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?