I have some data from a housing survey (sample size 13,500) and have found that the proportion of privately rented housing is slightly higher than the proportion of social housing. i want to test if the proportions are different. I found some advice at the link below indicating that I must take into account the dependence between the two proportions p and q.

http://www.pmean.com/04/MultinomialProportions.html

v = Var(p-q) = p(1-p)/n +q(1-q)/n +2pq/n

The 2pq/n term is the part that accounts for the dependence. To see if the difference p-q was signicantly different from zero I was going to test p-q/sqrt(v) against a t distribution but wasn't sure what the degrees of freedom would be.

Would it just be n-1?

Apologies if this is similar to another posting.