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 and do a sign test).
Before 8 7 6 9 7 10 8 6
After 6 5 8 6 9 8 10 7

Answer 1

To determine whether the new incentive plan improved the morale of the workers, we can use the sign test as a non-parametric statistical test. The sign test will help us assess whether there is a significant difference in the morale ratings before and after the introduction of the incentive plan.

### Step 1: Set Up Hypotheses

- **Null Hypothesis (\(H_0\))**: There is no difference in morale before and after the incentive plan. (The median difference is 0.)
- **Alternative Hypothesis (\(H_1\)**: There is a difference in morale before and after the incentive plan. (The median difference is not 0.)

### Step 2: Calculate Differences

We need to calculate the differences between the "Before" (B) and "After" (A) scores for each worker:
\[
\text{Difference} = A - B
\]

| Before | After | Difference (A - B) |
|--------|-------|---------------------|
| 8 | 6 | -2 |
| 7 | 5 | -2 |
| 6 | 8 | 2 |
| 9 | 6 | -3 |
| 7 | 9 | 2 |
| 10 | 8 | -2 |
| 8 | 10 | 2 |
| 6 | 7 | 1 |

### Step 3: Count Signs of the Differences

Next, we count the number of positives and negatives:

- **Negative Steps**: -2, -2, -3, -2 -> Total = 4
- **Positive Steps**: 2, 2, 2, 1 -> Total = 4
- **Zero Differences**: None

### Step 4: Conduct the Sign Test

The sign test uses the smaller of the number of positive and negative differences. Here, both are 4. Thus, we have \( N = 8 \) (total non-zero differences).

We find the critical value for a two-tailed test at \(\alpha = 0.025\). The critical value for a binomial distribution with \(N = 8\) at a significance level of \(\alpha\) can be found in binomial tables or calculated using binomial probabilities.

The critical region for the sign test at \(\alpha = 0.025\) is at least 1 or the values that fall below the 2.5% threshold on either tail (with 8 observations):

- This means we will reject \(H_0\) if the number of positive signs \( < 1 \) or the number of negative signs \( < 1 \).

Using binomial probabilities, we first calculate the probabilities of the number of successes (here, positive differences).

\[
P(X = k) = \binom{N}{k} p^k (1-p)^{N-k}
\]

Where:
- \(N = 8\) is the total number of differences
- \(k\) is the number of positive differences (successes)
- \(p = 0.5\)

For critical values, we will evaluate if the count of signs is less than or equal 1:

Calculating:
\[
P(X = 0) = \binom{8}{0} (0.5)^0 (0.5)^8 = \frac{1}{256} \approx 0.0039
\]
\[
P(X = 1) = \binom{8}{1} (0.5)^1 (0.5)^7 = 8 \times \frac{1}{256} = \frac{8}{256} = \frac{1}{32} \approx 0.03125
\]

Adding these gives:
\[
P(X \leq 1) = P(X=0) + P(X=1) \approx 0.0039 + 0.03125 = 0.03515
\]

Since this is greater than 0.025, we fail to reject the null hypothesis.

### Conclusion

At a significance level of \(\alpha = 0.025\), there is **not enough evidence to conclude** that the new incentive plan significantly improved morale in the manufacturing plant. The morale of the workers did not statistically change.