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.