# Confidence Intervals for the Binomial Distribution

Confidence Intervals for the Binomial Distribution

A confidence interval for estimating a parameter of a probability distribution must show two basic properties. First, it must contain the value of the parameter with a prescribed probability, and second, it must be as short as possible in order to be useful. Confidence intervals may be derived in different ways. In the case of a binomial distribution with trials and probability parameter , the conventional method for estimating uses the normal approximation and produces an interval centered at the point , where is the number of successes obtained in the trials.

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Another method, known as Wilson's score method, has some advantages over the conventional one. In general, it has shorter length and, for small values of , it is not centered at this value. A different approach, known as the Clopper–Pearson method, also shows this property, even though, in general, it produces results that differ from Wilson's method.

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These three methods are illustrated in this Demonstration, using in each case the coefficients that are commonly used to attain a probability of coverage of at least 95%. Move the sliders to observe the effect of different values of and on the position and length of the resulting intervals.

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