Sign test and Wilcoxon signed-rank test：練習問題

When performing a sign test for a single population median, which of the following assumptions must be met regarding the distribution of the population?

In a Wilcoxon signed-rank test for paired data, the calculated sum of the ranks of the positive differences is \(W_+ = 12\) and the sum of the ranks of the negative differences is \(W_- = 43\). For a sample size of \(n=10\) (with no zero differences), identify the correct test statistic \(T\) used for comparison with critical values in a two-tailed test.

A researcher uses the Wilcoxon rank-sum test to compare the medians of two independent groups, Group X (\(n_1 = 8\)) and Group Y (\(n_2 = 7\)). The sum of the ranks for Group X is calculated as \(R_1 = 82\). Calculate the value of the test statistic \(U\) for Group X, where \(U_1 = R_1 - \frac{n_1(n_1 + 1)}{2}\).

State the necessary assumption regarding the distribution of the underlying population for a Wilcoxon signed-rank test to be valid when testing a single population median.

When performing a Wilcoxon signed-rank test to investigate the difference between paired observations, explain why the assumption of symmetry in the distribution of differences is required, whereas it is not required for the sign test.

Explain the process of handling tied ranks in a Wilcoxon rank-sum test. If three observations in a combined dataset are tied for the 4th, 5th, and 6th positions, what rank value should be assigned to each of these three observations?

A fitness instructor wants to determine if a new 4-week flexibility program significantly increases the median reach of participants. The reach (in cm) of 9 participants was measured before and after the program. The differences (After - Before) were calculated as follows:

\( 2.5, \, 0.8, \, -1.2, \, 4.0, \, 3.2, \, 0.0, \, 1.5, \, -0.5, \, 2.1 \)

(a) State the necessary assumption about the distribution of differences for a Wilcoxon signed-rank test to be valid. (1 point)
(b) Calculate the test statistic \( W \) (the smaller of the sum of positive ranks and the sum of negative ranks) for these differences. (3 points)
(c) The critical value for a one-tailed test at the 5% significance level with \( n=8 \) is 5. Using your answer to part (b), determine whether there is sufficient evidence to suggest the program increases flexibility. (1 point)

A researcher is conducting a Wilcoxon signed-rank test to investigate whether a new sleep therapy increases the median hours of sleep for 12 patients. The differences (After - Before) in hours are:
1.2, 0.5, -0.1, 2.3, 0.0, 1.4, -0.4, 0.8, 1.9, 2.1, -0.2, 0.7.

(a) State the necessary assumption for this test to be valid.
(b) Calculate the test statistic \(T\) for a one-tailed test.
(c) Given the critical value for \(n=11\) at the 5% level is 13, state your conclusion regarding the therapy.