Step 1: Problem Setup:
The problem presents a partial ANOVA table, likely from a two-way classification (e.g., RBD or two-way ANOVA with replication). The goal is to compute the F-statistic to test for significant differences between "Service stations". This F-statistic is the ratio of the Mean Square for the factor (Service Stations) to the Mean Square for Error.
Step 2: Formula for F-statistic:
The F-statistic is calculated as follows:
\[ F = \frac{\text{MS(Treatment)}}{\text{MS(Error)}} = \frac{\text{SS(Treatment)}/\text{df(Treatment)}}{\text{SS(Error)}/\text{df(Error)}} \]
To compute this, we need the Sum of Squares for Error (SSE) and its degrees of freedom (dfE), derived by subtraction from the total.
Step 3: Detailed Calculation:
The ANOVA table lacks information about the Error term. This is calculated by subtraction using the following:
SS(Station) = 6810, df(Station) = 9
SS(Rating) = 400, df(Rating) = 4
SS(Total) = 9948, df(Total) = 49
Calculate SS(Error) and df(Error):
SS(Total) = SS(Station) + SS(Rating) + SS(Error)
\[ \text{SS(Error)} = \text{SS(Total)} - \text{SS(Station)} - \text{SS(Rating)} \]
\[ \text{SS(Error)} = 9948 - 6810 - 400 = 2738 \]
df(Total) = df(Station) + df(Rating) + df(Error)
\[ \text{df(Error)} = \text{df(Total)} - \text{df(Station)} - \text{df(Rating)} \]
\[ \text{df(Error)} = 49 - 9 - 4 = 36 \]
Calculate Mean Squares:
\[ \text{MS(Station)} = \frac{\text{SS(Station)}}{\text{df(Station)}} = \frac{6810}{9} = 756.67 \]
\[ \text{MS(Error)} = \frac{\text{SS(Error)}}{\text{df(Error)}} = \frac{2738}{36} \approx 76.056 \]
Calculate the F-statistic:
Testing for differences between service stations:
\[ F = \frac{\text{MS(Station)}}{\text{MS(Error)}} = \frac{756.67}{76.056} \approx 9.9488 \]
Which rounds to 9.95.
The calculated F-statistic is 9.95. Further verification and alternative scenarios were explored:
Source of variation | Sum of squares | Degrees of freedom
Service station | 6810 | 9
Rating | 400 | 4
Total | 9948 | 49
Assuming a two-way ANOVA without interaction:
SS(Error) = 9948 - 6810 - 400 = 2738.
df(Error) = 49 - 9 - 4 = 36.
MS(Station) = 6810/9 = 756.67.
MS(Error) = 2738/36 = 76.05.
F = 756.67 / 76.05 = 9.95. Matches option (C) in some answer keys.
Considering "Rating" as the error term, F = MS(Station)/MS(Rating) = (6810/9) / (400/4) = 756.67 / 100 = 7.56, which is not in the answer options.
Exploring other scenarios like nested designs or typos in the question yielded no matching F-statistic in the options.
Working backward from a possible answer (8.6) also led to inconsistencies.
The calculation leading to 9.95 is correct based on the given table.
Step 4: Conclusion:
Based on the standard two-way ANOVA calculation, the F-statistic is 9.95. The provided option (A) 8.6 appears to be incorrect based on the data given in the table.