Question

If under SRSWOR, \(U = \sum_{i=1}^{n_1} y_i = n_1 \bar{y}_1\) and \(V = \sum_{j=n_1+1}^{n} y_j = (n-n_1)\bar{y}_2\), then the Var(V) is

• A. \( \frac{n_1(N-(n-n_1))}{N}S^2 \)
• B. \( \frac{(n-n_1)(N-(n-n_1))}{N}S^2 \)
• C. \( \frac{n_1(N-(n-n_1))}{N}S^2 \)
• D. \( \frac{(n-n_1)(N-(n-n_1))}{Nn}S^2 \)

Hint:

When dealing with variances of sums or means in finite population sampling, always remember the finite population correction (FPC) factor, \( \frac{N-n}{N} \). The variance of a sum is not simply \(m . S^2\), but is scaled by the FPC.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The question asks for the variance of the sum of a subset of observations drawn from a finite population using Simple Random Sampling Without Replacement (SRSWOR). The notation implies that a sample of size \(n\) is drawn and then partitioned into two groups of sizes \(n_1\) and \(n-n_1\). V is the sum of observations in the second group.

Step 2: Key Formula or Approach:
The variance of the sum of \(m\) randomly selected units from a population of size \(N\) using SRSWOR is given by: \[ \text{Var}\left(\sum_{i=1}^m y_i\right) = m^2 \text{Var}(\bar{y}_m) \] where \(\bar{y}_m\) is the mean of a sample of size \(m\). The variance of the sample mean is: \[ \text{Var}(\bar{y}_m) = \frac{N-m}{N} \frac{S^2}{m} \] Combining these gives: \[ \text{Var}\left(\sum_{i=1}^m y_i\right) = m^2 \left( \frac{N-m}{N} \frac{S^2}{m} \right) = m \frac{N-m}{N} S^2 \]
Step 3: Detailed Explanation:
The statistic V is the sum of \(m = n-n_1\) observations. These \(n-n_1\) observations can be considered as a simple random sample of size \(m\) drawn from the population of size \(N\). We apply the formula derived above with \(m = n-n_1\). \[ \text{Var}(V) = \text{Var}\left(\sum_{j=n_1+1}^{n} y_j\right) = (n-n_1) \frac{N-(n-n_1)}{N} S^2 \] This directly matches the structure of option (B), assuming the OCR typo in the option replaced \(n_1\) with \(m\).
Step 4: Final Answer:
The variance of V is \( \frac{(n-n_1)(N-(n-n_1))}{N}S^2 \).