This Demonstration shows the confidence interval,

, for

based on random samples of size

from a normal population with mean

and standard deviation

, where

is the sample mean and

is the margin of error for a level

interval. There are two cases, corresponding to when

is assumed known, or is not known and is estimated by the standard deviation in the sample. For the known

case,

, where the critical value

is determined so that the area to the right of

is

. Similarly in the unknown

case,

, where

is the sample standard deviation and

is the critical value determined from a

-distribution with

degrees of freedom.

Five things to see in this Demonstration:

1. The width of the confidence interval increases as

increases.

2. The width of the confidence interval decreases as

increases.

3. For fixed

and

, the width of the confidence interval in the known

case is fixed, but it is stochastic when

is unknown due to the variation in the sample standard deviation,

. The stochastic property can be seen by varying the random seed when

unknown is selected.

4. The width of the confidence interval tends to be larger in the unknown

case but the difference decreases as

increases.

5. Running an animation varying the random seed, we can obtain an empirical estimate

of the coverage probability. Try slowing the animation down to get a large number of repetitions. The intervals are color coded: black when the interval covers

and red when it misses. The animation demonstrates the stochastic coverage probability of the interval.