Question

. Management decided to introduce an incentive plan to try to increase the productivity in a small manufacturing plant. The morale of the workers was measured before and after the introduction of the new plan. A scale of 1 to 10 was used, where 1 is a very low morale and 10 a very high morale. Can we say that the new incentive plan did improve the morale of the workers? ( Use α=0.025).

Before 7 6 5 8 6 9 7 5
After 5 4 9 7 8 9 11 8

Conduced a Wilcoxon signed – rank sum test.

Answer 1

To perform a Wilcoxon signed-rank test on the given data, we first need to compute the differences between the paired observations (After - Before) and then analyze these differences.

### Step 1: Calculate the differences
For each pair of before and after values, we calculate the difference \( D_i = \text{After}_i - \text{Before}_i \).

| Before | After | Difference \( D = \text{After} - \text{Before} \) |
|--------|-------|---------------------------------------------------|
| 7 | 5 | -2 |
| 6 | 4 | -2 |
| 5 | 9 | 4 |
| 8 | 7 | -1 |
| 6 | 8 | 2 |
| 9 | 9 | 0 |
| 7 | 11 | 4 |
| 5 | 8 | 3 |

### Step 2: Rank the absolute differences
Next, we take the absolute values of the differences, ignore any zeros, and rank them.

| Difference | Absolute Difference | Rank |
|------------|--------------------|------|
| -2 | 2 | 4.5 |
| -2 | 2 | 4.5 |
| 4 | 4 | 7 |
| -1 | 1 | 1 |
| 2 | 2 | 4.5 |
| 0 | 0 | N/A |
| 4 | 4 | 7 |
| 3 | 3 | 6 |

The ranks of the absolute differences are assigned as follows:
- For absolute differences of the same value, we assign the average rank.

The ranked differences with signs:

| Sign | Rank |
|------|------|
| - | 4.5 |
| - | 4.5 |
| + | 7 |
| - | 1 |
| + | 4.5 |
| - | N/A |
| + | 7 |
| + | 6 |

### Step 3: Calculate the test statistic
Next, we compute \( W \), the sum of the ranks for the positive differences (or negative differences). Here, we will choose to sum the ranks for the positive differences:

Positive ranks: \( 7 + 4.5 + 6 \)
\( W_+ = 7 + 4.5 + 6 = 17.5 \)
Negative ranks: \( 4.5 + 4.5 + 1 = 10 \)
Thus, \( W_- = 10 \).

The test statistic for Wilcoxon signed-rank test is given by:

\[
W = \min(W_+, W_-) = \min(17.5, 10) = 10
\]

### Step 4: Determine the critical value
For \( n = 7 \) (number of non-zero differences) and \( \alpha = 0.025 \), we refer to the Wilcoxon signed-rank table or use software to find the critical value:

The critical value \( W \) at \( n = 7 \) and \( \alpha = 0.025 \) (two-tailed) is usually around 2.

### Step 5: Conclusion
Since our computed \( W = 10 \) is greater than \( 2 \), we fail to reject the null hypothesis. This indicates that there is not enough evidence to suggest that the new incentive plan improved the morale of the workers at a significance level of \( \alpha = 0.025 \).

### Summary
Based on the Wilcoxon signed-rank test, we conclude that there is no statistically significant increase in worker morale after the implementation of the new incentive plan.