Question

In a completely randomized design with 6 treatments each replicated 3 times, the F-ratio would have degrees of freedom equal to

• A. F(5,12)
• B. F(5,18)
• C. F(6,12)
• D. F(6,18)

Hint:

Always verify that the sum of treatment degrees of freedom ($v-1$) and error degrees of freedom ($N-v$) equals the total degrees of freedom ($N-1$). Here, $5 + 12 = 17$, which is $18 - 1$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In a Completely Randomized Design (CRD), we evaluate treatment effects by partitioning the total degrees of freedom into treatment and experimental error components.
Key Formula or Approach:
For a CRD with $v$ treatments and total observations $N$:
\[ df_{\text{Treatment}} = v - 1 \]
\[ df_{\text{Error}} = N - v \]
The test statistic follows an $F$-distribution with degrees of freedom:
\[ F \sim F(df_{\text{Treatment}}, df_{\text{Error}}) \]

Step 2: Detailed Explanation:
We are given:
Number of treatments, $v = 6$
Number of replications per treatment, $r = 3$
Calculate the total number of experimental units ($N$):
\[ N = v \cdot r = 6 \cdot 3 = 18 \]
Calculate the numerator degrees of freedom (treatment):
\[ df_1 = v - 1 = 6 - 1 = 5 \]
Calculate the denominator degrees of freedom (error):
\[ df_2 = N - v = 18 - 6 = 12 \]
Thus, the $F$-ratio has degrees of freedom equal to $F(5, 12)$.

Step 3: Final Answer
The correct option is (A).