Question

In case of a randomized block design with 5 treatments, the mean sum of squares due to error is 2 and the F-value for testing significance of treatment differences is 12.5, then the treatment sum of squares will be :

• A. 25
• B. 50
• C. 75
• D. 100

Hint:

Always remember that a "Mean Square" is simply a "Sum of Squares" divided by its corresponding "Degrees of Freedom". Working backwards from the Mean Square to the Sum of Squares by multiplying by degrees of freedom is a common exam requirement.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In an Analysis of Variance (ANOVA) for a Randomized Block Design (RBD), the total variation is partitioned into sources due to treatments, blocks (replications), and error.
The F-statistic for testing the significance of treatment differences is the ratio of the Mean Sum of Squares for Treatments (MST) to the Mean Sum of Squares for Error (MSE).
Key Formula or Approach:
The relationships between sum of squares, degrees of freedom, and mean squares are: \[ F = \frac{\text{MST}}{\text{MSE}} \] \[ \text{MST} = F \times \text{MSE} \] Where: - $\text{MST} = \frac{\text{SST}}{df_{\text{treatment}}}$
- $\text{SST} = \text{MST} \times df_{\text{treatment}}$
- Degrees of freedom for treatments ($df_{\text{treatment}}$) is $t - 1$, where $t$ is the number of treatments.

Step 2: Detailed Explanation:
Given values from the problem:
- Number of treatments ($t$) = 5
- Mean Sum of Squares due to error ($\text{MSE}$) = 2
- F-value for treatments ($F$) = 12.5
First, calculate the degrees of freedom for treatments: \[ df_{\text{treatment}} = t - 1 = 5 - 1 = 4 \] Second, find the Mean Sum of Squares for Treatments (MST) using the F-ratio formula: \[ \text{MST} = F \times \text{MSE} \] \[ \text{MST} = 12.5 \times 2 = 25 \] Third, calculate the Treatment Sum of Squares (SST): \[ \text{SST} = \text{MST} \times df_{\text{treatment}} \] \[ \text{SST} = 25 \times 4 = 100 \] Thus, the treatment sum of squares is 100.

Step 3: Final Answer:
The calculated Treatment Sum of Squares (SST) is 100, which matches Option (D).