December 18, 2016

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.