In a field of barley the grain, y_i, and the grain plus straw, x_i, were weighted for each of a large number of sampling units located at random over the field. the total produce (grain plus straw) of the whole field was also weighed. the following data were obtained: c_yy = 1.13, c_yx = 0.78, c_xx = 1.11. compute the gain in precision by estimating the grain yield of the field from the ratio of grain to produce instead of from the mean yield of grain unit. it requires 20 min to cut, thresh, and weigh the grain on each unit, 2 min to weigh the straw on each unit, and 2 hr to collect and weigh the total produce of the field. how many units must be taken per field in order that the ratio estimate may be more economical than the mean per unit?