I've been looking into different ways of calculating confidence intervals and I came across methods using the normal distribution, t-distribution and chi square distribution.

I want to find the confidence intervals for a binomial distribution with a high sample size (around N=5000). I am wary of using a normal approximation to the binomial distribution not because it shouldn't be continuous but that it shouldn't have negative values (which the normal distribution does).

I have two questions

1. Is the t-distribution always better than a normal distribution for finite values of N (or rather N less than the population size)? I realise at N=5000 the difference is negligible but I am wondering about this from a theory point of view.

2. Is their any distribution that always gives the best estimate of the confidence interval for a sample smaller than the population?