非參數檢定：練習題

In a Wilcoxon rank-sum test comparing two independent samples of sizes \(n_1 = 8\) and \(n_2 = 10\), the sum of the ranks for the first sample is \(W_1 = 92\). Calculate the value of the Mann-Whitney \(U\) statistic associated with the first sample, where \(U_1 = W_1 - \frac{n_1(n_1+1)}{2}\).

A Wilcoxon rank-sum test is conducted to compare two populations with sample sizes \(n_1 = 12\) and \(n_2 = 15\). Under the null hypothesis, find the mean \(\text{E}(W)\) and the standard deviation \(\text{SD}(W)\) of the rank-sum \(W\) for the first sample.

A researcher uses the Wilcoxon rank-sum test to compare two independent samples, A and B. Sample A has 10 observations and Sample B has 12 observations. The sum of the ranks for Sample A is calculated to be \(W_A = 85\). Calculate the value of the test statistic \(U_A\) used in the Mann-Whitney U test equivalent.

In a Wilcoxon signed-rank test with \(n=20\) non-zero differences, the sum of the positive ranks is \(T^+ = 60\). If the test is two-tailed at a 5% significance level, calculate the mean and variance of the test statistic \(T\) under the null hypothesis to determine the \(z\)-score for a normal approximation.

A sign test is performed on 12 pairs of data to test a null hypothesis of no difference against a two-tailed alternative. The number of positive differences is 2. Using a 5% significance level, determine the critical value from the binomial distribution \(B(12, 0.5)\) and state the conclusion.

A Wilcoxon rank-sum test is performed to compare two samples of sizes \(n_1 = 8\) and \(n_2 = 10\). If the sum of ranks for the first sample is \(W_1 = 102\), calculate the value of the Mann-Whitney statistic \(U_1\) and state its expected value under the null hypothesis.