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244 CHAPTER 7 POINT ESTIMATION OF PARAMETERS

mean brightness, but the standard deviation is assumed to be compressive strength 5500 psi and the standard deviation
identical to that for type A. A random sample of n 25 tubes is 100 psi. Find the probability that the sample mean
compressive strength exceeds 4985 psi.
of each type is selected, and X B X A is computed. If B
equals or exceeds A , the manufacturer would like to adopt 7-54. A normal population has a known mean 50 and
type B for use. The observed difference is x B x A 3.5. known variance 2. A random sample of n 16 is se-
2
What decision would you make, and why? lected from this population, and the sample mean is x 52.
7-45. The elasticity of a polymer is affected by the concen- How unusual is this result?
tration of a reactant. When low concentration is used, the true 7-55. A random sample of size n 16 is taken from a nor-
mean elasticity is 55, and when high concentration is used the mal population with 40 and 5. Find the probability
2
mean elasticity is 60. The standard deviation of elasticity is 4, that the sample mean is less than or equal to 37.
regardless of concentration. If two random samples of size 16
7-56. A manufacturer of semiconductor devices takes a
are taken, find the probability that X high X low
2 .
random sample of 100 chips and tests them, classifying each
chip as defective or nondefective. Let X i 0 if the chip is
Supplemental Exercises
nondefective and X i 1 if the chip is defective. The sample
7-46. Suppose that a random variable is normally distrib- fraction defective is
2
uted with mean and variance , and we draw a random
X 1 X 2 p X 100
ˆ
sample of five observations from this distribution. What is the P
joint probability density function of the sample? 100
7-47. Transistors have a life that is exponentially distributed ˆ
P
What is the sampling distribution of the random variable ?
with parameter . A random sample of n transistors is taken.
7-57. Let X be a random variable with mean and variance
What is the joint probability density function of the sample? 2
. Given two independent random samples of sizes n 1 and n 2 ,
7-48. Suppose that X is uniformly distributed on the interval
with sample means X 1 and X 2 , show that
from 0 to 1. Consider a random sample of size 4 from X. What
is the joint probability density function of the sample?
X aX 1 11 a2X 2 , 0 a 1
7-49. A procurement specialist has purchased 25 resistors
from vendor 1 and 30 resistors from vendor 2. Let X 1,1 ,

X 1,2 , p , X 1,25 represent the vendor 1 observed resistances, is an unbiased estimator for . If X 1 and X 2 are independent,
which are assumed to be normally and independently distrib- find the value of a that minimizes the standard error of . X
uted with mean 100 ohms and standard deviation 1.5 ohms. 7-58. A random variable x has probability density function
Similarly, let X 2,1 , X 2,2 , p , X 2,30 represent the vendor 2 ob-
served resistances, which are assumed to be normally and in- 1 2 x
f 1x2 3 x e , 0 x , 0
dependently distributed with mean 105 ohms and standard 2
deviation of 2.0 ohms. What is the sampling distribution of
? Find the maximum likelihood estimator for .
X 1 X 2
7-59. Let f 1x2 x 1 , 0 , and 0 x 1.
7-50. Consider the resistor problem in Exercise 7-49. What ˆ n
? Show that n 1ln w i 1 X i 2 is the maximum likelihood
is the standard error of X 1 X 2
estimator for .
7-51. A random sample of 36 observations has been drawn
11 2
from a normal distribution with mean 50 and standard devia- 7-60. Let f 1x2 11 2x n , 0 1, and 0 x .
ˆ
tion 12. Find the probability that the sample mean is in the Show that 11 n2 g i 1 ln1X i 2 is the maximum likelihood
ˆ

interval 47 X 53 . estimator for and that is an unbiased estimator for .
7-52. Is the assumption of normality important in Exercise
7-51? Why?
7-53. A random sample of n 9 structural elements is
tested for compressive strength. We know that the true mean