Understanding Our Quantitative World
Transcript of Understanding Our Quantitative World
Understanding Our Quantitative World
AMS / MAA TEXTBOOKS VOL 6
Janet Andersen and Todd Swanson
Understanding our Quantitative World
10.1090/text/006
© 2005 byThe Mathematical Association of America (Incorporated)
Library of Congress Control Number 2004113543
e-ISBN 978-1-61444-125-0
Paperback ISBN 978-0-88385-738-0
Printed in the United States of America
Understanding our Quantitative World
Janet Andersen
Hope College
and
Todd Swanson
Hope College
Published and Distributed by
The Mathematical Association of America
Council on Publications
Roger Nelsen, Chair
Classroom Resource Materials Editorial Board
Zaven A. Karian, Editor
William Bauldry Stephen B Maurer
Gerald Bryce Douglas Meade
George Exner Judith A. Palagallo
William J. Higgins Wayne Roberts
Paul Knopp Kay Somers
CLASSROOM RESOURCE MATERIALS
Classroom Resource Materials is intended to provide supplementary classroom material
for students—laboratory exercises, projects, historical information, textbooks with unusual
approaches for presenting mathematical ideas, career information, etc.
101 Careers in Mathematics, 2nd edition edited by Andrew Sterrett
Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein
Calculus Mysteries and Thrillers, R. Grant Woods
Combinatorics: A Problem Oriented Approach, Daniel A. Marcus
Conjecture and Proof, Miklos Laczkovich
A Course in Mathematical Modeling, Douglas Mooney and Randall Swift
Cryptological Mathematics, Robert Edward Lewand
Elementary Mathematical Models, Dan Kalman
Environmental Mathematics in the Classroom, edited by B. A. Fusaro and P. C. Kenschaft
Essentials of Mathematics, Margie Hale
Exploratory Examples for Real Analysis, Joanne E. Snow and Kirk E. Weller
Geometry From Africa: Mathematical and Educational Explorations, Paulus Gerdes
Identification Numbers and Check Digit Schemes, Joseph Kirtland
Interdisciplinary Lively Application Projects, edited by Chris Arney
Inverse Problems: Activities for Undergraduates, Charles W. Groetsch
Laboratory Experiences in Group Theory, Ellen Maycock Parker
Learn from the Masters, Frank Swetz, John Fauvel, Otto Bekken, Bengt Johansson, and
Victor Katz
Mathematical Connections: A Companion for Teachers and Others, Al Cuoco
Mathematical Evolutions, edited by Abe Shenitzer and John Stillwell
Mathematical Modeling in the Environment, Charles Hadlock
Mathematics for Business Decisions Part 1: Probability and Simulation (electronic
textbook), Richard B. Thompson and Christopher G. Lamoureux
Mathematics for Business Decisions Part 2: Calculus and Optimization (electronic
textbook), Richard B. Thompson and Christopher G. Lamoureux
Ordinary Differential Equations: A Brief Eclectic Tour, David A. Sanchez
Oval Track and Other Permutation Puzzles, John O. Kiltinen
A Primer of Abstract Mathematics, Robert B. Ash
Proofs Without Words, Roger B. Nelsen
Proofs Without Words II, Roger B. Nelsen
A Radical Approach to Real Analysis, David M. Bressoud
She Does Math!, edited by Marla Parker
Solve This: Math Activities for Students and Clubs, James S. Tanton
Student Manual for Mathematics for Business Decisions Part 1: Probability and Simu-
lation, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic
Student Manual for Mathematics for Business Decisions Part 2: Calculus and Optimiza-
tion, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic
Teaching Statistics Using Baseball, Jim Albert
Understanding our Quantitative World, Janet Andersen and Todd Swanson
Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go,
Annalisa Crannell, Gavin LaRose, Thomas Ratliff, Elyn Rykken
MAA Service Center
P.O. Box 91112
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Preface
Philosophy
Understanding our Quantitative World is our approach to quantitative literacy. This book
is intended for a general education mathematics course and addresses the question “What
mathematical skills and concepts are useful for informed citizens?” We believe that it
is important for students to practice applying mathematical reasoning and concepts to
material they are likely to encounter outside academia. Therefore, the text is rich in
documented examples taken from sources such as the public media. While we include
questions asking students to perform simple calculations, many of the questions focus on
using mathematics correctly to interpret information. The topics fall into three categories:
interpreting graphs, interpreting simple functions, and interpreting statistical information.
Our goals are for students to:
� Realize that mathematics is a useful tool for interpreting information.
� See mathematics as a way of viewing the world that goes far beyond memorizingformulas.
� Become comfortable using and interpreting mathematics so that they will voluntarilyuse it as a tool outside of academics.
The text is written in a conversational tone, beginning each section by setting the
mathematics within a context and ending each section with an application. Mathematical
concepts are explored in multiple representations including verbal, symbolic, graphical,
and tabular. The questions at the end of each section are called Reading Questions
because we expect students to be able to answer most of these after carefully reading
the text. Requiring students to read the text before class and to attempt to answer the
reading questions allows us to spend class time highlighting key concepts and correcting
misconceptions. Having students read the text also emphasizes the importance of becoming
a self-learner.
The focus of the course is the Activities and Class Exercises found at the end of each
chapter. These activities are taken from public sources such as newspapers, magazines,
and the Web. Doing these activities demonstrates to students that they can use mathematics
as a tool in interpreting the world they encounter. Students spend most of their time in
vii
viii Understanding our Quantitative World
class working in groups on the activities. Rather than having students passively listen,
our approach requires students to read, discuss, and apply mathematics.
Students are required to have access to some type of technology such as a graphing
calculator or spreadsheet.
The National Science Foundation grant (Grant DUE-9652784) that supported this
course also supported the development of two science courses at Hope College, Pop-
ulations in a Changing Environment and The Atmosphere and Environmental Change.
Connecting this mathematics course with two general education science courses has
allowed us to use mathematics as an effective tool in the context of environmental
questions and thereby strengthen the students’ mathematical understanding.
Annotated Table of Contents
1. Functions. Four representations of functions (symbolic, graphical, tabular, and verbal)
are emphasized. Specialized vocabulary (such as domain and range) is introduced.
Examples include the stock market, population of the U.S., and the cost of Internet
services. Group activities include cell phone rates and credit card bills.
2. Graphical Representations of Functions. Correct interpretation of graphical infor-
mation is emphasized, particularly with regards to shape and labels. The concepts of
increasing/decreasing and concavity are introduced. Instruction on using the calculator to
construct graphs is included in the appendix. Group activities focus on analyzing a variety
of graphs from magazines, newspapers, and non-mathematical textbooks.
3. Applications of Graphs. The connections between and the meaning of the graphs of
y D f .x/, y D f .x C a/, y D f .x/ C a, y D f .ax/, and y D af .x/ are emphasized.
This is introduced via the context of a motion detector graph of time versus distance.
Group activities include working with a motion detector and converting baby weight
charts from English units to metric units.
4. Displaying Data. The emphasis in this section is on visual display of data. Histograms,
scatterplots, and xy-line graphs are included. In the appendix, students receive instruction
on using the calculator to graph data in each of these formats. Group activities include
looking at arm span versus height and data given from the American Film Association
on “best movies.”
5. Describing Data: Mean, Median, and Standard Deviation. Concepts underlying
one variable statistics are emphasized. These include ideas of center (e.g., median and
mean) and ideas of spread (e.g. standard deviation and quartiles). The emphasis is on the
difference between median and mean, particularly with skewed data. Normal distributions
are also introduced. Instruction on using the calculator to compute one-variable statistics
is included in the appendix. Group activities include salary versus winning percentage of
basketball teams and looking at house prices.
6. Multivariable Functions and Contour Diagrams. Commonly occurring multivariable
functions (such as computing the payment on a car loan) and commonly occurring contour
maps (such as weather and topological maps) are emphasized. Treating a multivariable
Preface ix
function as a single variable function by holding all but one input constant is also included.
This allows the students to connect some of the ideas in this section with those encountered
earlier in the text. Group activities include a contour map of Mount Rainier and looking
at car loans.
7. Linear Functions. The emphasis is on translating a situation with a constant rate of
change into the mathematical concept of a line. There is also an emphasis on the concept
that only two pieces of information—a starting point and a rate of change—are necessary
to determine a line. This section ends by showing that proportional changes (such as unit
conversions) can be thought of as linear functions. Group activities include working with
a motion detector and looking at an electric bill.
8. Regression and Correlation. Students are introduced to the concept of using linear
regression and correlation to determine if two variables exhibit a linear relationship.
Calculator instructions for these are included in the appendix. Other types of regression
(e.g. exponential) are introduced in later sections. Group activities include Olympic race
data and atmospheric carbon dioxide data.
9. Exponential Functions. The concept of an exponential function is introduced via the
idea of doubling. Exponential functions are contrasted with linear functions. In particular,
the idea of a constant rate of change versus a constant growth factor is emphasized. This
section also explores vertical and horizontal shifts of exponential functions, connecting
with the ideas introduced in Applications of Graphs. Group activities include a cooling
experiment and looking at prices of DVDs.
10. Logarithmic Functions. Logarithms are emphasized as functions that compute the
magnitude of a number. Only base 10 logarithms are used. Properties of logarithms and
using logarithms to solve simple exponential equations are included. Group activities
include working with decibels and verifying Bedford’s law on the occurrence of numbers
in print.
11. Periodic Functions. Periodic functions are introduced as a way of modeling cyclic
behavior. The behavior of a clock and a swing are used to motivate the concepts. Sine
and cosine are defined in terms of the circular definitions. The concepts of amplitude and
period are related to the ideas of shifting functions introduced earlier in the text. Group
activities include an experiment with sound waves and looking at the seasonal change in
the amount of daylight per day.
12. Power Functions. Power functions are the last type of function covered in the text and
are introduced graphically. Behavior of polynomials with even and odd positive integer
exponents is contrasted. Positive rational exponents are also included. Group activities
include Kepler’s law of planetary motion and looking at the wingspan of birds.
13. Probability. The basic concepts of counting and determining simple probabilities are
introduced. Systemic ways of listing (or counting) all possible outcomes are emphasized.
Multi-stage experiments and expected value are included. Group activities include codes
for garage door openers and roulette.
14. Random Samples. This chapter emphasizes how to set up a random sample and why
this is desirable. The concepts of variability, bias, and confidence intervals are included.
x Understanding our Quantitative World
Group activities include looking at phone-in surveys and simulating a “capture-recapture”
experiment.
Each of these readings is a single unit on the topic. The goal is to give students an
intuitive sense of the mathematical concept so they can adequately interpret (rather than
necessarily create) mathematics. In addition to the readings, we have also written four to
eight group activities for each section, of which we typically assign two to four.
Acknowledgments
We are grateful for the support we have received throughout the project from Hope
College. In particular, we are thankful to our colleagues for their encouragement and
advice throughout this long process. A special thanks goes to Rolland Swank, Darin
Stephenson, Kate Vance, Dyana Harrelson, Mary DeYoung, and Mike Catalano for field-
testing our manuscript.
This text was written with support from the National Science Foundation (Grant DUE-
9652784). We are thankful for this support.
We are also very thankful to our student assistants, Matt Youngberg, Benjamin Freeburn,
Mark Thelen, Melissa Sulok, Sarah Kelly, Todd Timmer, Andrea Spaman, and Kelly Joos
for their outstanding work and assistance. Their help has been a valuable part of this
project.
We are thankful to the Classroom Resource Materials Editorial Board of the MAA. We
are especially thankful to Sheldon Gordon and Zaven Karien for providing their advice
and encouragement in the final editing process of our manuscript.
Finally, we wish to thank the staff at the MAA, including Elaine Pedreira and Beverly
Ruedi, for their excellent work in producing this book.
Janet Andersen
Todd Swanson
Contents
Preface vii
1 Functions 1
Reading Questions for Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Functions: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Graphical Representations of Functions 21
Reading Questions for Graphical Representations of Functions . . . . . . . . . . . 32
Graphical Representations of Functions: Activities and Class Exercises . . . . . 35
3 Applications of Graphs 45
Reading Questions for Applications of Graphs . . . . . . . . . . . . . . . . . . . . . . . 59
Applications of Graphs: Activities and Class Exercises . . . . . . . . . . . . . . . . . 63
4 Displaying Data 67
Reading Questions for Displaying Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Displaying Data: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . . 76
5 Describing Data: Mean, Median, and Standard Deviation 85
Reading Questions for Describing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Describing Data: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . . 100
6 Multivariable Functions and Contour Diagrams 105
Reading Questions for Multivariable Functions . . . . . . . . . . . . . . . . . . . . . . 115
Multivariable Functions: Activities and Class Exercises . . . . . . . . . . . . . . . . 119
7 Linear Functions 123
Reading Questions for Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Linear Functions: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . 140
8 Regression and Correlation 145
Reading Questions for Regression and Correlation . . . . . . . . . . . . . . . . . . . . 152
Regression and Correlation: Activities and Class Exercises . . . . . . . . . . . . . . 155
xi
xii Understanding our Quantitative World
9 Exponential Functions 161
Reading Questions for Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . 179
Exponential Functions: Activities and Class Exercises . . . . . . . . . . . . . . . . . 184
10 Logarithmic Functions 195
Reading Questions for Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . 205
Logarithmic Functions: Activities and Class Exercises . . . . . . . . . . . . . . . . . 206
11 Periodic Functions 213
Reading Questions for Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Periodic Functions: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . 226
12 Power Functions 233
Reading Questions for Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
Power Functions: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . 247
13 Probability 255
Reading Questions for Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
Probability: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 266
14 Random Samples 273
Reading Questions for Random Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
Random Samples: Activities and Class Exercises . . . . . . . . . . . . . . . . . . . . . 282
Appendix: Instructions for the TI-83 Graphing Calculator 287
Index 301
About the Authors 303
Appendix: Instructions for the TI-83
Graphing Calculator
Graphing a function
You can use your TI-83 to construct an xy-plot of almost any function. This process
will be illustrated by graphing the function y D 2x C 3 and the function y D x2. (This
information is in Chapter 3 of the TI-83 Guidebook.)
1. The first step in graphing a function is to define the function so that the calculator
knows what you want to graph. Press MODE and select Func on the fourth line down
(if it is not already selected). You can do this by using the arrow keys. When the
cursor is on top of the word Func, press the ENTER key. Your calculator display
should look like the one shown below. If not, use M or O to get to the correct line
and use B or C to highlight the correct word. Once you are in the correct mode
for graphing, press the QUIT key (this is 2nd followed by MODE ).
2. Press the Y= button (the leftmost blue button directly under the screen). This is the
screen where you enter the function you wish to graph. If any of the Plot1, Plot2, or
Plot3 items are highlighted, clear these by using the arrow keys to place the cursor
over the appropriate word and pressing the ENTER key. If there are any functions
already defined, place the cursor to the immediate right of the equal sign and press
the CLEAR button. Your screen should now be empty and resemble the following.
287
288 Understanding our Quantitative World
3. Enter the function y D 2x C 3 into your calculator by moving the cursor to the right
of the equal sign for Y1 and pressing 2 X,T,�,n + 3 .
4. We also need to set an appropriate window for viewing the graph. Press the WINDOW
button (the second blue button from the left). Xmin is the smallest input for x that
will be displayed on the graph, Xmax is the largest input for x that will be displayed
on the graph, Xscl defines the distance between tick marks on the horizontal axis.
Ymin is the smallest output for y that will be displayed on the graph, Ymax is the
largest output for y that will be displayed on the graph, Yscl defines the distance
between tick marks on the vertical axis, and Xres tells the calculator at which pixels
it should evaluate the function in order to draw the graph. Xres = 1 means the
calculator will evaluate the function at every pixel. For now, leave all of window
items defined with their default values. (These are shown in the following screen.
These can also be obtained by pressing ZOOM 6 .) Press the GRAPH key (blue button
furthest to the right) to display the graph.
5. Now enter the second graph, y D x2, by pressing the Y= button, moving the cursor
to Y2, pressing X, T, �, n and X2 . Press GRAPH to display both graphs.
Appendix: Instructions for the TI-83 Graphing Calculator 289
6. We are interested in finding the coordinates of the left-most point where the graphs
intersect. To do this, we first want to zoom in closer to the point. Press the ZOOM
key (the blue button in the middle) and select 2: Zoom In by using the arrow keys
to highlight your selection and pressing ENTER . This will return you to your graph.
Place the cursor (using the arrow keys) over the point of intersection and press
ENTER . The calculator will re-draw the graph using the point where you placed the
cursor as the center of the screen. If you desire, repeat the process to zoom in even
closer.
7. To find the coordinates of the intersection point, press 2nd then TRACE . This selects
the CALC function. Choose the 5:intersect option. The calculator will ask for the
first curve. The curser should be on the line and you need to just hit enter. It will
then ask for the second curve. The curser should now be on the parabola and again
hit enter. It will finally ask for a guess. Just use the arrow keys to put the curser
approximately at the intersection point and hit enter. You should now see that the
coordinates of the intersection are .�1; 1/.
290 Understanding our Quantitative World
Finding x- and y -intercepts
We can use the calculator to find the x- and y-intercepts of a function given symbolically.
We will demonstrate this with the functions y D x3 C 2x2 � 7x � 6.
1. Enter the function in your calculator by pressing Y= X ^ 3 C 2 X ^ 2 �7 X � 6 . Graph it using the viewing window Xmin=-5, Xmax=5, Ymin=-10,
Ymax=10.
The y-intercept is where the function crosses the y-axis. Since this occurs when
x D 0, we can see by substituting zero for x that y D 03 C 2 � 02 � 7 � 0 � 6 D �6.
But as a check, let us use the calculator to find the y-intercept.
2. Press 2nd then TRACE . This selects the CALC function.
The first option on the list is 1:Value. This calculates the output or y-value for any
given input or x-value. Select this option and, when asked to give a x-value, enter
zero.
Notice that the y-value is �6. You can use this option to find the corresponding
y-value (output) for any x-value (input). If you are in the TRACE mode, you can do
the same thing by entering an x-value at any time.
Appendix: Instructions for the TI-83 Graphing Calculator 291
3. We now want to calculate the x-intercepts. These occur where the graph of the
function crosses (or touches) the x-axis. These are also called the zeros or roots of
the function because y D 0 when a function touches the x-axis. So the x-intercepts
are the values of x such that 0 D x3 C2x2 �7x �6. We will illustrate this by finding
the coordinates for the leftmost x-intercept. Press the 2nd and TRACE keys to select
the CALC option. Chose the 2:zero option and press ENTER . [Note: you may have
to use M and O to select the correct function.] Once selected, the calculator asks
you for the left-bound.
This is asking you to give a lower (left) bound for the zero you are seeking. Use
the C and B keys to move the cursor to the left of the leftmost zero and press
ENTER .
The calculator now asks you for the right bound. Use the B to move the cursor on
the function slightly to the right. Press ENTER .
The calculator now asks for a guess. The calculator is asking you to move the cursor
close to the x-intercept. Move the cursor close to the leftmost point where the graph
crosses the x-axis and press ENTER .
292 Understanding our Quantitative World
The coordinates of the x-intercept are shown at the bottom of the screen. [Note: the
y value may be something very, very small like 1E-12 instead of zero. 1E-12 is the
number 1 � 10�12 D 0:000000000001.]
Constructing a Histogram
You can use your TI-83 to construct a histogram of the height data found in Table 4.2 on
p. 68. All 50 pieces of data can either be entered into one list or the data can be entered
in the form of a frequency distribution. We will illustrate the second method since it
requires fewer keystrokes. To do this, we will input the first column of Table 4.2 (the
heights) into L1 and the second column (the frequencies) into L2. (This information can
be found on pages 12-2 and 12-32 of the TI-83 Guidebook.)
1. To open the statistical list editor, press STAT and then select 1:Edit from the menu.
Press ENTER . Your cursor should now be at the first entry in L1.
2. If L1 and L2 are empty, you can input the data. (If L1 and L2 are not empty, they
need to be cleared first. To do this, press M CLEAR ENTER .) Enter the data by
inputting 6 0 ENTER 6 1 ENTER , and so on. In a similar fashion, input the
frequencies into L2. To get the cursor from L1 to L2, press B .
Appendix: Instructions for the TI-83 Graphing Calculator 293
3. To graph the histogram, you need to go to the STAT PLOT menu. To do this, press
2nd Y= . Now select Plot 1 by pressing ENTER . On this menu, we want the plot
turned On, and the Type to be a histogram. (To do this, use the arrow keys and the
ENTER button.) We also want the XList to be L1 and the Freq: to be L2. (L1 is
2nd 1 and L2 is 2nd 2 .)
4. We need to set an appropriate window to view our histogram. First, make sure all
other graphs and statistical plots are turned off. The easiest way to get a window that
is close to what we want is to press ZOOM 9 . This is the ZoomStat feature. You
should now see a histogram on your screen.
5. This default will not group our data into integer groupings (i.e., 60; 61; 62; : : : ; 72).
To do this, press WINDOW O O 1 . This makes Xscl=1 and causes the x-axis to
be divided into integer increments. Also, since the largest frequency is 9, our window
needs to be a little larger. Therefore, use the arrow keys and set Ymax=10. Press
GRAPH to view the new histogram. Notice that the TI-83 does not put labels on the
axes.
294 Understanding our Quantitative World
Constructing a Scatterplot
To use your TI-83 to construct scatterplots, you first need to enter your two-variable data
into two lists. We demonstrate this by constructing a scatterplot of the height and arm
span data found in Table 4.3 on p. 69. (This information can be found on page 12-3 of
the TI-83 Guidebook.)
1. Using the height and arm span data from Table 4.3, enter the height data in L1 and
the arm span data in L2 using the same method described earlier in this appendix
for entering the data for the histogram.
2. To set up the graph for the scatterplot, first go to the STAT PLOT menu. On this
menu select Plot 1 and make sure the plot is turned On. The Type should be a
scatterplot (the first one listed), the XList should be L1, 2nd 1 , and the YList
should be L2, 2nd 2 . We have chosen the Mark to be the +.
3. To set an appropriate window to view the scatterplot, press ZOOM 9 . This is the
ZoomStat feature. You should now see a scatterplot on your screen. (Note: You
should make sure all other graphs and statistical plots are turned off.)
Appendix: Instructions for the TI-83 Graphing Calculator 295
Constructing an xy -line
To use your TI-83 to construct an xy-line, you first need to enter your two-variable data
into two lists. We will demonstrate this by constructing an xy-line of the natural gas data
found in the following table. (This information can be found on page 12-31 of the TI-83
Guidebook.)
1. The cost of natural gas per month is given in the following table. To make an xy-line
of these data, input the month data in L1 and the cost of the gas in L2. Make sure
any previous data is deleted.
Month Gas Month Gas
1 $27.73 13 $32.15
2 $19.73 14 $16.48
3 $11.30 15 $12.92
4 $11.76 16 $12.42
5 $12.81 17 $12.92
6 $23.96 18 $15.49
7 $34.16 19 $29.34
8 $50.85 20 $57.57
9 $75.87 21 $58.15
10 $75.29 22 $59.62
11 $72.73 23 $53.95
12 $45.44 24 $43.60
2. To set up the graph for the xy-line, go to the STAT PLOT menu. On this menu
select Plot 1 and make sure the plot is turned On. The Type should be an xy-line
(the second one listed), the XList should be L1 and the YList should be L2. We
have chosen the Mark to be the +.
296 Understanding our Quantitative World
3. To set an appropriate window to view our xy-line, press ZOOM 9 . This is the
ZoomStat feature. You should now see the xy-line on your screen. (Note: You
should again make sure all other graphs and statistical plots are turned off.)
Computing One Variable Statistics
To use your TI-83 to compute mean, median, and standard deviation, you first need to
enter your data into a list. We will demonstrate this with the following set of numbers:
3; 6; 9; 2; 3; 6; 7; 8:
1. To open up the statistical editor, press STAT and select 1:Edit from the menu. Press
ENTER . Your cursor should now be at the first entry in L1. If L1 is empty, you can
just proceed to input the data. (If L1 is not empty, it needs to be cleared first. To do
this, press M CLEAR ENTER .)
2. Once the data is entered, press STAT and move the cursor over to theCALC menu by
pressing B . Press ENTER to set up the calculator for finding statistics for 1-variable
data.
Appendix: Instructions for the TI-83 Graphing Calculator 297
3. To get the calculator to calculate values for L1, you need to input L1 after 1-Var
Stats. Do this by pressing 2nd 1 .2 Now pressing ENTER will give you a list of
statistics. The first item on your list, x D 5:5, is the mean, the fifth item on your list,
�x D 2:397915762 is the standard deviation, and if you scroll down by pressing Oyou will find, Med D 6, which is the median. You can also find values for Q1 (the
first quartile) and Q3 (the third quartile).
The meaning of each of the 1-Var Stats are:
x meanPx sum of the data pointsPx2 sum of the squares of the data points
Sx standard deviation for a sample
�x standard deviation
n number of items in your data set
mi nX smallest value in your data set
Q1 first quartile
Med median
Q3 third quartile
maxX largest value in your data set
[Note: The standard deviation we defined in this book, �x, is for a population rather
than a sample. When calculating the standard deviation for a sample, Sx, you divide
the sum of the squared differences by n � 1 rather than by n.]
4. You can also use your calculator to find these same statistics if the data are given in
a frequency distribution. Suppose we wanted to find the mean of the following data
representing heights, in inches, of a group of students.
Height 66 67 68 69 70 71 72
Frequency 2 4 7 8 6 2 4
To do this, put the heights in L1 and the frequencies in L2. Once the data is entered,
press STAT and move the cursor over to the CALC menu by pressing B and then
ENTER . To get the calculator to calculate values for L1 and L2 you need to input
L1,L2 after 1-Var Stats. Do this by pressing 2nd 1 , 2nd 2 . Now pressing
ENTER will give you a list of statistics.
2 The calculators default list is L1, so if your list of numbers is in L1, you do not really need to enter L1.
However if it is any other list, you must identify that particular list.
298 Understanding our Quantitative World
Computing the Regression Equation and the Correlation
To use your TI-83 to compute the linear regression equation and the correlation, you first
need to enter your data into two lists. We will demonstrate this with the data from the
following table.
Height 152 160 165 168 173 173 180 183
Arm Span 159 160 163 164 170 176 175 188
1. Before you enter the data in the calculator, you need to make sure it is set in the
right mode to calculate correlation.3 This is done by turning the “diagnostic” on.
To do this press 2nd 0 . This is the catalog. It is a list of calculator commands
in alphabetical order. Toggle down to DiagnosticOn by holding down O . Then
press ENTER ENTER and the calculator will now be in the proper mode to calculate
correlation.
2. To open up the statistical editor, press STAT and select 1:Edit from the menu by
pressing ENTER . Input the height data inL1 and the arm span data inL2. If necessary,
first clear lists L1 and L2 by using the arrow keys to scroll the cursor to the top of
the list and then press CLEAR ENTER .
3 This step is specifically for the TI-83. If you are using a TI-82, skip to number 2.
Appendix: Instructions for the TI-83 Graphing Calculator 299
3. Once the data is entered, press STAT and move the cursor over to the CALC menu
by pressing B . Now press O three times so that the cursor is on 4:LinReg(ax+b).
4. Press ENTER to get back to the home screen. To get the calculator to compute
the regression equation and correlation for L1 and L2 you need to input L1,L2 after
1-Var Stats.4 By having the lists in the order L1,L2, your calculator will make the L1
list (height) the input or independent variable and the L2 list (arm span) the output
or dependent variable when it calculates the regression equation. To calculate the
regression equation press 2nd 1 , 2nd 2 . Pressing ENTER will give you the
slope and y-intercept for the regression equation and the correlation. The regression
equation for this set of data can be written as y D 0:874575119x C 21:35316111.
The correlation is 0:9043440664.
5. To graph the regression line along with a scatterplot of the data, you can type in the
equation manually or use a short-cut. The short-cut allows you to automatically have
the regression equation stored as a function so you can easily graph it. To do this, go
back to the previous step after you have LinReg(ax+b) L1,L2 on your screen. You
4 The calculators default list is L1 and L2, so if your list of numbers is in L1 and L2, you do not really need
to enter L1,L2. However if it is in any other lists, you must identify those particular lists.
300 Understanding our Quantitative World
now need to insert ,Y1 after LinReg(ax+b) L1,L2. To do this press , VARS BENTER ENTER . Your screen should look like the following picture on the left. Now
by pressing ENTER , the calculator will calculate the regression equation and store
this equation in Y1. Graph this along with the scatterplot by following the directions
given earlier in this appendix.
Finding an exponential regression equation
1. Press STAT then choose Calc.
2. Choose 0: ExpReg.
3. Type L1,L2 if these are the two lists containing your data. Also include the variable
Y1 so the calculator will store the exponential function as Y1 (which makes it easier
to graph). Do this by pressing VARS and then choosing Y-VARS, 1:Function, then
1:Y1.
4. Press return. The calculator will give you the values of the y-intercept and growth
factor. [Note: The letters for the constants that the calculator uses are opposite those
used in the reading.]
5. Graph the scatterplot together with the exponential function given by the calculator.
Finding a power regression equation
1. Press STAT then choose Calc.
2. Choose A: PwrReg.
3. Type L1,L2 if these are the two lists containing your data. Also include the variable
Y1 so the calculator will store the power function as Y1 (which makes it easier to
graph). Do this by pressing VARS and then choosing Y-VARS, 1:Function, then
1:Y1.
4. Press return. The calculator will give you the values of the coefficient and the
exponent for your power function. [Note: The letters for the constants that the
calculator uses are different from those used in the reading.]
5. Graph the scatterplot together with the power function given by the calculator.
Index
Acceleration of a car, 31
Alternating current, 229
Amplitude, 216
Association, 69
Baby weights, 254
Base, 164
Baseball salaries, 101
Basketball players statistics, 82
Basketball salaries, 80
Beach ball, 248
Benford’s law, 206
Bias, 277
Biorhythms, 228
Boxes, 282
Boyle’s Law, 120
Briggs, Henry, 200
Calorie content, 95
Carbon dioxide, 187
Carbon emissions, 151
Car loan, 121
Cartesian coordinate system, 21
Cell phone rates, 18
Cell phone vs. Phone card 19
Chicken bacteria, 184
Columbia House 19
Complement, 259
Concavity, 25
Confidence interval, 274
Confidence level, 274
Contour curves, 109
Cooling, 192
Correlation, 149
Cosine, 219
Currency exchange rates, 140
Dance injuries, 38
Daylight, 230
Decibels, 203
Decreasing function, 25
Dice, 271
Discover Card 15
Domain, 7
DVD player, 65
Earthquakes, 210
Electric bill, 79, 142
Enrollment, 160
Event, 257
Exam grades, 102
Expected value, 263
Exponential function, 164
Exponents, properties of, 234
Fat percentages, 17
Ferris wheel, 222
Fifth-Third Bank Run, 79
Fish populations, 119
Frequency distribution, 67
Function, 3
Function notation, 10
Garage door, 267
Gasoline prices, 42, 76, 136
General counting method, 257
Graphs, poor, 36
Growth chart, 102
Growth factor, 162
Heads or tails, 184
Height vs. Weight, 63
Heights of couples, 78
Histogram, 68
Horizontal asymptote, 167
House prices, 72
Increasing function 25
Independent events, 260
Indy 500, 185
Inflation, 177
Intercepts, 27
Internet costs, 11
Interquartile range, 91
Kepler, Johannes, 242
Koebel, Master, 136
Land area vs. population, 249
Least-squares regression line, 147
Linear function, 124
Logarithm function, 196
Lotto, 266
Magnitude, 195
Mathematical predictions, 16
Mean, 85
Mean of a probability distribution, 261
301
302 Understanding our Quantitative World
Measurement, 156
Median, 85
Medical testing, 271
Mile, running the, 157
Miles per gallon, 143
Modeling clay, 120
Modeling functions, 5
Motion detector, 65, 140
Mount Rainier, 119
Movies, 252
Multistage experiment, 259
Multivariable function, 105
Museum attendance, 39
Napier, John, 200
Newspaper search, 37
Normal curve, 93
Olympic 100-meter run, 155
Origin, 21
Outlier, 87
Overweight Americans, 37
Ozone layer, 159
Parameter, 274
Pendulums, 247
Percentile, 91
Period, 213, 216
Periodic function, 213
pH, 208
Phone rates, 15
Piecewise functions, 10
Pixels, 27
Planets, 242
Pneumonia graph, 35
Population, 186, 273, 282
Power function, 234
Power regression, 240
Prison population, 39, 190
Probability distribution, 261
Probability of an event, 258
Proportion, estimating a 284
Quartile, 91
Race, 64
Radians, 221
Range, 7, 89
Rational number, 235
Recycling, 285
Regression, 145
Religion polls, 279
Roulette, 271
Sample, 273
Sample space, 257
Scatterplot, 69
Seed variability, 208
Shifting functions, 49, 173
Simple random sample, 276
Sine, 219
Skewed, 88
Sleep cycles, 41
Slope, 125
Slope-intercept form of a line, 127
Smoking trends, 158
Sound, 226
Species introduced to Great Lakes, 40
Sports, 269
Standard deviation, 89, 102
Statistic, 274
Stratified random sample, 277
Streaks, Joe DiMaggio, 263
Surface area and volume, 251
Textbook prices, 80
Thermostat, 57
Thrillers, 100 best, 100
Time management, 284
Topographical maps, 108
Trend line, 71
Tuition rates, 16
Tuition vs. inflation, 189
Variability, 277
Variance, 89
Vertical line test, 4
Viewing window, 28
Weather maps, 110, 112
Weather trends, 42
Wind chill, 107
x-intercept, 27
xy-line, 71
y-intercept, 27, 125
About the Authors
Janet Andersen has been a member of the Hope College Mathematics Department since1991, Director of the Pew Midstates Science and Mathematics Consortium since 2002,
Chair of the Mathematics Department from 2000 to 2004, GEMS (General Education
Mathematics & Science courses) Coordinator from 1996 to 2001, and Director of General
Education from 1998 to 2000. She taught high school in East Texas for four years before
attending graduate school at the University of Minnesota.
She has been the Principal Investigator for three NSF curriculum grants. The second
grant, awarded in 1997, led to the development of a general education mathematics course
tied to two general education science courses at Hope College. Her co-author, Todd
Swanson (Mathematics) collaborated with her on this project, along with, Ed Hansen
(Geological and Environmental Science), and Kathy Winnent-Murray (Biology). The
materials from the mathematics course are contained in Understanding our Quantitative
World. The first NSF grant, awarded in 1993, resulted in Projects for Precalculus and
Precalculus: A Study of Functions and Their Applications. Her third grant, awarded in
2000, resulted in the development of a co-taught mathematical biology course. She also
enjoys being with her family, contra dancing, playing Euro board games, and reading
mysteries.
Todd Swanson received a BS in mathematics from Grand Valley State University in
1985 and then taught high school mathematics for two years. He received an MA in
mathematics from Michigan State University in 1989 (where he received the Excellence
in Teaching Award for Senior Graduate Students). He has taught at the college level since
1989 and has been at Hope College since 1995.
His other books, Projects for Precalculus (published in 1997 and awarded the
Innovative Programs Using Technology Award) and Precalculus: A Study of Functions
and Their Applicationswere co-authored with Janet Andersen (Hope College) and Robert
Keeley (Calvin College).
Much of Todd Swanson's teaching time at Hope is devoted to Introductory Statistics.
He has written numerous laboratories that involve the incorporation of Minitab and are
aimed at trying to get students to understand the concepts while exploring real world data.
He has also taught liberal arts mathematics, precalculus, calculus, mathematics education
courses, and an introduction to writing proofs. Outside of work he can be found working
around the house, transporting one of his children to soccer or baseball practice, and
participating in some outdoor activity.
303
This book is intended for a general education mathematics course. The
authors focus on the topics that they believe students will likely encounter
after college. These topics fall into the two main themes of functions and
statistics. After the concept of a function is introduced and various repre-
sentations are explored, specific types of functions (linear, exponential,
logarithmic, periodic, power, and multi-variable) are investigated. These
functions are explored symbolically, graphically, and numerically and are
used to describe real world phenomena. On the theme of statistics, the
authors focus on different types of statistical graphs and simple descrip-
tive statistics. Linear regression, as well as exponential and power regres-
sion, is also introduced. Simple types of probability problems as well as
the idea of sampling and confidence intervals are the last topics covered
in the text. The book is written in a conversational tone. Each section
begins by setting the mathematics within a context and ends with an
application. The questions at the end of each section are called “Reading
Questions” because students are expected to be able to answer most
of these after carefully reading the text. “Activities and Class Exercises”
are also found at the end of each section. These activities are taken from
public sources such as newspapers, magazines, and the Internet. Doing
these activities demonstrates that mathematics can be used as a tool in
interpreting quantitative information encountered in everyday life. The
text assumes that students will have access to some type of technology
such as a graphing calculator.
AMS / MAA TEXTBOOKS