### The Variance 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 standard deviation \( \sigma \) at a specified confidence level.

The distribution can be chosen with a list box; the options are

In each case, the appropriate parameters and the sample size can be varied with scroll bars. The probability density function of the selected distribution is shown in blue in the first graph, as well as a bar, centered at the distribution mean and extending \( \sigma \) to the left and right.

The confidence level can be selected from a list box, as can the type of interval--two sided, upper bound, or lower bound. The pivot variable \(V\) has the chi-square distribution with \(n - 1\) degrees of freedom. The probability density function and the critical values of \(V\) are shown in blue in the second graph.

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

On each run, the sample density and the confidence interval are shown in red in the first graph, and the value of \(V\) is shown in red in the second graph. Note that the confidence interval contains \( \sigma \) in the first graph if and only if \(V\) falls between the critical values in the second graph. The third graph shows the proportion of successes and failures in red. The first table records the sample standard deviation \( S \), the lower and upper confidence bounds \(L\) and \(R\), the pivot variable \(V\), and \(I\). Finally, the second table gives the theoretical and empirical probability density functions of \(I\).