I have two methods, $\mathcal{X}$ and $\mathcal{Y}$, and want to determine which one produces lower responses. I figured to estimate $P(X<Y)$ where $X$ is the random variable representing the responses of method $\mathcal{X}$, whereas $Y$ denotes the responses of method $\mathcal{Y}$. Now, the methods have been tested on the same group of $l$ subjects for $n$ times. This means that for each subject $s_i$ I have the responses $x_{i,1},x_{i,2},\ldots,x_{i,n}$ for method $\mathcal{X}$ and $y_{i,1},y_{i,2},\ldots,y_{i,n}$ for $\mathcal{Y}$.
I know that I can estimate $P(X<Y)$ for a particular subject $s_i$ as follows: $$p_i = \frac{U_i}{n^2}$$ where $U_i$ is the Mann-Whitney U statistic for $x_{i,1},x_{i,2},\ldots,x_{i,n}$ and $y_{i,1},y_{i,2},\ldots,y_{i,n}$.
My question is:
Is it correct to estimate $P(X < Y)$ as the average of $p_i$, i.e. as follows?$$\frac{1}{l}\sum_{i=1}^{l}p_i=\frac{1}{l}\sum_{i=1}^{l}\frac{U_i}{n^2}=\frac{1}{n^2l}\sum_{i=1}^{l}U_i$$