Chapter 6.1, Problem 16E

Suppose the true average growth μ of one type of plant during a 1-year period is identical to that of a second type, but the variance of growth for the first type is σ², whereas for the second type the variance is 4σ². Let X₁,…, X_m be m independent growth observations on the first type [so E(X_i) = μ, V(X_i) = σ²], and let Y₁,..., Y_n be n independent growth observations on the second type [E(Y_i) = μ, V(Y_i) = 4σ²].

1. a. Show that the estimator $\hat{\mu }\text{ = }\delta \bar{X}\text{ + (1 - }\delta \text{)}\bar{Y}$ is unbiased for μ (for 0 < δ < 1, the estimator is a weighted average of the two individual sample means).
2. b. For fixed m and n, compute V( $\hat{\mu }$), and then find the value of δ that minimizes V( $\hat{\mu }$). [Hint: Differentiate V( $\hat{\mu }$) with respect to δ.]