Comparison of domain means. Suppose the population may be divided into two groups, with respective sizes N1 and N2 and population means 1U and 2U. The overall population mean is U = (N1 1U + N2 2U)/N, with N = N1 + N2. Let xi = 1 if observation unit i is in group 1, and xi = 0 if it is in group 2. The weight for observation unit i is wi.
Show that B1 = 1U − 2U and B0 = 2U. Also show that

and Ḃ0 = 2.