### The Mean Estimation Experiment

#### Description

The experiment is to select a random sample of size \(n\) from a specified distribution, and then to construct an approximate confidence interval for the mean \(\mu\). The distribution can be chosen with a list box; the options are

In each case, the appropriate parameters and the sample size and confidence level can be varied with scroll bars. The probability density function and mean of the selected distribution are shown in blue in the first graph.

The type of interval--two sided, upper bound, or lower bound can be selected with a list box. The interval can be constructed assuming either that the distribution standard deviation \(\sigma\) is known or unknown. In the first case the pivot variable has the standard normal distribution; in the second case the pivot variable has the student \(t\) distribution with \(n - 1\) degrees of freedom. The probability density function of the pivot variable and the critical values are shown in blue in the second graph.

Random variables \(L\) and \(R\) denote the left and right endpoints of the confidence interval and \(I\) indicates the event that the confidence interval contains the distribution mean. The theoretical probability density function of \(I\) is shown in blue in the third graph.

On each run, the sample density function and the confidence interval are shown in red in the first graph, and the value of the pivot variable is shown in red in the second graph. Note that the confidence interval contains the mean in the first graph if and only if the pivot variable falls between the critical values in the second graph. The third graph shows the proportion of successes and failures in red. The record table records the sample mean \( M \), the confidence bounds \(L\) and \(R\), the pivot variable (\(Z\) or \(T\)), and \(I\) on each run. The distribution table gives the probability density function and empirical density function of \(I\).