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 s2, whereas for the second type, the variance is 4s2. let x1, x2, . . , xm be m independent growth observations on the first type (so e[xi] = µ, v [xi] = s2), and let y1, y2, . . , yn be n independent growth observations on the second type (so e[yi] = µ, v [yi] = 4s2).

a. show that for any d between 0 and 1, the estimator µˆ = dx ¯ + (1 - d)y ¯ is unbiased for µ.

b. for fixed m and n, compute v [µˆ], and then find the value of d that mini- mizes v [µˆ]. (hint: differentiate v [µˆ] with respect to d.)