Ch 7.3 - 38E

Chapter 7, Problem 38E

(choose chapter or problem)

Two different plasma etchers in a semiconductor factory have the same mean etch rate \(\mu\). 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 \(\sigma_{1}^{2}\), and for machine 2, it is \(\sigma_{2}^{2}=a \sigma_{1}^{2}\). Suppose that we have \(n_{1}\) independent observations on etch rate from machine 1 and \(n_{2}\) independent observations on etch rate from machine 2.

(a) Show that \(\hat{\mu}=\alpha \bar{X}_{1}+(1-\alpha) \bar{X}_{2}\) is an unbiased estimator of \(\mu\) for any value of α between zero and one.

(b) Find the standard error of the point estimate of \(\mu\) in part (a).

(c) What value of α would minimize the standard error of the point estimate of \(\mu\)?

(d) Suppose that \(a=4\) and \(n_{1}=2 n_{2}\). What value of α would you select to minimize the standard error of the point estimate of \(\mu\)? How “bad” would it be to arbitrarily choose

\(\alpha=0.5\) in this case?

Equation Transcription:

Text Transcription:

mu

sigma_1^2

sigma_2^2=a sigma_1^2

n_1

n_2

mu hat=alpha X bar_1+(1-alpha)X bar_2

mu

alpha

mu

alpha

mu

a=4

n_1=2n_2

alpha

mu

alpha=0.5