In using the STR estimator ySTR as an estimator of '1". it may turn out advantageous to find an allocation and a sample size that minimizes the variance in\" [ySTR], for a xed cost (I. In other words, the allowed cost C for the surveyr is xed, and we are looking for the best distribution of resources that maximizes the information about '1". The optimal distribution in this case always remains : Mai/J3 Z \"Fr/ TH=E ii) If V (ySTR) = Vis fixed, show that the appropriate choice of n is : E NKOk/ VCK NKOK VCk N E N n= V + E N2 iii) A company wishes to obtain information on the efficiency of a machine it produces. It asks respondents to rate the equipment on a numerical scale. The cost per interview and the approximate variances of the ratings and the number of elements for three strata are given by (c1 = $9, 0^2 = 2.25, N1 = 112), (c2 = $25, 0^2 = 3.24, N2 = 68) and (c3 = $36, 0^2 = 3.24, N3 = 39). The company wants to give an estimate of the average score while respecting the condition V(ySTR) = 0.1. Determine the sample size n which makes it possible to reach this bound, and find the appropriate distribution