multinomial distribution and degrees of freedom

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.

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.


TS Contributor
First of all I assume that you are testing \( H_0: p = q \) and propose the test statistic

\( \frac{\hat{p} - \hat{q} - (p - q)} {\hat{v}} \)

With Slutsky theorem, we know that \( \hat{v} \to v \) and therefore the proposed test statistics should have a normal distribution under the null hypothesis.

Even when the sample size is not very large, I doubt that t-distribution will give a better approximation than the normal distribution; unless you have done some exploratory data analysis to find out that the data has a heavy tail and etc, but I do not think that is the case too.
Thank you for your email and help with this. And yes, that is my null hypothesis and test statistic. I did not look for heavy tails. I had just assumed that with an estimated standard error, the correct thing to do was use a t-test. If the distribution did look to have heavy tails is there a way of finding out the deg of freedom?
Would it just be n-2 as there are 2 estimated paramters; p and q. I will check out Slustky in any case. Thank you again for your help. hughm