1. Lecture Slides Elementary Statistics Twelfth Edition
and the Triola Statistics Series
by Mario F. Triola

2. Chapter 13 Nonparametric Statistics
13-1 Review and Preview
13-2 Sign Test
13-3 Wilcoxon Signed-Ranks Test for Matched Pairs
13-4 Wilcoxon Rank-Sum Test for Two Independent Samples
13-5 Kruskal-Wallis Test
13-6 Rank Correction
13-7 Runs Test for Randomness

3. Key Concept
The Wilcoxon signed-ranks test involves the conversion of the sample data to ranks.
This test can be used for two different applications:
Testing a claim that a population of matched pairs has the property that the matched pairs have differences with a median equal to zero.
Testing a claim that a single population of individual values has a median equal to some claimed value.

4. Wilcoxon Signed-Ranks Test Requirements
The data are a simple random sample.
The population of differences has a distribution that is approximately symmetric, meaning that the left half of its histogram is roughly a mirror image of its right half.
(There is no requirement that the data have a normal distribution.)

5. Notation T = the smaller of the following two sums:
1. The sum of the positive ranks of the nonzero differences d
2. The absolute value of the sum of the negative ranks of the nonzero differences d

6. Test Statistic for the Wilcoxon Signed-Ranks Test for Matched Pairs
For n ≤ 30, the test statistic is T.
For n > 30, the test statistic is:

7. P-Values / Critical Values
P-values: P-values are often provided by technology, or can be found using the z test statistic and Table A-2.
Critical Values:
For n ≤ 30, the critical T value is found in Table A-8.
2. For n > 30, the critical z values are found in Table A-2.

8. Procedure for Finding the Value of the Test Statistic
Step 1: For each pair of data, find the difference d by subtracting the second value from the first. Keep the signs, but discard any pairs for which d = 0.
Step 2: Ignore the signs of the differences, then sort the differences from lowest to highest and replace the differences by the corresponding rank value. When differences have the same numerical value, assign to them the mean of the ranks involved in the tie.

9. Procedure for Finding the Value of the Test Statistic
Step 3: Attach to each rank the sign difference from which it came. That is, insert those signs that were ignored in step 2.
Step 4: Find the sum of the absolute values of the negative ranks. Also find the sum of the positive ranks.
Step 5: Let T be the smaller of the two sums found in Step 4. Either sum could be used, but for a simplified procedure we arbitrarily select the smaller of the two sums.

10. Procedure for Finding the Value of the Test Statistic
Step 6: Let n be the number of pairs of data for which the difference d is not 0.
Step 7: Determine the test statistic and critical values based on the sample size, as shown above.
Step 8: When forming the conclusion, reject the null hypothesis if the sample data lead to a test statistic that is in the critical region that is, the test statistic is less than or equal to the critical value(s). Otherwise, fail to reject the null hypothesis.

11. Example
The first two rows of Table 13-4 include taxi times for a sample of American Airlines Flight 21. Use the sample data to test the claim that there is no difference between taxi-out and taxi-in times. Use the Wilcoxon signed-ranks test and a 0.05 level of significance.

12. Example - Continued
Requirement Check: The data are from a simple random sample. The differences should be symmetric, and though the histogram does not support this, we have only 11 differences and the issue is not too extreme.

13. Example - Continued
The claim is of no difference between taxi-in and taxi-out times, so the hypotheses are:
Using the 8-step procedure described earlier, the test statistic is T = 11.

14. Example - Continued
The sample size is n = 11, so the critical value is found in Table A-8. Using a 0.05 level of significance, the critical value is found to be 11.
We should reject the null hypothesis if the test statistic T is less than or equal to the critical value.
Because the test statistic of T = 11 equals the critical value, we reject the null hypothesis.
We conclude that the taxi-out and taxi-in times do not appear to be about the same.

15. Claims about the Median
of a Single Population
Make one simple adjustment:
When testing a claim about the median of a single population, create matched pairs by pairing each sample value with the claimed value of the median. The preceding procedure can then be used.