The experiment is to select a random sample of size \(n\) from a continuous distribution, and then to perform a hypothesis test about the quantile \(m\) at a specified order \( p_0 \in (0, 1) \), 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 input controls.
The order of the quantile \( p_0 \), the significance level \( \alpha \), and the null hypothesized value \(m_0\) of the quantile can be selected with input controls. The true quantile \( m \) of order \( p_0 \) and the quantile level \(p\) of \(m_0\) are shown. The probability density function of the distribution and the quantile \(m\) are shown in blue in the first graph; \(m_0\) is shown in green.
The test statistic \(Y\) is the number of sample values less than or equal to \(m_0\). Under the null hypothesis, \(Y\) has the binomial distribution with parameters \(n\) and \(p_0\). The probability density function of this distribution and the critical values 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 value of the test statistics \(Y\) is shown in red in the second graph.Random variable \(R\) indicates the event that the null hypothesis is rejected. On each update, \(Y\) and \(R\) are recorded in the data table. Note that the null hypothesis is rejected (\(R = 1\)) if and only if the test statistic \(Y\) falls outside of the critical values. Finally, the empirical probability density of \(R\) is shown in red in the distribution graph and recorded in the distribution table.