Let's assume I have a finite population $N$ and that I draw two samples from it $n_1$, and $n_2$. A single observation can belong to both samples.
Calculating the averages for each sample and the confidence intervals is simple enough. I want to know can I combine the two samples somehow to obtain a tighter confidence intervals on the average.
Based on the comments, I want to clarify. The data does not contain enough information to identify the observation. I am given the average, the variance and the number of observations for each sample.