To analyze the processing times using ANOVA (Analysis of Variance), we first need to organize the data and calculate the necessary components of the ANOVA table.
Step 1: Organize the Data
From the data provided, we extract the following:
| Temperature | Block 1 | Block 2 | Block 3 | Row Mean | |--------------|---------|---------|---------|-----------| | Low | 30 | 27 | 27 | (30+27+27)/3 = 28 | | Middle | 26 | 26 | 23 | (26+26+23)/3 = 25 | | High | 25 | 22 | 22 | (25+22+22)/3 = 23 |
Step 2: Calculate the Grand Mean
To calculate the grand mean \( G \):
\[ G = \frac{(30 + 27 + 27 + 26 + 26 + 23 + 25 + 22 + 22)}{9} = \frac{ 24 + 27 + 29 + 25 + 22 + 22 + 22 + 29}{9} = \frac{ 1 63 }{9} \approx 24.33 \]
Step 3: Calculate Sums of Squares
- Total Sum of Squares (SST):
\[ SST = \sum (X_{ij} - G)^2 \]
Calculating SST:
\[ SST = (30 - G)^2 + (27 - G)^2 + (27 - G)^2 + (26 - G)^2 + (26 - G)^2 + (23 - G)^2 + (25 - G)^2 + (22 - G)^2 + (22 - G)^2 \]
- Treatment Sum of Squares (SSTr):
\[ SSTr = n \sum (\overline{X_j} - G)^2 \]
Where \( \overline{X_j} \) is the mean of each treatment row.
- For Low Temperature:
\[ \overline{X_1} = \frac{30 + 27 + 27}{3} = 28 \] \((28 - G)^2\)
- For Middle Temperature:
\[ \overline{X_2} = \frac{26 + 26 + 23}{3} = 25 \] \((25 - G)^2\)
- For High Temperature:
\[ \overline{X_3} = \frac{25 + 22 + 22}{3} = 23 \] \((23 - G)^2\)
After that, calculate SSTr as:
\[ SSTr = 3((28 - G)^2 + (25 - G)^2 + (23 - G)^2) \]
- Blocks Sum of Squares (SSB):
This will be calculated similarly to Treatments, by using the means of each block.
Step 4: Calculate the Degrees of Freedom
- df Treatment: \( k - 1 \) where \( k \) is the number of treatments (3).
- df Block: \( b - 1\) where \( b \) is the number of blocks (3).
- df Residual: Total observations - number of groups = \( N - k - b + 1 \).
Step 5: Calculate Mean Squares
- MS Treatments: \(\frac{SSTr}{df Treatment} \)
- MS Blocks: \(\frac{SSB}{df Block}\)
- MS Residual: \(\frac{SSR}{df Residual}\)
Step 6: Calculate the F Statistic
- Treatments F Statistic: \( F_t = \frac{MS Treatments}{MS Residual} \)
- Blocks F Statistic: \( F_b = \frac{MS Blocks}{MS Residual} \)
Completing the ANOVA Table
Fill in the calculated values into the ANOVA table as provided. If calculations for sums of squares yield particular values, ensure to place those in the ANOVA table with respective degrees of freedom and mean squares as calculated.
The complete filling of the ANOVA table will depend on specific numerical calculations provided in the earlier steps.
ANOVA Table Example
| Source | SS | df | MS | F | |------------------|------|----|--------|--------| | Treatments | SSTr | 2 | MSTr | Ft | | Blocks | SSB | 2 | MSB | Fb | | Residuals | SSR | 4 | MSR | - | | Total | SST | 8 | - | - |
I recommend performing the arithmetic calculations for SST, SSTr, SSB based on the data to fill in the table completely. Each of the steps detailed above needs to be followed through for accurate calculations.