Test statistics t=p^x−p^yp^0(1−p^0)(1nx+1ny)t=\frac{\hat p_x - \hat p_y}{\sqrt{\hat p_0(1-\hat p_0)(\frac{1}{n_x} + \frac{1}{n_y})}} t=p^0(1−p^0)(nx1+ny1)p^x−p^y follows N(0,1)\mathcal{N}(0,1)N(0,1) distribution under H0H_0H0 in large sample, i.e. np0(1−p0)>5np_0(1-p_0)\gt 5np0(1−p0)>5, and p^0=nxp^x+nyp^ynx+ny\hat p_0=\frac{n_x\hat p_x + n_y\hat p_y}{n_x+n_y} p^0=nx+nynxp^x+nyp^y reject H0H_0H0 if t>zαt\gt z_\alphat>zα in (i)