I have built a logistic regression model in R and though the result appears to be satisfactory to some degree, there is one question I have not be able to address. I am not sure if my approach is at all correct.
I know that the overall purpose of the logistic model is to predict the probability of success for a binary random variable. From the same logistic model is it possible to know the probability for a given proportion? Say for instance we are interested in working out the probability of being admitted to a school, and we use a logistic regression to estimate that probability, which will obviously depends on some independent variables. My question is with the same model, is it possible to estimate the proportion of student that will be admitted in that particular school?
I have tried to use the Binomial probability to answer that but for some reason it doesn't give me the right answer. How I went about it. If we know the probability of success, then we also know that the Dependent variable follows a Binomial Distribution with Parameter N and P, where N is the Number of trials and P is the probability of success which can be estimate from the logistic model. So if I want the probability that the proportion is less or equal to say p then I thought this is equivalent to the probability that in N trials there are less or equal N*p success with the probability of success being P .
PS I know that the probability of success is an overall rate/proportion but what I am interested in is to find out the probability for a certain group. The group characteristic could be included in the model but that would be limiting in my case. Actually I am more interested in finding the probability for a proportion from the logistic model rather than the success of a certain group.