Stats: Estimating the Proportion
Since varies little for small changes in p, the substitution of for p and for q produces little error in calculating the exact value.
The assumption that the normal distribution is a good approximation for the binomial distribution is OK when n is 'sufficiently large' ..... How large 'sufficiently large' is depends on p ...... The rule of thumb is np > 5 and n(1-p) > 5. So you won't know for sure whether n is sufficiently large since you don't know p. But you can use to get an idea as to whether n was 'sufficiently large'.
If n is small, I have some thoughts but no time at the moment. I'll continue later unless someone else does.