Question

For the given ANOVA table:  

\[ \begin{array}{|l|c|c|} \hline \textbf{Source of variation} & \textbf{Sum of squares} & \textbf{Degrees of freedom} \\ \hline \text{Service station} & 6810 & 9 \\ \text{Rating} & 400 & 4 \\ \text{Total} & 9948 & 49 \\ \hline \end{array} \] 

The test statistic to test that there is no significant difference between the service stations is: 

• A. 8.6
• B. 12.95
• C. 9.95
• D. 6.85

Hint:

When given an incomplete ANOVA table, the first step is always to find the missing values for Sum of Squares and Degrees of Freedom by using the additivity property: the components (e.g., Treatment, Block, Error) must sum up to the Total.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The problem provides a partial ANOVA table for what appears to be a two-way classification (e.g., RBD or two-way ANOVA with replication). We need to calculate the F-test statistic to test for a significant difference between "Service stations". The F-statistic is the ratio of the Mean Square for the factor of interest to the Mean Square for Error.

Step 2: Key Formula or Approach:
The F-statistic for testing the effect of a factor (e.g., Treatment) is: \[ F = \frac{\text{MS(Treatment)}}{\text{MS(Error)}} = \frac{\text{SS(Treatment)}/\text{df(Treatment)}}{\text{SS(Error)}/\text{df(Error)}} \] We first need to find the Sum of Squares for Error (SSE) and its degrees of freedom (dfE) by subtraction from the total.

Step 3: Detailed Explanation:
The ANOVA table has missing information for the Error term. We can find it by subtraction. Let SS(Station) = 6810, df(Station) = 9. Let SS(Rating) = 400, df(Rating) = 4. Let SS(Total) = 9948, df(Total) = 49. Find SS(Error) and df(Error): The total sum of squares is partitioned as: SS(Total) = SS(Station) + SS(Rating) + SS(Error). \[ \text{SS(Error)} = \text{SS(Total)} - \text{SS(Station)} - \text{SS(Rating)} \] \[ \text{SS(Error)} = 9948 - 6810 - 400 = 2738 \] The degrees of freedom are also partitioned: 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: We are testing the difference between service stations, so this is our "treatment" of interest. \[ F = \frac{\text{MS(Station)}}{\text{MS(Error)}} = \frac{756.67}{76.056} \approx 9.9488 \] This value is very close to 9.95. There seems to be an error in my calculation or the provided options/solution. Let me re-read the table. Source of variation | Sum of squares | Degrees of freedom Service station | 6810 | 9 Rating | 400 | 4 Total | 9948 | 49 The structure implies 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. This is option (C). Let me check if I misinterpreted the table. What if "Rating" is the error term? This would be unusual naming. In that case, F = MS(Station)/MS(Rating) = (6810/9) / (400/4) = 756.67 / 100 = 7.56. Not an option. What if the design is nested? Unlikely. What if the question made a typo and "Rating" was meant to be "Error"? Let's assume the question meant SS(Error)=400 and df(Error)=4. F = MS(Station)/MS(Error) = (6810/9) / (400/4) = 756.67 / 100 = 7.57. Still not matching. What if SS(Error)=6810 and SS(Station)=400? Then F = (400/4) / (6810/9) = 100/756.67<1. There must be a typo in the numbers. Let's work backwards from the answer 8.6. If F = 8.6, then MS(Station)/MS(Error) = 8.6. (6810/9) / (SS(Error)/df(Error)) = 8.6. 756.67 / MS(Error) = 8.6 ⇒ MS(Error) = 756.67 / 8.6 = 88. If MS(Error) = 88, and df(Error)=36, then SS(Error)=8836 = 3168. This would make SS(Total) = 6810+400+3168 = 10378. Not 9948. The calculation leading to 9.95 is arithmetically correct based on the table. The provided answer key (A) seems to be incorrect.
Step 4: Final Answer:
Based on a standard two-way ANOVA decomposition, the calculated F-statistic is 9.95. Option (A) 8.6 appears to be incorrect based on the provided data.