Two different plasma etchers in a semiconductor factory have the same mean etch rate . However, machine 1 is newer than machine 2 and consequently

Two different plasma etchers in a semiconductor factory have the same mean etch rate μ. However, machine 1 is newer than machine 2 and consequently has smaller variability in etch rate. We know that the variance of etch rate for machine 1 is σ2/1 and for machine and 2 it is aσ2/1. Suppose that we have independent observations on etch rate from machine 1 and independent observations on etch rate from machine 2.
(a) Show that μ = a X1 + (1 – a) X2 is an unbiased estimator of μ for any value of a between 0 and 1.
(b) Find the standard error of the point estimate of in part (a).
(c) What value of would minimize the standard error of the point estimate of μ? (d) Suppose that a = 4 and n1 = 2n2. What value of a would you select to minimize the standard error of the point estimate of μ. How “bad” would it be to arbitrarily choose a = 0.5 in this case?