The quantity ˆB used in ratio estimation is sometimes called the ratio-of-means estimator. An alternative that has been proposed is the mean-of-ratios estimator: Let bi = yi/xi for unit i; then the mean-of-ratios estimator is ¯b = 1 n i∈S bi. a Doyou think the mean-of-ratios estimator is appropriate for the data in Example 4.5? Why, or why not? b Show that, for the ratio-of-means estimator ˆB, tx ˆB = ty when the entire population is sampled (i. e., S = U). c Give an example to show that it is possible to have tx ¯b = ty when the entire population is sampled. d Define Sbx = 1 N − 1 N i=1 (bi − ¯bU)(xi − ¯xU). Show that for an SRS of size n, the bias of ¯b as an estimator of B is E[¯b − B] = −(N − 1)Sbx tx . As a consequence, if Sbx = 0 the bias does not decrease as n increases. e (Requires linear model theory.) Showthat ¯b is the weighted least squares estimator of β under the model Yi = βxi + εi