With a systematic sample having a single randomly selected starting point, the expansion estimator ? = Ny is design-unbiased for the population total r, but no design-unbiased estimator of var() exists. Assume for simplicity that each possible sample (primary unit) has the same number of secondary units, that is, M; = m for all i. With the single starting point the sample has only 1 primary units, and y, the sample mean of the primary unit totals, is the sum of the m secondary units in the primary unit selected. N is the number of possible starting points (the number of possible systematic samples), so the total number of secondary units in the population is M = Nm. Now consider a model-based approach, in which the population Y-values are assumed to be independent, identically distributed random variables, each with mean B and variance y. Under the model, show that is (model) unbiased for the population total, find the mean square prediction error E(f- ), and find an unbiased estimator of E(f-T) (Hint: The notation may be simplified to avoid double subscripts by relabeling the M population from I to M, with Y, representing the Y-value of the jth secondary unit. The population total can be partitioned into sample and nonsample parts as r=EY, =E,Y,+E,Y,) Nm secondary units in the