In Simple Random Sampling with replacement (SRSWR) the variance... | Agrigyan

Derivation of sample mean variance under SRSWR:
In SRSWR, sample draws are independent identically distributed with variance $\text{Var}(y_i) = \sigma^2$.

• Correct: The variance of the sample mean is:
  $$\text{Var}(\bar{y}_n) = \frac{\sigma^2}{n}$$
  Since population variance $\sigma^2 = \frac{N - 1}{N} S^2$, substituting this relation gives:
  $$\text{Var}(\bar{y}_n) = \frac{N - 1}{nN} S^2$$
• Incorrect: $\frac{N - n}{nN} S^2$ is the variance under Simple Random Sampling Without Replacement (SRSWOR), where $\frac{N-n}{N}$ is the finite population correction factor.
• Incorrect: Lacks the sample size $n$ in the denominator.
• Incorrect: Lacks $n$ in the denominator and applies FPC incorrectly.