# Influence of sample estimates on confidence interval estimation

Say I am trying to predict an election (where there are 2 candidates) by sampling N likely voters at random from some underlying large population. Suppose my estimate of the probability of a voter voting for candidate A is p and for candidate B it is (1-p). Then my confidence interval centered at the mean is :

alpha * sqrt(p * (1-p)/N)

Intuitively why should the length of the confidence interval depend on the estimate of p? If my sample size is very small and I get p = 0, the confidence interval is of length 0, why does this make sense?

There are two answers to this. The first is that variance of the binomial is a function of $p$. Both the MVUE and the MLE would be 0 in this case. You are likely using a standard confidence interval, but any interval that covers your parameter at least $\alpha$ percent of the time is a valid interval. You could create an interval $$p\pm\sqrt{\frac{.25}{n}}$$ and this would always cover the parameter $\alpha$ percent of the time as the number of repetitions went to infinity.