Let X be a binomial random variable with n trials and probability p of success....

Let X be a binomial random variable with n trials and probability p of success.

This analysis is the basis of the sign test, a typical application of which would be something like this: An experimental drug is to be evaluated on laboratory rats. In n pairs of litter mates, one animal is given the drug and the other is given a placebo. A physiological measure of benefit is made after some time has passed.

Let X be the number of pairs for which the animal receiving the drug benefited more than its litter mate. A simple model for the distribution of X if there is no drug effect is binomial with p = .5. This is then the null hypothesis that must be made untenable by the data before one could conclude that the drug had an effect.