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1 2 3 4 5 6 7 8 9 10
Advanced Mathematical Concepts
Chapter 14 Resource Masters
XXX
11 10 09 08 07 06 05 04
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . . vii-x
Chapter 14 Assessment
Chapter 14 Test, Form 1A . . . . . . . . . . . 625-626
Chapter 14 Test, Form 1B . . . . . . . . . . . 627-628
Chapter 14 Test, Form 1C . . . . . . . . . . . 629-630
Chapter 14 Test, Form 2A . . . . . . . . . . . 631-632
Chapter 14 Test, Form 2B . . . . . . . . . . . 633-634
Chapter 14 Test, Form 2C . . . . . . . . . . . 635-636
Chapter 14 Extended Response
Assessment . . . . . . . . . . . . . . . . . . . . . . . 637
Chapter 14 Mid-Chapter Test . . . . . . . . . . . . 638
Chapter 14 Quizzes A & B . . . . . . . . . . . . . . 639
Chapter 14 Quizzes C & D. . . . . . . . . . . . . . 640
Chapter 14 SAT and ACT Practice . . . . 641-642
Chapter 14 Cumulative Review . . . . . . . . . . 643
Unit 4 Review . . . . . . . . . . . . . . . . . . . . 645-646
Unit 4 Test . . . . . . . . . . . . . . . . . . . . . . . 647-650
Lesson 14-1
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 609
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 611
Lesson 14-2
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 612
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 614
Lesson 14-3
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 615
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 617
Lesson 14-4
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 618
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 620
SAT and ACT Practice Answer Sheet,
10 Questions . . . . . . . . . . . . . . . . . . . . . . . A1
SAT and ACT Practice Answer Sheet,
20 Questions . . . . . . . . . . . . . . . . . . . . . . . A2
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A3-A16
Lesson 14-5
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 621
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 623
© Glencoe/McGraw-Hill
iii
Advanced Mathematical Concepts
A Teacher’s Guide to Using the
Chapter 14 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the
resources you use most often. The Chapter 14 Resource Masters include the core
materials needed for Chapter 14. These materials include worksheets, extensions,
and assessment options. The answers for these pages appear at the back of this
booklet.
All of the materials found in this booklet are included for viewing and printing in
the Advanced Mathematical Concepts TeacherWorks CD-ROM.
Vocabulary Builder Pages vii-x include a
Practice There is one master for each lesson.
student study tool that presents the key
vocabulary terms from the chapter. Students are
to record definitions and/or examples for each
term. You may suggest that students highlight or
star the terms with which they are not familiar.
These problems more closely follow the
structure of the Practice section of the Student
Edition exercises. These exercises are of
average difficulty.
When to Use These provide additional
practice options or may be used as homework
for second day teaching of the lesson.
When to Use Give these pages to students
before beginning Lesson 14-1. Remind them to
add definitions and examples as they complete
each lesson.
Enrichment There is one master for each
lesson. These activities may extend the concepts
in the lesson, offer a historical or multicultural
look at the concepts, or widen students’
perspectives on the mathematics they are
learning. These are not written exclusively
for honors students, but are accessible for use
with all levels of students.
Study Guide There is one Study Guide
master for each lesson.
When to Use Use these masters as
reteaching activities for students who need
additional reinforcement. These pages can also
be used in conjunction with the Student Edition
as an instructional tool for those students who
have been absent.
© Glencoe/McGraw-Hill
When to Use These may be used as extra
credit, short-term projects, or as activities for
days when class periods are shortened.
iv
Advanced Mathematical Concepts
Assessment Options
Intermediate Assessment
The assessment section of the Chapter 14
Resources Masters offers a wide range of
assessment tools for intermediate and final
assessment. The following lists describe each
assessment master and its intended use.
Chapter Tests
•
Forms 1A, 1B, and 1C Form 1 tests contain
multiple-choice questions. Form 1A is
intended for use with honors-level students,
Form 1B is intended for use with averagelevel students, and Form 1C is intended for
use with basic-level students. These tests are
similar in format to offer comparable testing
situations.
Forms 2A, 2B, and 2C Form 2 tests are
composed of free-response questions. Form
2A is intended for use with honors-level
students, Form 2B is intended for use with
average-level students, and Form 2C is
intended for use with basic-level students.
These tests are similar in format to offer
comparable testing situations.
The Extended Response Assessment
includes performance assessment tasks that
are suitable for all students. A scoring rubric
is included for evaluation guidelines.
Sample answers are provided for
assessment.
© Glencoe/McGraw-Hill
•
Four free-response quizzes are included to
offer assessment at appropriate intervals in
the chapter.
•
The SAT and ACT Practice offers
continuing review of concepts in various
formats, which may appear on standardized
tests that they may encounter. This practice
includes multiple-choice, quantitativecomparison, and grid-in questions. Bubblein and grid-in answer sections are provided
on the master.
•
The Cumulative Review provides students
an opportunity to reinforce and retain skills
as they proceed through their study of
advanced mathematics. It can also be used
as a test. The master includes free-response
questions.
Answers
All of the above tests include a challenging
Bonus question.
•
A Mid-Chapter Test provides an option to
assess the first half of the chapter. It is
composed of free-response questions.
Continuing Assessment
Chapter Assessments
•
•
v
•
Page A1 is an answer sheet for the SAT and
ACT Practice questions that appear in the
Student Edition on page 939. Page A2 is an
answer sheet for the SAT and ACT Practice
master. These improve students’ familiarity
with the answer formats they may encounter
in test taking.
•
The answers for the lesson-by-lesson
masters are provided as reduced pages with
answers appearing in red.
•
Full-size answer keys are provided for the
assessment options in this booklet.
Advanced Mathematical Concepts
Chapter 14 Leveled Worksheets
Glencoe’s leveled worksheets are helpful for meeting the needs of every
student in a variety of ways. These worksheets, many of which are found
in the FAST FILE Chapter Resource Masters, are shown in the chart
below.
•
Study Guide masters provide worked-out examples as well as practice
problems.
•
Each chapter’s Vocabulary Builder master provides students the
opportunity to write out key concepts and definitions in their own
words.
•
Practice masters provide average-level problems for students who
are moving at a regular pace.
•
Enrichment masters offer students the opportunity to extend their
learning.
Five Different Options to Meet the Needs of
Every Student in a Variety of Ways
primarily skills
primarily concepts
primarily applications
BASIC
AVERAGE
1
Study Guide
2
Vocabulary Builder
3
Parent and Student Study Guide (online)
© Glencoe/McGraw-Hill
4
Practice
5
Enrichment
vi
ADVANCED
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Reading to Learn Mathematics
Vocabulary Builder
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 14.
As you study the chapter, complete each term’s definition or description.
Remember to add the page number where you found the term.
Vocabulary Term
Found
on Page
Definition/Description/Example
arithmetic mean
back-to-back bar graph
bar graph
bimodal
box-and-whisker plot
class interval
class limit
class mark
cumulative frequency distribution
frequency distribution
(continued on the next page)
© Glencoe/McGraw-Hill
vii
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Reading to Learn Mathematics
Vocabulary Builder (continued)
Vocabulary Term
Found
on Page
Definition/Description/Example
frequency polygon
hinge
histogram
inferential statistics
interquartile range
leaf
level of confidence
line plot
mean
mean deviation
measure of central tendency
(continued on the next page)
© Glencoe/McGraw-Hill
viii
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Reading to Learn Mathematics
Vocabulary Builder (continued)
Vocabulary Term
Found
on Page
Definition/Description/Example
measure of variability
median
median class
mode
normal curve
normal distribution
outlier
percentile
population
quartile
random sample
(continued on the next page)
© Glencoe/McGraw-Hill
ix
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
Reading to Learn Mathematics
14
Vocabulary Builder (continued)
Vocabulary Term
Found
on Page
Definition/Description/Example
range
semi-interquartile range
standard deviation
standard error of the mean
stem
stem-and-leaf plot
three-dimensional bar graph
variance
whisker
© Glencoe/McGraw-Hill
x
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
14-1
Study Guide
The Frequency Distribution
A frequency distribution is a convenient system for organizing large
amounts of data. A number of classes are determined, and all values in a class
are tallied and grouped together. The most common way of displaying frequency
distributions is by using a type of bar graph called a histogram.
Example The number of passengers who boarded planes at 36 airports
in the United States in one year are shown below.
30,526
30,372
26,623
22,722
16,287
15,246
14,807
14,117
14,054
13,547
12,916
12,616
11,906
11,622
11,489
10,828
10,653
10,008
9703
9594
9463
9348
9125
8572
7300
6772
6549
6126
5907
5712
5287
4848
4832
4820
4750
4684
Source: U.S. Department of Transportation
a. Find the range of the data.
The range of the data is 30,526 4684 or 25,842.
b. Determine an appropriate class interval.
An appropriate class interval is 4500 passengers,
beginning with 4500 and ending with 31,500. There
will be six classes.
c. Name the class limits and the class marks.
The class limits are the upper and lower values in
each interval, or 4500, 9000, 13,500, 18,000, 22,500,
27,000, and 31,500. The class marks are the averages
of the class limits of each interval, or 6750, 11,250,
15,750, 20,250, 24,750, and 29,250.
d. Construct a frequency distribution of the data.
Use tallies to determine the number of passengers in
each interval.
Number of Passengers
Tallies
Frequency
4500-9000
13
9000-13,500
13
13,500-18,000
6
18,000-22,500
0
22,500-27,000
2
27,000-31,500
2
e. Draw a histogram of the data.
Label the horizontal axis with the
class limits. The vertical axis should
be labeled from 0 to a value that will
allow for the greatest frequency.
Draw the bars side by side so that the
height of each bar corresponds to its
interval’s frequency.
© Glencoe/McGraw-Hill
609
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
14-1
Practice
The Frequency Distribution
Determine which class intervals would be appropriate for the data below.
Explain your answers.
1. 25, 32, 18, 99, 43, 16, 29, 35, 36, 34,
21, 33, 26, 26, 17, 40, 22, 38, 16, 19
a. 1
b. 10
c. 2
2. 111, 115, 130, 200, 234, 98, 115, 72
305, 145, 87, 63, 245, 285, 256, 302
a. 25
b. 10
c. 30
3. Meteorology The average wind speeds recorded at various weather
stations in the United States are listed below.
Station
Speed
(mph)
Albuquerque
Baltimore
Station
8.9
Anchorage
9.1
Speed
(mph)
Station
Speed
(mph)
7.1
Atlanta
9.1
10.4
Boston
12.5
Chicago
Dallas-Ft. Worth
10.8
Honolulu
11.3
Indianapolis
9.6
Kansas City
10.7
Las Vegas
9.3
Little Rock
7.8
Los Angeles
6.2
Memphis
8.8
Miami
9.2
Minneapolis– St. Paul
10.5
Philadelphia
9.5
New Orleans
8.1
New York City
9.4
Phoenix
6.2
Seattle
9.0
Source: National Climatic Data Center
a. Find the range of the data.
b. Determine an appropriate class interval.
c. What are the class limits and the class marks?
d. Construct a frequency distribution of the data.
e. Draw a histogram of the data.
e.
© Glencoe/McGraw-Hill
610
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
14-1
Enrichment
Misuses of Statistics
Statistics can be misleading. Two graphs for the same set of data can
look very different from each other. Compare the following graphs.
Notice that the two graphs show the same data, but the spacing in
the vertical and horizontal scales differs. Scales can be cramped or
spread out to make a graph that gives a certain impression.
1. Which graph would you use to give the impression that the
unemployment rate dropped dramatically from 1982 to 1990?
2. Suppose that a car company claims, “75% of people surveyed say
that our car is better than the competition.” If only four people
were surveyed, how many people thought that this company’s car
was better?
Suppose an advertiser claims that 90% of all the cars of one brand sold in the last
10 years are still on the road.
3. If 10,000 cars were sold, how many are still on the road?
4. If 1000 cars were sold, how many are still on the road?
5. Find an example to show how you think averages could be used in
a misleading way.
6. A survey of a large sample of people who own small computers
revealed that 85% of the people thought the instruction manuals
should be better written. A manufacturer of small computers
claimed that it surveyed many of the same people and found that
all of them liked their manuals. Discuss the possible discrepancy
in the results.
© Glencoe/McGraw-Hill
611
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
14-2
Study Guide
Measures of Central Tendency
The mean is found by adding the values in a set of data and dividing
the sum by the number of values in that set. In other words, if a set
of data has n values given by Xi such that i is an integer and 1 i n, then the arithmetic mean X can be found as follows.
n
X n1 Xi
i=1
The median of a set of data is the middle value. If there are two
middle values, the median is the mean of the two middle values.
The mode of a set of data is the most frequent value. Some sets
have multiple modes, and others have no mode.
Example 1
Find the mean of the set {13, 18, 21, 14, 16, 19, 25, 17}.
sum of the values in the set of data
X number of values in the set
13 + 18 + 21 + 14 + 16 + 19 + 25 + 17
X 8
14
3 or 17.875
X
8
The mean of the set of data is 17.875.
Example 2
The table at the right shows the number of
households without a telephone in 1990.
a. Find the mean of the data.
Since there are 11 states listed in the table,
n 11.
11
Xi 111 (131,600 313,100 270,200 1
11i=1
195,700 36,500 140,900 40,400 25,100 67,500 106,400 70,800)
The mean is about 127,109.
b. Find the median of the data.
To find the median, order the data. Since all
the numbers are multiples of 100, you can
order the set by hundreds.
State
Alaska
Number of
Households
131,600
California
313,100
Florida
270,200
Georgia
195,700
Iowa
36,500
Kentucky
140,900
Minnesota
40,400
Nevada
25,100
New Mexico
67,500
Oklahoma
106,400
West Virginia
70,800
Source: U.S. Census Bureau
251 365 404 675 708 1064 1316 1409 1957 2702 3131
Since there are an odd number of values, the
median is the middle value, or 106,400.
c. Find the mode of the data.
Since all elements in the set of data have the
same frequency, there is no mode.
© Glencoe/McGraw-Hill
612
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
14-2
Practice
Measures of Central Tendency
Find the mean, median, and mode of each set of data.
1. {15, 42, 26, 39, 93, 42}
2. {32, 12, 61, 94, 73, 62, 94, 35, 44, 52}
3. {152, 697, 202, 312, 109, 134, 116}
4. {18, 6, 22, 33, 19, 34, 14, 54, 12, 22, 19}
5. A shoe store employee sets up a display by placing shoeboxes in
10 stacks. The numbers of boxes in each stack are 5, 7, 9, 11, 13,
10, 9, 8, 7, and 5.
a. What is the mean of the number of boxes in a stack?
b. Find the median of the number of boxes in a stack.
c. If one box is removed from each stack, how will the mean and
median be affected?
Find the mean, median, and mode of the data represented by each
stem-and-leaf plot.
6.
Stem Leaf
2 2 4 4 7
3 1 3 4
4 5 6 8
5 9
2/2 220
7.
Stem Leaf
9 0113
10 1 3 5 6
11 3 4 6 8
9/0 90
9. Medicine A frequency distribution for
the number of patients treated at 50 U.S.
cancer centers in one year is given at
the right.
a. Use the frequency chart to find the
mean of the number of patients
treated by a cancer center.
b. What is the median class of the
frequency distribution?
8.
Stem Leaf
1 1 2 9
2 3 3 5
3 2
4 0
5 4 5 6 8 9
1/1 1.1
Patients
Number
of Cancer
Centers
500–1000
26
1000–1500
14
1500–2000
6
2000–2500
0
2500–3000
2
3000–3500
0
3500–4000
2
Source: U.S. News Online
© Glencoe/McGraw-Hill
613
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
14-2
Enrichment
The Harmonic Mean
The harmonic mean H is a useful measure of central tendency in
special cases of averaging rates.
Example
Recently Kendra and Bill took a trip of 370 miles
and shared the driving. Kendra drove two hours
at a rate of 30 mph and then drove the next
110 miles on a freeway at 55 mph. Then Bill
drove the next two hours at 50 mph and he drove
the last 100 miles on a freeway at 55 mph. What
was the average speed of each driver?
Kendra drove the same length of time on both portions
of her driving, so her average speed is the mean of the
two rates. Her average speed was
30 55
2
or 42.5 mph.
On the other hand, Bill drove the same distance on
both portions of his driving, but the two lengths of time
varied. Actually, the time he drove was
100
50
100
,
55
or
approximately 3.82 hours. His average speed was
200
,
3.82
or about 52.4 mph.
Bill’s average speed also may be found by using the
formula for the harmonic mean as follows.
Let n number of rates xi where 1 i n.
H n
n
1
__
xi
i1
We apply the formula to Bill’s speeds.
2
H ______
1
1
50
55
H 52.4 mph
The mean, also called the arithmetic mean, is used when equal
times are involved. When equal distances are involved, the
harmonic mean is used.
Find the harmonic mean of each set of data. Round each answer to the nearest
hundreth.
1. {3, 4, 5, 6}
2. {5, 10, 15, 20, 25}
3. Bev, Phyllis, and Gordon competed in a 375-mile relay race. Bev drove 40 mph,
Phyllis drove 50 mph, and Gordon drove 60 mph. If each drove 125 miles, find the
average driving speed of the contestants.
© Glencoe/McGraw-Hill
614
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
14-3
Study Guide
Measures of Variability
If a set of data has been arranged in order and the median is found,
the set of data is divided into two groups. If the median of each group
is found, the data is divided into four groups. Each of these groups is
called a quartile, and the quartile points Q1, Q2, and Q3 denote the
breaks for each quartile. The interquartile range is the difference
between the first quartile point and the third quartile point.
Example 1
Month
The table shows the average monthly
temperatures for San Diego in 1997.
a. Find the interquartile range of the
temperatures and state what it
represents.
First, order the data from least to greatest
and identify Q1, Q2, and Q3.
57.4 57.9 58.0 61.6 62.5 64.0
67.4 68.7 68.7 69.3 72.9 75.5
For this set of data, the quartile points Q1, Q2,
and Q3 are not members of the set. Instead,
Q2 is the mean of the middle values of the set.
Thus, Q1 59.8, Q2 65.7, and Q3 69.0.
The interquartile range is 69.0 59.8, or 9.2.
This means that half the average monthly
temperatures are within 9.2F of each other.
b. Find the semi-interquartile range of the
temperatures.
9.2
, or 4.6.
The semi-interquartile range is 2
Example 2
Temperature
(°F)
Jan.
58.0
Feb.
57.9
March
61.6
April
62.5
May
68.7
June
67.4
July
69.3
Aug.
72.9
Sept.
75.5
Oct.
68.7
Nov.
64.0
Dec.
57.4
Source: National Climatic
Data Center
Find the mean deviation of the temperatures in
Example 1.
There are 12 temperatures listed, and the mean is 112 Xi,
or 65.325.
MD 112 Xi 65.325
MD n1 Xi X
MD 112(75.5 65.325 72.9 65.325 . . . 57.4 65.325)
MD 112(10.175 7.575 . . . 7.925) or about 5.092
The mean deviation of the temperatures is about 5.092.
This means that the temperatures are an average of
about 5.092F above or below the mean temperature of
65.325F.
© Glencoe/McGraw-Hill
615
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
14-3
Practice
Measures of Variability
Find the interquartile range and the semi-interquartile range of
each set of data. Then draw a box-and-whisker plot.
1. 43, 26, 92, 11, 8, 49, 52, 126, 86, 42, 63, 78, 91, 79, 86
2. 1.6, 9.8, 4.5, 6.2, 8.7, 5.6, 3.9, 6.8, 9.7, 1.1, 4.7, 3.8, 7.5, 2.8, 0.1
Find the mean deviation and the standard deviation of each
set of data.
3. 146, 289, 121, 146, 212, 98, 86, 153, 128, 136, 181, 142
4. 1592, 1486, 1479, 1682, 1720, 1104, 1486, 1895, 1890, 2687, 2450
5. Sociology The frequency distribution
at the right shows the average life
expectancy for males and females in
15 European Union countries in 1994.
a. Find the mean of the female life
expectancy.
b. Find the mean of the male life
expectancy.
c. What is the standard deviation
of the female life expectancy?
Life
Expectancy
(years)
Male
Female
71.573.0
3
0
73.074.5
9
0
74.576.0
2
0
76.077.5
1
0
77.579.0
0
3
79.080.5
0
8
80.582.0
0
4
Frequency
Source: Department of Health and Children, Ireland
d. What is the standard deviation
of the male life expectancy?
© Glencoe/McGraw-Hill
616
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
14-3
Enrichment
Percentiles
The table at the right shows test scores and their
frequencies. The frequency is the number of people
who had a particular score. The cumulative frequency
is the total frequency up to that point, starting at the
lowest score and adding up.
Example 1
What score is at the 16th percentile?
A score at the 16th percentile means the
score just above the lowest 16% of the scores.
16% of the 50 scores is 8 scores.
The 8th score is 55.
The score just above this is 56.
Thus, the score at the 16th percentile is 56.
Score
Frequency
Cumulative
Frequency
95
1
50
90
2
49
85
5
47
80
6
42
75
7
36
70
8
29
65
7
21
60
6
14
55
4
8
50
3
4
45
1
1
Notice that no one had a score of 56 points.
Use the table above to find the score at each percentile.
1. 42nd percentile ______
2. 70th percentile ______
3. 33rd percentile ______
4. 90th percentile ______
5. 58th percentile ______
6. 80th percentile ______
Example 2
At what percentile is a score of 75?
There are 29 scores below 75.
Seven scores are at 75. The fourth of these seven is the
midpoint of this group.
Adding 4 scores to the 29 gives 33 scores.
33 out of 50 is 66%.
Thus, a score of 75 is at the 66th percentile.
Use the table above to find the percentile of each score.
7. a score of 50 ______
8. a score of 77 ______
9. a score of 85 ______
10. a score of 58 ______
11. a score of 62 ______
12. a score of 81 ______
© Glencoe/McGraw-Hill
617
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
14-4
Study Guide
The Normal Distribution
A normal distribution is a frequency distribution that often occurs when
there is a large number of values in a set of data. The graph of a normal
distribution is a symmetric, bell-shaped curve known as a normal curve.
The tables below give the fractional parts of a normally distributed set of
data for selected areas about the mean. The letter
 t represents the number
of standard deviations from the mean, that
 is, X t. P represents the
fractional part that lies in the interval X t.
t
P
t
P
t
P
t
P
t
P
t
P
0.0
0.000
0.6
0.451
1.2
0.770
1.7
0.911
2.2
0.972
2.7
0.993
0.1
0.080
0.7
0.516
1.3
0.807
1.8
0.929
2.3
0.979
2.8
0.995
0.2
0.159
0.8
0.576
1.4
0.838
1.9
0.943
2.4
0.984
2.9
0.996
0.3
0.236
0.9
0.632
1.5
0.866
1.96
0.950
2.5
0.988
3.0
0.997
0.4
0.311
1.0
0.683
1.6
0.891
2.0
0.955
2.58
0.990
3.5
0.9995
0.5
0.383
1.1
0.729
1.65 0.900
2.1
0.964
2.6
0.991
4.0
0.9999
Example 1
Air passengers traveling through Atlanta have an
average layover of 82 minutes with a standard
deviation of 7.5 minutes. Sketch a normal curve
that represents the frequency of layover times.
First, find the values defined by the standard deviation
in
 a normal distribution.

1 82 1(7.5) or 74.5
X 1 82 1(7.5) or 89.5
X


2 82 2(7.5) or 67
X 2 82 2(7.5) or 97
X


X 3 82 3(7.5) or 104.5
X 3 82 3(7.5) or 59.5
Then, sketch the general shape
of a normal curve. Replace the
horizontal scale with the values
you have calculated.
Example 2
Find the upper and lower limits of an interval about the
mean within which 15% of the values
of a set of normally

distributed data can be found if X 725 and 4.
Use the tables above to find the value of t that most closely
approximates P 0.15. For t 0.2, P 0.159. Choose t 0.2.


Now find the limits. X t 725 0.2(4) X 725, t 0.2, 4
724.2 and 725.8
The interval in which 15% of the data lies is 724.2–725.8.
© Glencoe/McGraw-Hill
618
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
14-4
Practice
The Normal Distribution
A set of 1000 values has a normal distribution. The mean of
the data is 120, and the standard deviation is 20.
1. How many values are within one standard deviation of the mean?
2. What percent of the data is between 110 and 130?
3. What percent of the data is between 90 and 110?
4. Find the interval about the mean that includes 90% of the data.
5. Find the interval about the mean that includes 77% of the data.
6. Find the limit below which 90% of the data lie.
7. Dog Breeding The weights of full-grown German shepherds at
the City View Kennels are normally distributed. The mean weight
is 86 pounds, and the standard deviation is 3 pounds. Skipper, a
full-grown German shepard, weighs 79 pounds.
a. What percent of the full-grown German shepherds at City View
Kennels weigh more than Skipper?
b. What percent of the full-grown German shepherds at City View
Kennels weigh less than Skipper?
© Glencoe/McGraw-Hill
619
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
14-4
Enrichment
Shapes of Distribution Curves
Graphs of frequency distributions can be described as either
symmetric or skewed.
In a distribution skewed to the right, there are a larger number of
high values. The long “tail” extends to the right.
In a distribution skewed to the left, there are a larger number of low
values. The long “tail” extends to the left.
For each of the following, state whether the distribution is symmetric or skewed. If
it is skewed, tell whether it is skewed to the right or to the left.
1.
2.
3.
4.
5.
6.
A vertical line along the median divides the area under a frequency curve in half.
7. Where is the median in a symmetric
distribution?
© Glencoe/McGraw-Hill
8. Where is the median in a skewed
distribution?
620
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
14-5
Study Guide
Sample Sets of Data
In statistics, the word population refers to an entire set of items or
individuals in a group. Rarely will 100% of a population be
accessible as a source of data. Therefore, researchers usually select a
random sample of the population to represent the entire
population. Because discrepancies are common in random samples,
researchers often take many samples and assume that the sample
mean is near its true population mean. The standard deviation of
the distribution of the sample means is known as the standard
error of the mean.
Standard Error
of the Mean
If a sample of data has N values and is the standard deviation, the
standard error of the mean x is
x .
N
Example 1
A sample of data has 4500 values and a standard deviation
of 12. What is the standard error of the mean?
For the sample, N 4500. Find x.
2 , or about 0.179
x 1
N
5
0
4
0
The standard error of the mean for the set of data is
approximately 0.179.
Example 2
The daily calorie consumption of people in the United
States is normally distributed. A team of nutritionists takes
a sample of 250 people and records their daily calorie
consumption. From this sample, the average daily calorie
consumption is 2150 with a standard deviation of 60
calories per day. Determine the interval about the sample
mean that has a 1% level of confidence.
A 1% level of confidence means that there is less than a 1%
chance that the true mean differs from the sample mean by a
certain amount. A 1% level of condifence is given when P 99%.
When P 0.99, t 2.58.
Find x.
x 60 or about 3.795
Find the range.
X
2150 2.58(3.795)
X 2140.2089 and 2159.7911
2
5
0
Thus, the probability is 99% that the true mean is within the
interval of 2140.2089 calories per day to 2159.7911 calories per
day.
© Glencoe/McGraw-Hill
621
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
14-5
Practice
Sample Sets of Data
Find the standard error of the mean for each sample. Then find
the interval about the sample mean that has a 1% level of
confidence and the interval about the sample mean that has a
5% level of confidence.
1. 50, N 100, X 250
2. 4, N 64, X 100
3. 2.6, N 250, X 50
4. 4.3, N 375, X 110
The table below shows a frequency distribution of the time in
minutes required for students to wash a car during a car wash
fundraiser. The distribution is a random sample of 250 cars. Use
the table for Exercises 5-10.
Number of Minutes
5
6
7
8
9
10
Frequency
2
4
5
1
8
5
5. What is the mean of the data in the frequency distribution?
6. Find the standard deviation of the data.
7. Find the standard error of the mean.
8. Find the interval about the sample mean such that the
probability is 0.90 that the true mean lies within the interval.
9. Find the interval about the sample mean such that the
probability is 0.95 that the true mean lies within the interval.
10. Determine the interval about the sample mean that has a
1% level of confidence.
© Glencoe/McGraw-Hill
622
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
14-5
Enrichment
Binomial Expansion Coefficients
The picture at the right shows a device often used to
illustrate a normal probability distribution. The
device is filled with small steel marbles. The marbles
roll past a series of hexagonal obstacles, collecting at
the bottom in each of nine columns.
It can be shown that the number of paths from A to G
is 1, A to J is 1, A to H is 3, and A to I is 3. For
example, H can be reached by the way of E. Hence
the number of paths to H is the sum of the number of
paths to D and the number of paths to E. Likewise
the number of paths to any point can be found by
adding the number of paths to points diagonally
above it. This is precisely the method by which the
numbers in Pascal’s triangle are obtained.
The numbers in Pascal’s triangle are the coefficients
in the expansion of (x y)n where n is any positive
integer. Therefore, the probability of a marble falling
in any given column is proportional to the coefficient
of the corresponding term in the binomial expansion
of a power. The power is a whole number equal to the
row being considered. For example, in the
illustration above the columns are in the eighth row.
Thus the probability of a marble falling in the third
column is proportional to the coefficient of the third
term in the binomial expansion of (x y)8.
The figure above at the right has equally-spaced
vertical segments whose lengths are proportional
to the numbers in the eighth row of Pascal’s
triangle. A smooth curve connecting the tops of
these segments suggests the probable distribution
of marbles in the column. Notice the similarity of
this curve to the normal distribution curve.
Solve.
1. Draw a smooth curve connecting the tops of
the segments whose lengths are
proportional to the coefficients in the
expansion of (x y)12.
2. A teacher decided to mark 64 tests with the grades A, B, C, D, and
F (A highest) in proportion to the coefficients in the
expansion of (x y)4. How many tests received each grade?
© Glencoe/McGraw-Hill
623
Advanced Mathematical Concepts
blank
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 Test, Form 1A
Write the letter for the correct answer in the blank at the
right of each problem.
The playing times of 20 songs on a top-hits radio station are
recorded in the chart below. Use the chart for Exercises 1-6.
Playing Time (minutes:seconds)
3:32
3:24
2:54
3:07
4:52
3:45
2:39
3:09
3:34
3:26
3:35
4:17
4:03
3:52
5:10
4:59
3:07
4:00
3:07
2:56
1. In a stem-and-leaf plot of this data where 332 represents 3:32,
which of the following stems has the fewest number of leaves?
A. 25
B. 30
C. 45
D. 51
1. ________
2. In a frequency distribution of this data, how many data values are in
the class 3:00–3:30?
A. 11
B. 5
C. 6
D. 7
2. ________
3. In a histogram of this data, which bar would have the greatest height?
A. 3:00–3:15
B. 3:15–3:30
C. 3:30–3:45
D. 3:45–4:00
3. ________
4. What is the mean of the data?
A. 3:29
B. 3:48
4. ________
C. 3:67
D. 3:40
5. What is the median of the data?
A. 3:55
B. 3:33
C. 3:20
5. ________
D. 3:07
6. What is the mode of the data?
A. 3:07
B. 3:33
D. None of these
6. ________
C. 3:40
7. Find the value of x so that the mean of 5x, 32 x, x 9, x is 1.
A. 2
B. 12
C. 12
D. 2
7. ________
8. Find the mean of the data
represented by the stem-and-leaf
plot at the right.
A. 2.2
B. 2.5
C. 2.6
D. 2.7
8. ________
stem leaf
1 0 0 1 4 6 8 8
2 2 2 5 6 7 7
3 0 0 1 5
1|0 1.0
For Exercises 9 and 10, use the frequency distribution below.
Amount
0–8
Frequency
Amount
Frequency
18
24–32
11
8–16
15
32–40
9
16–24
16
40–48
6
9. Estimate the mean of the data.
A. 20.07
B. 19.57
C. 20.25
10. Estimate the median of the data.
A. 19
B. 18.75
C. 20
© Glencoe/McGraw-Hill
625
9. ________
D. 18.64
10. ________
D. 18.25
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 Test, Form 1A (continued)
For Exercises 11–13, use the data in the table below.
Average Monthly Temperatures in New Braunfels, Texas (Fahrenheit degrees)
Jan.
Feb.
Mar.
April
May
June
July
Aug.
Sept.
Oct.
Nov.
Dec.
50.7
54.5
61.7
68.7
75.0
81.3
84.0
84.0
79.0
70.0
59.9
52.7
Source: WorldClimate
11. Find the mean deviation of the temperatures.
A. 11.84
B. 12.36
C. 10.47
11. ________
D. 12.15
12. Find the standard deviation of the temperatures.
A. 11.84
B. 12.36
C. 10.47
D. 12.15
12. ________
13. What values are used to create a box-and-whisker plot for the data?
A. 50.7, 54.5, 69, 79, 84
B. 50.7, 57.2, 69.35, 80.15, 84
C. 50.7, 54.5, 68.7, 79, 84
D. 50.7, 54.5, 69.35, 79, 84
13. ________
For Exercises 14-16, a set of 750 values has a normal distribution
with a mean of 12.5 and a standard deviation of 0.36.
14. What percent of the data is between 12.25 and 12.75?
A. 38.3%
B. 51.6%
C. 20.1%
D. 45.1%
14. ________
15. Find the interval about the mean within which 45% of the data lie.
A. 12.28–12.72 B. 12.32–12.68 C. 12.15–12.85 D. 12.49–12.51
15. ________
16. Find the probability that a value selected at random from this
data is between 11.67 and 13.33.
A. 98.4%
B. 57.6%
C. 97.9%
D. 83.0%
16. ________
In a random sample of 30 tires of the same type, it is found that the average
life span of a tire is 36,200 miles with a standard deviation of 3800 miles.
17. Find the standard error of the mean.
17. ________
A. 126.67
B. 693.78
C. 9.53
D. 587.24
18. Find the interval about the sample mean that has a 1% level of
confidence.
A. 26,396–46,004
B. 35,506–36,894
C. 33,425–38,975
D. 34,410–37,990
18. ________
19. Find the interval about the sample mean such that the probability
is 0.75 that the mean number lies within the interval.
A. 35,402–36,998
B. 31,830–40,570
C. 35,436–36,964
D. 35,367–37,033
19. ________
20. Find the probability that the mean of the population will be less
than 560 miles from the mean of the sample.
A. 80.7%
B. 57.6%
C. 1.3%
D. 8.1%
20. ________
Bonus Find the probability that the true mean is between
35,000 and 36,000.
A. 67.5%
B. 33.75%
C. 57.35%
D. 83.8%
© Glencoe/McGraw-Hill
626
Bonus: ________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 Test, Form 1B
Write the letter for the correct answer in the blank at the right of
each problem.
The playing times for 20 movies are recorded in the chart below.
Use the chart for Exercises 1–6.
Playing Time of Movies (minutes)
102
128
123
132
104
95
109
121
108
124
92
140
117
102
124
115
113
89
111
108
1. In a stem-and-leaf plot of this data where 102 represents 102,
which stem has the greatest number of leaves?
A. 9
B. 10
C. 11
D. 12
1. ________
2. In a frequency distribution of this data, how many data values
are in the class 130–140?
A. 0
B. 1
C. 2
D. 3
2. ________
3. In a histogram of this data, which bar would have the greatest height?
A. 105–110
B. 110–115
C. 115–120
D. 120–125
3. ________
4. What is the mean of the data?
A. 112.85
B. 112
4. ________
C. 110.91
D. 124
5. What is the median of the data?
A. 115
B. 113
C. 112
6. What is the mode of the data?
A. 102
B. 108
5. ________
D. 111
6. ________
C. 124
D. All of the above
7. Find the value of x so that the mean of {x, x 2, 2x 1, 1.4x} is 6.
A.
389
B. 5
C.
8. Find the mean of the data
represented by the stem-and-leaf
plot at the right.
A. 85.5
B. 82.45
C. 86.23
D. 86.38
123
7. ________
D. None of these
stem leaf
7 6 7 9
8 0 0 1 3 3 3 5 5 6 6 8
9 0 1 1 1 2 5 6 9
7|6 76
8. ________
For Exercises 9 and 10, use the frequency distribution below.
Amount
Frequency
Amount
Frequency
15–20
21
30–35
10
20–25
46
35–40
7
25–30
13
40–45
3
9. Estimate the mean of the data.
A. 24.75
B. 24.25
C. 25.25
10. Estimate the median of the data.
A. 22.07
B. 23.15
C. 28.15
© Glencoe/McGraw-Hill
627
9. ________
D. 23.25
10. ________
D. 27.93
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 Test, Form 1B (continued)
For Exercises 11–13, use the data in the table below.
Numbers of Cars Rented Each Month in the U.S. (thousands)
Jan.
Feb.
Mar.
April
May
June
July
Aug.
Sept.
Oct.
Nov.
Dec.
0.7
0.6
0.9
1.2
1.3
1.5
1.8
1.7
1.1
0.9
0.9
1.2
11. Find the mean deviation of the data.
A. 0.5
B. 0.4
C. 0.3
11. ________
D. 0.2
12. Find the standard deviation of the data.
A. 0.36
B. 0.43
C. 0.41
D. 0.38
12. ________
13. What values are used to create a box-and-whisker plot for the data?
A. 0.6, 0.9, 1.3, 1.3, 1.8
B. 0.6, 0.9, 1.15, 1.4, 1.8
C. 0.6, 0.9, 1.2, 1.5, 1.8
D. 0.6, 0.9, 1.15, 1.5, 1.8
For Exercises 14–16, a set of 300 values has a normal distribution
with a mean of 50 and a standard deviation of 5.
14. What percent of the data is between 45 and 55?
A. 38.3%
B. 50%
C. 68.3%
D. 95.5%
13. ________
14. ________
15. Find the interval about the mean within which 90% of the data lie.
A. 40–60
B. 42.5–57.5
C. 38.75–61.25 D. 41.75–58.25
15. ________
16. Find the probability that a value selected at random from this
data is between 49.5 and 50.5.
A. 8%
B. 9.2%
C. 9.8%
D. 7.66%
16. ________
In a random sample of 700 refreshment-dispensing machines, it is
found that an average of 8.1 ounces is dispensed with a standard
deviation of 0.75 ounce.
17. Find the standard error of the mean.
A. 0.0107
B. 0.0011
C. 0.2833
D. 0.0283
17. ________
18. Find the interval about the sample mean that has a 1% level
of confidence.
A. 8.045–8.155
B. 8.027–8.173
C. 6.165–10.035
D. 8.053–8.147
18. ________
19. Find the interval about the sample mean such that the probability
is 0.90 that the mean number lies within the interval.
A. 8.045–8.155
B. 8.027–8.173
C. 6.165–10.035
D. 8.053–8.147
19. ________
20. Find the probability that the mean of the population will be less
than 0.085 ounce from the mean of the sample.
A. 30%
B. 99.7%
C. 49.9%
D. 25%
20. ________
Bonus Find the probability that the true mean is between
8.157 and 8.185.
A. 2.1%
B. 4.45%
C. 0.4%
D. 0.2%
© Glencoe/McGraw-Hill
628
Bonus: ________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 Test, Form 1C
Write the letter for the correct answer in the blank at the right of each problem.
The number of cars sold by 20 salespeople in one week are recorded
in the chart below. Use the chart for Exercises 1-6.
Number of Cars
10
7
6
9
7
3
5
6
8
4
8
2
7
5
7
9
11
5
7
10
1. In a frequency distribution of this data, how many data values are
in the class 4-6?
A. 3
B. 7
C. 4
D. 5
1. ________
2. In a histogram of this data, which bar would have the greatest height?
A. 4–6
B. 6–8
C. 8–10
D. 10–12
2. ________
3. In a histogram of this data, which bar would have the least height?
A. 2–4
B. 4–6
C. 6–8
D. 8–10
3. ________
4. What is the mean of the data?
A. 6.5
B. 6.8
4. ________
C. 7
D. 7.2
5. What is the median of the data?
A. 6.8
B. 6
C. 7
5. ________
D. 7.5
6. What is the mode of the data?
A. 5
B. 8
D. 7
6. ________
C. 6
7. Find the value of x so that the mean of {x, 2x, 3x, 4x} is 5.
D. 2.5
A. 2
B. 2
C. 12
7. ________
8. Find the mean of the data
represented by the stem-and-leaf
plot at the right.
A. 194
B. 193.8
C. 193
D. 194.2
8. ________
stem leaf
18 5 9
19 1 3 3 5 6 8 8
20 0
18|5 185
For Exercises 9 and 10, use the frequency distribution below.
Amount
Frequency
Amount
Frequency
$0–$10
15
$30–$40
28
$10–$20
21
$40–$50
20
$20–$30
26
$50–$60
12
9. Estimate the mean of the data.
A. $29.34
B. $32.21
C. $34.34
10. Estimate the median of the data.
A. $25
B. $29.62
C. $45
© Glencoe/McGraw-Hill
629
9. ________
D. $59.62
10. ________
D. $39.62
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 Test, Form 1C (continued)
For Exercises 11-13, use the data in the table below.
Average Rainfall in China Flat, California (inches)
Jan.
Feb.
Mar.
April
May
June
July
8.8
6.6
5.9
3.0
2.5
0.9
0.1
Aug. Sept.
0.0
0.6
Oct.
Nov.
Dec.
3.4
5.9
9.0
Source: WorldClimate
11. Find the mean deviation of the data.
A. 3.20
B. 3.89
C. 3.13
11. ________
D. 2.79
12. Find the standard deviation of the data.
A. 3.20
B. 3.89
C. 3.13
D. 2.79
12. ________
13. What values are used to create a box-and-whisker plot for the data?
A. 0, 0.6, 3, 5.9, 9
B. 0, 0.9, 3.2, 6.3, 9
C. 0, 0.75, 3.2, 6.25, 9
D. 0, 0.6, 3.2, 5.9, 9
For Exercises 14-16, a set of data has a normal distribution with
a mean of 120 and a standard deviation of 10.
14. What percent of the data is between 110 and 130?
A. 38.3%
B. 50%
C. 68.3%
D. 95.5%
13. ________
14. ________
15. Find the interval about the mean within which 90% of the data lie.
A. 94.2–145.8
B. 103.5–136.5
C. 113.68–126.32
D. 100.4–139.6
15. ________
16. Find the probability that a value selected at random from this data
is between 100 and 140.
A. 99.9%
B. 90%
C. 99%
D. 95.5%
16. ________
In a random sample of 1000 exams, the average score was
500 points with a standard deviation of 80 points.
17. Find the standard error of the mean.
A. 15.81
B. 2.53
C. 3.58
D. 8
17. ________
18. Find the interval about the sample mean that has a 1% level of
confidence.
A. 499–501
B. 479–520
C. 495–505
D. 493–507
18. ________
19. Find the interval about the sample mean such that the probability
is 0.90 that the mean number lies within the interval.
A. 499–501
B. 495–505
C. 496–504
D. 368–632
19. ________
20. Find the probability that the mean score of the population will be
less than five points from the mean score of the sample.
A. 95.5%
B. 38.3%
C. 98.8%
D. 62.5%
20. ________
Bonus Find the probability that the true mean is between
495 and 500.
A. 95%
B. 47.75%
C. 95.5%
D. 68.3%
© Glencoe/McGraw-Hill
630
Bonus: ________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 Test, Form 2A
Fifty randomly–selected people going to a science fiction movie
were asked their age. The results are recorded in the chart below.
Use the chart for Exercises 1-6.
Ages of Science-Fiction Moviegoers (years)
17
42
21
78
16
21
31
29
29
16
49
19
81
16
69
69
18
31
22
14
21
75
42
78
18
41
22
16
18
80
42
42
42
16
16
21
19
18
44
18
22
14
49
17
16
18
18
18
17
23
1. List the stems that would be used in a stem-and-leaf plot of
the data.
1. __________________
2. Find the range of the data.
2. __________________
3. Make a histogram of the data.
3.
4. Find the mean of the data.
4. __________________
5. Find the median of the data.
5. __________________
6. Find the mode of the data.
6. __________________
7. Find the value of x so that the mean of
{2x, 13 x, 52 x 3, x 2} is 8.
7. __________________
8. Find the mean of the data below.
stem leaf
51 2 5 6 7 9
52 0 2 3 5 8 8 9 9
53 1 1 4
8. __________________
51|2 5120
For Exercises 9 and 10, use the frequency distribution below.
Class
Frequency
Class
Frequency
600–615
2
675–690
2
615–630
8
690–705
38
630–645
16
705–720
24
645–660
3
720–735
7
660–675
5
735–750
3
9. Estimate the mean of the data.
9. __________________
10. Estimate the median of the data.
10. __________________
© Glencoe/McGraw-Hill
631
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 Test, Form 2A (continued)
The table below gives the save percentages for some goalies
at a certain point in a recent NHL season. Use the table for
Exercises 11–13.
Save Percentages
0.940
0.937
0.930
0.929
0.926
0.925
0.925
0.923
0.923
0.923
0.921
0.920
0.919
0.918
0.916
0.914
0.911
0.911
0.910
0.910
0.909
0.907
0.907
0.904
0.903
0.903
0.902
0.900
0.898
0.895
11. Find the mean deviation of the percentages.
11. __________________
12. Find the standard deviation of the percentages.
12. __________________
13. Make a box-and-whisker plot of the percentages.
13.
For Exercises 14 –16, a set of 1000 values has a normal
distribution with a mean of 400 and a standard deviation
of 30.
14. What percent of the data is between 385 and 415?
14. __________________
15. Find the interval about the mean within which 60% of
the data lie.
15. __________________
16. Find the probability that a value selected at random from
this data is greater than 350.
16. __________________
A random sample of 225 homes showed an average of
5.2 clocks in each home. The standard deviation was 0.8.
17. Find the standard error of the mean.
17. __________________
18. Find the interval about the sample mean that has a
1% level of confidence.
18. __________________
19. Find the interval about the sample mean such that the
probability is 0.75 that the mean number lies within
the interval.
19. __________________
20. Find the probability that the mean of the population will
be less than 0.10 from the mean of the sample.
20. __________________
Bonus Find the probability that the true mean is
between 5 and 5.1.
© Glencoe/McGraw-Hill
632
Bonus: __________________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 Test, Form 2B
Fifty students recorded the number of hours that the television
was on in their homes during one week. The results are given in
the chart below. Use the chart for Exercises 1–6.
Weekly Television Hours (to the nearest hour)
54
28
9
15
3
54
35
32
0
34
72
57
62
33
58
23
57
53
24
27
36
63
34
58
53
13
12
75
66
57
18
53
53
46
77
26
32
42
43
88
44
71
22
57
45
73
44
11
45
34
1. List the leaves for stem 1 in a stem-and-leaf plot of the data. 1. __________________
2. List the class marks for the intervals 0–20, 20–40, 40–60,
60–80, and 80–100 in a frequency distribution of the data.
2. __________________
3. Make a histogram of the data.
3.
4. Find the mean of the data.
4. __________________
5. Find the median of the data.
5. __________________
6. Find the mode of the data.
6. __________________
7. Find the value of x so that the mean of
{3x 3, x 5, 3x, 2x 7} is 9.
7. __________________
8. Find the mean of the data below.
stem leaf
0 4 7 8 9 9
1 0 1 2 3 3 4 5 6 6 8 9
2 1 2 5 8
0|4 4
8. __________________
For Exercises 9 and 10, use the frequency distribution below.
Class
Frequency
Class
Frequency
325–375
8
575625
10
375–425
10
625675
6
425–475
30
675725
2
475–525
20
725775
1
525–575
12
775825
1
9. Estimate the mean of the data.
9. __________________
10. Estimate the median of the data.
10. __________________
© Glencoe/McGraw-Hill
633
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 Test, Form 2B (continued)
The table below gives the percent pay raise for 13 employees. Use
the table for Exercises 11–13.
Percent Pay Raise
3.2%
4.4%
4.1%
3.8%
1.5%
2.4%
3.3%
1.7%
9.2%
4.5%
4.2%
5.1%
4.6%
11. Find the mean deviation of the percentages.
11. __________________
12. Find the standard deviation of the percentages.
12. __________________
13. Make a box-and-whisker plot of the percentages.
13.
For Exercises 14-16, a set of data has a normal distribution
with a mean of 120 and a standard deviation of 10.
14. What percent of the data is between 100 and 140?
14. __________________
15. Find the interval about the mean within which 80% of
the data lie.
15. __________________
16. Find the probability that a value selected at random
from this data is between 105 and 135.
16. __________________
In a random sample of 256 people, it was found that each person
ate fast food an average of 2.6 times per week with a standard
deviation of 0.4.
17. Find the standard error of the mean.
17. __________________
18. Find the interval about the sample mean that has a 1%
level of confidence.
18. __________________
19. Find the interval about the sample mean such that the
probability is 0.80 that the mean number lies within the
interval.
19. __________________
20. Find the probability that the mean of the population
will be less than 0.1 from the mean of the sample.
20. __________________
Bonus Find the probability that the true mean is
between 2.55 and 2.575.
© Glencoe/McGraw-Hill
634
Bonus: __________________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 Test, Form 2C
The speeds of 50 cars in a 70 mile-per-hour zone are recorded
in the chart below. Use the chart for Exercises 1–6.
Speed (miles per hour)
66
69
70
67
74
88
71
65
68
73
69
72
67
69
68
68
68
58
69
67
72
65
72
59
63
73
72
65
66
67
58
63
73
66
77
66
62
52
63
81
64
71
72
67
65
73
64
71
65
64
1. List the stems that would be used in a stem-and-leaf plot
of the data.
1. __________________
2. List the class marks for the intervals 50–60, 60–70, 70–80,
and 80–90 in a frequency distribution of the data.
2. __________________
3. Make a histogram of the data.
3.
4. Find the mean of the data.
4. __________________
5. Find the median of the data.
5. __________________
6. Find the mode of the data.
6. __________________
7. Find the value of x so that the
mean of {x, 3x, 2x 1, 2x 5} is 15.
7. __________________
8. Find the mean of the data below.
stem leaf
6 4 5 5 5 6 7 8 8 8 8 9
7 0 0 0 1 1 1 2 2 2 2 3 7
8 0 2
6|4 6.4
8. __________________
For Exercises 9 and 10, use the frequency distribution below.
Class
Frequency
Class
Frequency
0-10
3
50-60
7
10-20
6
60-70
5
20-30
8
70-80
4
30-40
7
80-90
1
40-50
6
90-100
3
9. Estimate the mean of the data.
9. __________________
10. Estimate the median of the data.
10. __________________
© Glencoe/McGraw-Hill
635
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 Test, Form 2C (continued)
The table below gives the number of traffic tickets issued
per day over a 20-day period by a police officer. Use the table
for Exercises 11–13.
Number of Traffic Tickets Issued
7
12
10
8
7
12
15
10
10
7
14
6
10
12
9
8
8
2
9
12
11. Find the mean deviation of the data.
11. __________________
12. Find the standard deviation of the data.
12. __________________
13. Make a box-and-whisker plot of the data.
13.
For Exercises 14–16, a set of data has a normal distribution with a
mean of 8 and a standard deviation of 1.4.
14. What percent of the data is between 7 and 9?
14. __________________
15. Find the interval about the mean within which 90% of
the data lie.
15. __________________
16. Find the probability that a value selected at random from
this data is between 7.3 and 8.7.
16. __________________
In a random sample of 100 band students, it was found that each
student practiced an average of 10.5 hours per week with a
standard deviation of 1.4 hours.
17. Find the standard error of the mean.
17. __________________
18. Find the interval about the sample mean that has a
5% level of confidence.
18. __________________
19. Find the interval about the sample mean such that the
probability is 0.90 that the mean number lies within
the interval.
19. __________________
20. Find the probability that the mean of the population will
be less than 15 minutes from the mean of the sample.
20. __________________
Bonus Find the probability that the true mean is
between 10.2 and 10.6 hours.
© Glencoe/McGraw-Hill
636
Bonus: __________________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 Open-Ended Assessment
Instructions: Demonstrate your knowledge by giving a clear,
concise solution to each problem. Be sure to include all
relevant drawings and justify your answers. You may show
your solution in more than one way or investigate beyond
the requirements of the problem.
1. The table and graph below show the distribution of grades for an English test.
English Scores
Class
Limits
Class
Mark
Frequency
62.5–67.5
65
1
67.5–72.5
70
2
72.5–77.5
75
3
77.5–82.5
80
5
82.5–87.5
85
7
87.5–92.5
90
9
92.5–97.5
95
7
97.5–102.5
100
3
a. Are the grades on the test
normally distributed? Why or why not?
b. Which measure of central tendency (mean, median, or mode)
will be the greatest? Why?
c. Which measure of central tendency will be the least? Why?
2. The combined test scores for all of the advanced mathematics classes in a school are
normally distributed. The mean score is 85, and the standard deviation is 10. There
are 200 students in the classes.
a. Those who had scores above 100 were given a grade of A. How many students
received an A? Explain your reasoning.
b. What are the mode and median for the set of scores? How do you know?
c. Those who had scores between 80 and 90 were given a grade of C. How many
students received a C? Explain your reasoning.
d. If the teacher changes the range for the grade of C to scores from 75 to 85, will
there be an increase or decrease in the number of C grades? Explain your
reasoning.
3. Consider the statement “Given two sets of data, the mean of the combination of the
two sets equals the mean of the means.” Provide an example that disproves the
statement. In what situation is the statement true?
© Glencoe/McGraw-Hill
637
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 Mid-Chapter Test (Lessons 14-1 through 14-3)
The amount of money 13 families spent on food in one year
is recorded in the table below. Use the table for Exercises 1–7.
Food Expenses (thousands of dollars)
4.5
5.2
6.5
2.9
2.7
4.6
3.9
6.0
4.7
4.2
5.2
4.6
7.2
1. Organize the data into a frequency distribution.
1. __________________
2. List the stems that would be used in a stem-and-leaf
plot of the data.
2. __________________
3. Make a box-and-whisker plot of the data.
3.
4. Find the mean of the data.
4. __________________
5. Find the mode of the data.
5. __________________
6. What is the mean deviation for the data?
6. __________________
7. What is the standard deviation for the data?
7. __________________
The weights of baseball players on the Chicago White Sox 1999 roster
are recorded in the table below. Use the table for Exercises 8–10.
Weight (in pounds)
Frequency
160–180
6
180–200
13
200–220
8
220–240
3
240–260
2
Source: Yahoo! Sports
8. Estimate the mean weight of the players.
8. __________________
9. Estimate the median weight of the players.
9. __________________
10. Estimate the standard deviation of the weights.
© Glencoe/McGraw-Hill
638
10. __________________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14, Quiz A
(Lessons 14-1 and 14-2)
Use the data in the table below for each exercise.
28
16
37
31
21
26
35
29
12
24
28
34
32
26
19
25
35
31
28
26
19
22
30
24
19
29
33
1. Find the range and determine an appropriate class interval. 1. __________________
2. Find the class marks.
2. __________________
3. Draw a histogram of the data.
3.
4. Draw a stem-and-leaf plot of the data.
4.
5. Find the mean, median, and mode of the data.
5. __________________
stem
leaf
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
Chapter 14, Quiz B (Lesson 14-3)
14
Use the data in the table below for Exercises 1-14.
Heights of Women Basketball Players on the Houston Comets 1999 Roster (inches)
71
70
67
74
77
75
66
72
74
76
73
79
Source: WNBA
1. Find the mean deviation of the heights.
1. __________________
2. Find the interquartile range for the data.
2. __________________
3. Find the semi-interquartile range for the data.
3. __________________
4. What is the standard deviation of the heights?
4. __________________
5. Make a box-and-whisker plot of the data.
5.
© Glencoe/McGraw-Hill
639
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14, Quiz C (Lesson 14-4)
A set of 900 values has a normal distribution with a mean
of 150 and a standard deviation of 8.
1. What percent of the data is between 134 and 166?
1. __________________
2. How many values are within one standard deviation of
the mean?
2. __________________
3. How many values fall in the interval between one and two
standard deviations of the mean?
3. __________________
4. Find the probability that a value selected at random from
the data will be greater than 162.
4. __________________
5. Find the interval about the mean that includes 90%
of the data.
5. __________________
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14, Quiz D (Lesson 14-5)
In a sample of 100 adults, the average time each adult kept
a car was 6.2 years. The standard deviation was 1.1 years.
1. Find the standard error of the mean.
1. __________________
2. Find the interval about the sample mean that has a
5% level of confidence.
2. __________________
3. Find the interval about the sample mean that has a
1% level of confidence.
3. __________________
4. Find the probability that the mean of the population will
be less than 0.1 year from the mean of the sample.
4. __________________
5. Find the probability that the mean of the population will
be less than 0.5 year from the mean of the sample.
5. __________________
© Glencoe/McGraw-Hill
640
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 SAT and ACT Practice
After working each problem, record the
correct answer on the answer sheet
provided or use your own paper.
Multiple Choice
1. Use the graph below to determine how
many more items were sold in January
than in May.
A 5250
B 3500
C 1750
D 4250
E 1000
5. Of the 50 students in a class, exactly
30 are women. What percent of the
students are men?
A 20%
B 30%
C 40%
D 50%
E 60%
6. Five is what percent of 2?
A 2.5%
B 25%
C 40%
D 250%
E 400%
7.
2. In the graph below, which of the
following could be the percent change
from the number of widgets built in
1985 to the number built in 1990?
Long Distance Rates
From
City A
First
Minute
Each
Additional
Minute
to City B
$0.55
$0.15
to City C
$0.25
$0.05
Based on the table above, what is the
cost of a 30-minute call from City B to
City C?
A $4.90
B $5.05
C $7.30
D $7.55
E It cannot be determined from the
information given.
A
B
C
D
E
150% decrease
67% decrease
60% decrease
67% increase
300% increase
8.
Attendance
Age
Number of
Students
14
15
16
17
5
6
10
10
Based on the table above, what is the
mean age of the students?
A 14.6
B 15.0
C 15.8
D 16.5
E 16.8
3. If 5 more than x is 2 less than y, what
is y in terms of x?
A x3
B y7
C y3
D x7
E x7
9. Point A is on line m. If two points, B
and C, are each placed to the right of
point A so that AB 2AC, what will
AC
?
be the value of BC
4. If 5x 3y 23 and x and y are
positive integers, then y can equal
which of the following?
A 3
B 4
C 5
D 6
E 7
A 1
B 2
1
2
D 14
C
E It cannot be determined from the
information given.
© Glencoe/McGraw-Hill
641
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14 SAT and ACT Practice
10. In ABC below, which of the following
could be a value of y?
A 2
B 4
C 6
D 8
E 12
11. Mosin’s monthly expenses are $1375
per month and are distributed as
shown in the pie chart below. How
much does he spend for “other”
expenses each month?
A $206.25
B $137.50
C $68.75
D $13.75
E $10
17–18. Quantitative Comparison
A if the quantity in Column A is
greater
B if the quantity in Column B is
greater
C if the two quantities are equal
D if the relationship cannot be
determined from the information
given
Column A
17.
Mean of 101,
202, and 303
Column B
Mean of 101, 202,
303, and 10
18. Set X {1, 2, 2, 3, 3, 3, 4, 4, 4, 4}
13. Which of the following ratios is equal
to the ratio of 14 to 4?
B 4 to 14
A 13 to 3
Mode of Set X
C 18 to 2
D 12 to 2
E None of these
14. A car traveling 60 miles per hour for 30
minutes covers the same distance
as a car traveling 20 miles per hour for
how many hours?
B 1
A 23
© Glencoe/McGraw-Hill
15. Use the chart below to determine the
number of boxes of cabbage sold by a
farmer in 1998.
A 1000
B 4000
C 8000
D 10,000
E 40,000
16. Use the chart above to determine how
many more boxes of cabbage were sold
in 1999 than in 2000.
A 250
B 500
C 5000
D 1000
E 2500
12. If Mosin’s monthly expenses are $1225
per month and are distributed as
shown in the pie chart above, how
much more does he spend on rent than
on food?
A $68.75
B $72.50
C $7.25
D $61.25
E $612.50
C 112
E 13
(continued)
D 3
Median of Set X
19–20. In a class of 250 students, 4 are
running for the position of class
president. Every student in the
class voted exactly once. The votes
were distributed as shown below.
Candidate Number of Votes
D.J.
75
Belinda
20
Darius
45
Lou Ann
x
19. Grid-In How many votes did Lou Ann
get?
20. Grid-In What was the median
number of votes per candidate?
642
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
14
Chapter 14, Cumulative Review (Chapters 1-14)
1. Write the ordered triple that represents CD for C(5, 0, 1)
and D(3, 2, 6).
1. __________________
2. Write an equation of the cosine function with amplitude 6,
phase shift 0, vertical shift 0, and period 12.
2. __________________
3. Write the polynomial equation of least degree with roots
7i and 7i.
3. __________________
4. Find the area of A BC to the nearest tenth if c 11.4,
B 31.6 and C 120.3.
4. __________________
3
5. Find 8i.
5. __________________
6. Write in general form the equation of a parabola whose
focus is at (5, 1) and whose directrix is x 3.
6. __________________
1 3 in exponential form.
7. Write log5 125
7. __________________
8. Find the first three iterates of the function
ƒ(z) z2 z i if z0 i.
8. __________________
9. The serial number for a product is formed from the
digits 1, 2, and 3 and the letters A and B. No letters
or numbers are repeated. What is the probability
that the serial number ends in 2A given that it
ends in a letter?
9. __________________
10. A set of data has a normal distribution with a mean
of 16 and a standard deviation of 0.3. What percent
of the data is in the interval 15.216?
10. __________________
© Glencoe/McGraw-Hill
643
Advanced Mathematical Concepts
Blank
NAME _____________________________ DATE _______________ PERIOD ________
UNIT
4
Unit 4 Review, Chapters 12-14
Solve.
1. Find the 20th term in the arithmetic
sequence for which a1 3 and d 2.
2. Find the sum of the first nine terms of
the geometric series 2 4 8 . . . .
3. Write an arithmetic sequence that
has two arithmetic means
between 3 and 9.
24. ƒ(x) 1x, x0 2
Find the first three iterates of the
function ƒ(z) 2z 3i for each initial
value.
26. z0 3 i
25. z0 i
Find each limit or state that the limit
does not exist.
n 1
6. lim n
7. lim 42n
8. lim n2
2
(4n 5)(n 3)
Find the sum of each series or state that
the sum does not exist.
9. 1 1 1 . . .
10.
2
4
1
1 4 11
6
...
Determine whether each series is
convergent or divergent.
11. 1 4 7 10 . . .
12.
13.
Write each expression in expanded form
and then find the sum.
14.
2k
15.
(3a 6)
a2
17.
k1
8
16.
Use mathematical induction to prove
that each proposition is valid for all
positive integral values of n.
27. 3 9 15 . . . (6n 3) 3n2
28. 5 n 1 is divisible by 4
Find each value.
29. P(9, 6)
30. P(7, 4)
31. C(8, 2)
32. C(6, 5)
Solve.
33. The letters a, b, c, d, and e are to be
used to form 5-letter patterns. How
many patterns can be formed if
repetitions are not allowed?
6 2 23 29 . . .
31 32 33 . . .
1
2
3
5
Find the first three iterates of each
function using the given initial value. If
necessary, round your answers to the
nearest hundredth.
22. ƒ(x) 3x 1, x0 1
23. ƒ(x) x2 5, x0 2
4. Find the sixth term of the geometric
sequence 13, 145, 1765, . . . .
4n 1
5. lim 3n
Use the first five terms of the exponential
series and a calculator to approximate
each value to the nearest hundredth.
21. e0.3
20. e1.72
7
34. How many different 5-member teams
can be formed from 10 players?
k1
6
35. How many different ways can the letters of the word color be arranged?
(2k2 1)
312 k
k0
36. From a group of 3 men and 5 women,
how many different committees of 2
men and 2 women can be formed?
Find the designated term of each
binomial expansion.
37. How many different ways can 8 keys
be arranged on a circular key ring?
18. 4th term of (3x 1)9
19. 7th term of (x 2y)12
© Glencoe/McGraw-Hill
645
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
UNIT
4
Unit 4 Review, Chapters 12-14 (continued)
State the odds of an event occurring,
given the probability of the event.
1
39. 38. 27
14
40. 121
56. A population is normally distributed
with a mean of 60 and a standard
deviation of 5. What is the probability
that a randomly selected value will be
greater than 65?
57. A set of data is normally distributed
with a mean of 70 and a standard
deviation of 6. What is the probability
that a randomly selected value lies
between 65 and 75?
3
41. 13
Solve.
42. Three cards are drawn at random
from a standard deck of 52 cards.
What is the probability that all three
are clubs?
43. Find the probability of getting a sum
of 6 on the first throw of two number
cubes and a sum of 2 on the second
throw.
44. Find the probability of getting a sum
of 7 or 9 on a single throw of two
number cubes.
45. One card is drawn from a standard
deck of 52 cards. What is the
probability that it is a king if it
is known to be a face card?
Solve.
58. Find the interval about the mean
within which 77% of the values of a
set of normally distributed data can
67 and 3.2.
be found if X
59. Find the interval about the mean
within which 16% of the values of a
set of normally distributed data can
0.25 and 0.12.
be found if X
Renee is a forward on her school’s
soccer team. The probability of her
making a goal is 14. Find each probability
if Renee makes 5 attempts on the goal.
46. P(3 goals)
47. P(at least 2 goals)
The test scores for Ms. Humphrey’s
humanities class are listed below.
89 95
77 99
65
65
70
89
77
72
82
80
66
42
69
76
91
86
82
77
48.
49.
50.
51.
52.
53.
Find the standard error of the mean for
each sample.
60. 8, N 100
61. 3.23, N 30
62. 5, N 38
63. 12.3, N 89
Solve.
64. A set of data of size N 100 is normally
distributed with a mean of 50 and a
standard deviation of 5. Determine the
interval about the sample mean that
has a 1% level of confidence.
Find the range of the data.
Find the mean of the data.
Find the median of the data.
Find the mode of the data.
Find the mean deviation of the data.
Find the semi-interquartile range of
the data.
54. Find the standard deviation of the data.
65. A set of data of size N 30 is
normally distributed with a mean of
100 and a standard deviation of 3.3.
Determine the interval about the
sample mean that has a 1% level
of confidence.
Solve.
55. A set of data is normally distributed
with a mean of 30 and a standard
deviation of 7. What percent of the
data is between 9 and 51?
66. A set of data of size N 36 is
normally distributed with a mean of
20 and a standard deviation of 4.2.
Determine the interval about the
sample mean that has 5% level
of confidence.
© Glencoe/McGraw-Hill
646
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
UNIT
4
Unit 4 Test, Chapters 12-14
1. Express 23 29 35 41 47 using sigma notation.
1. __________________
2. A set of data is normally distributed with a mean of 16 and
a standard deviation of 0.3. What percent of the data is
between 15.2 and 16?
2. __________________
3. Use mathematical induction to prove that
n(n 1)(2n
1) for
12 22 32 . . . n2 6
all positive integral values of n.
3. __________________
, 1, 1, . . . .
4. Find the twelfth term of the geometric sequence 11
6 8 4
4. __________________
5. There are 21 wrapped packages in a grab bag at an office
holiday party. Five of the packages contain $20 bills,
7 packages contain $5 bills, and 9 packages contain $1 bills.
What is the probability that the first two people will
choose packages with $20 bills inside?
5. __________________
....
6. Find the sum of the infinite series 1 15 21
5
6. __________________
7. Find P(5, 2).
7. __________________
8. True or false: Choosing an entrée and choosing an appetizer
from a dinner menu are independent events.
8. __________________
9. How many ways can 9 keys be arranged on a circular
key ring?
9. __________________
10. Determine whether the series 1 215 315 415 . . .
is convergent or divergent.
10. __________________
11. What is the probability of getting an even number on a
single roll of a number cube if you roll a 3 or greater?
11. __________________
12. How many eight-letter patterns can be formed from the
letters of the word circular?
12. __________________
© Glencoe/McGraw-Hill
647
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
UNIT
4
Unit 4 Test, Chapters 12-14 (continued)
13. Use the ratio test to determine whether the series
1 12 13 . . . is convergent or divergent.
3
3
3
13. __________________
14. Find the first three iterates of the function
ƒ(z) z2 z i, if the initial value is i.
14. __________________
15. The probability of being named captain of the basketball
team is 15. What are the odds of being named captain of
the team?
15. __________________
16. Eight out of ten people surveyed prefer to observe Veterans 16. __________________
Day on November 11 rather than on the second Monday of
November. Use the Binomial Theorem to determine the
probability that each of the first three people surveyed
prefer to observe Veterans Day on November 11.
17. Use the Binomial Theorem to expand (x 2y) 3.
17. __________________
18. Find the probability of tossing 2 heads on 3 tosses of a
fair coin.
18. __________________
19. Find n for the arithmetic sequence for which an 129,
a1 15, and d 6.
19. __________________
20. The set of class marks in a frequency distribution is
{25.5, 35.5, 45.5, 55.5}. Find the class interval and
the class limits.
20. __________________
21. Use the first five terms of the exponential series and a
calculator to approximate the value of e0.67 to the
nearest hundredth.
21. __________________
22. Find the first three iterates of the function ƒ(x) x2 2,
if the initial value is 2.
22. __________________
n2 1 , or state that the limit does not exist.
23. Find lim n
23. __________________
n→
© Glencoe/McGraw-Hill
648
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
UNIT
4
Unit 4 Test, Chapters 12-14 (continued)
24. Twenty slips of paper are numbered 1 to 20 and placed
24. __________________
in a box. What is the probability of drawing a number that is
odd or a multiple of 5?
25. Find C(10, 4).
25. __________________
The table below shows the number of gallons of heating oil delivered
to a residential customer each December from 1986 to 1993. Use
the table for Exercises 26-28.
Year
Gallons of Oil
1986
1987
1988
1989
1990
1991
1992
1993
42
61
53
59
53
51
75
100
26. Make a stem-and-leaf plot of the data.
26. __________________
27. Find the mean, median, and mode of the data.
27. __________________
28. Find the interquartile range and the semi-interquartile
range of the data.
28. __________________
29. Two cards are drawn at random from a standard deck of
52 cards. What is the probability that both cards
are queens?
29. __________________
6
as a fraction.
30. Write 0.3
30. __________________
31. Find the 50th term in the arithmetic sequence 1, 5, 9, . . . .
31. __________________
32. A set of data is normally distributed with a mean
of 50 points and a standard deviation of 10 points. What
percent of the data is greater than 60?
32. __________________
© Glencoe/McGraw-Hill
649
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
UNIT
4
Unit 4 Test, Chapters 12-14 (continued)
33. A red number cube and a white number cube are rolled.
Find the probability that the red number cube shows a 2,
given that the sum showing on the two number cubes
is less than or equal to 5.
33. __________________
34. Find the standard error of the mean for a sample in which 34. __________________
3.6, N 100, and X 36. Then use t 2.58 to find the
interval about the sample mean that has a 1% level of
confidence. Round your answer to the nearest hundredth.
The table below shows a frequency distribution of the number
of pickup trucks sold at 85 truck dealerships in Maine over an
18-month period.
Number of
Trucks Sold
7090
90110
110130
130150
150170
170190
Number of
Dealerships
2
11
39
17
9
7
35. Find the interval about the sample mean such that the
probability is 0.90 that the true mean lies within the
interval. (When P 90%, t 1.65.)
35. __________________
A set of data is normally distributed with a mean of 500
and a standard deviation of 40.
36. What percent of the data is between 460 and 540?
36. __________________
37. Find the probability that a value selected at random is
less than 420.
37. __________________
Suppose that the respondents in a survey of 100 teenagers watch
an average of 10 hours of television per week. The standard
deviation of the sample is 2.5 hours.
38. Find the standard error of the mean.
38. __________________
39. What is the interval about the sample mean that has a
1% level of confidence?
39. __________________
40. Find the interval about the sample mean that gives a 90%
chance that the true mean lies within the interval.
40. __________________
© Glencoe/McGraw-Hill
650
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
SAT and ACT Practice Answer Sheet
(10 Questions)
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
© Glencoe/McGraw-Hill
A1
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
SAT and ACT Practice Answer Sheet
(20 Questions)
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
© Glencoe/McGraw-Hill
A2
Advanced Mathematical Concepts
© Glencoe/McGraw-Hill
6.2
A3
9.5
Phoenix
New Orleans
Memphis
Las Vegas
Honolulu
Boston
Anchorage
Station
6.2
8.1
8.8
9.3
11.3
12.5
7.1
Seattle
New York City
Miami
Little Rock
Indianapolis
Chicago
Atlanta
Station
9.0
9.4
9.2
7.8
9.6
10.4
9.1
Speed
(mph)
© Glencoe/McGraw-Hill
Wind Speed (mph)
6.0–7.0
7.0–8.0
8.0–9.0
9.0–10.0
10.0–11.0
11.0–12.0
12.0–13.0
610
Tallies Frequency
兩兩
2
兩兩
2
兩兩兩
3
兩兩兩兩 兩兩兩
8
兩兩兩兩
4
兩
1
兩
1
e. Draw a histogram of the data.
d. Construct a frequency distribution of the data.
e.
Advanced Mathematical Concepts
Sample answer: 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0;
6.5, 7.5, 8.5, 9.5, 10.5, 11.5, 12.5
c. What are the class limits and the class marks?
Sample answer: 1
Speed
(mph)
b. Determine an appropriate class interval.
a. Find the range of the data. 6.3
Source: National Climatic Data Center
Philadelphia
10.5
Los Angeles
Minneapolis– St. Paul
10.8
10.7
Kansas City
9.1
Dallas-Ft. Worth
8.9
Baltimore
Speed
(mph)
Albuquerque
Station
3. Meteorology The average wind speeds recorded at various weather
stations in the United States are listed below.
1. 25, 32, 18, 99, 43, 16, 29, 35, 36, 34, 2. 111, 115, 130, 200, 234, 98, 115, 72
21, 33, 26, 26, 17, 40, 22, 38, 16, 19
305, 145, 87, 63, 245, 285, 256, 302
a. 25 yes; 10 classes
a. 1 no; too many classes
b. 10 yes; 8 classes
b. 10 no; too many classes
c. 2 no; too many classes
c. 30 yes; 9 classes
© Glencoe/McGraw-Hill
611
Advanced Mathematical Concepts
3. If 10,000 cars were sold, how many are still on the road? 9,000
4. If 1000 cars were sold, how many are still on the road? 900
5. Find an example to show how you think averages could be used in
a misleading way. See students’ work.
6. A survey of a large sample of people who own small computers
revealed that 85% of the people thought the instruction manuals
should be better written. A manufacturer of small computers
claimed that it surveyed many of the same people and found that
all of them liked their manuals. Discuss the possible discrepancy
in the results. See students’ work.
2. Suppose that a car company claims, “75% of people surveyed say
that our car is better than the competition.” If only four people
were surveyed, how many people thought that this company’s car
was better? 3 people
Suppose an advertiser claims that 90% of all the cars of one brand sold in the last
10 years are still on the road.
the first graph
1. Which graph would you use to give the impression that the
unemployment rate dropped dramatically from 1982 to 1990?
Notice that the two graphs show the same data, but the spacing in
the vertical and horizontal scales differs. Scales can be cramped or
spread out to make a graph that gives a certain impression.
Statistics can be misleading. Two graphs for the same set of data can
look very different from each other. Compare the following graphs.
Enrichment
Misuses of Statistics
14-1
Determine which class intervals would be appropriate for the data below.
Explain your answers.
Practice
NAME _____________________________ DATE _______________ PERIOD ________
The Frequency Distribution
14-1
NAME _____________________________ DATE _______________ PERIOD ________
Answers
(Lesson 14-1)
Advanced Mathematical Concepts
© Glencoe/McGraw-Hill
23; 19; 22 and 19
4. {18, 6, 22, 33, 19, 34, 14, 54, 12, 22, 19}
55.9; 56.5; 94
2. {32, 12, 61, 94, 73, 62, 94, 35, 44, 52}
A4
about 357.3;
330; 240
Stem Leaf
2 2 4 4 7
3 1 3 4
4 5 6 8
5 9
2/2 ⫽ 220
7.
about 103.42;
104; 91
Stem Leaf
9 0113
10 1 3 5 6
11 3 4 6 8
9/0 ⫽ 90
© Glencoe/McGraw-Hill
613
9. Medicine A frequency distribution for
the number of patients treated at 50 U.S.
cancer centers in one year is given at
the right.
a. Use the frequency chart to find the
mean of the number of patients
treated by a cancer center. 1210
b. What is the median class of the
frequency distribution? 500–1000
6.
6
0
2
0
2
1000–1500
1500–2000
2000–2500
2500–3000
3000–3500
3500–4000
Advanced Mathematical Concepts
Source: U.S. News Online
26
14
500–1000
Patients
about 3.59; 3.2; 2.3
Stem Leaf
1 1 2 9
2 3 3 5
3 2
4 0
5 4 5 6 8 9
1/1 ⫽ 1.1
Number
of Cancer
Centers
8.
Find the mean, median, and mode of the data represented by each
stem-and-leaf plot.
5. A shoe store employee sets up a display by placing shoeboxes in
10 stacks. The numbers of boxes in each stack are 5, 7, 9, 11, 13,
10, 9, 8, 7, and 5.
a. What is the mean of the number of boxes in a stack? 8.4
b. Find the median of the number of boxes in a stack. 8.5
c. If one box is removed from each stack, how will the mean and
median be affected? Each will decrease by 1 box.
246; 152; no mode
3. {152, 697, 202, 312, 109, 134, 116}
42.83; 40.5; 42
1. {15, 42, 26, 39, 93, 42}
30 ⫹ 55
ᎏ
2
or 42.5 mph.
100
ᎏ
50
⫹
100
ᎏ,
55
or about 52.4 mph.
We apply the formula to Bill’s speeds.
Let n ⫽ number of rates xi where 1 ⱕ i ⱕ n.
i⫽1
冱
n
ᎏᎏᎏ
n
1
__
xi
or
H ⬇ 52.4 mph
2
H ⫽ ______
1
1
ᎏ ⫹ ᎏ
50
55
H ⫽
Bill’s average speed also may be found by using the
formula for the harmonic mean as follows.
200
ᎏ,
3.82
approximately 3.82 hours. His average speed was
varied. Actually, the time he drove was
On the other hand, Bill drove the same distance on
both portions of his driving, but the two lengths of time
two rates. Her average speed was
Kendra drove the same length of time on both portions
of her driving, so her average speed is the mean of the
Recently Kendra and Bill took a trip of 370 miles
and shared the driving. Kendra drove two hours
at a rate of 30 mph and then drove the next
110 miles on a freeway at 55 mph. Then Bill
drove the next two hours at 50 mph and he drove
the last 100 miles on a freeway at 55 mph. What
was the average speed of each driver?
10.95
2. {5, 10, 15, 20, 25}
© Glencoe/McGraw-Hill
614
Advanced Mathematical Concepts
3. Bev, Phyllis, and Gordon competed in a 375-mile relay race. Bev drove 40 mph,
Phyllis drove 50 mph, and Gordon drove 60 mph. If each drove 125 miles, find the
average driving speed of the contestants. 48.65 mph
4.21
1. {3, 4, 5, 6}
Find the harmonic mean of each set of data. Round each answer to the nearest
hundreth.
The mean, also called the arithmetic mean, is used when equal
times are involved. When equal distances are involved, the
harmonic mean is used.
Example
The harmonic mean H is a useful measure of central tendency in
special cases of averaging rates.
Enrichment
The Harmonic Mean
14-2
Find the mean, median, and mode of each set of data.
Practice
NAME _____________________________ DATE _______________ PERIOD ________
Measures of Central Tendency
14-2
NAME _____________________________ DATE _______________ PERIOD ________
Answers
(Lesson 14-2)
Advanced Mathematical Concepts
© Glencoe/McGraw-Hill
A5
4.7, 2.35
© Glencoe/McGraw-Hill
about 1.16
d. What is the standard deviation
of the male life expectancy?
about 1.02
c. What is the standard deviation
of the female life expectancy?
b. Find the mean of the male life
expectancy. 73.85
616
5. Sociology The frequency distribution
at the right shows the average life
expectancy for males and females in
15 European Union countries in 1994.
a. Find the mean of the female life
expectancy. 79.85
334.84, 433.25
4
8
3
0
0
0
0
Female
Advanced Mathematical Concepts
Source: Department of Health and Children, Ireland
0
80.5⫺82.0
1
76.0⫺77.5
0
2
74.5⫺76.0
79.0⫺80.5
9
73.0⫺74.5
0
3
71.5⫺73.0
77.5⫺79.0
Male
Life
Expectancy
(years)
Frequency
4. 1592, 1486, 1479, 1682, 1720, 1104, 1486, 1895, 1890, 2687, 2450
37.1, 51.99
Find the mean deviation and the standard deviation of each
set of data.
3. 146, 289, 121, 146, 212, 98, 86, 153, 128, 136, 181, 142
2. 1.6, 9.8, 4.5, 6.2, 8.7, 5.6, 3.9, 6.8, 9.7, 1.1, 4.7, 3.8, 7.5, 2.8, 0.1
44, 22
Notice that no one had a score of 56 points.
Thus, the score at the 16th percentile is 56.
What score is at the 16th percentile?
A score at the 16th percentile means the
score just above the lowest 16% of the scores.
16% of the 50 scores is 8 scores.
The 8th score is 55.
The score just above this is 56.
3
1
50
45
28th
11. a score of 62 ______
1
4
8
14
21
29
36
42
47
49
50
Cumulative
Frequency
Advanced Mathematical Concepts
84th
12. a score of 81 ______
90th
9. a score of 85 ______
© Glencoe/McGraw-Hill
72nd
8. a score of 77 ______
16th
10. a score of 58 ______
6th
7. a score of 50 ______
617
Use the table above to find the percentile of each score.
33 out of 50 is 66%.
Thus, a score of 75 is at the 66th percentile.
Adding 4 scores to the 29 gives 33 scores.
At what percentile is a score of 75?
There are 29 scores below 75.
Seven scores are at 75. The fourth of these seven is the
midpoint of this group.
81
6. 80th percentile ______
Example 2
86
4. 90th percentile ______
71
5. 58th percentile ______
76
2. 70th percentile ______
4
55
6
7
65
60
8
70
6
80
7
5
85
75
2
1
95
90
Frequency
Score
66
3. 33rd percentile ______
66
1. 42nd percentile ______
Use the table above to find the score at each percentile.
Example 1
The table at the right shows test scores and their
frequencies. The frequency is the number of people
who had a particular score. The cumulative frequency
is the total frequency up to that point, starting at the
lowest score and adding up.
Enrichment
Percentiles
14-3
Find the interquartile range and the semi-interquartile range of
each set of data. Then draw a box-and-whisker plot.
1. 43, 26, 92, 11, 8, 49, 52, 126, 86, 42, 63, 78, 91, 79, 86
Practice
NAME _____________________________ DATE _______________ PERIOD ________
Measures of Variability
14-3
NAME _____________________________ DATE _______________ PERIOD ________
Answers
(Lesson 14-3)
Advanced Mathematical Concepts
© Glencoe/McGraw-Hill
A6
© Glencoe/McGraw-Hill
1.05%
619
Advanced Mathematical Concepts
b. What percent of the full-grown German shepherds at City View
Kennels weigh less than Skipper?
98.95%
a. What percent of the full-grown German shepherds at City View
Kennels weigh more than Skipper?
7. Dog Breeding The weights of full-grown German shepherds at
the City View Kennels are normally distributed. The mean weight
is 86 pounds, and the standard deviation is 3 pounds. Skipper, a
full-grown German shepard, weighs 79 pounds.
6. Find the limit below which 90% of the data lie. 146
96–144
5. Find the interval about the mean that includes 77% of the data.
87–153
4. Find the interval about the mean that includes 90% of the data.
3. What percent of the data is between 90 and 110? 24.15%
2. What percent of the data is between 110 and 130? 38.3%
683
1. How many values are within one standard deviation of the mean?
symmetric
symmetric
5.
2.
symmetric
skewed to
the left
6.
3.
skewed to
the right
skewed to
the right
© Glencoe/McGraw-Hill
in the middle of the range;
It is the same as the mean.
7. Where is the median in a symmetric
distribution?
620
Advanced Mathematical Concepts
to the left of the mean if
skewed to the right; to the
right of the mean if
skewed to the left.
8. Where is the median in a skewed
distribution?
A vertical line along the median divides the area under a frequency curve in half.
4.
1.
For each of the following, state whether the distribution is symmetric or skewed. If
it is skewed, tell whether it is skewed to the right or to the left.
In a distribution skewed to the left, there are a larger number of low
values. The long “tail” extends to the left.
In a distribution skewed to the right, there are a larger number of
high values. The long “tail” extends to the right.
Graphs of frequency distributions can be described as either
symmetric or skewed.
Enrichment
Shapes of Distribution Curves
14-4
A set of 1000 values has a normal distribution. The mean of
the data is 120, and the standard deviation is 20.
Practice
NAME _____________________________ DATE _______________ PERIOD ________
The Normal Distribution
14-4
NAME _____________________________ DATE _______________ PERIOD ________
Answers
(Lesson 14-4)
Advanced Mathematical Concepts
© Glencoe/McGraw-Hill
0.22; 109.43–110.57;
109.57–110.43
A7
4
2
Frequency
5
7
1
8
8
9
5
10
1.64
© Glencoe/McGraw-Hill
7.11376–8.80624 min
622
10. Determine the interval about the sample mean that has a
1% level of confidence.
7.31712–8.60288 min
Advanced Mathematical Concepts
9. Find the interval about the sample mean such that the
probability is 0.95 that the true mean lies within the interval.
7.4188–8.5012 min
8. Find the interval about the sample mean such that the
probability is 0.90 that the true mean lies within the interval.
7. Find the standard error of the mean. 0.328
6. Find the standard deviation of the data.
5. What is the mean of the data in the frequency distribution? 7.96
6
5
Number of Minutes
The table below shows a frequency distribution of the time in
minutes required for students to wash a car during a car wash
fundraiser. The distribution is a random sample of 250 cars. Use
the table for Exercises 5-10.
4. ␴ ⫽ 4.3, N ⫽ 375, 苶
X ⫽ 110
0.16; 49.59–50.41;
49.69–50.31
0.5; 98.71–101.29;
99.02–100.98
2. ␴ ⫽ 4, N ⫽ 64, 苶
X ⫽ 100
3. ␴ ⫽ 2.6, N ⫽ 250, X
苶 ⫽ 50
5; 237.1–262.9; 240.2–259.8
1. ␴ ⫽ 50, N ⫽ 100, 苶
X ⫽ 250
© Glencoe/McGraw-Hill
623
A ⫽ 4, B ⫽ 16, C ⫽ 24, D ⫽ 16, F ⫽ 4
Advanced Mathematical Concepts
1. Draw a smooth curve connecting the tops of
the segments whose lengths are
proportional to the coefficients in the
expansion of (x ⫹ y)12.
2. A teacher decided to mark 64 tests with the grades A, B, C, D,
and F (A highest) in proportion to the coefficients in the
expansion of (x ⫹ y)4. How many tests received each grade?
Solve.
The figure above at the right has equally-spaced
vertical segments whose lengths are proportional
to the numbers in the eighth row of Pascal’s
triangle. A smooth curve connecting the tops of
these segments suggests the probable distribution
of marbles in the column. Notice the similarity of
this curve to the normal distribution curve.
The numbers in Pascal’s triangle are the coefficients
in the expansion of (x ⫹ y)n where n is any positive
integer. Therefore, the probability of a marble falling
in any given column is proportional to the coefficient
of the corresponding term in the binomial expansion
of a power. The power is a whole number equal to
the row being considered. For example, in the
illustration above the columns are in the eighth row.
Thus the probability of a marble falling in the third
column is proportional to the coefficient of the third
term in the binomial expansion of (x ⫹ y)8.
It can be shown that the number of paths from A to G
is 1, A to J is 1, A to H is 3, and A to I is 3. For
example, H can be reached by the way of E. Hence
the number of paths to H is the sum of the number of
paths to D and the number of paths to E. Likewise
the number of paths to any point can be found by
adding the number of paths to points diagonally
above it. This is precisely the method by which the
numbers in Pascal’s triangle are obtained.
The picture at the right shows a device often used to
illustrate a normal probability distribution. The
device is filled with small steel marbles. The marbles
roll past a series of hexagonal obstacles, collecting at
the bottom in each of nine columns.
Enrichment
Binomial Expansion Coefficients
14-5
Find the standard error of the mean for each sample. Then find
the interval about the sample mean that has a 1% level of
confidence and the interval about the sample mean that has a
5% level of confidence.
Practice
NAME _____________________________ DATE _______________ PERIOD ________
Sample Sets of Data
14-5
NAME _____________________________ DATE _______________ PERIOD ________
Answers
(Lesson 14-5)
Advanced Mathematical Concepts
Chapter 14 Answer Key
Form 1A
Page 625
1.
2.
D
Form 1B
Page 626
11.
C
12.
A
3.
A
4.
D
6.
7.
8.
9.
2.
B
B
3.
D
4.
A
5.
C
B
15.
A
6.
D
16.
C
7.
B
8.
C
B
Page 628
11.
C
12.
A
13.
B
14.
C
15.
D
16.
A
B
14.
A
D
A
17.
B
17.
D
18.
D
18.
B
19.
D
19.
D
20.
B
9.
A
10.
B
B
20.
10.
1.
C
13.
5.
Page 627
B
D
Bonus:
© Glencoe/McGraw-Hill
Bonus:
B
A8
A
Advanced Mathematical Concepts
Chapter 14 Answer Key
Form 1C
Page 629
1.
2.
C
Page 631
11.
D
1. 1, 2, 3, 4, 6, 7, 8
12.
C
2.
B
3.
A
4.
B
5.
Form 2A
Page 630
C
6.
D
7.
A
8.
B
3.
13.
C
14.
C
15.
16.
B
Page 632
11. 0.0097
12. 0.0114
67
Sample
answer:
13.
4.
31.78
5.
21
14.
6.
18
15. 376–424
7.
5 ᎏ311ᎏ
5
38.3%
16. 95.55%
D
8. 5236.875
17. 0.0533
17.
B
18.
D
19.
C
18. 5.06–5.34
19. 5.14–5.26
20.
9.
A
10.
B
20.
Bonus:
© Glencoe/McGraw-Hill
A
9.
685
10.
697
94.3%
Bonus: 2.8%
B
A9
Advanced Mathematical Concepts
Chapter 14 Answer Key
Page 633
Form 2B
1. 1, 2, 3, 5, 8
2. 10, 30, 50, 70, 90
Page 634
11.
1.25%
12.
1.85%
Form 2C
Page 635
1.
5, 6, 7, 8
2. 55, 65, 75, 85
3. Sample answer:
Page 636
11.
2.3
12.
2.9
Sample answer:
13.
42.42
5.
44
6. 53 and 57
7.
15
4.
8.
14.
95.5%
15. 107-133
16.
13.
3.
86.6%
14.
4.
67.94
5.
67.5
6.
65, 67, 72
7.
8
8.
7.024
51.6%
15. 5.69–10.31
16.
38.3%
17.
0.14
14.5
17.
9.
497.5
10.
480
0.025
18. 2.5355–2.6645
18. 10.23 –10.77
19. 2.5675–2.6325
19. 10.27–10.73
20. 99.99%
20.
Bonus: 13.6%
© Glencoe/McGraw-Hill
A10
9.
44
10.
41.67
92.9%
Bonus:
74%
Advanced Mathematical Concepts
Chapter 14 Answer Key
CHAPTER 14 SCORING RUBRIC
Level
Specific Criteria
3 Superior
• Shows thorough understanding of the concepts
histogram, normal distribution, mean, median, mode,
central tendency, and standard deviation.
• Uses appropriate strategies to solve problems.
• Computations are correct.
• Written explanations are exemplary.
• Goes beyond requirements of problems.
2 Satisfactory,
with Minor
Flaws
• Shows understanding of the concepts histogram,
normal distribution, mean, median, mode, central,
tendency, and standard deviation.
• Uses appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are effective.
• Satisfies all requirements of problems.
1 Nearly
Satisfactory,
with Serious
Flaws
• Shows understanding of most of the concepts
histogram, normal distribution, mean, median, mode,
central tendency, and standard deviation.
• May not use appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are satisfactory.
• Satisfies most requirements of problems.
0 Unsatisfactory • Shows little or no understanding of the concepts
histogram, normal distribution, mean, median, mode,
central tendency, and standard deviation.
• May not use appropriate strategies to solve problems.
• Computations are incorrect.
• Written explanations are not satisfactory.
• Does not satisfy requirements of problems.
© Glencoe/McGraw-Hill
A11
Advanced Mathematical Concepts
Chapter 14 Answer Key
Open-Ended Assessment
Page 637
1a. No; it is not a symmetric bell curve.
1b. Mode; it is not affected by extreme
scores.
1c. Mean; it is most affected by extreme
scores.
100ᎏ
-85 1.5 standard deviations
2a. ᎏ
10
Of the scores, 0.866 lie within 1.5
standard deviation of the mean.
1.00-0.866
ᎏ, or 0.067, lie 1.5
Thus, ᎏ
2
standard deviations above the
mean. 0.067 200 13.4, so about
13 students received an A.
2b. They are both 85 because for a
normally distributed set of data
the mean, median, and mode
are equal.
2c. Of the grades, 0.383 are within
0.5 standard deviation of the mean.
0.383 200 76.6, so about 77
students received a C.
2d. A decrease; there are not as many
scores in the 75 to 80 range as there
are in the 85 to 90 range.
© Glencoe/McGraw-Hill
A12
3. Counterexample: Let the data sets be
{1} and {100, 100, 100, 100}. The mean
of the first data set is 1, and the mean
of the second data set is 100, so the
1 ᎏ
100 50.5.
mean of the means is ᎏ
2
The combination of the two data sets
is {1, 100, 100, 100, 100}, which has a
mean of 80.2. Thus, the statement is
not true in this case. The statement is
true, however, if the two data sets
苶 be the
have the same size, n. Let Z
mean of the combination of two data
sets of the same size, and let Xi and Yi
be the values in the two sets. Then
苶
Z
X1 X2 . . . Xn Y1 Y2 . . .Yn
ᎏᎏᎏᎏᎏ
2n
X X ...X Y Y ...Y
1
2
n
1
2
n
ᎏ12ᎏ 冢 ᎏᎏᎏᎏᎏ
冣
n
X X ...X
Y Y ...Y
1
2
n
1
2
n
ᎏ12ᎏ 冢ᎏᎏᎏ
ᎏᎏᎏ
冣
n
n
苶 ᎏ
X
苶
Y
ᎏ12ᎏ ( X
苶苶
Y) ᎏ
2
So, when the data sets are the same
size, the mean of the combination
data set equals the mean of
the means.
Advanced Mathematical Concepts
Chapter 14 Answer Key
Mid-Chapter Test
Page 638
1.
2.
Sample answer:
Costs Frequency
2–4
3
4–6
7
6–8
3
1. 25; Sample answer: 5
1.
95.5%
2.
615
3.
245
4.
6.7%
5.
136.8-163.2
Sample answer: 12.5,
2. 17.5, 22.5, 27.5, 32.5, 37.5
Sample answer:
2, 3, 4, 5, 6, 7
3.
3.
4.
Quiz C
Page 640
Quiz A
Page 639
4.
4.78
stem leaf
1 26999
2 1244566688899
3 011234557
1|2 ⫽ 12
5. 26.6; 28; 19, 26, and 28
5.
4.6 and 5.2
6.
0.95
7.
1.23
Quiz D
Page 640
Quiz B
Page 639
8.
198.75 lb
9.
195.4 lb
1.
about 3.03
1.
0.11
2.
5 in.
2.
5.984-6.416
3.
2.5 in.
3.
5.916-6.484
4.
63.2%
5.
100%
4.
about 3.72
5.
10.
21.8
© Glencoe/McGraw-Hill
A13
Advanced Mathematical Concepts
Chapter 14 Answer Key
SAT/ACT Practice
Page 641
1.
2.
C
B
Cumulative Review
Page 643
Page 642
10.
11.
1.
D
2. y 6 cos ᎏ␲6ᎏt
B
3.
E
12.
D
4.
D
13.
C
5.
C
14.
C
具2, 2, 7典
3.
x2 49 0
4.
18.6 units2
5.
2i
6. y2 2y 4x 17 0
6.
D
15.
D
7.
7.
E
16.
1ᎏ
53 ᎏ
125
E
8. 1, i, 1 2i
8.
C
17.
A
9.
9.
A
© Glencoe/McGraw-Hill
18.
A
19.
110
20.
60
10.
A14
ᎏ1ᎏ
8
49.6%
Advanced Mathematical Concepts
Unit 4 Answer Key
Unit 4 Review
1. 35
27. Proof: S
is defined as
n
3 9 15 ... (6n 3) 3n2.
Step 1: Verify that Sn is valid for n 1.
Since S1 3 and 3(1)2 3, the formula
is valid for n 1.
2. 1022
3. 3, 1, 5, 9
102
4
ᎏ
4. ᎏ
9375
Step 2: Assume that Sn is valid for n k
and derive a formula for n k 1.
Sk ⇒ 3 9 15 ... (6k 3) 3k2
Sk1 ⇒ 3 9 15 ... (6k 3) [6(k 1) 3]
3k2 [6(k 1) 3]
3k2 6k 3
3(k2 2k 1)
3(k 1)2
The formula gives the same result as
adding the (k 1) term directly. Thus, if
the formula is valid for n k, it is also
valid for n k 1. Since the formula is
valid for n 1, it is also valid for n 2.
Since it is valid for n 2, it is also valid
for n 3, and so on, indefinitely. Thus,
the formula is valid for all positive
integral values of n.
5. ᎏ43ᎏ
6. does not exist
7. 0
8. 4
9. 2
10. ᎏ43ᎏ
11. divergent
12. convergent
13. convergent
14. 2(1) 2(2) 2(3) 2(4) 2(5) or 30
15. [2(1)2 1] [2(2)2 1] [2(3)2 1] [2(4)2 1] [2(5)2 1] [2(6)2 1] [2(7)2 1] or 287
16. [3(2) 6] [3(3) 6] [3(4) 6] [3(5) 6] [3(6) 6] [3(7) 6] [3(8) 6] or 63
17. 3冢ᎏ21ᎏ冣0 3冢ᎏ21ᎏ冣1 3冢ᎏ21ᎏ冣2 3冢ᎏ21ᎏ冣3 38ᎏ
1
3冢ᎏ21ᎏ冣4 3冢ᎏ21ᎏ冣5 3冢ᎏ21ᎏ冣6 or ᎏ
64
18. 61,236x6
28.
Proof: Sn ⇒ 5 n 1 4r for some
integer r
Step 1: Verify that Sn is valid for n 1.
S1 51 1 or 4. Since 4 4 1,
Sn is valid for n 1.
Step 2: Assume that Sn is valid for n k
and show that it is also valid for
n k 1.
Sk ⇒ 5 k 1 4r for some integer r.
Sk 1 ⇒ 5 k1 1 4t for some integer t.
5 k 1 4r
5(5 k 1) 5(4r)
5 k1 5 20r
5 k1 1 20r 4
5 k1 1 4(5r 1)
Thus, 5k1 1 4t, where t (5r 1), is
an integer, and we have shown that if Sk
is valid, then Sk1 is also valid. Since Sn
is valid for n 1, it is also valid for
n 2, n 3, and so on, indefinitely.
Hence 5n 1 is divisible by 4 for all
positive integral values of n.
29. 60,480
30. 840
31. 28
32. 6
19. 59,136x6y6
20. 5.41
21. 0.74
38. ᎏ25ᎏ
1ᎏ
39. ᎏ
13
40. ᎏ29ᎏ
3ᎏ
41. ᎏ
10
1ᎏ
1
42. ᎏ
850
5ᎏ
43. ᎏ
1296
44. ᎏ15ᎏ
8
45. ᎏ13ᎏ
4ᎏ
5
46. ᎏ
512
4ᎏ
7
47. ᎏ
128
48. 57
49. 77.45
50. 77
51. 77
52. 9.595
53. 9
54. 12.65
55. 99.7%
56. 0.1585
57. 0.576
58. 63.1670.84
59. 0.2260.274
60. 0.8
61. about 0.59
62. about 0.81
22. 4, 13, 40
33. 120 patterns
63. about 1.30
23. 1, 4, 11
34. 252 teams
64. 48.71–51.29
35. 60 ways
65. 98.45–101.55
36. 30 committees
66. 18.628–21.372
24. ᎏ12ᎏ, 2, ᎏ12ᎏ
25. i, 5i, 13i
26. 6 i, 12 5i, 24 13i
37. 2520 ways
© Glencoe/McGraw-Hill
A15
Advanced Mathematical Concepts
Unit 4 Answer Key
Unit 4 Test
4
1.
(continued)
冱 (23 6k)
9. 20,160
k0
10. convergent
27. mean: 61.75;
median: 56;
mode: 53
2. 49.65%
11. 0.5
3.
28. 16; 8
Proof: Sn is defined as
n(n 1)(2n 1)
12 22 32 . . . n2 ᎏ6ᎏ
12. 10,080
1(1 1)(2 1 1) 1,
Since S1 1 and ᎏᎏᎏ
13. convergent
the formula is valid for n 1.
Step 2: Assume that Sn is valid for n k
and derive a formula for n k 1.
14. 1, i, 1 2i
Step 1: Verify that Sn is valid for n 1.
6
Sk ⇒
12
22
32
...
k(k 1)(2k 1)
ᎏᎏ
6
k2
Sk1 ⇒ 1 2 3 . . . k2 (k 1)2
2
2
2
k)(2k 1) 6(k2 2k 1)
ᎏᎏᎏᎏ
6
4ᎏ
30. ᎏ
11
15. ᎏ14ᎏ
31. 197
k(k 1)(2k 1)
ᎏ6ᎏ (k 1)2
(k2
1ᎏ
29. ᎏ
221
6ᎏ
4
16. ᎏ
125
2k3 9k2 13k 6
ᎏᎏᎏ
6
(k 1)(2k2 7k 6)
ᎏᎏᎏ
6
(k 1)(2k 3)(k 2)
ᎏᎏᎏ
6
(k 1)[(k 1) 1][2(k 1) 1]
ᎏᎏᎏᎏ
6
The formula gives the same result as
adding the (k 1) term directly. Thus, if
the formula is valid for n k, it is also
valid for n k 1. Since the formula is
valid for n 1, it is also valid for n 2.
Since it is valid for n 2, it is also valid
for n 3, and so on, indefinitely. Thus,
the formula is valid for all positive
integral values of n.
4. 128
17. x3 6x2y 12xy2 8y3
18. 0.375
32. 13.55%
33. ᎏ3ᎏ
10
19. 20
20. 10; 20.5, 30.5,
40.5, 50.5, 60.5
34. 0.36; 35.0736.93
21. 1.95
35. 128.05131.25
22. 2, 2, 2
36. 68.3%
23. does not exist
5. ᎏ1ᎏ
21
37. 0.0225
24. ᎏ53ᎏ
25. 210
6. ᎏ5ᎏ
4
7. 20
8. true
© Glencoe/McGraw-Hill
26.
38. 0.25
stem
leaf
4 2
5 1339
6 1
7 5
10 0
4|2 42
39. 9.35510.645
A16
Advanced Mathematical Concepts
40. about 9.5910.41
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BLANK
BLANK
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