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228 CHAPTER 7 POINT ESTIMATION OF PARAMETERS

7-8. Let three random samples of sizes n 1 20, n 2 10, 7-14. Suppose that X is the number of observed “successes”
and n 3 8 be taken from a population with mean and in a sample of n observations where p is the probability of
2 2 2 2 success on each observation.
variance . Let S 1 , S 2 , and S 3 be the sample variances.
ˆ
2
2
2
2
Show that S 120S 1 10S 2 8S 3 2 38 is an unbiased (a) Show that P X n is an unbiased estimator of p.
2
estimator of . (b) Show that the standard error of P ˆ is 1p11 p2 n.
2
7-9. (a) Show that g n i 1 1X i X 2 n is a biased estimator How would you estimate the standard error?
of 2 . 7-15. X 1 and S 1 2 are the sample mean and sample variance
(b) Find the amount of bias in the estimator. from a population with mean and variance 2 . 2 Similarly, X 2
(c) What happens to the bias as the sample size n increases? and S 2 2 are the sample mean and sample variance from a sec-
2
7-10. Let X 1 , X 2 , p , X n be a random sample of size n from ond independent population with mean 1 and variance 2 .
a population with mean and variance 2 . The sample sizes are n 1 and , respectively.
n 2
(a) Show that X 2 is a biased estimator for 2 . (a) Show that X 1 X 2 is an unbiased estimator of 1 2 .
(b) Find the amount of bias in this estimator. (b) Find the standard error of X 1 X 2 . How could you
(c) What happens to the bias as the sample size n increases? estimate the standard error?
7-11. Data on pull-off force (pounds) for connectors used in 7-16. Continuation of Exercise 7-15. Suppose that both pop-
2 2
an automobile engine application are as follows: 79.3, 75.1, ulations have the same variance; that is, 1 2 2 . Show
78.2, 74.1, 73.9, 75.0, 77.6, 77.3, 73.8, 74.6, 75.5, 74.0, 74.7, that
75.9, 72.9, 73.8, 74.2, 78.1, 75.4, 76.3, 75.3, 76.2, 74.9, 78.0, 2 2
1n 1 12 S 1 1n 2 12 S 2
2
75.1, 76.8. S p
(a) Calculate a point estimate of the mean pull-off force of all n 1 n 2 2
connectors in the population. State which estimator you 2
is an unbiased estimator of .
used and why.
7-17. Two different plasma etchers in a semiconductor fac-
(b) Calculate a point estimate of the pull-off force value that

tory have the same mean etch rate . However, machine 1 is
separates the weakest 50% of the connectors in the popu-
newer than machine 2 and consequently has smaller variability
lation from the strongest 50%.
in etch rate. We know that the variance of etch rate for machine
(c) Calculate point estimates of the population variance and
2 2 2 . Suppose that we have
the population standard deviation. 1 is 1 and for machine 2 it is 2 a 1
(d) Calculate the standard error of the point estimate found in n 1 independent observations on etch rate from machine 1 and n 2
independent observations on etch rate from machine 2.
part (a). Provide an interpretation of the standard error.
(a) Show that ˆ X 1 (1 ) X 2 is an unbiased estima-
(e) Calculate a point estimate of the proportion of all connec-
tor of for any value of between 0 and 1.
tors in the population whose pull-off force is less than

(b) Find the standard error of the point estimate of in part (a).
73 pounds.

(c) What value of would minimize the standard error of the
7-12. Data on oxide thickness of semiconductors are as

point estimate of ?
follows: 425, 431, 416, 419, 421, 436, 418, 410, 431, 433,
(d) Suppose that a 4 and n 1 2n 2 . What value of would
423, 426, 410, 435, 436, 428, 411, 426, 409, 437, 422, 428,
you select to minimize the standard error of the point esti-
413, 416.

mate of . How “bad” would it be to arbitrarily choose
(a) Calculate a point estimate of the mean oxide thickness for
0.5 in this case?
all wafers in the population.
7-18. Of n 1 randomly selected engineering students at ASU,
(b) Calculate a point estimate of the standard deviation of
oxide thickness for all wafers in the population. X 1 owned an HP calculator, and of n 2 randomly selected
(c) Calculate the standard error of the point estimate from engineering students at Virginia Tech X 2 owned an HP calculator.
Let p 1 and p 2 be the probability that randomly selected ASU and
part (a).
Va. Tech engineering students, respectively, own HP calculators.
(d) Calculate a point estimate of the median oxide thickness
for all wafers in the population. (a) Show that an unbiased estimate for p 1 p 2 is (X 1 n 1 )
(X 2 n 2 ).
(e) Calculate a point estimate of the proportion of wafers in
(b) What is the standard error of the point estimate in
the population that have oxide thickness greater than 430
part (a)?
angstrom.
(c) How would you compute an estimate of the standard error
7-13. X 1 , X 2 , p , X n is a random sample from a normal found in part (b)?
distribution with mean and variance 2 . Let X min and X max (d) Suppose that n 1 200, X 1 150, n 2 250, and X 2 185.
be the smallest and largest observations in the sample. Use the results of part (a) to compute an estimate of p 1 p 2 .
(a) Is 1X min X max 2 2 an unbiased estimate for ? (e) Use the results in parts (b) through (d) to compute an
(b) What is the standard error of this estimate? estimate of the standard error of the estimate.
(c) Would this estimate be preferable to the sample mean ?
X