## Confidence Intervals

is called a confidence interval with coverage probability if for all

If are i.i.d. normal random variables with mean and variance , then

1. has a normal distribution with mean and variance .
2. has -distribution with degrees of freedom.
3. and are independent.
4. has a t-distribution with degrees of freedom.

With the help of this theorem one can obtain confidence intervals for the normal distribution:

1. confidence interval for ( known):

2. confidence interval for ( unknown):

3. confidence interval for ( known):

4. confidence interval for ( unknown):

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.