A study is conducted to determine whether sunshine affects depression. Eight individuals are given a questionnaire measuring depression immediately following a run of 10 consecutive days when the sun shone for over 80% of the daylight hours. The same individuals have their depression measured immediately following 10 consecutive days without any sunshine. The following data are collected. The higher the score the greater the depression. Using the Wilcoxon signed ranks test to evaluate the data, the value of T obt is _________.
A) - 3
B) 3
C) 33
D) 4

Question

A study is conducted to determine whether sunshine affects depression. Eight individuals are given a questionnaire measuring depression immediately following a run of 10 consecutive days when the sun shone for over 80% of the daylight hours. The same individuals have their depression measured immediately following 10 consecutive days without any sunshine. The following data are collected. The higher the score the greater the depression.  Using the Wilcoxon signed ranks test to evaluate the data, the value of T obt is _________.
A) - 3 
B) 3 
C) 33 
D) 4

Quizplus · Accepted Answer

To use the Wilcoxon signed ranks test, we need to calculate the differences between each individual's depression scores during sunny days and cloudy days. Then, we rank the absolute values of these differences and calculate the sum of the ranks for the positive differences and for the negative differences separately. The smaller sum of the two is taken as the test statistic, denoted by T. In this case, the differences and absolute differences are:

|13 - 23| = 10,
|11 - 19| = 8,
|18 - 24| = 6,
|9 - 17| = 8,
|8 - 14| = 6,
|14 - 18| = 4,
|15 - 19| = 4,
|17 - 20| = 3.

Since there are only 8 individuals, we can see that there are no ties in the absolute differences. Ranking these absolute differences from smallest to largest, we get:

3, 4, 4, 6, 6, 8, 8, 10.

For the positive differences, we have ranks 6 and 7, summing to 13. For the negative differences, we have ranks 1-4, summing to 10. Therefore, the smaller sum is 10, which is our test statistic T. Since there are 8 individuals, we can consult a table of critical values of T for the Wilcoxon signed-ranks test (or use a software). At the 5% level of significance, the critical value for a two-tailed test with n=8 is 0. Therefore, we reject the null hypothesis if |T| > 0, which is the case here since T=10. The p-value is between 0.01 and 0.05. Conclusion: there is evidence to suggest that sunshine affects depression, since depression scores were significantly higher during cloudy days compared to sunny days.

A Study Is Conducted to Determine Whether Sunshine Affects Depression