Question

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?

Answer 1

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**.