Discussion
Consider a population of \(N = nk\) units with \(k \gt 1\). A sample of size \(n\) is obtained by the following method :
(i) A unit is drawn from the set of units labelled \(\{k, 2k, \ldots, N\}\) with probability \(1/n\) and is included in the sample.
(ii) A SRSWOR of size \(n - 1\) is then obtained from the remaining \(N - 1\) units in the population and these units are included in the sample.
Then,
Answer: C
The answer is inclusion probability is not the same for all the population units.
Let \(S = \{k, 2k, \ldots, N\}\), which has \(n\) units. A unit in S can enter at the first draw (probability \(1/n\)) or later, while a unit outside S enters only at the second stage:
\[ \begin{aligned} \pi_{S} &= \frac{1}{n} + \left(1 - \frac{1}{n}\right)\frac{n - 1}{N - 1} \\ \pi_{\text{other}} &= \frac{n - 1}{N - 1} \end{aligned} \]
Since \(N \gt n\), \(\pi_{S} \gt \pi_{\text{other}}\). The design is not systematic sampling, and because inclusion probabilities differ, the plain sample mean is biased. But all inclusion probabilities are positive, so the Horvitz-Thompson estimator is unbiased for the total, which rules out option B.
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