Two different plasma etchers in a semiconductor fac- tory have the same mean etch rate. However, machine 1 is newer than machine 2 and consequently has smaller variabil- ity in etch rate. We know that the variance of etch rate for machine I is of and for machine 2 it is dad. Suppose that we have n, independent observations on etch rate from machine 1 and 3 independent observations on etch rate from machine 2.

(a) Show that a X, + (1 -

a) X, is an unbiased estima- tor of a 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 a would minimize the standard error of the point estimate of p?

(d) Suppose that a 4 and n, 2, What value of a would you select to minimize the standard error of the point esti- mate of ? How "bad" would it be to arbitrarily choose a=0.5 in this case?