the form \(H_0: \mu(P)-\mu(Q)=0\), and the corresponding test statistic is given by 
 $$T_{m,n}=\frac{N^{1/2}(\bar{X}_m-\bar{Y}_n)}{\sqrt{\frac{N}{m}\sigma^2_m(X_1,\dots,X_m)+ \frac{N}{n}\sigma^2_n(Y_1,\dots,Y_n)}}$$  
 where \(\bar{X}_m\) and \(\bar{Y}_n\) are the sample means from population \(P\) and population \(Q\), respectively, and \(\sigma^2_m(X_1,\dots,X_m)\) is a consistent estimator of \(\sigma^2(P)$ when $X_1,\dots,X_m\) are i.i.d. from \(P\). Assume consitency also under \(Q\).
Difference of me