Is it possible to do a Binomial hypothesis test of 2 samples with widely different sample sizes? I have two samples I'd like to do a hypothesis test on. The sample proportions are:
$$
\hat{p}_1 = \frac{120}{4000}
$$
$$
\hat{p}_2 = \frac{9000}{300000}
$$
If I did a test of two proportions where the null is $p_1=p_2$, would there be issues because $n_1=4000$ and $n_2 = 300000$?
 A: Similarity in sample size between groups is not an assumption for any hypothesis test for proportions that I'm familiar with. You should be fine to run the test.
Note: if you are planning a study, the study will be better powered (there will be a higher probability of finding a result if there is one) if the sample sizes in the two groups are similar. You don't want to plan a study and recruit 10 people in one group and 1000 in another group.
A: There may be different issues here:

*

*very different sample sizes, here with one sample $75$ times the other: hypothesis tests for proportions will adjust for this automatically


*small proportions of successes, here of $0.03$: this can be an issue for small samples where different methods can sometimes give noticeably different results, but not really in this case with the large sample sizes


*large sample sizes, here in the thousands and hundreds of thousands: in general larger sample sizes are better, and the only risk is that you might get a statistically significant result which was very small and so not worth being concerned about - the answer to this is to consider a confidence interval for $p_1-p_2$ and in this example the 95% confidence interval for the difference would be approximately  $[-0.005,+0.005]$


*invented numbers, as here the proportions are exactly equal and this is unlikely to be seen with such large samples: if the two population proportions were in fact both $0.03$ then the probability of the two sample proportions being equal would be about $0.0005$, which is very small, but not something to worry about here as you presumably invented these numbers simply to illustrate the question.
So none of these issues need worry you.
