Question

An experiment is conducted in a Randomized block design with five treatments in four replications, the standard error of the difference between two treatments means are (MSE is the error mean square)

• A. $\sqrt{\frac{\text{MSE}}{2}}$
• B. $\sqrt{\frac{\text{MSE}}{4}}$
• C. $\sqrt{\frac{5\text{MSE}}{4}}$
• D. $\sqrt{\frac{5\text{MSE}}{2}}$

Hint:

The standard error of a single treatment mean is $\sqrt{\frac{\text{MSE}}{r}}$.
When comparing two independent means, the variance of their difference is the sum of their individual variances: $\frac{\text{MSE}}{r} + \frac{\text{MSE}}{r} = \frac{2\text{MSE}}{r}$. Taking the square root gives the formula for $SE_d$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To compare treatment means in design of experiments, we calculate the standard error of the difference ($SE_d$) between two treatment means.
This metric is crucial for performing post-hoc tests like the Critical Difference (CD) or Least Significant Difference (LSD).
Key Formula or Approach:
The standard error of the difference between two treatment means is given by: \[ SE_d = \sqrt{\frac{2 \times \text{MSE}}{r}} \] Where: - $\text{MSE}$ is the Mean Square due to Error.
- $r$ is the number of replications (blocks) for each treatment.

Step 2: Detailed Explanation:
From the problem description:
- Number of treatments ($t$) = 5
- Number of replications ($r$) = 4
Substitute $r = 4$ into the standard error of difference formula: \[ SE_d = \sqrt{\frac{2 \times \text{MSE}}{4}} \] Simplify the fraction inside the square root: \[ SE_d = \sqrt{\frac{\text{MSE}}{2}} \] This is the standard error of the difference between two treatment means.

Step 3: Final Answer:
The derived formula is $\sqrt{\frac{\text{MSE}}{2}}$, which corresponds to Option (A).