A Wilcoxon rank sum test for comparing two populations involves two independent samples of sizes 15 and 20. The un-standardized test statistic (that is the rank sum) is T = 210. The value of the standardized test statistic z is:
A) 14.0
B) 10.5
C) 6.0
D) -2.0
E) -3.2

Question

Quizplus · Accepted Answer

The standardized test statistic $z$ for a Wilcoxon rank sum test is calculated using the formula: $$z = \frac{T - E(T)}{\sqrt{Var(T)}}$$ where $T$ is the rank sum for one of the groups, $E(T)$ is the expected value of the rank sum under the null hypothesis, and $Var(T)$ is the variance of the rank sum under the null hypothesis. For two independent samples of sizes $n_1$ and $n_2$, the expected value and variance are given by: $$E(T) = \frac{n_1(n_1 + n_2 + 1)}{2}$$ $$Var(T) = \frac{n_1 n_2 (n_1 + n_2 + 1)}{12}$$ Given $n_1 = 15$ and $n_2 = 20$, we calculate: $$E(T) = \frac{15(15 + 20 + 1)}{2} = \frac{15(36)}{2} = 270$$ $$Var(T) = \frac{15 \times 20 \times (15 + 20 + 1)}{12} = \frac{15 \times 20 \times 36}{12} = 900$$ Thus, $$z = \frac{210 - 270}{\sqrt{900}} = \frac{-60}{30} = -2.0$$

A Wilcoxon Rank Sum Test for Comparing Two Populations Involves