is called a confidence interval with coverage probability if for all
If are i.i.d. normal random variables with mean and variance , then
With the help of this theorem one can obtain confidence intervals for the normal distribution:
When looking at proportions, that is to say
one can use the fact that has an approximate normal distribution with mean and variance . This leads to an approximate confidence interval
but, unfortunately, we do not know the exact value of . So we could replace it by its estimator , or solve the equation
with respect to and use the two solutions as the limits of our confidence interval.