Assume that the Y -values in the population are related to an auxiliary variable x through the ratio model Y_i = βx_i + i, for i = 1, 2,...,N, where the random errors i are independent with E( i) = 0 and var( i) = viσ2 R, in which the vi are known and β and σ2 R are unknown constants. A simple random sample of size n is selected. The best linear unbiased predictor of the population total τ is τˆ = Y_s + βs, where Ys is the total of the Y -values in the sample and xs is the total of the x-values not in the sample and

When the variance of Yi is proportional to xi, that is, v_i = x_i, the best linear unbiased estimator β is the ratio estimator 

Compute the value of  with this estimator with the data in Example 1.

(b) Show that when the variance of Yi is proportional   is the mean-of-the-ratios estimator

Compute the value of τˆ with this estimator with the data in Example 1.