Discussion
Let \(V_{SY}^{2}\) be the variance of the usual estimator in a systematic sampling design and \(V_{SRS}^{2}\) the variance of the sample mean in a SRSWOR design. It is known that the population has a linear trend, i.e., \(y_{i}\) is approximately equal to \(ic\), for a positive \(c\), \(i = 1, 2, \ldots, N\). Then :
Answer: B
The answer is \(V_{SY}^{2} \lt V_{SRS}^{2}\).
When the population has a linear trend, a systematic sample takes one unit from each successive stretch of \(k\) units, so it automatically spreads over the whole range, from small to large values. Its mean stays close to the population mean. A simple random sample can by chance cluster at one end of the trend.
The standard result (for example in Cochran's treatment of populations with linear trend) is that systematic sampling is more precise than SRS in this case, though less precise than stratified sampling with one unit per stratum. So the variances are not equal, and they can be compared.
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