Agricultural Statistics- ICAR JRF/SRF 2024

• Theory of Systematic Sampling:
• A 1-in-$k$ systematic sample selects units $i, i + k, i + 2k, \dots, i + (n - 1)k$. The statement writes $i + (k - 1)n$, which is incorrect. (False)
• A population of $N = nk$ units can be divided into $k$ systematic samples each of size $n$. Systematic sampling is equivalent to selecting 1 cluster at random from $k$ clusters of size $n$. (True)
• A systematic sample is more efficient than SRS if the units within the systematic sample are as heterogeneous as possible (i.e., high variation within systematic samples maximizes efficiency, corresponding to negative intraclass correlation $\rho$). (True)
• Systematic sampling is more efficient than SRS when variation within systematic samples is greater than overall variation, not less. (False)
• Conclusion: Statements (B) and (C) only are correct (Option 3).

Correct option: (3) (B) and (C) only.

• Yates' Missing Plot Technique in Randomized Block Design (RBD):
• Number of treatments ($p$) = 4
• Number of blocks ($q$) = 6
• Let $x$ be the single missing observation in treatment 4, block 6.
• From the table of known values:
• Total of known observations in treatment 4: $T' = 423.9$
• Total of known observations in block 6: $B' = 287.1$
• Grand total of all known observations: $G' = 2064.4$
• Yates' formula for estimating a single missing value in RBD:

• Substituting the values:

• Conclusion: The value of $x$ is 90.25 (Option 4).

Correct option: (4) 90.25

• Associacy Matrices in Two-Associate Class PBIBD:
• In a two-associate PBIBD, parameters satisfy the identity relations:

• Given $n_1 = 6$ and $n_2 = 2$:
• For $P^1$ (first associates):
• Row 1: $p_{11}^1 + p_{12}^1 = 6 - 1 = 5 \implies 3 + 2 = 5$
• Row 2: $p_{21}^1 + p_{22}^1 = n_2 = 2 \implies 2 + 0 = 2$
• Hence, $P^1 = \begin{pmatrix} 3 & 2 \\ 2 & 0 \end{pmatrix}$
• For $P^2$ (second associates):
• Row 1: $p_{11}^2 + p_{12}^2 = n_1 = 6 \implies 6 + 0 = 6$
• Row 2: $p_{21}^2 + p_{22}^2 = n_2 - 1 = 2 - 1 = 1 \implies 0 + 1 = 1$
• Hence, $P^2 = \begin{pmatrix} 6 & 0 \\ 0 & 1 \end{pmatrix}$
• Conclusion: Matches Option 2.

Correct option: (2) ana p?= (5 ° 21 ol 2 as 4 ana p?= (¢ e 20 o1