# Expectation of length of a confidence interval for a proportion

a) For a given significance-level $\alpha$, if we find $\mathop{\mathbb E}(b-a)$ such that $\mathbb P(a \leq X \leq b) \geq 1−\alpha$ for all possible values of $a$ and $b$, from the point of view of interpretability, what extra interpretability does the expected length buy us?

Say, if I had observed proportions $p_{1} ,p_{2},p_{2} ...p_{n}$ over different sample sizes and modeled them as binomial proportions and calculated confidence intervals (CIs) for each of them:

b) Would comparing any pair of expected lengths of these confidence intervals be analogous to the expected length of the confidence interval around the difference of binomial proportions?

c) How does the expected length of the CI compare with the standard error of the estimator- considering the intervals to be symmetric around the estimator?

• What is $X$ - ie what parameter is it? – probabilityislogic Feb 21 '12 at 7:58
• In a), X is open-ended. I agree-that inorder to integrate, we would require the sampling distribution of the statistic X. – hearse Feb 22 '12 at 2:02
• $X$ is not a statistic, $a$ and $b$ are statistics - these would be functions of the $p_i$ and other sample properties. $X$ would be a parameter which is not fully known from the sample data at hand (such as a population proportion). – probabilityislogic Feb 22 '12 at 2:39
• True. X would be an estimator of a parameter and has a sampling distribution while a,b are statistics. – hearse Feb 22 '12 at 2:55

Ceteris paribus, shorter CIs are preferable as more informative. Of course, for a symmetric CI based on the (asymptotic) standard error of the estimator, the length of the interval is a monotone transformation of that standard error. However, for skewed distributions such as those of the proportion estimates (for $p\neq1/2$), there are other options: you can compute the point estimate and its standard error on the log odds scale (where it would be pivotal, and won't depend on $p$ any more), form the CI, and then back-transform the endpoints of the CI to the original proportions scale. Such CI will likely have better small sample performance -- and be shorter, too.