To perform the Wilcoxon signed-rank test, we first need to calculate the differences between the paired observations (Result 1 - Result 2) and then rank the absolute values of these differences. The rank will be assigned in ascending order.
Step 1: Calculate Differences
Let's calculate the differences (Result 1 - Result 2) for the given data.
| Observation | Result 1 | Result 2 | Difference (Result 1 - Result 2) | |-------------|----------|----------|-----------------------------------| | 1 | 12 | 10 | 2 | | 2 | 6 | 6 | 0 | | 3 | 9 | 10 | -1 | | 4 | 15 | 13 | 2 | | 5 | 13 | 12 | 1 | | 6 | 17 | 14 | 3 | | 7 | 8 | 6 | 2 | | 8 | 13 | 13 | 0 | | 9 | 8 | 9 | -1 | | 10 | 10 | 11 | -1 |
Step 2: Absolute Differences
Next, we take the absolute values of these differences.
| Observation | Difference | Absolute Difference | |-------------|------------|---------------------| | 1 | 2 | 2 | | 2 | 0 | 0 | | 3 | -1 | 1 | | 4 | 2 | 2 | | 5 | 1 | 1 | | 6 | 3 | 3 | | 7 | 2 | 2 | | 8 | 0 | 0 | | 9 | -1 | 1 | | 10 | -1 | 1 |
Step 3: Rank the Absolute Differences
Now we rank the absolute differences, ignoring the zeros.
| Absolute Difference | Rank | |---------------------|------| | 1 | 1.5 | (two occurrences of 1, average rank for both) | 1 | 1.5 | | 2 | 3.5 | (three occurrences of 2, average rank for all three) | 2 | 3.5 | | 2 | 3.5 | | 3 | 6 |
Summary of Ranks
| Observation | Absolute Difference | Rank | |-------------|---------------------|------| | 1 | 2 | 3.5 | | 2 | 0 | | | 3 | 1 | 1.5 | | 4 | 2 | 3.5 | | 5 | 1 | 1.5 | | 6 | 3 | 6 | | 7 | 2 | 3.5 | | 8 | 0 | | | 9 | 1 | 1.5 | | 10 | 1 | 1.5 |
Step 4: Calculate Total of (-) Ranks
Now we sum the ranks for the negative differences:
- In our data set, the negative differences occurred for observations 3, 9, and 10.
- The ranks of the negative differences (ranked values):
Only negative ranks:
- For observation 3: Rank 1.5
- For observation 9: Rank 1.5
- For observation 10: Rank 1.5
Total negative ranks = 1.5 + 1.5 + 1.5 = 4.5
Result
Total of the (-) ranks = 4.5