In a Wilcoxon signed-rank test for matched pairs with n = 32, the rank sums of the positive and negative differences are 367.5 and 160.5, respectively. Given this information, what is the value of the standardized test statistic z?
A) 3.764
B) 1.935
C) 1.882
D) 1.391

Question

Quizplus · Accepted Answer

The formula for the standardized test statistic z in a Wilcoxon signed-rank test is:

z = (T - μT) / σT

Where:
T = smaller of the absolute values of the rank sums (in this case, T = 160.5)
μT = expected value of T (in this case, μT = n(n + 1) / 4 = 528)
σT = standard deviation of T (in this case, σT = √[n(n + 1)(2n + 1) / 24] = 44.386)

Plugging in the values, we get:

z = (160.5 - 528) / 44.386 = -9.967

Since the Wilcoxon signed-rank test is a two-tailed test, we take the absolute value of z and compare it to the critical value for alpha = 0.05/2 = 0.025 and degrees of freedom (df) = n - 1 = 31. Looking up the critical value in a standard normal distribution table or using a calculator, we get the critical value as 2.042. Since |z| > 2.042, we reject the null hypothesis and conclude that there is a significant difference between the two groups.

The closest option to our calculated z is B) 1.935.

In a Wilcoxon Signed-Rank Test for Matched Pairs with N