown.

We mainly test

t=dˉsdnt=\frac{\bar{d}}{\frac{s_d}{\sqrt{n}}}

which follows the tn−1t_{n-1} distribution under H0H_0, where dˉ=xˉ−yˉ\bar{d}=\bar{x}-\bar{y}, sds_d being the sample standard variance of {xi−yi}\{x_i-y_i\}

H0H_0 H1H_1 pp-value Decision Rule
μx−μy=0\mu_x-\mu_y=0 or ≤0\le 0 μx−μy>0\mu_x-\mu_y\gt 0 P(T>t)\mathbb{P}(T\gt t) t>tn−1,αt\gt t_{n-1,\alpha}