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 σ1 2 , and for machine 2, it is σ2 = σ 2 1 2 a . Suppose that we have n1 independent observations on etch rate from machine 1 and n2 independent observations on etch rate from machine 2. (a) Show that ˆ μ = +− α α X X ( ) 1 2 1 is an unbiased estimator of μ for any value of α between zero and one. (b) Find the standard error of the point estimate of μ in part (a). (c) What value of α would minimize the standard error of the point estimate of μ? (d) Suppose that a = 4 and n n 1 2 = 2 . What value of α would you select to minimize the standard error of the point estimate of μ? How “bad” would it be to arbitrarily choose α= . 0 5 in this case?
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 σ1
2
, and for machine 2, it is σ2 = σ 2 1
2 a . Suppose that we have
n1 independent observations on etch rate from machine 1 and n2
independent observations on etch rate from machine 2.
(a) Show that ˆ
μ = +− α α X X ( ) 1 2 1 is an unbiased estimator
of μ for any value of α between zero and one.
(b) Find the standard error of the point estimate of μ in part (a).
(c) What value of α would minimize the standard error of the
point estimate of μ?
(d) Suppose that a = 4 and n n 1 2 = 2 . What value of α would
you select to minimize the standard error of the point estimate of μ? How “bad” would it be to arbitrarily choose
α= . 0 5 in this case?
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