Can I use normal approximation for t-test? The question is asking us to compare two samples
Sample 1: n=10, p=0.3
sample 2: n=15, p=0.4
I know the normal approximation is valid when n is larger than 30.
However, the sample size is small so I choose a t-test for the test statistics.
Does the normal approximation is still valid for t-distribution?

I would say no. A nice feature of the t-test is that it has solid robustness to deviations from the normality assumption, so it is, in some sense, safe to chuck anything in there. This is nice, because it is common for distributions not to be normal. They are funky, awful, messy, hard to identify as coming from any particular family...
However, when you are dealing with a proportion, the distribution is set as binomial: either you get one outcome or the other. Absolutely everything is governed by the success probability. Therefore, you don't have to concede anything and rely on the robustness of the t-test; just use a test for proportions.
