Question One. Consider a population U of size N=3. U=(1,2,3). A simple random sample without replacement of size 2 is obtained from this population. Consider the following two estimators of the population mean, the sample mean - and =91+ 2, if..s..occurs 2 2 2 3 t= t, =1+ 23, if..s,..occurs 2 t, =12 + 3, if ..S, ..occurs Where S, = 1,2 $2 = 1,3 S3 = 2,3 i) Show that the two estimators are unbiased. ii) Show that var()) ~ var(t) for all parameter vectors y =( ) , , ) 2 , y. ) ER 3 such that y3 ( 3)2 - 3y, - y3) 20. iii) What conclusions can you make about as an estimator of Y ? Question Two. Two dentists A and B make a survey of the state of the teeth of 200 children in a village. i) Dr. A selects a simple random sample of 20 children and counts the number of decayed teeth for each child, with the following results. Number 8 10 of decayed teeth/chil d Number 8 4 2 of children Estimate the total number of decayed teeth in the village children using Dr A's results. Dr. B, using the same dental techniques, examines all 200 children, recording merely those who have no decayed teeth. He finds 60 children with no decayed teeth. Estimate the total number of decayed teeth in the village children. iii) Are the two estimates unbiased ? V) Which estimate do you expect to be more precise? Question Three. In nurseries that produce young trees for sale it is advisable to estimate, in late winter or early spring, how many healthy young trees are likely to be on hand, since this determines policy toward the solicitation and acceptance of orders. A study of sampling methods for the estimation of the total numbers of seedlings was undertaken by Johnson (1943). The data that