Given proportions $p_1$ and $p_2$ for samples of size $n_1$ and $n_2$ is the distribution of differences in sample means $n_1p_1$ and $n_1p_2$ normally distributed or $t$-distributed?
I am unsure because I "know"--in the sense of "statistics text books say"--that differences in proportions are normally distributed and differences in means are $t$-distributed.
I understand that differences in proportions are normally distributed because the individual events are Bernoulli-distributed--as are all categorical random variable put into the categories $A$ and ~$A$--and therefore binomially distributed in aggregate. This makes differences in proportions asymptotically normal.
I am less certain as to exactly why the difference in sample means are $t$-distributed, except that the sample variances are also normally distributed, meaning that the sample variance can underestimate or overestimate the population variance.
I used the wrong symbols when asking the question. $p_1$ and $p_2$ are supposed to be the sample proportions $\hat{p}_1$ and $\hat{p}_2$. Thus, I am interested in the distribution of the difference in sample means $n_1\hat{p}_1$ and $n_2\hat{p}_2$.