Suppose the true average growth m of one type of

Chapter 6, Problem 16

(choose chapter or problem)

Suppose the true average growth m of one type of plantduring a 1year period is identical to that of a secondtype, but the variance of growth for the first type is s2,whereas for the second type the variance is 4s2. LetX1, , Xm be m independent growth observations on thefirst type [so E(Xi) 5 m, V(Xi) 5 s2], and let Y1, , Ynbe n independent growth observations on the secondtype [E(Yi) 5 m, V(Yi) 5 4s2].a. Show that the estimator m 5 dX 1 (1 2 d)Y is unbiasedfor m (for 0 , d ,1, the estimator is a weightedaverage of the two individual sample means).b. For fixed m and n, compute V(m ), and then find thevalue of d that minimizes V(m ). [Hint: DifferentiateV(m ) with respect to d.]