Is a t-test applicable here in comparing two proportions? I have 3 categories of tissues examined by microscopy (A , B and C, 12 of each). I want to test whether the proportion of tissues staining by a particular dye, differs in the three categories. Each  of the 36 tissue specimens is stained and examined under the microscope. The data is collected in the form of proportion of cells staining positive for the dye in each tissue specimen.
Now, I understand that the proportions have a binomial distribution and t-test is probably not appropriate because the variance of proportions is known. But I don't know how to apply the z-test to detect the difference in proportion here, since the data collected itself is in the form of "proportion" for each specimen. Do I calculate the mean proportion ($p_A,p_B,p_C$ for each category) by averaging the recorded proportions under each observation and then apply the z-test?
As an alternative thought, am I not estimating the proportion from my study population by averaging the proportions themselves. Thus, since I am actually estimating the true population proportion from sample itself, would the assumptions of z-test hold here?


Now, I understand that the proportions have a binomial distribution

Independent Bernoulli trials have a binomial distribution, but proportions in general don't. Unless you have some reason to think that the stains are independent, a binomial distribution is not appropriate. If you're hypothesizing that the stains are independent and the samples within a category have the same population parameter p (i.e. IID within a type), then you can take the (weighted, if the number of cells in each sample is different) averages for each type. If you're not hypothesizing that, then in some sense you have a sample size of $12$ and no distribution. You could do a paired test, but the power of the test would be rather small.

and t-test is probably not appropriate because the variance of proportions is known.

I don't see how. The variance for binomial tests is calculated from p, which isn't known a priori. So that indicates t-test. But if you're assuming IID and the number of cells is large, then the difference between t-test and z-test is neglible.
