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 (the "confidence level"), and second, it must be as narrow as possible in order to be useful. Confidence intervals may be constructed in several ways, although in practice it is usually not possible to attain precisely the desired confidence level, contrary to common belief. This is illustrated in the present Demonstration for a binomial distribution with

trials and probability parameter

.

In this case, 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

*. *Another method, known as Wilson's score method, which is an enhancement of the former method, produces more narrow intervals at the same level of significance. A different approach, known as the Clopper–Pearson method, shares this same property, even though, in general, the resulting intervals are slightly different from Wilson's. The intervals obtained from these two methods are not necessarily centered at

.

Each of the three methods uses a particular coefficient which depends on

,

and the desired confidence level, and which determines the length of the resulting intervals and hence the probability of containing the value

. For a

confidence level, both the conventional and Wilson's methods use the standard normal distribution

quantile, while in the Clopper-Pearson case the coefficient is

.

In this Demonstration, the appropriate values of each coefficient are found using a simulation scheme with 10,000 replications of a binomial distribution experiment with

trials and probability

(for

, and

). By moving the sliders for each coefficient, you can find the values that produce confidence intervals with a significance level closest to the prescribed one and the shortest average length. The initial values are for the supposedly 95% intervals.