Statistical Methods & Experimental Designs

Step-by-step ANOVA computation:
Number of treatments $t = 5$, so treatment degrees of freedom is $df_T = t - 1 = 5 - 1 = 4$.
Given Error Mean Square $\text{MSE} = 2$, and $F = 12.5$.

• Incorrect: 25 is the Treatment Mean Square (MST), not the Sum of Squares.
• Incorrect: 50 corresponds to multiplying by 2.
• Incorrect: 75 corresponds to $df = 3$.
• Correct:
  $$F = \frac{\text{MST}}{\text{MSE}} \implies \text{MST} = F \times \text{MSE} = 12.5 \times 2 = 25$$
  $$\text{SST} = df_T \times \text{MST} = 4 \times 25 = 100$$

Derivation of Standard Error of Difference:
Number of replications $r = 4$.

• Correct: The standard error of the difference between two treatment means $\bar{y}_i$ and $\bar{y}_j$ is given by:
  $$\text{SE}_d = \sqrt{\text{Var}(\bar{y}_i - \bar{y}_j)} = \sqrt{\frac{\text{MSE}}{r} + \frac{\text{MSE}}{r}} = \sqrt{\frac{2\text{MSE}}{r}}$$
  Substituting $r = 4$:
  $$\text{SE}_d = \sqrt{\frac{2\text{MSE}}{4}} = \sqrt{\frac{\text{MSE}}{2}}$$
• Incorrect: $\sqrt{\text{MSE}/4}$ is the standard error of a single treatment mean $\text{SE}(\bar{y}_i) = \sqrt{\text{MSE}/r}$.
• Incorrect: 5 is the number of treatments, which does not appear in the numerator of $\text{SE}_d$.
• Incorrect: Invalid expression.

Analysis of RBD structural layout:

• Incorrect: 7 blocks of 5 plots would imply 7 replications and 5 treatments.
• Correct: In a standard Randomized Block Design (RBD), the number of blocks is equal to the number of replications ($r = 5$). Each block forms a complete replicate and must contain all treatments ($t = 7$). Thus, the 35 plots are grouped into 5 blocks of 7 plots each.
• Incorrect: 35 blocks of 1 plot each provides zero within-block replication.
• Incorrect: 1 block containing 35 plots corresponds to a Completely Randomized Design (CRD), not an RBD.