• Comparison of Variances in SRSWR and SRSWOR:
• Let $V_1$ be the variance of the sample mean based on all $n$ units under SRSWR:

• Let $V_2$ be the variance of the sample mean based on distinct units $d$ under SRSWR (using the Rao-Blackwellized estimator $\bar{y}_d$, where $E[d] \lt n$):

• Let $V_3$ be the variance of the sample mean under SRSWOR of the same fixed size $n$:

• Theoretical Ordering: Since repeated sampling adds redundant units in SRSWR, discarding duplicate units strictly improves efficiency ($V_1 \ge V_2$). Furthermore, a direct SRSWOR sample of fixed size $n$ always achieves at least as many distinct units as SRSWR, giving:

• Conclusion: The relationship that holds is $V_1 \ge V_2 \ge V_3$ (Option 1).

Correct option: (1) V42V22V3

• Estimation of Population Total in Simple Random Sampling:
• Population size: $N = 120$ households
• Sample size: $n = 4$ households
• Household sizes in sample: $y = \{6, 4, 7, 3\}$
• Sample mean persons per household:

• The unbiased estimator of the population total $\hat{Y}$ is:

• Conclusion: The estimate of the total number of people in the village is 600 (Option 3).

Correct option: (3) 600

• Inclusion Probability in Simple Random Sampling With Replacement (SRSWR):
• Population size: $N$, Sample size: $n$ draws.
• Probability that a specified unit $i$ is NOT selected on any single draw is:

• Because draws in SRSWR are independent with replacement, the probability that the unit is NOT selected in all $n$ draws is:

• Therefore, the probability that the specified unit is included at least once in the sample (first-order inclusion probability $\pi_i$) is the complement:

• Conclusion: The probability is $1 - [1 - (1/N)]^n$ (Option 3).

Correct option: (3) 4-[1-(4/N)P