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

Washington, DC 20090-1112

1-800-331-1MAA FAX: 1-301-206-9789

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.


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/.


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.


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 .


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 .


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.


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.)


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 +.


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.


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.


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.


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.


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


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

Understanding Our Quantitative World

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