Sure, let's analyze the given data and calculate the Spearman's Rank correlation coefficient for each combination of the production techniques. The Spearman's Rank correlation coefficient is a measure of the strength and direction of association between two ranked variables.
### Data Set:
We have the following data:
| Worker | Technique 1 | Technique 2 | Technique 3 |
|--------|-------------|-------------|-------------|
| A | 40 | 60 | 55 |
| B | 50 | 86 | 65 |
| C | 55 | 40 | 40 |
| D | 60 | 59 | 42 |
| E | 63 | 62 | 85 |
| F | 57 | 85 | 70 |
| G | 42 | 90 | 73 |
| H | 69 | 42 | 47 |
| I | 80 | 40 | 90 |
| S | 44 | | 64 |
(Note: Worker S does not have data for Technique 2, hence calculations with this entry will exclude this specific technique.)
To find the Spearman's Rank correlation coefficient, we follow these steps for each pair of techniques:
1. Rank the data for each technique.
2. Compute the difference in ranks for each pair of techniques.
3. Calculate the square of the differences.
4. Use the formula for Spearman's Rank correlation coefficient:
[tex]\[
\rho = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}
\][/tex]
where [tex]\( d_i \)[/tex] is the difference between ranks for each pair of observations and [tex]\( n \)[/tex] is the number of observations.
Given the calculations:
### Technique 1 and Technique 2:
- Spearman's Rank correlation coefficient: -0.5607 (approx.)
### Technique 1 and Technique 3:
- Spearman's Rank correlation coefficient: 0.2000 (approx.)
### Technique 2 and Technique 3:
- Spearman's Rank correlation coefficient: 0.3347 (approx.)
### Conclusion:
The Spearman's Rank correlation coefficient is highest between Technique 2 and Technique 3 with a value of 0.3347. Thus, the firm should consider this pair of production techniques as the most efficient based on their higher correlation.
In summary:
- Correlation coefficient for Technique 1 and Technique 2 is approximately -0.5607.
- Correlation coefficient for Technique 1 and Technique 3 is approximately 0.2000.
- Correlation coefficient for Technique 2 and Technique 3 is approximately 0.3347.
- Best combination of techniques with the highest Spearman's Rank correlation coefficient is Technique 2 and Technique 3.
