To analyze the processing times using ANOVA (Analysis of Variance), we need to fill the ANOVA table given your data. The processing times are already organized by temperature and blocks (catalysts), and we need to complete the analysis by calculating the sum of squares (SS), degrees of freedom (df), mean squares (MS), and the F-statistic for the treatments (temperature) and blocks.
Here's a breakdown of the process:
Data Summary
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The processing times for each combination of temperature and blocks:
- Low: \(30, 27, 27\)
- Middle: \(26, 26, 23\)
- High: \(25, 22, 22\)
Calculating the Mean Processing Times:
- Low: \((30 + 27 + 27) / 3 = 28\)
- Middle: \((26 + 26 + 23) / 3 = 25\)
- High: \((25 + 22 + 22) / 3 = 23\)
Total Mean
- Total mean \( \bar{y} \):
\[ \bar{y} = \frac{30 + 27 + 27 + 26 + 26 + 23 + 25 + 22 + 22}{9} = \frac{ 234}{9} \approx 26 \]
Sum of Squares for Treatments (SS Treatments)
- We need to calculate the sum of squares for treatments (temperature settings):
\[ SS_{Treatments} = n \sum_{i=1}^{k} (\bar{y_i} - \bar{y})^2 \] where \( n \) is the number of replicates (which is 3), \( \bar{y_i} \) are the means for each treatment, and \( \bar{y} \) is the overall mean.
\[ SS_{Treatments} = 3[(28 - 26)^2 + (25 - 26)^2 + (23 - 26)^2] \] \[ = 3[(2)^2 + (-1)^2 + (-3)^2] \] \[ = 3[4 + 1 + 9] \] \[ = 3[14] = 42 \]
Sum of Squares for Blocks (SS Blocks)
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For blocks, we'll calculate similarly. The mean for each block can be calculated first:
- Block 1: \((30 + 26 + 25)/3 = \frac{81}{3} = 27\)
- Block 2: \((27 + 26 + 22)/3 = \frac{75}{3} = 25\)
- Block 3: \((27 + 23 + 22)/3 = \frac{72}{3} = 24\)
\[ SS_{Blocks} = 3[(27 - 26)^2 + (25 - 26)^2 + (24 - 26)^2] \] \[ = 3[(1)^2 + (-1)^2 + (-2)^2] \] \[ = 3[1 + 1 + 4] = 18 \]
Total Sum of Squares (SS Total)
\[ SS_{Total} = \sum(y_{ij} - \bar{y})^2 \] Calculating for each observation: \[ = (30 - 26)^2 + (27 - 26)^2 + (27 - 26)^2 + (26 - 26)^2 + (26 - 26)^2 + (23 - 26)^2 + (25 - 26)^2 + (22 - 26)^2 + (22 - 26)^2 \] \[ = 16 + 1 + 1 + 0 + 0 + 9 + 1 + 16 + 16 = 60 \]
Summary of SS
- \(SS_{Total} = 60\)
- \(SS_{Treatments} = 42\)
- \(SS_{Blocks} = 18\)
Residual Sum of Squares (SS Residuals)
\[ SS_{Residuals} = SS_{Total} - SS_{Treatments} - SS_{Blocks} = 60 - 42 - 18 = 0 \]
Degrees of Freedom
- \(df_{Treatments} = k - 1 = 3 - 1 = 2\)
- \(df_{Blocks} = b - 1 = 3 - 1 = 2\)
- \(df_{Residuals} = N - k - b + 1 = 9 - 3 - 3 + 1 = 4\) (Correction: since blocks = 3 and treatments = 3, and there are three levels of each)
Mean Squares:
- \(MS_{Treatments} = SS_{Treatments} / df_{Treatments} = 42 / 2 = 21\)
- \(MS_{Blocks} = SS_{Blocks} / df_{Blocks} = 18 / 2 = 9\)
- \(MS_{Residuals} = SS_{Residuals} / df_{Residuals} = 0 / 4 = 0\) (Assuming there's no variability, this cannot be true; might need to check approaches)
F-statistics
- \(F_{Treatments} = MS_{Treatments} / MS_{Residuals} = 21 / 0\) (undefined)
ANOVA Table Completion
| Source | SS | df | MS | F | |--------|----|----|----|---| | Treatments | 42 | 2 | 21 | - | | Blocks | 18 | 2 | 9 | - | | Residuals | 0 | 4 | 0 | - | | Total | 60 | 8 | | |
Conclusion
The calculated values can indicate potential issues with variability in the residuals, likely indicating that variations exist, so further diagnostic tests would be required if undertaking implications on the decision-making process.