# Potential problem in using paired t-test or Wilcoxon if the two samples consist of multiple draws from same group?

I have an experiment that works loosely works as follows:

There are ten subjects in each treatment. There are two treatments, A and B. Each treatment has 8 rounds. In each round, each subject is selected to be of type 1 or 2 with equal probability each. Both types in each round (based on things irrelevant to my question) they submit a number $$x$$ (the two types are likely to submit different values). I want to compare the average $$x$$ from type 1 in the two treatments and see if there is a difference.

Since I'm comparing averages from a within-subject design, they are paired across the treatments so I'd like to use either a paired t-test or a Wilcoxon signed rank test, BUT the "average $$x$$" in each treatment comprises of 8 rounds (resamplings?) of answers from an unfixed random selection of the same ten subjects so my intuition tells me there may be some problems. Since the group of type 1's from each round (and each treatment) may differ, some subjects' choices may have larger weight on the average $$x$$ from a treatment than other subjects, and the makeup of this will be different in the other treatment.

This is a common experiment set-up so I'm sure there is a way to do it I just can't find details on it in any papers. Is it still ok to use a paired t-test or Wilcoxon signed rank test? Or is there a better test?