# Comparing two binomials with with partially matched sample

Imagine a scenario where I want to understand a population's support for a political candidate. I sample n=500 people randomly with replacement and ask them if they want to vote for candidate A or B. Assume (and ignore) that there is no null values and no option to not provide an answer. I want to determine if sentiment has changed 6 months later, and I have the original n=500 sample provide updated answers, but I also add a new random sample of n=600 with replacement (he indicates he wants to improve power to detect a change).

Since our research question is to see if support for a candidate increased, my colleague says that we can combine the two random samples and see if support has increased for the candidate from time A to time B.

What feels wrong to me here is that the second sample now is partially a matched pair to the first sample, and partially not. It feels weird to compare two samples where part of them are matched pairs between the two, and part of them are independent. He indicates that since they are both random samples, that it's fine to combine them.

What is the right answer here? I think we should instead sample a new random set of participants with a higher sample size and do an independent sample comparison, but I'm not sure if I'm thinking straight.