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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?

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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.

This will work because that is the largest possible confidence interval width for a binomial that would be used under ordinary circumstances. There is no "correct" confidence interval although different intervals have different properties. What matters is that you choose your interval calculation method prior to collecting your data.

The second is that you may not have sufficient power to detect the real parameter with a small sample size. Any estimate may be too noisy to use, so just getting a zero isn't that big of a deal compared to not having enough power to detect the real value.

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