Confidence intervals for a binomial proportion

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Abstract
Thirteen methods for computing binomial confidence intervals are compared based on their coverage properties, widths and errors relative to exact limits. The use of the standard textbook method, x/n ± 1.96√[(x/n) (1 − x/n)/n], or its continuity corrected version, is strongly discouraged. A commonly cited rule of thumb stating that alternatives to exact methods may be used when the estimated proportion p̂ is such that np̂ and n(1 − p̂) both exceed 5 does not ensure adequate accuracy. Score limits are easily calculated from closed from solutions to quadratic equations and can be used at all times. Based on coverage functions, the continuity corrected score method is recommended over exact methods. Its conservative nature should be kept in mind, as should the wider fluctuation of actual coverage that accompanies omission of the continuity correction.

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