The work deals with the problematic of precisely expressing confidence intervals for proportion estimation. The asymptotic Wald interval is practically useless for small sample sizes, so we provide three alternative constructions: the Wald, Agresti--Coull and Clopper--Pearson intervals.
Since an exact confidence interval does not exist for the binomial distribution, we can have at most a conservative one. We show that the Clopper--Pearson interval is conservative and express it in terms of quantiles of the beta distribution. The Clopper--Pearson interval is, on average, a lot more conservative than desired, so we suggest two alternative intervals that technically do not achieve the desired coverage, but are on average a lot closer to it than Wald or Clopper--Pearson.
In the second part, we compare intervals visually by graphing them with a computer program at various confidence levels. We compare nominal to actual coverage levels of each interval at different values of the proportion.
We observe the influence of rounding errors on the coverage probability, and calculate the number of decimal places to which we must round the Wald interval to achieve mean nominal coverage, and the Agresti--Coull interval to achieve conservative coverage.
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