With a systematic sample having a single randomly selected starting point, the expansion estimator τˆ = Ny is design-unbiased for the population total τ , but no design-unbiased estimator of var(τ ) exists. Assume for simplicity that each possible sample (primary unit) has the same number of secondary units, that is, Mi = m for all i. With the single starting point the sample has only n = 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 β and variance γ . Under the model, show that 
is (model) unbiased for the population total, find the mean square prediction error E( - τ )2, and find an unbiased estimator of E(τˆ - τ )2