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Perform the Wilcoxon signed rank sum test given the following paired

observations:


Data

1

2

3

4

5

6

7

8

9

10

Result 1

12

6

9

15

13

17

8

13

8

10

Result 2

10

6

10

13

12

14

6

13

9

11


Rank the absolute differences of the above paired data.

What is the value of the test statistic?

1 answer

To perform the Wilcoxon signed rank test on the given paired observations, we'll follow these steps:

  1. Calculate the differences between the paired samples.
  2. Calculate the absolute differences and rank them.
  3. Assign signs (+ or -) based on whether the difference is positive or negative.
  4. Calculate the test statistic (the sum of ranks for the positive or negative differences).

Step 1: Calculate the differences

Let's calculate the differences (Result 1 - Result 2):

  • \( 12 - 10 = 2 \)
  • \( 6 - 6 = 0 \)
  • \( 9 - 10 = -1 \)
  • \( 15 - 13 = 2 \)
  • \( 13 - 12 = 1 \)
  • \( 17 - 14 = 3 \)
  • \( 8 - 6 = 2 \)
  • \( 13 - 13 = 0 \)
  • \( 8 - 9 = -1 \)
  • \( 10 - 11 = -1 \)

Thus, the differences are:

\[ \begin{align*} D &= [2, 0, -1, 2, 1, 3, 2, 0, -1, -1] \end{align*} \]

Step 2: Calculate the absolute differences and rank them

Now, we take the absolute values of these differences and rank them. We will ignore the zero differences for ranking.

\[ \begin{align*} |D| &= [2, 0, 1, 2, 1, 3, 2, 0, 1, 1] \ & = [2, 1, 3, 2, 2, 1, 1] \quad (\text{ignoring } 0) \end{align*} \]

Now, we assign ranks:

  • The value 1 appears 3 times (average ranking for these = (2 + 3 + 4)/3 = 3).
  • The value 2 appears 3 times (average ranking for these = (5 + 6 + 7)/3 = 6).
  • The value 3 appears 1 time (rank = 8).

Thus the ranks for each difference (based on absolute values) are:

| Index | Difference | Absolute Difference | Rank | |-------|------------|---------------------|------| | 1 | 2 | 2 | 6 | | 2 | 0 | 0 | — | | 3 | -1 | 1 | 3 | | 4 | 2 | 2 | 6 | | 5 | 1 | 1 | 3 | | 6 | 3 | 3 | 8 | | 7 | 2 | 2 | 6 | | 8 | 0 | 0 | — | | 9 | -1 | 1 | 3 | | 10 | -1 | 1 | 3 |

Step 3: Assign signs and sum ranks

Now we assign the signs based on the original differences and sum the ranks for both positive and negative groups.

  • Positive differences:

    • 2 (Rank 6)
    • 2 (Rank 6)
    • 1 (Rank 3)
    • 3 (Rank 8)
    • 2 (Rank 6)

  • Negative differences:

    • -1 (Rank 3)
    • -1 (Rank 3)
    • -1 (Rank 3)

Sum of ranks

  1. Positive ranks:

    • \( 6 + 6 + 3 + 8 + 6 = 29 \)

  2. Negative ranks:

    • \( 3 + 3 + 3 = 9 \)

Step 4: Compute the test statistic

The test statistic \( T \) is the smaller of the two rank sums (for positive and negative differences). Thus,

\[ T = \min(29, 9) = 9 \]

Conclusion

The Wilcoxon signed rank test statistic \( T \) is 9.