Discussion
PPSWR sampling reduced to SRS if the probability of proportion to size i.e. \(p_{i}\) is :
Answer: C
PPSWR reduces to SRS (with replacement) when \(p_{i} = 1/N\).
In PPS sampling with replacement, unit \(i\) is chosen on each draw with probability \(p_{i}\) proportional to its size measure. If every unit has the same size, all \(p_{i}\) are equal, and since they sum to 1 over the N units, \(p_{i} = 1/N\). Every draw is then equally likely to pick any unit, which is simple random sampling with replacement, and the Hansen-Hurwitz estimator becomes \(N\bar{y}\).
\(1/n\) and \(n/N\) do not sum to 1 over N units (\(n/N\) is the inclusion probability under SRSWOR), and \(p_{i} = 1\) would always select the same unit.
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