Page 209 - Applied statistics and probability for engineers

Section 5-4/Linear Functions of Random Variables 187

Let X , X , . . . , X denote the ill volumes of the 10 cans. The average ill volume (denoted as X) is a normal random

1 2 10
variable with . 0 1 2
0
E X ( ) = 12 .1 and V X ( ) = = .001
10
Consequently, ⎡
−
(
P X <12) = P ⎢ X − μ X < 12 12 1 . ⎤ ⎥
.
⎣ σ X 0 001 ⎦
= ( 3 16) = .
P Z < − .
0 00079
EXERCISES FOR SECTION 5-4

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.
5-62. X and Y are independent, normal random variables (a) If a particular lamp is made up of these two inks only, what
with E(X) = 0, V(X) = 4, E(Y) = 10, and V(Y) = 9. is the probability that the total ink thickness is less than
Determine the following: 0.2337 mm?
(
(a) E 2 + ) (b) V 2 + ) (b) A lamp with a total ink thickness exceeding 0.2405 mm
(
X
X
Y 3
Y 3
(
(
(c) P 2 + 3 Y <30) (d) P 2 + 3 Y < 40) lacks the uniformity of color that the customer demands.
X
X
5-63. X and Y are independent, normal random variables with Find the probability that a randomly selected lamp fails to
(
(
(
E X ( ) = 2 ,V X) = 5 ,E Y) = 6 ,and V Y) = .8 meet customer specii cations.
Determine the following: 5-68. The width of a casing for a door is normally distributed
(
(a) E 3 + ) (b) V 3 + ) with a mean of 24 inches and a standard deviation of 1/8 inch. The
(
X
X
Y
Y
2
2
3 (
3 (
(c) P X + 2 Y <18) (d) P X + 2 Y < 28) width of a door is normally distributed with a mean of 23 7/8 inches
5-64. Suppose that the random variable X represents the and a standard deviation of 1/16 inch. Assume independence.
length of a punched part in centimeters. Let Y be the length of (a) Determine the mean and standard deviation of the difference
the part in millimeters. If E(X) = 5 and V(X) = 0.25, what are between the width of the casing and the width of the door.
the mean and variance of Y? (b) What is the probability that the width of the casing minus
5-65. A plastic casing for a magnetic disk is composed of the width of the door exceeds 1/4 inch?

two halves. The thickness of each half is normally distributed (c) What is the probability that the door does not it in the casing?
with a mean of 2 millimeters, and a standard deviation of 0.1 5-69. An article in Knee Surgery Sports Traumatology,
millimeter and the halves are independent. Arthroscopy [“Effect of Provider Volume on Resource Utili-
(a) Determine the mean and standard deviation of the total zation for Surgical Procedures” (2005, Vol. 13, pp. 273–279)]
thickness of the two halves. showed a mean time of 129 minutes and a standard deviation
(b) What is the probability that the total thickness exceeds 4.3 of 14 minutes for ACL reconstruction surgery for high-volume
millimeters? hospitals (with more than 300 such surgeries per year). If a
5-66. Making handcrafted pottery generally takes two high-volume hospital needs to schedule 10 surgeries, what are
major steps: wheel throwing and i ring. The time of wheel the mean and variance of the total time to complete these sur-
throwing and the time of i ring are normally distributed ran- geries? Assume that the times of the surgeries are independent
dom variables with means of 40 minutes and 60 minutes and and normally distributed.
standard deviations of 2 minutes and 3 minutes, respectively. 5-70. An automated i lling machine i lls soft-drink cans,
(a) What is the probability that a piece of pottery will be i nished and the standard deviation is 0.5 luid ounce. Assume that the i ll

within 95 minutes? volumes of the cans are independent, normal random variables.
(b) What is the probability that it will take longer than 110 minutes? (a) What is the standard deviation of the average i ll volume
5-67. In the manufacture of electroluminescent lamps, sev- of 100 cans?

eral different layers of ink are deposited onto a plastic substrate. (b) If the mean ill volume is 12.1 oz, what is the probability that
The thickness of these layers is critical if specii cations regard- the average ill volume of the 100 cans is less than 12 oz?

ing the inal color and intensity of light are to be met. Let X and (c) What should the mean ill volume equal so that the prob-

Y denote the thickness of two different layers of ink. It is known ability that the average of 100 cans is less than 12 oz is
that X is normally distributed with a mean of 0.1 mm and a stand- 0.005?
ard deviation of 0.00031 mm, and Y is also normally distributed (d) If the mean ill volume is 12.1 oz, what should the standard

with a mean of 0.23 mm and a standard deviation of 0.00017 mm. deviation of ill volume equal so that the probability that

Assume that these variables are independent. the average of 100 cans is less than 12 oz is 0.005?