Discussion
Under systematic sampling :
Answer: D
Under systematic sampling, the sample mean is an unbiased estimator of the population mean (in the usual case where the population size is a multiple of the sample size).
With \(N = nk\), linear systematic sampling chooses a random start \(r\) from 1 to \(k\) and takes units \(r, r + k, r + 2k, \ldots\). Each of the \(k\) possible samples has probability \(1/k\), and every unit lies in exactly one of them, so every unit has inclusion probability \(1/k = n/N\). Averaging the \(k\) possible sample means gives the population mean exactly.
The other statements are false. The first unit is chosen at random with equal probabilities, not by pps, and the rest follow deterministically, not by SRS. Each unit has an equal chance, \(n/N\), of selection. And the variance of the systematic mean can be larger than that of SRSWOR, for example when the population has a periodic pattern with period \(k\).
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