Tan S.T. Finite Mathematics for the Managerial, Life, and Social Sciences
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c. The probability that a bag selected at random weighs between 49 and
51 pounds is given by
APPLIED EXAMPLE 3
College Admissions The grade-point average
(GPA) of the senior class of Jefferson High School is normally distributed
with a mean of 2.7 and a standard deviation of 0.4. If a senior in the top 10% of
his or her class is eligible for admission to any of the nine campuses of the state
university system, what is the minimum GPA that a senior should have to ensure
eligibility for university admission?
Solution
Let X denote the GPA of a randomly selected senior at Jefferson High
School, and let x denote the minimum GPA that will ensure his or her eligibility
for admission to the university. Since only the top 10% are eligible for admission,
x must satisfy the equation
P(X x) .1
Using Formula (18), Section 8.5, with 2.7 and 0.4, we find that
This is equivalent to the equation
Why?
Consulting Table 2, Appendix D, we find that
Upon solving for x, we obtain
Thus, to ensure eligibility for admission to one of the nine campuses of the state
university system, a senior at Jefferson High School should have a minimum of
3.2 GPA.
Approximating Binomial Distributions
As mentioned in the last section, one important application of the normal distribution
is that it provides us with an accurate approximation of other continuous probability
distributions. Here we show how a binomial distribution may be approximated by a
suitable normal distribution. This technique leads to a convenient and simple solution
to certain problems involving binomial probabilities.
Recall that a binomial distribution is a probability distribution of the form
P(X x) C(n, x)p
x
q
nx
x
0, 1, 2, . . . ,
n
(19)
⬇ 3.2
x 11.282 10.4 2 2.7
x 2.7
0.4
1.28
P aZ
x 2.7
0.4
b .9
P 1X a 2 P aZ
a m
s
bP 1X x2 P aZ
x 2.7
0.4
b .1
.9544
.9772 .0228
P 1Z 22 P 1Z 22
P 12 Z 22
P a
a m
s
Z
b m
s
b P a
49 50
0.5
Z
51 50
0.5
b
P 1a X b 2 P 149 X 51 2
472 8 PROBABILITY DISTRIBUTIONS AND STATISTICS
87533_08_ch8_p417-482 1/30/08 10:10 AM Page 472
(See Section 8.4.) For small values of n, the arithmetic computations of the binomial
probabilities may be done with relative ease. However, if n is large, then the work
involved becomes prodigious, even when tables of P(X x) are available. For
example, if n 50, p .3, and q .7, then the probability of ten or more successes
is given by
To see how the normal distribution helps us in such situations, let’s consider a
coin-tossing experiment. Suppose a fair coin is tossed 20 times and we wish to com-
pute the probability of obtaining 10 or more heads. The solution to this problem may
be obtained, of course, by computing
P(X 10) P(X 10) P(X 11)
. . .
P(X 20)
The inconvenience of this approach for solving the problem at hand has already been
pointed out. As an alternative solution, let’s begin by interpreting the solution in terms
of finding the area of suitable rectangles of the histogram for the distribution associ-
ated with the problem. We may use Formula (19) to compute the probability of obtain-
ing exactly x heads in 20 coin tosses. The results lead to the binomial distribution dis-
played in Table 13.
Using the data from the table, we next construct the histogram for the distribution
(Figure 30). The probability of obtaining 10 or more heads in 20 coin tosses is equal
to the sum of the areas of the shaded rectangles of the histogram of the binomial dis-
tribution shown in Figure 31.
Next, observe that the shape of the histogram suggests that the binomial distribu-
tion under consideration may be approximated by a suitable normal distribution. Since
the mean and standard deviation of the binomial distribution are given by
2.24
112021.521.52
s 1npq
12021.52 10
m np
50!
10! 40!
1.32
10
1.72
40
50!
11! 39!
1.32
11
1.72
39
50!
50! 0!
1.32
50
1.72
0
P 1X 10 2 P 1X 10 2 P 1X 11 2 P 1X 50 2
8.6 APPLICATIONS OF THE NORMAL DISTRIBUTION 473
FIGURE 30
Histogram showing the probability of obtaining x
heads in 20 coin tosses
TABLE 13
Probability Distribution
xP(X ⫽ x)
0 .0000
1 .0000
2 .0002
3 .0011
4 .0046
5 .0148
6 .0370
7 .0739
8 .1201
9 .1602
10 .1762
11 .1602
12 .1201
..
..
..
20 .0000
123456789
10 12 14 16 18 20
.15
.10
.05
x
123456789
10 12 14 16 18 20
.15
.10
.05
x
FIGURE 31
The shaded area gives the probability of obtaining 10
or more heads in 20 coin tosses.
87533_08_ch8_p417-482 1/30/08 10:10 AM Page 473
respectively (see Section 8.4), the natural choice of a normal curve for this purpose is
one with a mean of 10 and standard deviation of 2.24. Figure 32 shows such a normal
curve superimposed on the histogram of the binomial distribution.
The good fit suggests that the sum of the areas of the rectangles representing
P(X 10), the probability of obtaining 10 or more heads in 20 coin tosses, may be
approximated by the area of an appropriate region under the normal curve. To deter-
mine this region, let’s note that the base of the portion of the histogram representing
the required probability extends from x 9.5 on, since the base of the leftmost rect-
angle in the shaded region is centered at x 10 and the base of each rectangle has
length 1 (Figure 33). Therefore, the required region under the normal curve should
also have x 9.5. Letting Y denote the continuous normal variable, we obtain
Use the table of values of
Z
.
The exact value of P(X 10) may be found by computing
P(X 10) P(X 11)
. . .
P(X 20)
in the usual fashion and is equal to .5881. Thus, the normal distribution with suitably
chosen mean and standard deviation does provide us with a good approximation of the
binomial distribution.
In the general case, the following result, which is a special case of the central limit
theorem, guarantees the accuracy of the approximation of a binomial distribution by a
normal distribution under certain conditions.
.5871
P 1Z 0.222
⬇ P 1Z 0.222
P
1X a 2
P
aZ
a m
s
b ⬇ P aZ
9.5 10
2.24
b
P 1Y 9.52
P 1X 10 2⬇ P 1Y 9.5 2
474 8 PROBABILITY DISTRIBUTIONS AND STATISTICS
2468
10 12 14 16 18
x
FIGURE 32
Normal curve superimposed on the his-
togram for a binomial distribution
2468
10 12 14 16 18 20
x
FIGURE 33
P(X 10) is approximated by the area
under the normal curve.
87533_08_ch8_p417-482 1/30/08 10:10 AM Page 474
Note
It can be shown that if both np and nq are greater than 5, then the error result-
ing from this approximation is negligible.
Applications Involving Binomial Random Variables
Next, we look at some applications involving binomial random variables.
APPLIED EXAMPLE 4
Quality Control An automobile manufacturer
receives the microprocessors used to regulate fuel consumption in its auto-
mobiles in shipments of 1000 each from a certain supplier. It has been estimated
that, on the average, 1% of the microprocessors manufactured by the supplier are
defective. Determine the probability that more than 20 of the microprocessors in
a single shipment are defective.
Solution
Let X denote the number of defective microprocessors in a single
shipment. Then X has a binomial distribution with n 1000, p .01, and
q .99, so
Approximating the binomial distribution by a normal distribution with a mean of
10 and a standard deviation of 3.15, we find that the probability that more than
20 microprocessors in a shipment are defective is given by
Where
Y
denotes the normal random variable
In other words, approximately 0.04% of the shipments containing 1000 micro-
processors each will contain more than 20 defective units.
APPLIED EXAMPLE 5
Heart Transplant Survival Rate The proba-
bility that a heart transplant performed at the Medical Center is successful
(that is, the patient survives 1 year or more after undergoing the surgery) is .7.
Of 100 patients who have undergone such an operation, what is the probability
that
a. Fewer than 75 will survive 1 year or more after the operation?
b. Between 80 and 90, inclusive, will survive 1 year or more after the operation?
.0004
⬇ P 1Z 3.332
⬇ P
1Z 3.332
P 1X a 2 P aZ
a m
s
b P aZ
20.5 10
3.15
b
P 1X 20 2⬇ P 1Y 20.5 2
⬇ 3.15
s 111000 2 1.01 2 1.99 2
m 110002 1.01 2 10
THEOREM 1
Suppose we are given a binomial distribution associated with a binomial exper-
iment involving n trials, each with a probability of success p and probability of
failure q. Then, if n is large and p is not close to 0 or 1, the binomial distribution
may be approximated by a normal distribution with
8.6 APPLICATIONS OF THE NORMAL DISTRIBUTION 475
m np
and
s 1npq
87533_08_ch8_p417-482 1/30/08 10:10 AM Page 475
Solution
Let X denote the number of patients who survive 1 year or more after
undergoing a heart transplant at the Medical Center; then X is a binomial ran-
dom variable. Also, n 100, p .7, and q .3, so
Approximating the binomial distribution by a normal distribution with a mean
of 70 and a standard deviation of 4.58, we find, upon letting Y denote the associ-
ated normal random variable:
a. The probability that fewer than 75 patients will survive 1 year or more is
given by
Why?
b. The probability that between 80 and 90, inclusive, of the patients will survive
1 year or more is given by
P
(
a
X
b
)
Note:
P
(
Z
4.48) ⬇ 1
.0192
1 .9808
P 1Z 4.482 P 1Z 2.072
⬇ P 12.07 Z 4.482
P a
a m
s
Z
b m
s
b ⬇ P a
79.5 70
4.58
Z
90.5 70
4.58
b
⬇ P
179.5 Y 90.52
P 180 X 90 2
.8365
⬇ P 1Z 0.982
P 1X b 2 P aZ
b m
s
b ⬇ P aZ
74.5 70
4.58
b
P
1X 752⬇ P 1Y 74.52
⬇ 4.58
s 11100 2 1.7 2 1.3 2
m 11002 1.7 2 70
476 8 PROBABILITY DISTRIBUTIONS AND STATISTICS
1. The serum cholesterol levels in milligrams/decaliter (mg/dL)
in a current Mediterranean population are found to be nor-
mally distributed with a mean of 160 and a standard devia-
tion of 50. Scientists at the National Heart, Lung, and Blood
Institute consider this pattern ideal for a minimal risk of heart
attacks. Find the percentage of the population having blood
cholesterol levels between 160 and 180 mg/dL.
2. It has been estimated that 4% of the luggage manufactured
by The Luggage Company fails to meet the standards
established by the company and is sold as “seconds” to
discount and outlet stores. If 500 bags are produced, what
is the probability that more than 30 will be classified as
“seconds”?
Solutions to Self-Check Exercises 8.6 can be found on
page 478.
8.6 Self-Check Exercises
8.6 Concept Questions
1. What does the central limit theorem allow us to do?
2. Suppose a binomial distribution is associated with a bino-
mial experiment involving n trials, each with a probability
of success p and probability of failure q, and suppose n and
p satisfy the other conditions given in the central limit the-
orem. What are the formulas for and that can be used
to approximate this binomial distribution by a normal dis-
tribution?
87533_08_ch8_p417-482 1/30/08 10:10 AM Page 476
1. M
EDICAL
R
ECORDS
The medical records of infants deliv-
ered at Kaiser Memorial Hospital show that the infants’
lengths at birth (in inches) are normally distributed with a
mean of 20 and a standard deviation of 2.6. Find the prob-
ability that an infant selected at random from among those
delivered at the hospital measures
a. More than 22 in.
b. Less than 18 in.
c. Between 19 and 21 in.
2. F
ACTORY
W
ORKERS
’ W
AGES
According to the data released
by the Chamber of Commerce of a certain city, the weekly
wages of factory workers are normally distributed with a
mean of $600 and a standard deviation of $50. What is the
probability that a factory worker selected at random from
the city makes a weekly wage
a. Of less than $600? b. Of more than $760?
c. Between $550 and $650?
3. P
RODUCT
R
ELIABILITY
TKK Products manufactures 50-, 60-,
75-, and 100-watt electric light bulbs. Laboratory tests show
that the lives of these light bulbs are normally distributed
with a mean of 750 hr and a standard deviation of 75 hr.
What is the probability that a TKK light bulb selected at ran-
dom will burn
a. For more than 900 hr?
b. For less than 600 hr?
c. Between 750 and 900 hr?
d. Between 600 and 800 hr?
4. E
DUCATION
On average, a student takes 100 words/minute
midway through an advanced court reporting course at the
American Institute of Court Reporting. Assuming that the
dictation speeds of the students are normally distributed
and that the standard deviation is 20 words/minute, what is
the probability that a student randomly selected from the
course can take dictation at a speed
a. Of more than 120 words/minute?
b. Between 80 and 120 words/minute?
c. Of less than 80 words/minute?
5. IQ
S
The IQs of students at Wilson Elementary School
were measured recently and found to be normally distrib-
uted with a mean of 100 and a standard deviation of 15.
What is the probability that a student selected at random
will have an IQ
a. Of 140 or higher? b. Of 120 or higher?
c. Between 100 and 120? d. Of 90 or less?
6. P
RODUCT
R
ELIABILITY
The tread lives of the Super Titan
radial tires under normal driving conditions are normally
distributed with a mean of 40,000 mi and a standard devi-
ation of 2000 mi. What is the probability that a tire selected
at random will have a tread life of more than 35,000 mi?
Determine the probability that four tires selected at random
still have useful tread lives after 35,000 mi of driving.
(Assume that the tread lives of the tires are independent of
each other.)
7. F
EMALE
F
ACTORY
W
ORKERS
’ W
AGES
According to data
released by the Chamber of Commerce of a certain city, the
weekly wages (in dollars) of female factory workers are
normally distributed with a mean of 575 and a standard
deviation of 50. Find the probability that a female factory
worker selected at random from the city makes a weekly
wage of $550 to $650.
8. C
IVIL
S
ERVICE
E
XAMS
To be eligible for further considera-
tion, applicants for certain civil service positions must first
pass a written qualifying examination on which a score of
70 or more must be obtained. In a recent examination it
was found that the scores were normally distributed with a
mean of 60 points and a standard deviation of 10 points.
Determine the percentage of applicants who passed the
written qualifying examination.
9. W
ARRANTIES
The general manager of the service depart-
ment of MCA Television has estimated that the time that
elapses between the dates of purchase and the dates on
which the 50-in. plasma TVs manufactured by the com-
pany first require service is normally distributed with a
mean of 22 mo and a standard deviation of 4 mo. If the
company gives a 1-yr warranty on parts and labor for these
TVs, determine the percentage of these TVs manufactured
and sold by the company that will require service before
the warranty period runs out.
10. G
RADE
D
ISTRIBUTIONS
The scores on an economics exami-
nation are normally distributed with a mean of 72 and a
standard deviation of 16. If the instructor assigns a grade of
A to 10% of the class, what is the lowest score a student
may have and still obtain an A?
11. G
RADE
D
ISTRIBUTIONS
The scores on a sociology examina-
tion are normally distributed with a mean of 70 and a stan-
dard deviation of 10. If the instructor assigns A’s to 15%,
B’s to 25%, C’s to 40%, D’s to 15%, and F’s to 5% of the
class, find the cutoff points for grades A–D.
12. H
IGHWAY
S
PEEDS
The speeds (in mph) of motor vehicles
on a certain stretch of Route 3A as clocked at a certain
place along the highway are normally distributed with a
mean of 64.2 mph and a standard deviation of 8.44 mph.
What is the probability that a motor vehicle selected at ran-
dom is traveling at
a. More than 65 mph? b. Less than 60 mph?
c. Between 65 and 70 mph?
In Exercises 13–24, use the appropriate normal distribu-
tions to approximate the resulting binomial distributions.
13. A coin is weighted so that the probability of obtaining a
head in a single toss is .4. If the coin is tossed 25 times,
what is the probability of obtaining
a. Fewer than 10 heads?
b. Between 10 and 12 heads, inclusive?
c. More than 15 heads?
8.6 APPLICATIONS OF THE NORMAL DISTRIBUTION 477
8.6 Exercises
87533_08_ch8_p417-482 1/30/08 10:10 AM Page 477
14. A fair coin is tossed 20 times. What is the probability of
obtaining
a. Fewer than 8 heads? b. More than 6 heads?
c. Between 6 and 10 heads inclusive?
15. S
PORTS
A marksman’s chance of hitting a target with each
of his shots is 60%. (Assume that the shots are independent
of each other.) If he fires 30 shots, what is the probability
of his hitting the target
a. At least 20 times? b. Fewer than 10 times?
c. Between 15 and 20 times, inclusive?
16. S
PORTS
A basketball player has a 75% chance of making a
free throw. (Assume that the throws are independent of
each other.) What is the probability of her making 100 or
more free throws in 120 trials?
17. Q
UALITY
C
ONTROL
The manager of C & R Clothiers, a
manufacturer of men’s dress shirts, has determined that 3%
of C & R’s shirts do not meet with company standards and
are sold as “seconds” to discount and outlet stores. What is
the probability that, in a production run of 200 shirts, fewer
than 10 will be classified as “seconds”?
18. T
ELEMARKETING
Jorge sells magazine subscriptions over
the phone. He estimates that the probability of his making
a sale with each attempt is .12. What is the probability of
Jorge making more than 10 sales if he makes 80 calls?
19. I
NDUSTRIAL
A
CCIDENTS
Colorado Mining and Mineral has
800 employees engaged in its mining operations. It has
been estimated that the probability of a worker meeting
with an accident during a 1-yr period is .1. What is the
probability that more than 70 workers will meet with an
accident during the 1-yr period?
20. Q
UALITY
C
ONTROL
PAR Bearings is the principal supplier
of ball bearings for the Sperry Gyroscope Company. It has
been determined that 6% of the ball bearings shipped are
rejected because they fail to meet tolerance requirements.
What is the probability that a shipment of 200 ball bearings
contains more than 10 rejects?
21. D
RUG
T
ESTING
An experiment was conducted to test the
effectiveness of a new drug in treating a certain disease.
The drug was administered to 50 mice that had been previ-
ously exposed to the disease. It was found that 35 mice
subsequently recovered from the disease. It has been deter-
mined that the natural recovery rate from the disease is 0.5.
a. Determine the probability that 35 or more of the mice
not treated with the drug would recover from the dis-
ease.
b. Using the results obtained in part (a), comment on the
effectiveness of the drug in the treatment of the disease.
22. L
OAN
D
ELINQUENCIES
The manager of Madison Finance
Company has estimated that, because of a recession year,
5% of its 400 loan accounts will be delinquent. If the man-
ager’s estimate is correct, what is the probability that 25 or
more of the accounts will be delinquent?
23. C
RUISE
S
HIP
B
OOKINGS
Because of late cancellations, Nep-
tune Lines, an operator of cruise ships, has a policy of ac-
cepting more reservations than there are accommodations
available. From experience, 8% of the bookings for the
90-day around-the-world cruise on the S.S. Drion, which
has accommodations for 2000 passengers, are subse-
quently canceled. If the management of Neptune Lines has
decided, for public relations reasons, that 99% of all
booked passengers will obtain accommodation on the ship,
determine the largest number of reservations that should be
taken for this cruise on the S.S. Drion.
24. T
HEATER
B
OOKINGS
Preview Showcase, a research firm,
screens pilots of new TV shows before a randomly selected
audience and then solicits their opinions of the shows.
Based on past experience, 20% of those who receive com-
plimentary tickets are “no-shows.” The theater has a seat-
ing capacity of 500. Management has decided, for public
relations reasons, that 99% of all ticket holders will be
seated. How many tickets should the company send out to
prospective viewers for each screening?
478
8 PROBABILITY DISTRIBUTIONS AND STATISTICS
1. Let X be the normal random variable denoting the serum
cholesterol levels in mg/dL in the current Mediterranean
population under consideration. Then, the percentage of
the population having blood cholesterol levels between
160 and 180 mg/dL is given by P(160 X 180). To
compute P(160 X 180), we use Formula (16), Section
8.5, with 160, 50, a 160, and b 180. We find
Thus, approximately 15.5% of the population has blood
cholesterol levels between 160 and 180 mg/dL.
.1554
.6554 .5000
P
1Z 0.42 P 1Z 0 2
P
10 Z 0.42
P
1160 X 1802 P a
160 160
50
Z
180 160
50
b
8.6 Solutions to Self-Check Exercises
87533_08_ch8_p417-482 1/30/08 10:10 AM Page 478
2. Let X denote the number of substandard bags in the pro-
duction. Then X has a binomial distribution with n 500,
p .04, and q .96, so
Approximating the binomial distribution by a normal
distribution with a mean of 20 and standard deviation of
4.38, we find that the probability that more than 30 bags in
the production of 500 will be substandard is given by
s 115002 1.042 1.962⬇ 4.38
m 15002
1.042 20
or approximately 0.8%.
.0082
P
1Z 2.402
⬇ P
1Z 2.402
⬇ P
aZ
30.5 20
4.38
b
Where
Y
denotes the
normal random variable
P 1X 302⬇ P 1Y 30.5 2
SUMMARY OF PRINCIPAL FORMULAS AND TERMS 479
Summary of Principal Formulas and Terms
CHAPTER 8
FORMULAS
1. Mean of n numbers
2. Expected value E(X) x
1
p
1
x
2
p
2
. . .
x
n
p
n
3. Odds in favor of E occurring
4. Odds against E occurring
5. Probability of an event occurring
given the odds
6. Variance of a random variable Var(X) p
1
(x
1
)
2
p
2
(x
2
)
2
. . .
p
n
(x
n
)
2
7. Standard deviation of a random variable
8. Chebychev’s inequality
9. Probability of x successes in nC(n, x)p
x
q
nx
Bernoulli trials
10. Binomial random variable:
Mean
Variance
Standard deviation
s
X
1npq
Var 1X2 npq
m E
1X2 np
1
1
k
2
P 1m ks X m ks 2
s 1Var
1X2
a
a b
P 1E
c
2
P 1E2
P
1E2
P 1E
c
2
x
x
1
x
2
p
x
n
n
TERMS
random variable (418) average (mean) (427) Bernoulli (binomial) trial (453)
finite discrete random variable (419) expected value (428) binomial experiment (453)
infinite discrete random variable (419) median (435) binomial random variable (455)
continuous random variable (419) mode (435) binomial distribution (455)
probability distribution of a random variable (419) variance (441) probability density function (462)
histogram (421) standard deviation (443) normal distribution (463)
87533_08_ch8_p417-482 1/30/08 10:10 AM Page 479
Fill in the blanks.
1. A rule that assigns a number to each outcome of a chance
experiment is called a/an ______ variable.
2. If a random variable assumes only finitely many values,
then it is called ______ discrete; if it takes on infinitely
many values that can be arranged in a sequence, then it is
called ______ discrete; if it takes on all real numbers in an
interval, then it is said to be ______.
3. The expected value of a random variable X is given by the
______ of the products of the values assumed by the random
variable and their associated probabilities. For example, if
X assumes the values 2, 3, and 4 with associated proba-
bilities , , and , then its expected value is ______ .
4. a. If the probability of an event E occurring is P(E), then
the odds in favor of E occurring are ______.
b. If the odds in favor of an event E occurring are a to b,
then the probability of E occurring is ______ .
1
4
1
4
1
2
5. Suppose a random variable X takes on the values x
1
, x
2
,
. . . , x
n
with probabilities p
1
, p
2
, . . . . , p
n
and has a mean
of . Then the variance of X is ______, and the standard
deviation of X is ______ .
6. In a binomial experiment, the number of trials is ______,
there are exactly ______ outcomes in each trial, the proba-
bility of “success” in each trial is the ______, and the trials
are ______ of each other.
7. A probability distribution that is associated with a continu-
ous random variable is called a/an______ probability distri-
bution. Such a probability distribution is defined by a/an
______ ______ ______ whose domain is the ______ of values
taken on by the random variable associated with the exper-
iment.
8. A binomial distribution may be approximated by a/an
______ distribution with np and if n is
______ and p is not close to ______ or ______ .
s 1npq
1. Three balls are selected at random without replacement
from an urn containing three white balls and four blue balls.
Let the random variable X denote the number of blue balls
drawn.
a. List the outcomes of this experiment.
b. Find the value assigned to each outcome of this experi-
ment by the random variable X.
c. Find the probability distribution of the random variable
associated with this experiment.
d. Draw the histogram representing this distribution.
2. A man purchased a $25,000, 1-yr term-life insurance pol-
icy for $375. Assuming that the probability that he will live
for another year is .989, find the company’s expected gain.
3. The probability distribution of a random variable X is
shown in the following table:
xP(X x)
0.1
1.1
2.2
3.3
4.2
5.1
a. Compute P(1 X 4).
b. Compute the mean and standard deviation of X.
4. A binomial experiment consists of four trials in which the
probability of success in any one trial is .
a. Construct the probability distribution for the experiment.
b. Compute the mean and standard deviation of the proba-
bility distribution.
2
5
In Exercises 5–8, let Z be the standard normal variable.
Make a rough sketch of the appropriate region under the
standard normal curve and find the probability.
5. P(Z 0.5) 6. P(Z 0.75)
7. P(0.75 Z 0.5) 8. P(0.42 Z 0.66)
In Exercises 9–12, let Z be the standard normal variable.
Find z if z satisfies the given value.
9. P(Z z) .9922 10. P(Z z) .1469
11. P(Z z) .9788
12. P(z Z z) .8444
In Exercises 13–16, let X be a normal random variable
with 10 and 2. Find the value of the given prob-
ability.
13. P(X 11) 14. P(X 8)
15. P(7 X 9) 16. P(6.5 X 11.5)
17. T
RAFFIC
A traffic survey of the speeds of vehicles traveling
along a stretch of Hampton Road between 4 p.m. and
6 p.m. yielded the following results:
Speed, mph 30–34 35–39 40–44 45–49 50–54
Probability .07 .28 .42 .18 .05
Find the average speed of the vehicles.
Hint: Assume that all speeds lying within a group interval take the
midvalue of that group.
18. E
XPECTED
P
ROFIT
A buyer for Discount Fashions, an outlet
for women’s apparel, is considering buying a batch of cloth-
480
8 PROBABILITY DISTRIBUTIONS AND STATISTICS
Concept Review Questions
CHAPTER 8
Review Exercises
CHAPTER 8
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ing for $64,000. She estimates that the company will be able
to sell it for $80,000, $75,000, or $70,000 with probabilities
of .30, .60, and .10, respectively. Based on these estimates,
what will be the company’s expected gross profit?
19. S
PORTS
If the probability that a bowler will bowl a strike is
.7, what is the probability that he will get exactly two
strikes in four attempts? At least two strikes in four
attempts? (Assume that the attempts to bowl a strike are
independent of each other.)
20. H
EIGHTS OF
W
OMEN
The heights of 4000 women who par-
ticipated in a recent survey were found to be normally dis-
tributed with a mean of 64.5 in. and a standard deviation of
2.5 in. What percentage of these women have heights of
67 in. or greater?
21. H
EIGHTS OF
W
OMEN
Refer to Exercise 20. Use Cheby-
chev’s inequality to estimate the probability that the height
of a woman who participated in the survey will fall within
2 standard deviations of the mean—that is, that her height
will be between 59.5 and 69.5 in.
22. V
OLKSWAGEN
’
S
R
EVENUE
The revenue of Volkswagen (in
billions of Euros) for the five quarters beginning with the
first quarter of 2003 are summarized in the following table:
2003 2004
Quarter Q1 Q2 Q3 Q4 Q1
Revenue 20.6 22.1 21.3 21.8 21.9
Find the average quarterly revenue of Volkswagen for the
five quarters in question. What is the standard deviation?
Source: Company Reports
23. M
ARITAL
S
TATUS OF
W
OMEN
The number of single women
(in thousands) between the ages of 20 and 44 in the United
States in 1998 is given in the following table:
Age 20–24 25–29 30–34 35–39 40–44
Women 6178 3689 2219 1626 1095
Find the mean and the standard deviation of the given data.
Hint: Assume that all values lying within a group interval take the
middle value of that group.
Source: U.S. Census Bureau
24. Q
UALITY
C
ONTROL
The proprietor of a hardware store will
accept a shipment of ceramic wall tiles if no more than 2 of
a random sample of 20 are found to be defective. What is
the probability that he will accept shipment if exactly 10%
of the tiles in a certain shipment is defective?
25. D
RUG
E
FFECTIVENESS
An experimental drug has been found
to be effective in treating 15% of the people afflicted by a
certain disease. If the drug is administered to 800 people
who have this disease, what are the mean and standard
deviation of the number of people for whom the drug can
be expected to be effective?
26. Q
UALITY
C
ONTROL
Dayton Iron Works manufactures steel
rods to a specification of 1-in. diameter. These rods are
accepted by the buyer if they fall within the tolerance lim-
its of 0.995 and 1.005. Assuming that the diameter of the
rods is normally distributed about a mean of 1 in. and has
a standard deviation of 0.002 in., estimate the percentage
of rods that will be rejected by the buyer.
27. E
XPECTED
S
ALES
The division manager of a company
claims that 80% of her sales representatives will make or
exceed their sales quota in the following month. Assuming
that the manager’s assessment is correct, what is the prob-
ability that the sales quota will be made or exceeded by
a. Four of the six sales representatives?
b. At least four of the sales representatives?
28. O
N
-T
IME
A
RRIVALS
Diane, who commutes regularly be-
tween Los Angeles and San Francisco for business, esti-
mates that the probability that her flight will arrive on time
or earlier is .92. Assuming that her assessment is accurate,
what is the probability that her flight will arrive on time or
earlier
a. In three of her next five flights?
b. In at least three of her next five flights?
29. C
OIN
T
OSSES
A coin is biased so that the probability of it
landing heads is .6. If the coin is tossed 100 times, what is
the probability that heads will appear more than 50 times
in the 100 tosses?
30. Q
UALITY
C
ONTROL
A division of Solaron Corporation man-
ufactures photovoltaic cells for use in the company’s solar
energy converters. It is estimated that 5% of the cells man-
ufactured are defective. In a batch of 200 cells manufac-
tured by the company, what is the probability that it will
contain at most 20 defective units?
BEFORE MOVING ON . . . 481
1. The values taken on by a random variable X and the fre-
quency of their occurrence are shown in the following
table. Find the probability distribution of X.
x 3 20 123
Frequency of
Occurrence 4 8 20 24 16 8
2. The probability distribution of the random variable X is
shown in the following table. Find (a) P(X 0) and (b)
P(4 X 1).
x 4 3 10 1 3
P(X x) .06 .14 .32 .28 .12 .08
Before Moving On . . .
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