Question

In Simple Random Sampling with replacement (SRSWR) the variance of $\bar{y_n$ (with standard notation) is given by :}

• A. $\frac{N-1}{nN}S^2$
• B. $\frac{N-n}{nN}S^2$
• C. $\frac{N-1}{N}S^2$
• D. $\frac{N-n}{N}S^2$

Hint:

Remember that SRSWR does not have a finite population correction (fpc) term like $(N-n)$. Thus, any formula containing $(N-n)$ is for SRSWOR, making it easy to eliminate incorrect options.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In sampling theory, we analyze the behavior of the sample mean ($\bar{y}_n$) under different sampling designs.
Simple Random Sampling with Replacement (SRSWR) and Simple Random Sampling without Replacement (SRSWOR) are the two fundamental methods.
Key Formula or Approach:
The population variance $\sigma^2$ and the population mean square $S^2$ are related by: \[ S^2 = \frac{N}{N-1} \sigma^2 \implies \sigma^2 = \frac{N-1}{N} S^2 \] The variance of the sample mean ($\bar{y}_n$) under SRSWR is defined as: \[ V(\bar{y}_n)_{\text{SRSWR}} = \frac{\sigma^2}{n} \]

Step 2: Detailed Explanation:
Substitute the relation for $\sigma^2$ in terms of $S^2$ into the variance of the sample mean formula: \[ V(\bar{y}_n)_{\text{SRSWR}} = \frac{\left(\frac{N-1}{N}\right)S^2}{n} \] Rearrange the terms: \[ V(\bar{y}_n)_{\text{SRSWR}} = \frac{N-1}{nN} S^2 \] For comparison, the variance of the sample mean under SRSWOR is: \[ V(\bar{y}_n)_{\text{SRSWOR}} = \frac{N-n}{nN} S^2 \]

Step 3: Final Answer:
The variance under SRSWR corresponds to Option (A).