Page 278 - Applied statistics and probability for engineers

256 Chapter 7/Point Estimation of Parameters and Sampling Distributions

(d) Calculate a point estimate of the median oxide thickness (d) Suppose that a = 4 and n 1 = 2 n 2 . What value of α would
for all wafers in the population. you select to minimize the standard error of the point esti-
(e) Calculate a point estimate of the proportion of wafers in mate of μ? How “bad” would it be to arbitrarily choose
the population that have oxide thickness of more than 430 α = .5 in this case?
0
angstroms. 7-39. Of n 1 randomly selected engineering students at ASU, X 1
7-36. Suppose that X is the number of observed “successes” owned an HP calculator, and of n 2 randomly selected engineering
in a sample of n observations where p is the probability of suc- students at Virginia Tech, X 2 owned an HP calculator. Let p 1 and
cess on each observation. p 2 be the probability that randomly selected ASU and Virginia
ˆ
(a) Show that P = X / n is an unbiased estimator of p. Tech engineering students, respectively, own HP calculators.
(
)
(b) Show that the standard error of P ˆ is p 1− ) (a) Show that an unbiased estimate for p 1 − p 2 is (X 1 / n 1 =
p / n. How
)
(X 2 / n 2 .
would you estimate the standard error?
(b) What is the standard error of the point estimate in part (a)?
2
7-37. X 1 and S 1 are the sample mean and sample variance (c) How would you compute an estimate of the standard error
and variance σ .
2
from a population with mean μ 1 1 Similarly, found in part (b)?
2
X 2 and S 2 are the sample mean and sample variance from a (d) Suppose that n 1 = 200, X 1 150= , n 2 = 250, and X 2 = 185.
2
second independent population with mean μ 2 and variance σ 2 . Use the results of part (a) to compute an estimate of p 1 − p 2.
The sample sizes are n 1 and n 2 , respectively. (e) Use the results in parts (b) through (d) to compute an esti-
(a) Show that X 1 − X 2 is an unbiased estimator of μ − μ 2 .
1
mate of the standard error of the estimate.
(b) Find the standard error of X 1 − X 2 . How could you estimate 7-40. Suppose that the random variable X has a lognormal dis-
the standard error? tribution with parameters θ = 1.5 and ω = 0.8. A sample of size
(c) Suppose that both populations have the same variance; that n = 15 is drawn from this distribution. Find the standard error of
2
2
2
is, σ = σ = σ . Show that the sample median of this distribution with the bootstrap method
2
1
2 ( n 1 1) S 1 +( n 2 1) 2 using n = 200 bootstrap samples.
−
−
2
B
S p = S 2
n 1 + n 2 − 2 7-41. An exponential distribution is known to have a mean of 10.
You want to ind the standard error of the median of this distri-
is an unbiased estimator of σ . bution if a random sample of size 8 is drawn. Use the bootstrap
2
7-38. Two different plasma etchers in a semiconductor method to ind the standard error, using n = 100 bootstrap samples.
B
factory have the same mean etch rate μ. However, machine 1 is 7-42. Consider a normal random variable with mean 10 and
newer than machine 2 and consequently has smaller variability standard deviation 4. Suppose that a random sample of size 16 is
in etch rate. We know that the variance of etch rate for machine drawn from this distribution and the sample mean is computed.
2
2
2
a
1 is σ 1 , and for machine 2, it is σ = σ 1 . Suppose that we have We know that the standard error of the sample mean in this case
2
n 1 independent observations on etch rate from machine 1 and n 2 is σ / n = σ / 16 = . 1 Use the bootstrap method with n = 200
B
independent observations on etch rate from machine 2. bootstrap samples to ind the standard error of the sample mean.
(a) Show that ˆ μ = α X 1 + 1 ) α X 2 is an unbiased estimator Compare the bootstrap standard error to the actual standard error.
( −
of μ for any value of α between zero and one. 7-43. Suppose that two independent random samples (of size
(b) Find the standard error of the point estimate of μ in part (a). n and n ) from two normal distributions are available. Explain
2
1
(c) What value of α would minimize the standard error of the how you would estimate the standard error of the difference in
point estimate of μ? sample means X 1 − X 2 with the bootstrap method.
7-4 Methods of Point Estimation
The deinitions of unbiasedness and other properties of estimators do not provide any guid-
ance about how to obtain good estimators. In this section, we discuss methods for obtaining
point estimators: the method of moments and the method of maximum likelihood. We also
briely discuss a Bayesian approach to parameter estimation. Maximum likelihood esti-
mates are generally preferable to moment estimators because they have better eficiency
properties. However, moment estimators are sometimes easier to compute. Both methods
can produce unbiased point estimators.
7-4.1 Method of Moments
The general idea behind the method of moments is to equate population moments, which
are deined in terms of expected values, to the corresponding sample moments. The
population moments will be functions of the unknown parameters. Then these equations
are solved to yield estimators of the unknown parameters.