Model Answer
0 min readIntroduction
In operations management, understanding and optimizing workforce productivity is crucial for organizational success. Statistical tools like ANOVA are frequently employed to analyze variations in output and identify significant differences between groups. ANOVA allows us to determine whether observed differences are due to real variations or simply random chance. This question presents a scenario where we need to assess if workers and machines exhibit differing levels of productivity using a given dataset. The application of hypothesis testing at a 5% significance level will help us draw statistically sound conclusions regarding worker and machine performance.
Hypothesis Formulation
We need to test two sets of hypotheses:
(i) Workers do not differ with regard to mean productivity
- Null Hypothesis (H0): μ1 = μ2 = μ3 = μ4 (The mean productivity of all four workers is the same).
- Alternative Hypothesis (H1): At least one μi is different from the others (The mean productivity of at least one worker is different).
(ii) Mean production is same for different machines
- Null Hypothesis (H0): μA = μB = μC = μD = μE (The mean productivity of all five machines is the same).
- Alternative Hypothesis (H1): At least one μi is different from the others (The mean productivity of at least one machine is different).
ANOVA Calculations and Interpretation
Since the actual data table is not provided in the prompt, we will assume that the ANOVA calculations have been performed and the results are summarized in the following tables (these would normally be generated using statistical software like SPSS, R, or Excel). We will focus on interpreting the results.
ANOVA Table for Workers
| Source of Variation | Sum of Squares (SS) | Degrees of Freedom (df) | Mean Square (MS) | F-statistic | P-value |
|---|---|---|---|---|---|
| Between Workers | [Calculated Value] | 3 | [Calculated Value] | [Calculated Value] | [Calculated Value] |
| Within Workers | [Calculated Value] | 16 | [Calculated Value] | ||
| Total | [Calculated Value] | 19 |
ANOVA Table for Machines
| Source of Variation | Sum of Squares (SS) | Degrees of Freedom (df) | Mean Square (MS) | F-statistic | P-value |
|---|---|---|---|---|---|
| Between Machines | [Calculated Value] | 4 | [Calculated Value] | [Calculated Value] | [Calculated Value] |
| Within Machines | [Calculated Value] | 15 | [Calculated Value] | ||
| Total | [Calculated Value] | 19 |
Critical F-value: At a 5% significance level, with 3 and 16 degrees of freedom (for workers) and 4 and 15 degrees of freedom (for machines), we would determine the critical F-value from an F-distribution table. Let's assume, for illustrative purposes, that the critical F-value for workers is 3.24 and for machines is 3.68.
Decision Making
We compare the calculated F-statistic with the critical F-value for each scenario:
(i) Workers
If the calculated F-statistic for workers is greater than 3.24, we reject the null hypothesis and conclude that there is a significant difference in mean productivity between the workers. If the F-statistic is less than or equal to 3.24, we fail to reject the null hypothesis and conclude that there is no significant difference.
(ii) Machines
If the calculated F-statistic for machines is greater than 3.68, we reject the null hypothesis and conclude that there is a significant difference in mean productivity between the machines. If the F-statistic is less than or equal to 3.68, we fail to reject the null hypothesis and conclude that there is no significant difference.
The P-value in the ANOVA table also provides a direct indication. If the P-value is less than 0.05, we reject the null hypothesis.
Conclusion
Based on the ANOVA analysis, we can determine whether significant differences exist in the mean productivity of the workers and the machines. The decision to reject or fail to reject the null hypotheses depends on comparing the calculated F-statistics with the critical F-values (or examining the P-values). This information is vital for optimizing production processes, assigning tasks effectively, and identifying potential areas for improvement in workforce training or machine maintenance. Further investigation, such as post-hoc tests, may be necessary to pinpoint which specific workers or machines differ significantly if the overall ANOVA test is significant.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.