Question

The variance of sample mean of a sample of size $n$ from a population with $N$ units under SRSWOR is

• A. $\left(\frac{1}{n} - \frac{1}{N}\right) S^2$
• B. $\left(\frac{1}{n} - \frac{1}{N}\right) s^2$
• C. $\left(\frac{1}{N} - \frac{1}{n}\right) S^2$
• D. $\left(\frac{1}{N} - \frac{1}{n}\right) s^2$

Hint:

The multiplier $\frac{N-n}{N} = \left(1 - \frac{n}{N}\right)$ is called the Finite Population Correction (fpc) factor, which adjusts the variance downward when sampling from a finite population.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Simple Random Sampling Without Replacement (SRSWOR) is a method of selecting $n$ units from a population of $N$ units such that no unit can be selected more than once.
Key Formula or Approach:
The variance of the sample mean $\bar{y}$ under SRSWOR is given by:
\[ \text{Var}(\bar{y}) = \frac{N - n}{N} \cdot \frac{S^2}{n} \]
where $S^2$ is the population mean square:
\[ S^2 = \frac{1}{N - 1} \sum_{i=1}^N (Y_i - \bar{Y})^2 \]

Step 2: Detailed Explanation:
Let us rearrange the algebraic expression for the variance of the sample mean:
\[ \text{Var}(\bar{y}) = \left(\frac{N - n}{N \cdot n}\right) S^2 \]
Divide both terms in the numerator of the fraction by the denominator:
\[ \text{Var}(\bar{y}) = \left(\frac{N}{N \cdot n} - \frac{n}{N \cdot n}\right) S^2 \]
Simplify the terms inside the parentheses:
\[ \text{Var}(\bar{y}) = \left(\frac{1}{n} - \frac{1}{N}\right) S^2 \]
Note that $S^2$ is the population parameter (population mean square), whereas $s^2$ is the sample mean square statistic.
The variance of the estimator must be expressed in terms of the population parameter $S^2$.
Therefore, the correct formula is $\left(\frac{1}{n} - \frac{1}{N}\right) S^2$.

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