ABSTRACT
Ta.king an extreme case, consider the universe of six households with respective sizes 4, 4, 4, 5, 5, a.nd 5. The reader will verify that the variance per unit in this universe is u2 = 0.25. If we were to draw a. simple random sample of 4 households with replacement from this universe, an unbiased estimator of the total size is Ny (where N = 6 andy is the sample mean, based on n = 4 sample units), with the sampling variance N 2 u2 /n = 2.25. Suppose, however, that the universe is subdivided into two strata, the first with the three households each of size 4, and the second with the other three households each of size 5; and an srs with replacement of 2 units is to be drawn from each stratum. Then, a.s we shall see later, an unbiased estimator of the total size is (N1y1 +N2y2), where N1 = 3, N2 = 3, being respectively the total number of units in the two strata., and y1 a.nd y2 the respective stratum means based on n1 = 2 a.nd n2 = 2 sample units; the sampling variance ofthis unbiased estimator of the total is NluUn1 + N?uVn2 = 0, a.s the variance per unit in the two strata. u~ = u~ = 0.
