The Variance Test Experiment

]> The Variance Test Experiment

Description

The experiment is to select a random sample of size $n$ from a selected distribution and then test a hypothesis about the standard deviation $σ$ at a specified significance level. The distribution can be selected from a list box; the options are the normal, gamma, and uniform distributions. In each case, the appropriate parameters and the sample size $n$ can be varied with scroll bars.

The significance level can be selected with a scroll bar. The type of test: two-sided, left-sided, or right-sided. can be selected from a list box. The boundary point $σ 0$ between the null and alternative hypotheses can be varied with a scroll bar. The probability density function of the distribution, as well as $μ$ and $σ$ , and $σ 0$ are shown graphically.

The test can be constructed under the assumption that the distribution mean is known or unknown. In the first case, the test statistic has the chi-square distribution with $n$ degrees of freedom; in the second case the test statistics has the chi-square distribution with $n 1$ degrees of freedom. The probability density function and the critical values of the test statistic $V$ are shown in the second graph in blue.

On each update, the sample density function is shown in red in the first graph and the sample values are recorded in the sample table. The sample standard deviation $S$ is shown in red in the first graph and the value of the test statistic $V$ is shown in red in the second graph. Random variable $I$ indicates the event that the null hypothesis is rejected. On each update, $S 2$ , $V$ , and $I$ are recorded in the data table. Note that the null hypothesis is rejected ( $I 1$ ) if and only if the test statistic $V$ falls outside of the critical values. Finally, the empirical density function of $I$ is shown in red in the last graph and recorded in the last table.