1. No category

Common Sense Mathematics
Instructor’s Manual
Ethan D. Bolker
Maura Mast
December 20, 2014
c 2014 Ethan Bolker, Maura Mast
2
Contents
About Common Sense Mathematics
9
A class on GPAs
19
A class about runs
21
Page by page comments on the text
25
1 Calculating on the Back of an Envelope
27
1.1
Billions of phone calls? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
1.2
How many seconds? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
1.3
Heartbeats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
1.4
Calculators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
1.5
Millions of trees? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
1.6
Carbon footprints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
1.7
Kilo, Mega, Giga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
1.8
Comments on the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2 Units and Unit Conversions
45
2.1
Rate times time equals distance . . . . . . . . . . . . . . . . . . . . . . . . .
45
2.2
The MPG illusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
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2.3
Converting currency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.4
Unit pricing and crime rates . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.5
The metric system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.6
Working on the railroad . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.7
Scientific notation, powers of 10, milli and micro . . . . . . . . . . . . . . . .
46
2.8
Carpeting and paint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.9
Comments on the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3 Percentages, Sales Tax and Discounts
55
3.1
Don’t drive and text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.2
Red Sox ticket prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.3
Patterns in percentage calculations . . . . . . . . . . . . . . . . . . . . . . .
55
3.4
The one-plus trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.5
Two raises in a row . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.6
Sales tax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.7
20 percent off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.8
Successive discounts
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.9
Percentage points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.10 Comments on the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
4 Inflation
73
4.1
Red Sox ticket prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
4.2
Inflation is a rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
4.3
More than 100%
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
4.4
The consumer price index . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
4.5
How much is your raise worth?
74
c 2014 Ethan Bolker, Maura Mast
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4.6
The minimum wage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
4.7
Comments on the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
5 Average Values
79
5.1
Average test score . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.2
Grade point average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.3
Improving averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.4
The consumer price index . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.5
New car prices fall . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.6
An averaging paradox
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.7
Comments on the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
6 Income Distribution – Excel, Charts and Statistics
87
6.1
Salaries at Wing Aero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
6.2
What if? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
6.3
Using software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
6.4
Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
6.5
Bar charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
6.6
Pie charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
6.7
Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
6.8
Mean, median, mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
6.9
Computing averages from histograms . . . . . . . . . . . . . . . . . . . . . .
88
6.10 Percentiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
6.11 The bell curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
6.12 Margin of error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
6.13 Bimodal data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
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c 2014 Ethan Bolker, Maura Mast
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6.14 Comments on the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Electricity Bills and Income Taxes – Linear Functions
89
97
7.1
Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
7.2
Reading your electricity bill . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
7.3
Linear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
7.4
Linear functions in Excel . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
7.5
Comparing electricity bills . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
7.6
Energy and power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
7.7
Taxes: sales, social security, income . . . . . . . . . . . . . . . . . . . . . . .
98
7.8
Comments on the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
8 Climate Change – Linear Models
109
8.1
Climate change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.2
The greenhouse effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.3
How good is the linear model? . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.4
Regression nonsense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.5
Comments on the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
9 Compound Interest – Exponential Growth
117
9.1
Money earns money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.2
Using Excel to explore exponential growth . . . . . . . . . . . . . . . . . . . 117
9.3
Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.4
Doubling times and half lives . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.5
Exponential models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
9.6
Comments on the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
c 2014 Ethan Bolker, Maura Mast
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10 Borrowing and Saving
127
10.1 Credit card interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
10.2 Can you afford a mortgage? . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
10.3 Saving for college or retirement . . . . . . . . . . . . . . . . . . . . . . . . . 127
10.4 Annual and effective percentage rates . . . . . . . . . . . . . . . . . . . . . . 127
10.5 Instantaneous compounding . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
10.6 Comments on the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
11 Probability – Counting, Betting, Insurance
131
11.1 Equally likely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
11.2 Odds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
11.3 Raffles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
11.4 State lotteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
11.5 The house advantage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
11.6 One time events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
11.7 Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
11.8 Sometimes the numbers don’t help at all . . . . . . . . . . . . . . . . . . . . 132
11.9 Comments on the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
12 Break the Bank – Independent Events
12.1 A coin and a die
135
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
12.2 Repeated coin flips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
12.3 Double your bet? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
12.4 Cancer clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
12.5 The hundred year storm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
12.6 Improbable things happen all the time . . . . . . . . . . . . . . . . . . . . . 136
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12.7 Comments on the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
13 How Good Is That Test?
139
13.1 UMass Boston enrollment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
13.2 False positives and false negatives . . . . . . . . . . . . . . . . . . . . . . . . 140
13.3 Screening for a rare disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
13.4 Trisomy 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
13.5 The prosecutor’s fallacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
13.6 The boy who cried “Wolf” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
13.7 Comments on the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Index
c 2014 Ethan Bolker, Maura Mast
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8
About Common Sense Mathematics
The art of teaching requires the instructor to guide his student to work
independently to discover principles for himself, and in time to acquire the power
of principles to the manifold situations which may confront him.
James Brander Matthews (1852 – 1929)
Introduction
We understand that if you’re one of those people who skips the instructions when assembling
the new porch furniture, or the manual when you get a new cell phone or the on line help for
a software application, you may never read this.1 That said, we’ll try to make it worth your
while.
Both you and your students should understand from the start that this is not a traditional
math text for a traditional math course. We try to make that clear in the Preface, and in
the first few classes at the beginning of each semester. Students rarely believe us. Many
complain half way through the semester that this math course isn’t like any other they’ve
taken. Where are the formulas? As instructors we often find it hard to believe too. Since
we’re mathematicians, we’re tempted to think (semi)formal mathematics is both more useful
and more important than it really is. That comforts us, since it’s something we know how to
teach 2 so we fall back on it when the real quantitative reasoning issues seem too messy and
frustrating to tackle.
There is real mathematical content here – lots of overlap with what you find in most
quantitative reasoning and some liberal arts texts, perhaps with an occasional favorite
topic missing, often something more advanced that you might not expect to find. But the
mathematics is embedded in discussions of issues most of which most students find genuinely
interesting.3
1
2
3
Please pardon the self-referential paradox.
Or at least think we know how to teach.
But you can’t please all of the people all of the time . . .
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There’s much to gain from the fact that this isn’t a regular mathematics course like calculus
or linear algebra, or even college algebra, where some set of topics must be covered to prepare
students for the next course. At UMass Boston there is no next course – students are here
just to satisfy the quantitative reasoning requirement. So we are (rather ambitiously) trying
to prepare students for life. Learning to think a few ideas through is more important than
exposure-for-the-record to lots of ideas.
The text presents each topic in the context of quotations from the media to be understood.
Only occasionally we thought it better to make up a problem in order to present a particular
idea. In principle, you should replace many of the stories in the text with current ones as
you teach. That’s difficult; we do it ourselves less often than we’d like. There are, after all,
advantages to teaching from the text: the material there is class tested. You avoid the risk of
having to muddle through an example you thought would work – until you tried it the first
time. Moreover, the students have something to read to go along with the class discussion.
Save your ingenuity for new homework exercises based on current local news stories.
Unfortunately, many of the instructors teaching quantitative reasoning are underpaid adjuncts
stitching together multiple jobs to eke out a living. Perhaps it’s unfair to ask them to spend
the extra time it can take to teach from our text. We hope that the increased satisfaction
can be its own reward. Fortunately, it does become easier with time.
Constructing a syllabus
The semester at UMass Boston is fourteen weeks long, which suggests that a chapter a week is
about the right pace. At that pace we find it impossible to cover all of each chapter. Instead
we choose a few sections or topics that we think will particularly interest the particular group
of students that semester, and think about them thoroughly. Sometimes we build a class
around one of the exercises.
The chapters fall naturally into three groups.
• Chapters 1-4 deal with numbers a few at at time. The central concepts are estimation,
working with units, absolute and relative change and percentages. The first chapter, on
Fermi problems, sets the tone for the entire text.
• In Chapters 5-10 we work with sets of of numbers. Chapter 5 on averages introduces
weighted averages as a more useful concept than the simple mean. It also provides a
bridge to Chapter 6, where we use Excel to study the mean, median and mode for real
data sets, to ask “what-if” questions and to draw histograms. The following chapters
introduce linear and exponential functions in useful real world contexts, focusing on
implementing and graphing them in Excel rather than on more traditional algebraic
c 2014 Ethan Bolker, Maura Mast
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CONTENTS
treatments. The algebra does enter surreptitiously in the form of cell references in
Excel. We think a spreadsheet program rather than a graphing calculator for more
advanced calculations is a much better long term time investment for the students.
• Chapters 11-13 cover probability. The first starts with dice and coins where counting
outcomes solves problems and ends with insurance and extended warranties where
probabilities are essentially statistical. The next two chapters address the frequency of
rare events like runs and the ease with which you can construct misleading arguments
based on the misuse of conditional probability – of course all done without formal
definitions.
Where’s the math?
Answer: embedded in the real applications. You will find (among other things, and not
necessarily in order of presentation or importance)
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Estimation and mental arithmetic
Scientific notation
Unit conversions and the metric system
Weighted averages
Descriptive statistics (mean, median, mode)
The normal distribution
Linear equations and linear models
Exponential equations and exponential models
Regression
Elementary probability
Independent events
Bayes’ theorem
Logical thinking
The geometry of areas and volumes
Some mathematics you might expect is missing – primarily because it fails the “should your
students remember this ten years from now?” test:
•
•
•
•
•
Formal logic and set theory (and most other formal mathematics)
Laws of exponents (“laws” of just about any kind)
Quadratic equations
The algebra of polynomials
Traditional “word problems”
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Homework exercises
so much depends
upon
a red wheel
barrow
glazed with rain
water
beside the white
chickens.
That’s William Carlos Williams’ The Red Wheelbarrow. 4 We quote it here for the “depends.”
We believe every interesting question has the same answer: “it depends” – if you can look up
the answer or find it with easy arithmetic the question isn’t interesting. So answers to exercises
call for complete sentences – even complete paragraphs. Just circling the (presumably correct)
numerical answer isn’t sufficient. Each answer should effectively restate the question. We
suggest that students write enough prose so that they can use their homeworks to study for
the exams without having to return to the text to reread the questions. We return corrected
homeworks promptly, and post solutions (available in the solutions manual!) that model
what we expect from them. We encourage students to turn in word-processed documents.
Since writing mathematics in those documents is tedious, we suggest that they leave blank
space for the calculations and equations and fill them in legibly by hand.
There are many more exercises in the text than you can use in any one semester. There are
more ideas for exercises in this manual.
We find that weekly homework tends to work best. We’ve taken to spending some of the
time flipping the classroom: having students start homework exercises in groups in class,
asking for help when they need it. We always encourage students to start each assignment
when it’s posted, so that they can ask questions before the due date.
Most homework exercises relate to material already taught. We also find it helpful to assign
some on topics we haven’t yet reached in class. Students are then much better able to
participate in discussions.
Goals
Teaching from Common Sense Mathematics is more work than teaching from a standard text
with problems at the end of each section that test the mastery of basic techniques rather
than the understanding of important ideas.
4
http://www.poetryarchive.org/poetryarchive/singlePoem.do?poemId=7350
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We’ve tried to provide some help making those ideas explicit. At the start of each chapter
you’ll find that chapter’s goals – the essential ideas we hope the students will begin to master,
and the particular mathematics needed for that task.
To help you construct assignments, we’ve tagged the exercises to suggest which goals they
tend to address and which sections they relate to – often across chapters.
Instructors using Common Sense Mathematics have found that their students often need
practice with routine arithmetic and algebra in order to come to grips with the ideas that are
the focus of the text. To address that need, we have included a few review exercises at the
ends of some of the chapters. This kind of review is often best addressed with problems with
online software support. We recommend that you use them sparingly, and only as necessary –
that is, after the need is clear. We sometimes ask students to read the review problems and
decide for themselves whether they need to work some and turn them in for feedback.
If you assign review problems first, before the applications in the text, your students may
think this is a math course like any other – formal skills to master with no thought as to
their usefulness or meaning.
Exams
We usually schedule two hour long examinations and a final exam. It’s hard to make much
use of ten minute in class quizzes to test the kinds of analysis we are trying to teach.
Our exams take place in a computer lab. We allow “open everything” – textbook, class
notes, internet, but, of course no texting or email asking for help. When students react with
pleasant surprise to this announcement we point out that this is exactly what they will have
available when they tackle real problems in life – which is what we want to teach and test.
We remind them that allowing access to all these resources means we can ask harder problems
than they may be used to in an exam. In fact it is difficult to construct exams that truly test
the use of common sense mathematics.
Here is the preamble to our first exam:
General guidelines
• When you’ve solved a problem (perhaps on scrap paper), write the answer
out neatly on the paper with the problem (you can use the other side too).
Don’t just circle a number. Show all units, and write complete sentences. If
you’ve used any technology, say so.
• The purpose of this course is to help you learn how to use quantitative
reasoning principles to solve real problems that matter to you. An exam
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can’t test that well because you must answer the questions quickly. Here’s
a compromise. For homework for Thursday, rethink your answers. If you
can write better ones, submit them. (Don’t redo problems you got right the
first time.) I will correct both the exam and the resubmissions. Getting a
problem right the second time isn’t worth as much as getting it right the
first time, but it can make a difference in your grade. The exam is posted
on the course web page.
Work independently. You can email me with questions, but don’t consult with
friends or classmates or tutors.
• Google (and the internet), calculators, class notes and the text are all OK.
Make sure you acknowledge any help of this kind. But take care. Time
spent searching the web or shuffling through notes is time you’re not using
to answer the questions. Of course you can’t use the computer to exchange
email with your classmates during the exam. No text messages either, please.
• Remember to show only the number of significant digits (precision) in your
answer justified by the numbers you start with and the estimates you make.
Remember to use the equal sign only between numbers that are equal, not as
a substitute for words that explain what the numbers mean and what you
are doing.
The first question on the exam is
1. (5 points) Read the general guidelines - particularly the first two about
the form your answers should take, and the chance to improve your answers
between now and Thursday. Write “I understand the instructions” here for
a free 5 points.
Sad note: All students write a correct answer to this first question but many have not read
or understood the instructions!
Vocabulary
From G. K. Chesterton’s The Scandal of Father Brown:
Father Brown laid down his cigar and said carefully: “It isn’t that they can’t
see the solution. It is that they can’t see the problem.” 5
5
http://gutenberg.net.au/ebooks02/0201031.txt
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Teaching from early drafts of Common Sense Mathematics we found that students often had
so much trouble with vocabulary that they couldn’t even get to the quantitative reasoning.
Here’s a blog entry that addresses that issue, from the sixth class of the semester.
I spent almost all of the rest of the class working the Exercise on Goldman Sachs
bonuses (I promised that on Tuesday). I knew that the difficulty was in reading
the words around the numbers more than in the manipulations themselves. I was
surprised at how important just plain vocabulary problems were. In particular,
some students thought “consistent” meant “the same from year to year” (which
Goldman Sachs’ data aren’t) and not (when applied to numbers) something like
“fit together the way they should.” Later some people didn’t quite grasp that
“salaries plus bonuses”, “compensation” and “what GS paid employees” were all
referring to the same quantity.
The same kind of problem came up in a previous class about the meaning
of “wholesale.” Since I can’t anticipate all the words students might not know
(both in the course and after they leave) I hope I’ve convinced them that they
can’t make sense of paragraphs with numbers in them unless they check out the
meanings of words they’re unsure of. That said, I will try not to use fancy ones
too often.
One student called the need to think about both the words and the numbers
a perfect storm. I hope not.
...
Notes later. When I described today’s class to my highly educated wife she
said she’d have had the same trouble as some of the students with the meaning I
attached to “consistent.” She did agree after we looked it up in our (hard copy)
dictionary that I’d used it correctly – but that it was unreasonable of me to
assume my students would have been able to. She suggested I bring a dictionary
to class, and a thesaurus too. I pointed out that we already have a dictionary
in class – on line – and that we should have used it right then and there. At
dictionary.com the first meaning is
1. agreeing or accordant; compatible; not self-contradictory:
His views and actions are consistent.
The “not self-contradictory” would have cleared things up right away.
The term paper
We assign one each semester; students choose a topic that they care about, with some
guidance from us about what kinds of topics are suitable. We allow students to work together
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CONTENTS
in pairs if they wish. Some instructors allow, encourage or require students to study some
appropriate topic in a group and present to the class.
We pace the students through this significant project by asking them to submit several
topic ideas about halfway through the semester. Then we give individual feedback on each
student’s choice. A few weeks later we require an outline or sketch that poses the questions
to be answered (with guesses as to the answers) and identifies data sources. Near the end of
the semester students turn in a draft.6 The final version is due at the final exam.
Here are some excerpts from term paper instructions we offer our students.
One of the important parts of this course is the term paper. Yes, you didn’t
expect a term paper in a math course. But this one is about quantititative
reasoning about things that matter in the real world. Your paper will give you a
chance to practice that.
You will choose a topic, find some data and quantitative information about it,
perhaps form a hypothesis, explore “what-if” questions, make estimates, analyze
data, and draw conclusions. In other words, you will use many of the techniques
and ideas of this course to make a quantitative analysis of a topic that interests
to you.
If you are going to use lots of data from the web to do your analyses (sports
statistics, poverty rates) you should not be typing it into Excel one number at
a time. Many websites let you download tables in csv format. “csv” stands for
“comma separated values” – and Excel can load those files. Even if csv is not
available there are tricks that let you cut data and then paste it into Excel. If
you show me your data source I can help with that.
You may work with a classmate and submit a joint paper.
What should I write about?
The best way to do well in this assignment is to write about something that
really matters to you. Here are some ideas suggested by classmates from previous
semesters. (This is not a list for you to choose from, it’s a guide as to the kinds
of topics that might - or might not - work.)
• Business plans.
What would it take to open a beauty salon? A bike store? A photography
business? Can my garage band make enough money to support me? Can
my rugby club or softball league break even sponsoring a tournament?
Each of these questions led to a good paper. The authors had to collect
information (often from personal or job experience or a friend in the business),
build a spreadsheet, ask what-if questions and analyze the outcomes. They
were successful because they had access to the data and enough knowledge
of the activity to make sense of it.
6
You may want to arrange some peer review.
c 2014 Ethan Bolker, Maura Mast
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Your business plan probably has two parts. The first is the estimate for the
startup costs. The second is the estimate of the cash flow in and out once
the business is up and running. I strongly suggest you focus on the second
part. For startup costs, just imagine you will have to borrow the money, and
put the monthy loan payment down as an expense in your monthly cash flow
spreadsheet. You can vary that amount to see how much you could afford
to borrow.
Whatever your business (dog training, personal fitness, growing marijuana,
...) you should search on line for business plans in your kind of business.
They will give you some idea of the kinds of things you need to consider. Of
course real plans will call for a lot more detail than you can provide in your
paper.
• Can I afford to buy a house?
This is a common question and a common topic. Sometimes it works, but
most of the time it doesn’t. Much more than the cost of a mortgage is
involved. The best papers start by imagining lifestyles and family structure
and particular communities to live in and trying to quantify those in some
sense before plugging in numbers. There are templates on line that help
with the ongoing costs of home ownership.
• Sports.
There are lots of numbers on the sports pages. Students (mostly guys) really
care about them. That’s a good place to begin. But it’s only a beginning.
I’ve never seen a successful paper that tries to answer questions like “do the
teams with the highest salaries win the most?” or “are superstars worth the
big bucks?” I have seen a few good sports papers. If you want to try one
you have to start with smaller questions. And you must be careful to find
real data to think about. You can’t use the paper just to sound off about
your own firm opinions.
• Personal budgets.
This sometimes works and sometimes doesn’t. To do it well you have to
collect data on your actual income and expenses over a reasonable period of
time, estimate things you can’t pin down exactly, take into account large
expenses that don’t happen every week or month, build a spreadsheet and
then ask and answer reasonable “what-if” questions. There are many web
sites that offer Excel spreadsheets you can personalize and fill in to create
your budget. Look at several and find one that matches your needs. Don’t
try to build your own from scratch.
• Transportation
Can I afford to buy a car? Is it better to drive to school or take the T?
Often asked, usually not answered well. I’ve suggested to students tackling
it that they try to quantify the parts of the decision that aren’t monetary:
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time, convenience, . . . but no one has taken the suggestion. You should also
consider several alternatives – occasional taxis, zipcar – not just a simple
comparison of commuting costs.
• Current events topics.
You can do a good paper on a current controversy only if you phrase the
questions narrowly enough. One common error is to write a paper that’s just
a platform for expressing your own opinions, perhaps quoting experts with
whom you agree. I’ve seen that done on the legalization of marijuana, on the
incidence of rape or domestic violence, and on the cost of incarceration (from
a criminal justice major). You can’t do justice to global warming in a paper
for this course. You probably can’t do justice to energy independence. You
might be able to manage income inequality. On any of these topics you’d
do well to argue both sides of an issue, using data to support contradictory
opinions before you come to a conclusion.
• Sharing a paper with another professor
If there’s a topic that would work well in another course you are taking you
can consider writing about it if you clear that in advance with me and the
other instructor.
c 2014 Ethan Bolker, Maura Mast
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A class about GPAs
This posting from our teaching blog shows how we build a class around a couple of homework
problems. The students find these particular problems relevant – they care about their
GPAs but most don’t really understand how GPA is calculated or what they can do to
improve theirs. We use group work and encourage discussion; as this blog posting shows, this
sometimes leads to new or different approaches we hadn’t expected.
Class 10 – Thursday, October 3, 2013
7
From Maura:
I filled in for Ethan today, who couldn’t make it. He gave me two problems to have the class
work on, so that’s what we did. Both problems were about weighted averages. I passed them
out and asked them to work in groups to solve them. (They are in the book: Exercise 5.7.4
(page 117) and Exercise 5.7.8 (page 118) .)
The first problem built on what the class had done with GPA calculations before the exam.
The essential problem is that a student has 55 credits and a GPA of 1.8 (this is a cumulative
GPA – should have emphasized that). The student will take 12 credits and needs to raise the
GPA above 2.0 to avoid academic suspension. What semester GPA does the student need to
earn?
While they had done an example last week on calculating semester GPA, the students for the
most part weren’t able to make the leap to using cumulative and semester GPA information
together. Some got stuck on the number of courses that could make up 12 credits and what
individual grades the student should earn. I suggested they keep it simple and just think of
a 12-credit course. The other issue was that quite a few groups proposed a semester GPA
of 2.2 and argued that since (1.8+2.2)/2 = 2.0, that GPA should work. The tutor and I
talked them through the importance of weighting the 1.8 GPA by the 55 credits and how
they would do that. Once the groups heard that, they got the right idea. Most of the groups
used algebra to solve the problem, but a few did the guess-and-check approach. Both are
valid approaches. The algebra approach has the advantage of giving the answer as the lowest
possible GPA the student needs to earn. As for guess-and-check, many students took a 4.0
7
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19
CONTENTS
semester GPA and established that this would raise the overall GPA above 2.0. Well, yes,
but that’s not too realistic for a student who is on academic probation. I encouraged them
to refine their guess to get a bit closer to the minimum GPA – most settled on 3.0 as close
enough. I asked the groups to put their answers on the board and then we talked them
through. While all three groups used the algebra approach, one group used percentages to
represent the weights. This was for part (c), where the student takes 6 credits. The group
argued that 55 out of 61 credits represents just over 90% of the credits, while 6 out of 61
represents just under 10%. Then they finished the calculation. I liked this approach as it
illustrated very clearly how much weight is placed on the 1.8 GPA and should help students
see how the 1.8 GPA is is pulling the overall GPA down.
This exercise took a lot longer than I expected – almost 45 minutes. Part of the time was
spent talking about what they think they would do in that situation. Would they try 12
credits or focus on only 6? The answers varied, with good reasoning for both sides. I told
them that I’m the one who sends out the probation and suspension letters and my experience
is that it’s better for students to focus on a small number of classes and do well. I then told
them that if they repeat a course, the grade for the repeat is what’s included in their GPA
and the first grade is taken out of their GPA. They seemed surprised to hear this. The point
is that if you are selective and careful about courses you repeat, you can raise your GPA
fairly quickly.
Some of them had already moved on to the second question so I gave them some more time
on that. This is a paradoxical one. We have two students, Alice and Bob. Alice’s semester
GPA is better than Bob’s in both the fall and spring semesters, but overall Bob has the higher
cumulative GPA. The students were asked to invent courses and grades that illustrated this
paradox. It’s hard to imagine how this could be, until you start to take it apart. The initial
approach of many of the students was to keep everything the same in the two semesters – for
example, Alice takes 15 credits each semester and earns a 3.7 GPA each semester, while Bob
takes 12 credits each semester and earns a 3.5 each semester. With this approach, Alice’s
cumulative GPA will always be higher than Bob’s. To give Bob the edge, he needs to have a
lot of credits (that is, a large weight) with a GPA that is higher than one of Alice’s GPAs.
And Alice needs to have a lot of credits associated to that GPA. The trick is that Bob’s fall
semester GPA could be higher than Alice’s spring semester GPA. When we talked it through
in class, people protested that this wasn’t allowed. But in fact it’s legal and the only way to
give Bob a higher cumulative GPA. One student put an example on the board for us and
we could see how the weights made it work. As an extreme example, I encouraged them to
think of Bob taking 15 credits with a high GPA in one semester and only 1 credit with a low
GPA in the other semester. Balance Alice’s credits and GPA accordingly, and Bob will end
up the winner.
It was fun to revisit this group, several weeks after the beginning of the semester. It’s a good
group and I was impressed again at how well they were engaged with the material.
c 2014 Ethan Bolker, Maura Mast
20
A class about runs
If you can find time in your schedule when you’re teaching Chapter 12 this class exercise
may help your students learn a lot about Poisson processes – runs happen.
Everyone understands that the 50% chance of heads when flipping a coin once doesn’t mean
that heads and tails will alternate. But they often think, (subconsciously) that when they
see a lots of heads in a long sequence something happens to help the tails “catch up.”
We’ve designed an in-class experiment that demonstrates this belief and counter it.
Here’s an outline:
• Don’t describe the purpose of the experiment before you do it!
• Give each student a sheet of paper with an 8 × 8 grid of empty squares.
• Ask each student to imagine flipping an imaginary coin over and over again (64 times),
filling in his or her grid with “H” or “T” depending on how the imaginary coin falls.
Make sure the students understand that they’re to fill in the squares in “reading order”
– left to right in a row, then moving to the start of the next row. Some students will
want to fill in random squares – don’t let that happen. If you see students pondering
while filling in their grids you can encourage them to flip their mental coins faster.
• When all the grids are done, ask each to count the number of runs of four: four heads
in a row, or four tails in a row. Sequences don’t respect the ends of rows – ask students
to imagine that all 64 squares were arranged in a single line.
This fragment
H T T T T H H H
H H T ...
contains three runs – one of tails starting at the second position, two of heads starting
at the sixth and seventh positions.
It may take a while to make this counting process clear.
21
CONTENTS
• Query the class to build this data summary at the blackboard.8
number of runs
0
1
2
...
9
number of students
• Calculate the average number of runs – a good opportunity to review s.
• Calculate the expected number of runs per students. It’s not hard to convince the class
that the probability for n random flips to be all heads or all tails is just 1/2n (you can
count all the cases for n = 1, , 2, 3, 4. There are 61 places on each grid where runs of
four might start, so the expected number of runs is 61/8 ≈ 7.6.
You will (almost surely) find your experimental value significantly smaller.
• Now count the runs by column rather than by row, and retabulate. You will find many
more, although in our experience not usually enough to reach the expected average.
• Discuss.
The first time most students find very few runs of four. Why? You expect about the same
number of heads and tails. Your intuition says don’t put too many of each in a row. Your
vision of a fair coin is that it should alternate between H and T fairly often.
When you fill in by rows you switch more often than a real coin would. When you look by
columns you’re comparing what’s there to what you wrote 8 or so flips before. You didn’t
remember that far back when you were writing, so you get a more random distribution of Hs
and Ts and, as happens in nature, we will get runs of Hs and Ts.
The key point here is that runs do happen. We think of the probability of getting heads on a
coin toss as being 12 , and so we expect that we will not have a long run of heads, but the
rules of probability say that this will happen occasionally. You need to be aware that it is a
real possibility.
The Ideas section in The Boston Globe on Sunday, June 17, 2012 the carried a story on
creating a computer by harnessing human social behavior. 9 One quote caught our eye:
8
We know there are few blackboards these days. It’s probably a greenboard or a whiteboard.
https://secure.pqarchiver.com/boston-sub/access/2689315841.html?FMT=FT&FMTS=ABS:
FT&type=current&date=Jun+17
9
c 2014 Ethan Bolker, Maura Mast
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CONTENTS
We asked a couple hundred people to complete a string of 1’s and 0’s, and asked
them to make it ”as random as possible.” As it happens, people are fairly bad at
generating random numbers-there is a broad human tendency to suppose that
strings must alternate more than they do. And what we found in our Mechanical
Turk survey was exactly this: Predictably, people would generate a nonrandom
number. For example, faced with 0, 0, there was about a 70 percent chance the
next number would be 1.
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CONTENTS
c 2014 Ethan Bolker, Maura Mast
24
Page by page comments on the text
The pages that follow offer comments on the text, pedagogical tips and suggestions for the
class and sketches or outlines for possible extra exercises. When those exercises have solutions,
they appear here.
It’s generated from the TEX source for Common Sense Mathematics so that we can edit it
where it’s relevant. When page numbers there change the page references here change too.
For class-by-class stories about teaching from Common Sense Mathematics, visit our teaching
blog at http://quantitativereasoning.net/ethan-bolkers-quantitative-reasoning-teaching-blo
25
CONTENTS
c 2014 Ethan Bolker, Maura Mast
26
Chapter 1
Calculating on the Back of an
Envelope
1.1
Billions of phone calls?
This chapter and the next are closely related. You might want to combine them when
planning classes and homework assignments. It was often hard to decide which exercises
belonged here, which there.
Real Fermi problems ask for estimates from scratch. We concentrate instead on developing
Fermi problem techniques to verify claims in the media. That’s both easier for students,
and follows directly from our focus on working with numbers in the news – educating our
students to be consumers rather than producers of quantitative information.
The text in this Chapter stresses techniques: quick mental arithmetic, counting zeroes, some
scientific notation and the metric prefixes.
Note that many Fermi problems that used to require estimation skills now succumb to a web
query. For example, it’s easy to find a reliable count of the number of kindergarten teachers
in the United States.
There are many more exercises in this chapter than you can assign in any single semester.
Some of them can form the basis for a discussion that will fill a whole class period. You can
consider using those instead of the examples in the text.
27
1.2. HOW MANY SECONDS?
1.2
How many seconds?
This section makes an excellent class early in the semester. Tallying the number of students
who think millions, billions and trillions is a good place to start.
In class we call this “curly arithmetic”, since ≈ replaces =. The students then cheerfully use
the phrase in their homework.
1.3
Heartbeats
Heartbeats in a lifetime: Consider asking the students to take their pulses and report the
results. Collect the data so you can make them available in a spreadsheet for the class to
play with later when you introduce Excel. You might want to introduce median and mode
quickly – but don’t spend arithmetic time now on the mean.
When we first asked if two estimates of the same quantity were “consistent” we discovered
that many of the students didn’t know the word meant. Students’ limited vocabularies turn
out to be a source of confusion in many quantitative reasoning problems. It’s important to
be aware of that, and to encourage dictionary use. See the Vocubalary section above.
1.4
Calculators
When introducing the Google calculator, stress the advantage of a command line (search box)
over mouse clicks on buttons – faster, easier to fix mistakes, cut and paste to and from other
documents.
When converting heartbeats per day to heartbeats per minute the class will probably want
to go directly to “divide by 24, then divide by 60.” It may be worth a little blackboard time
to write it out with units to prepare for unit conversions later on.
1.5
Millions of trees?
1.6
Carbon footprints
When we teach this section in a course (often as the first class) we divide the students into
teams of three (since each table in the computer lab happens to have seats for three). Each
c 2014 Ethan Bolker, Maura Mast
28
1.7. KILO, MEGA, GIGA
team starts on one of the seven estimates with instructions to move on to the next one when
done, circling back to google search if they reach the end of the list. The teams start in
different places, so after about half an hour the class has found two or three estimates for
each of the seven tasks. The different answers for each task should have the same order of
magnitude. If they don’t, we try to figure out why not.
1.7
Kilo, Mega, Giga
1.8
Comments on the Exercises
13 We haven’t done this problem with a class yet. If you do, let us know what happened.
The author is probably right about the number. Does that mean that 1/6 of the population
are criminals? (Some of the fingerprints on file are for people who’ve died, and some are for
foreigners - but probably very few are of living children.)
The real story is probably that they have lots of fingerprints on file for folks who aren’t
criminals - e.g. people who have registered with selective service.
50 This is a particularly provocative example since almost everyone’s first guess – including
ours – is that it’s an exaggeration.
Extra (ideas for) Exercises
1.8
Exercise 1.8.59. [U][N] Pig-Crate Politics
Why a state that has almost seven times as many pigs as people should play
a bigger role in determining who gets to be president than one (New Jersey) that
has, say, 1,000 times as many people as pigs is another story, more about our
dysfunctional political system than raising animals. 1
Exercise 1.8.60. [U][N] How many backspaces?
From
http://blog.stephenwolfram.com/2012/03/the-personal-analytics-of-my-life/
1
http://www.nytimes.com/2014/12/03/opinion/christies-pig-crate-politics.html
29
c 2014 Ethan Bolker, Maura Mast
1.8. COMMENTS ON THE EXERCISES
For many years, I.ve captured every keystroke I’ve typed, now more than
100 million of them . . .
There are all kinds of detailed facts to extract: like that the average fraction
of keys I type that are backspaces has consistently been about 7% (I had no idea
it was so high!).
Lots of other stuff here too.
Exercise 1.8.61. [U][Section 1.1] [Goal 1.1] [Goal 1.4] [Goal 1.5] Astronauts’ Crowd Boils
Down to Simple Arithmetic
Mike Royko wrote a column with that headline for the Chicago Daily News on June 17, 1965.
Here are the first paragraphs.
The astronauts had hardly set foot in Chicago Monday when Jack Reilly, the
mayor’s official estimator of crowds, announced that 1,000,000 Chicagoans had
turned out to greet the space heroes.
A few hours later, Reilly got all excited and announced that he had been
wrong. Two million people had turned out.
Later in the afternoon, Mayor Daley got even more excited than Reilly, and
took over the crowd-estimating function, although he kept Reilly on the city
payroll. Daley said 2,500,000 had turned out to greet the space heroes.
They would have gone on like a pair of drunken pinochle players, shouting
“million, billion, trillion, zillion” at each other, had not the astronauts left town.
There is no question that it was an impressive turnout. It may have been,
as Reilly (or was it Daley?) said: The biggest welcoming reception in Chicago’s
history.
But was it 2,500,000? Or 2,000,000? Or even 1,000,000, which now sounds
like almost nothing.
I have taken a flight at finding the answer by using some fundamental and
highly questionable arithmetic, which is more than Reilly or the mayor did.
The astronauts were viewed in three basic areas: The airport, along the
Kennedy Expressway, and on the parade route downtown. Working backwards,
which is the way I like to work, let us start with the downtown parade route.
The city department of streets tells me that the average downtown sidewalk is
about 15 feet wide. The average city block is about 450 feet long.
For the entire 31 blocks, this would provide about 420,000 square feet of
sidewalk space. Let us assume that all of the spectators were dangerously skinny –
half-starved, in fact – and let us cram one into every square foot. It is impossible,
but let’s try it. That gives us 420,000 people.
Then there are those who watched from the office buildings. It is impossible
to be accurate, but let’s be generous and estimate all of the buildings to be 20
c 2014 Ethan Bolker, Maura Mast
30
1.8. COMMENTS ON THE EXERCISES
stories high, with about 50 windows fronting the street on a block. And let’s put
three people in every window, which is also overdoing it.
This gives us 180,000 window-spectators along the parade route.
Working back to the expressways, there are 31 overpasses, about 450 feet long,
with sidewalks about 12 feet wide. By using the same principle of one beanpole
per square foot, we have another 341,000 people, most of whom wouldn’t see
anything but someone else’s neck.
Then there are those who sat on the slopes of the expressway, where there
are slopes, or scrambled up to hang onto the guard rails where the expressway is
elevated.
We’ll exaggerate wildly and say that there were 3,000 every mile. This gives
us another 60,000 or so.
There were only a thousand or two at the airport, according to witnesses.
The whole thing adds up to about 1,000,000 people.
And since most people need more than one square foot of space, and there
weren’t three people in every window, and the overpasses weren’t that impossibly
jammed, it appears that there were considerably fewer than 1,000,000 out for the
astronauts.
Had there been 2,000,000 or 2,500,000 as Reilly and the mayor suggested, the
death toll among the weak and the small would have been too terrible to even
think about, even in this column.
It appears, then, that a more accurate guess of the turnout would be in the
neighborhood of 500,000 or 700,000.
(a) Comment on Royko’s assumptions. Check his arithmetic.
(b) Is Reilly’s first claim of 1,000,000 greeters a reasonable order of magnitude estimate?
(c) What was the population of the Chicago metropolitan area when this column was
written? How can you use that information to provide an alternate critique of the Mayor’s
claims?
(d) Estimate the size of the largest crowd that could view a parade in your home town.
Exercise 1.8.62. [U][R] [Section 1.3] Goldman Settles With S.E.C. for $550 Million
That’s what The New York Times reported on July 15, 2010. How much is that per person
in the United States?
Exercise 1.8.63. [U][C][G][Section 1.3][Goal 1.1] [Goal 1.2] [Goal 1.5] [Goal 1.8] Nuclear
bombs
“Kiloton” and “Megaton” are terms you commonly hear when nuclear bombs are being
discussed. In that context the “ton” refers not to 2,000 pounds, but to the explosive yield of
a ton of TNT.
31
c 2014 Ethan Bolker, Maura Mast
1.8. COMMENTS ON THE EXERCISES
The material in the exercise on the power of nuclear bombs makes a good class, combining
history, physics and quantitative reasoning about an important subject many current college
students know nothing about.
(a) What was the explosive yield of the only (two) atomic bombs ever used in war?
(b) How does the explosive yield of the hydrogen bombs in the current arsenals of the
United States and Russia (and other countries) compare to that of those first atomic
bombs?
(c) Estimate the destructive power of the world’s current stockpile of nuclear weapons, in
terms easier to grasp than kilotons or megatons or gigatons or . . . .
Exercise 1.8.64. [G][N][Section 1.6] Recycling
The February 28, 2009 issue of The Economist has enough information on waste and recycling
to generate as many Fermi problems as you can imagine. And there are ideas there that
could lead to interesting possible term papers, if you still need ideas.
http://www.economist.com/opinion/displaystory.cfm?story_id=13184704&CFID=48248106&
CFTOKEN=72987889
In this special report
*
*
*
*
*
*
*
*
*
Talking rubbish
You are what you throw away
Down in the dumps
Modern landfills
The science of waste
The value of recycling
Waste and money
Tackling waste
Sources and acknowledgments
http://www.economist.com/opinion/displaystory.cfm?story_id=13135349
Exercise 1.8.65. [U] [Section 1.5] Cheerio.
Here’s an exercise and its solution from a student.
2
(Edward McConaghy, Fall 2012)
c 2014 Ethan Bolker, Maura Mast
32
2
1.8. COMMENTS ON THE EXERCISES
How many boxes of Cheerios are sold each year?
If 1 person out of 30 (10,483,000 out of U.S. Population: 314,490,000) had
a 2oz bowl (8 servings per box) of Cheerios 3 times a week during a year, that
would equal 220,143,000 pounds or boxes (Cheerios happen to come in one pound
boxes).
Critique the solution.
Exercise 1.8.66. [U][N][Section 1.5] I.R.S.’s Taxpayer Advocate Calls for a Tax Code
Overhaul
From The New York Times, January 9, 2013:
In her legally required annual report to Congress, the national taxpayer
advocate, Nina E. Olson, estimated that individuals and businesses spend about
6.1 billion hours a year complying with tax-filing requirements. That adds up
to the equivalent of more than three million full-time workers, or more than the
number of jobs on the entire federal government’s payroll. 3
(a) Check the math: does 6.1 billion hours a year “add up to the equivalent of more than
three million full-time workers”?
(b) How many federal employees are there? Estimate or use the web to find out. Compare
your answer to three million.
(c) The IRS estimates how long it takes an individual to prepare and file annual income
taxes. They may also give an estimate for how much time a business needs to comply
with tax-filing requirements. Find these estimates and see if they help you verify Olson’s
claim of 6.1 billion hours.
1.8
Exercise 1.8.67. [U][Section 1.1][Goal 1.1] [Goal 1.2] [Goal 1.5] No strings
January 2011: The inner flap of the Celestial Seasonings Mint Magic Tea box tells this story:
Ever wonder why . . .
no string and tag?
Our unique pillow style tea bag is the result of our commitment to doing what’s
best for the environment. Because these natural fiber tea bags don’t need strings,
tags, staples or individual wrappers we’re able to save more than 3.5 million
pounds of waste from entering landfills every year!
3
www.nytimes.com/2013/01/09/business/irss-taxpayer-advocate-calls-for-a-tax-code-overhaul.
html
33
c 2014 Ethan Bolker, Maura Mast
1.8. COMMENTS ON THE EXERCISES
(a) Is that claim reasonable?
(b) If you find it unreasonable, or a close call, compose a letter to send to the company
explaining your puzzle and asking perhaps humorously but certainly politely for an
explanation or a correction.
(c) If your letter really pleases you, consider sending it to Celestial Seasons – you can find
an address at http://www.celestialseasonings.com.
Exercise 1.8.68. [S][Section 1.2][Goal 1.1] [Goal 1.2] [Goal 1.4] [Goal 1.6] Too many plastic
bags.
An article from October 3, 2007 in The Washington Post about how paper and plastic bags
impact the environment, stated that: “Worldwide, an estimated 4 billion plastic bags end up
as litter each year. Tied end to end, the bags could circle the Earth 63 times.”
(a) Are the numbers “4 billion plastic bags” and “circle the Earth 63 times” consistent?
(b) Use an estimate of the population of the United States and some common sense to
estimate how many plastic bags are used in the United States each year. Use your answer
to show that the 4 billion plastic bag claim is too small by several orders of magnitude.
(c) Confirm your U.S. estimate with a web search.
(a) Are the numbers “4 billion plastic bags” and “circle the Earth 63 times” consistent?
Suppose a plastic bag is about a foot long. There are about 5,000 feet in a mile, so four
billion feet is 800,000 miles. I know that the distance around the earth at the equator
is 25,000 miles. Then 800,000/25,000 = 800/25 = 32. So my estimate says the bags
would go around the equator 32 times, not 63. If I started with the assumption that bags
were two feet long I’d get the answer the article claims. But the number is in the right
ballpark.
(b) Use an estimate of the population of the United States and some common sense to
estimate how many plastic bags are used in the United States each year. Use your answer
to show that the 4 billion plastic bag claim is too small by several orders of magnitude.
Off the top of my head: 300 million people each using 10 plastic bags a week for 50 weeks
would mean 150,000 million plastic bags a year for the US. That’s 150 billion, which is
two orders of magnitude more than the 4 billion the article claims for the whole world.
And I think 10 bags per week is a low estimate.
Since I am uncomfortable claiming the newspaper was wrong, I will check my work by
reasoning backwards from the article’s number. Say we use a quarter of the plastic bags
in the world (even though we’re just 1/20th of the population). That would be 1 billion
per year for us, or just over three per year per person (since 3 * 300 million is just short
of a billion). I am sure we use more than three bags per person per year!
c 2014 Ethan Bolker, Maura Mast
34
1.8. COMMENTS ON THE EXERCISES
(c) Confirm your U.S. estimate with a web search.
I googled
How Many Plastic Bags do we Use Each Year?
and found this 2008 estimate from http://www.natural-environment.com/blog/2008/
01/10/how-many-plastic-bags-do-we-use-each-year/:
A common estimate is that global consumption of plastic bags is over 500
billion plastic bags annually. Yes that’s 500,000,000,000 plastic bags used per
year. In other words, that’s almost 1 million plastic bags used per minute.
That means the Washington Post estimate really was two orders of magnitude too small!
Lots of other web sources support this conclusion. From http://www.salon.com/news/
feature/2007/08/10/plastic_bags:
Every year, Americans throw away some 100 billion plastic bags after
they’ve been used to transport a prescription home from the drugstore or a
quart of milk from the grocery store. It’s equivalent to dumping nearly 12
million barrels of oil.
...
Only 1 percent of plastic bags are recycled worldwide – about 2 percent in
the U.S.
...
In the U.S., one company buys half of the used plastic bags available on
the open market in the United States, using about 1.5 billion plastic bags per
year. That’s Trex, based in Winchester, Va., which makes composite decking
out of the bags and recycled wood. It takes some 2,250 plastic bags to make a
single 16-foot-long, 2-inch-by-6-inch plank.
From http://www.copperwiki.org/index.php/Plastic_bags:
A million plastic bags are being consumed every minute across the globe.
This amounts to 150 bags a year for every person on earth.
Exercise 1.8.69. [N][Section 1.2][Goal 1.1] [Goal 1.2] [Goal 1.4] Losing lots.
An article in The Economist on September 10, 2011 noted that
Walmart . . . has moved on to health, with a campaign that has already
caused associates to lose a combined 200,000 lbs of weight. 4
4
http://www.economist.com/node/21528436
35
c 2014 Ethan Bolker, Maura Mast
1.8. COMMENTS ON THE EXERCISES
Put that number in context. Can you make sense of it?
Exercise 1.8.70. [U][Section 1.3] [Goal 1.1] [Goal 1.2] [Goal 1.4] [Goal 1.5] Sharp increase
in pollution predicted
In The Boston Globe on April 11, 2009 reporter Matt Viser wrote that
Marc Draisen, executive director of the Metropolitan Area Planning Council,
said the MBTA plan “would be nothing short of a disaster for the Greater Boston
area.”
He estimated that the reduction in services would result in 25 million new
automobile trips per year with 119 million additional miles traveled and 6 million
extra gallons of gas consumed. 5
(a) How long (on average) is each of the new automobile trips?
(b) How many gallons of gas (on average) will each trip use?
(c) What is the average fuel economy (in miles/gallon and in gallons/(100 miles)) for
these trips?
(d) Is Draisin’s estimate of 25 million new automobile trips reasonable?
[See the back of the book for a hint.] Consider estimating how many trips per week per
person this represents.
Exercise 1.8.71. [U] [Section 1.5] Snapple facts
6
(a) What are “Snapple facts”?
(b) Find some Snapple facts on the web and figure out whether they are believable. You
might start here: http://www.popculturemadness.com/Trivia/Snapple-Facts.html
Exercise 1.8.72. [U][N] [Section 1.6] The Emperor of All Identities
On December 18, 2012, Pamela Jones Harbour wrote in The New York Times that
[Google] creates as much data in two days – roughly 5 exabytes – as the world
produced from the dawn of humanity until 2003, according to a 2010 statement
by Eric Schmidt, the company’s chairman. 7
5
http://www.boston.com/news/local/massachusetts/articles/2009/04/11/plan_to_cut_t_
services_draws_ire_criticism/
6
(Robert Tagliani, Fall 2012)
7
http://www.nytimes.com/2012/12/19/opinion/why-google-has-too-much-power-over-your-private-life.
html
c 2014 Ethan Bolker, Maura Mast
36
1.8. COMMENTS ON THE EXERCISES
Exercise 1.8.73. [U][Section 1.3][Goal 1.2] [Goal 1.4] [Goal 1.5] [Goal 1.8] Stolen coins.
In 2007 the Massachusetts Bay Transportation Authority (MBTA, “the T”) switched from
tokens to Charlie Cards and Charlie Tickets.
That year a T retiree was charged with stealing coins and tokens. The Boston Globe reported
that when investigators went to his home “. . . they found more than $40,000 in coins and
tokens . . . stashed in 17 plastic jugs, each large enough to hold 5 gallons.” 8
(a) Estimate how many coins and tokens it would take to total $40,000.
(b) Check your estimate by thinking about whether those coins and tokens would fit into
17 five gallon jugs.
(c) Estimate the weight of these coins and tokens.
Exercise 1.8.74. [U][N][Section 1.3][Goal 1.1] [Goal 1.2] [Goal 1.4] [Goal 1.8] Keep the cap
From Joan Wickersham’s op-ed in The Boston Globe, March 4, 2011, written as an open
letter to a nonprofit environmental group:
This week you also sent me your magazine, as well as another solicitation for
money. Taken all together, you sent me a total of 1.25 pounds of stuff in the past
week alone. Browsing your website, I found a handy carbon footprint calculator,
which I was able to use to see how much your mailings may be contributing to
global warming.
Assuming you shower all your 1,000,000 members with this same 1.25 pounds
of attention when they pay their annual dues, you fill at least three Boeing 747s
with thank-you gifts and mailings, thus creating, by a conservative estimate,
between 1,800 and 2,000 tons of carbon dioxide. 9
Verify Wickersham’s estimates in the second paragraph.
Exercise 1.8.75. [U][Section 1.3][Goal 1.1] [Goal 1.2] Up and down the ladder.
An April 21, 2011 article in The Boston Globe reported that
Volker Kraft’s apple sapling sported just 18 eggs when he first decorated it
for Easter in 1965. Decades later, the sturdy tree is festooned with 9,800 eggs,
with artful decorations that include sequins and sea shells.
8
http://www.boston.com/news/globe/city_region/breaking_news/2007/03/mbta_police_t_r.
html
9
http://www.boston.com/bostonglobe/editorial_opinion/oped/articles/2011/03/04/keep_the_
cap/
37
c 2014 Ethan Bolker, Maura Mast
1.8. COMMENTS ON THE EXERCISES
...
Kraft needs two weeks and countless trips up and down his ladder to hang
the eggs. 10
The trips aren’t countless, even if Kraft didn’t count them. Estimate how many trips in how
many hours.
Exercise 1.8.76. [U][Section 1.3][Goal 1.2] [Goal 1.4] [Goal 1.8]
shatterproof pint glasses
Britons toast to new
The Design Council, a British charity that champions great design, announced in February
2010 that they were designing a new beer glass that would be shatter-proof. The blog
described their work and stated that
Alcohol related violence has fallen by 33% since 1997, yet there remain 87,000
violent incidents involving glassware each year, which can leave victims badly
injured ... as well as costing the NHS (National Health Service) an estimated 2.7
billion pounds. 11
(a) When this design was proposed, 2.7 billion pounds was equivalent to about $4.3 billion.
Using either currency, estimate how much each alcohol-related glass attack costs the
National Health Service.
(b) Are the numbers 87,000 attacks and $4.3 billion reasonable?
Exercise 1.8.77. [U][S][Section 1.3][Goal 1.1] [Goal 1.2] The 18 million second rule.
The cartoon in Figure 1.1 appeared in The Boston Globe on Sunday, November 6, 2011.
12
Here’s the text from the two frames:
10
http://www.boston.com/news/world/europe/articles/2011/04/21/tree_in_germany_boasts_
9800_eggs/
11
http://www.designcouncil.org.uk/our-work/challenges/security/design-out-crime/
alcohol-related-crime/
12
Since the actual cartoon will be too expensive to reproduce in the published version of Common Sense
Mathematicswe’ve quoted the captions here. You can see the original at http://www.grimmy.com/comics.
php?sel_dt=2011-11-06
c 2014 Ethan Bolker, Maura Mast
38
1.8. COMMENTS ON THE EXERCISES
Grimm:
Mother Goose:
DO WE BELIEVE
IN THE EIGHTEEN
MILLION ONE
HUNDRED AND
FORTY THOUSAND
SECOND RULE?
GROSS, SOMEBODY MUST’VE
DROPPED THIS PIE
BEHIND THE SOFA WHEN
I HAD THAT BRIDGE PARTY
HERE LAST MONTH.
(a) What is the five second rule?
(b) Check Grimm’s (approximate) arithmetic. Is he in the right ballpark?
Figure 1.1: The 18 million second rule
(a) What is the five second rule?
According to wikipedia, http://en.wikipedia.org/wiki/Five-second_rule, a reasonable reference for this kind of thing,
A common superstition, the five-second rule states that food dropped on
the ground will not be contaminated with bacteria if it is picked up within five
seconds of being dropped. Some may earnestly believe this assertion, whereas
other people employ the rule as a polite social fiction to prevent their having
to forgo eating something that dropped.
(b) Check Grimm’s (approximate) arithmetic. Is he in the right ballpark?
The Google calculator says that
39
c 2014 Ethan Bolker, Maura Mast
1.8. COMMENTS ON THE EXERCISES
one month = 2 629 743.83 seconds
so Grimm’s arithmetic is way off. 2.7 million seconds and 18 million seconds are not of
the same order of magnitude.
Exercise 1.8.78. [N][Section 1.5] If you love nature, move to the city
Edward L. Glaeser wrote that op-ed piece in The Boston Globe on February 10, 2011.
Here’s just one good quotation.
In much of the country, cars are the biggest carbon emitters, and density
determines driving. Households in areas with more than 10,000 people per square
mile average 687 gallons of gas per year, while households in areas with fewer than
1,000 people per square mile average 1,164 gallons of gas per year. Even among
driving families, there is a big difference between gas usage in denser places, like
Brookline, and far-flung suburbs. 13
(a) The first part of the quote talks about households in areas with more than 10,000
people per square mile. Estimate the number of households in an area with (about)
10,000 people. Then estimate the number of households in an area with (about) 1,000
people.
(b) Use your work above to estimate the total amount of gasoline used the households in
the area with about 10,000 people per square mile. Then estimate the total amount of
gas used by the households in the area with 1,000 people per square mile.
(c) Which region emits more carbon – the denser region or the less dense region?
(d) Why do people in denser places use less gas?
(e) Figure out (or look up) the carbon footprint of an average car using a gallon of gasoline.
Use that to calculate the carbon footprint of the two different regions that is related to
car usage.
Exercise 1.8.79. [Section 1.7][Goal 1.7] Explore the Powers of Ten web site http://www.
powersof10.com/. Be sure to watch the film. We showed the film “Powers of 10” in class
once but our students didn’t find it as interesting as we do. Much as we like it, we haven’t
used it since.
Exercise 1.8.80. [S][Section 1.6][Goal 1.1] [Goal 1.2] A self-checkout way of life
The global research company IHL group reported in 2008 that
13
http://www.boston.com/bostonglobe/editorial_opinion/oped/articles/2011/02/10/if_you_
love_nature_move_to_the_city/
c 2014 Ethan Bolker, Maura Mast
40
1.8. COMMENTS ON THE EXERCISES
(t)he average American woman could lose 4.1 lbs. a year simply from resisting
the urge to purchase impulse items such as chocolate candies, chips and soda once
they are in the checkout line . . . According to the study, self-checkout systems
have a dramatic impact on the purchase of impulse items at checkout. Impulse
purchases among women drop 32.1 percent...when self-checkout is used instead of
a staffed checkout. 14
Think about the assertion at the beginning of the problem: that the average American woman
could lose 4.1 lbs. a year by not buying items such as candy in the checkout line.
Is that assertion reasonable?
A small package of M&Ms weighs 1.69 ounces and contains about 240 calories, according to
their website. 15
Now I need to know how calories in food are converted to pounds of weight on a person.
The website convertunits.com tells me that 4.1 pounds (of weight) is equivalent to 14,350
calories.
If a shopper (I really question why it’s a woman in the article, but that’s a different issue)
buys a bag of M&Ms each week, that’s about 50 × 250 = 12, 500 calories, so it’s close enough.
A different calculation with chips would be worthwhile for comparison, but I usually see
candy at the checkout, not chips.
Exercise 1.8.81. [U][N][W][Goal 1.1] [Goal 1.2] [Section 1.6] Scientist’s game helps map
the brain
On July 16, 2012 Carolyn Johnson reported in The Boston Globe that
When Sebastian Seung read that each day people around the world spend
600 years collectively playing Angry Birds, he saw not a huge waste, but a big
opportunity. . . .
By Seung’s calculations, tracing all the neural connections in a cubic millimeter
of brain would take one person working around the clock 100,000 years. Aided
by the computer programs his lab has been building, that task would be slightly
more doable, requiring 1,000 years of work. . . .
But, he said, “if we were 1 percent as fun [as Angry Birds], we could do this
in a year.” 16
14
http://www.ihlservices.com/ihl/press_detail.cfm?PressReleaseID=57/
http://www.mms.com/us/product/milk
16
http://bostonglobe.com/metro/2012/07/15/mit-scientists-crowdsource-effort-map-connections-brain/
v1CEwhwl90GDjKzL6oTC4L/story.html
15
41
c 2014 Ethan Bolker, Maura Mast
1.8. COMMENTS ON THE EXERCISES
Exercise 1.8.82. [N][Section 1.6] The Nine Billion Names of God
Read that classic short science fiction story by Arthur C. Clarke at http://downlode.org/
Etext/nine_billion_names_of_god.html.
Exercise 1.8.83. [N] No Food Safety in These Numbers
On June 21, 2011 Mark Bittman wrote in The New York Times about budget cuts in food
safety programs. The article is at http://opinionator.blogs.nytimes.com/2011/06/21/
no-food-safety-in-these-numbers
There’s lots of food for quantitative thought here.
Exercise 1.8.84. [S][Goal 1.1] [Goal 1.2] [Goal 1.5] [Section 1.6] How much television?
The web page http://www.shirky.com/herecomeseverybody/2008/04/looking-for-the-mouse.
html claims that U.S. adults spend 200 billion hours a year watching TV. 17
On July 22, 2010, Hiawatha Bray wrote in The Boston Globe:
My trouble is I don’t watch enough television. If I burned through the
national average, watching 35 hours a week, I would probably love Hulu Plus, the
new pay-to-play video service. 18
Do these two estimates of the amount of time we spend watching TV agree?
I will convert the data in the first paragraph to hours per person per week, and see whether
the answer is about 35.
If I start with the estimate that 2/3 of the 300 million people in the United States are adults,
then 200 billion hours per year divided by 200 million adults means 1000 hours per year per
adult. There are about 50 weeks in a year, so that becomes 20 hours per adult per week.
Bray’s estimate is nearly twice that. They do have the same order of magnitude, but when
counting hours in front of a TV set the difference between 20 hours per week and 35 hours
per week matters.
Bray’s figure is more recent, by two years. Maybe people are watching a lot more television
now, but I don’t think that explains the discrepancy.
17
You can see an interesting visual depiction of this information at http://www.informationisbeautiful.
net/2010/cognitive-surplus-visualized/
18
http://www.boston.com/business/technology/articles/2010/07/22/tv_gurus_will_enjoy_
hulus_new_pay_service/
c 2014 Ethan Bolker, Maura Mast
42
1.8. COMMENTS ON THE EXERCISES
I think that with a lot more digging around on the web I could find more information about
the amount of television adults watch, but what I’ve said so far is good enough for this
question.
Exercise 1.8.85. [N][Goal 1.1] [Goal 1.4] [Goal 1.5] [Goal 1.6] [Goal 1.8] [Section 1.7] Global
data storage calculated at 295 exabytes
From the BBC on February 11, 2011: http://www.bbc.co.uk/news/technology-12419672
Mankind’s capacity to store the colossal amount of information in the world
has been measured by scientists.
The study, published in the journal Science, calculates the amount of data
stored in the world by 2007 as 295 exabytes.
That is the equivalent of 1.2 billion average hard drives.
The researchers calculated the figure by estimating the amount of data held
on 60 technologies from PCs and and DVDs to paper adverts and books.
“If we were to take all that information and store it in books, we could cover
the entire area of the US or China in 13 layers of books,” Dr Martin Hilbert of
the University of Cancelhern California told the BBC’s Science in Action.
...
The same information stored digitally on CDs would create a stack of discs
that would reach beyond the moon, according to the researchers.
Exercise 1.8.86. [U][S][Section 1.7][Goal 1.7] Metric history.
(a) The metric prefix Kilo was invented along with the metric system. When was that?
(b) The need for larger numbers grew with time. When were the prefixes Mega . . . Zetta
invented?
(a) The metric prefix Kilo was invented along with the metric system. When was that?
1795, in France, right after the French Revolution.
(b) The need for larger numbers grew with time. When were the prefixes Mega . . . Zetta
invented?
The wikipedia entry at http://en.wikipedia.org/wiki/SI_prefix refers to meetings
of the General Conference on Weights and Measures which certified mega, giga and tera
in 1960, peta and exa in 1975 and zetta and yotta in 1991.
Exercise 1.8.87. [U][N][Section 1.7][Goal 1.7] [Goal 1.8] [Section 1.7] Rush For 1940 Census
Data Jams Archives’ Website
43
c 2014 Ethan Bolker, Maura Mast
1.8. COMMENTS ON THE EXERCISES
On April 2, 2012 National Public Radio’s All Things Considered aired an interview with
Susan Cooper, head of publicity for the National Archives, about the release on the web of
data from the 1940 census.
COOPER: This is our first online release of census material and we released
16 terabytes of census material and this is, by far, the largest release of digital
material that the National Archives has ever had.
SIEGEL: What was your past record?
COOPER: We released 250 megabytes of Nixon grand jury transcripts, and
so there’s a huge leap from 250 megabytes to 16 terabytes. 19
Unfortunately, these bytes are microfilm images of census forms. That means they’re not
searchable. For our quantitative reasoning purposes it means we can’t use the number of
bytes to estimate the amount of information collected.
19
http://www.npr.org/2012/04/02/149866207/rush-for-1940-census-data-jams-archives-website
c 2014 Ethan Bolker, Maura Mast
44
Chapter 2
Units and Unit Conversions
2.1
Rate times time equals distance
The formal introduction to units might be a place to discuss the distinction between number
and amount – discrete vs. continuous measurement, and the accompanying grammatical
questions. It makes sense to say “I bought five pounds of potatoes” or “I bought five potatoes”
– note the different units. You can’t say just “I bought five.” But be careful. Students may
find this both pedantic and distracting.
45
2.2. THE MPG ILLUSION
2.2
The MPG illusion
2.3
Converting currency
2.4
Unit pricing and crime rates
2.5
The metric system
2.6
Working on the railroad
2.7
Scientific notation, powers of 10, milli and micro
2.8
Carpeting and paint
The bad news is that this carpeting problem is another artifial one with easy numbers. The
good news is that it’s accompanied by some practical advice for painting.
2.9
Comments on the Exercises
37 The problem on the Arizona border fence is interesting because it’s political, the computations are easy and the answers don’t make sense.
Extra (ideas for) Exercises
2.9
Exercise 2.9.58. [Section 2.3][Goal 2.1] [Goal 2.2] Gasoline in Europe
Compare gasoline taxes in several European countries with those in the state you live in.
Then compare the cost of the gasoline itself.
Use the web to find the European tax rates. Then convert units as needed. Subtract the
taxes from the price at the pump to find the cost of the gasoline.
c 2014 Ethan Bolker, Maura Mast
46
2.9. COMMENTS ON THE EXERCISES
Exercise 2.9.59. [G][Goal 2.4] [Section 2.8] What does it cost to go green?
One way to save energy is to install solar panels to cover all or part of your roof. Solar
panels are expensive, but there are some clever ways to reduce or eliminate the cost. In this
project, you will act as a consultant to a homeowner who is thinking of installing solar panels.
Your job is to figure out how much of the roof will be covered and what the options are for
paying for it. Make sure you look at the option of the homeowner paying for the panels
up front, using rebates and credits, and calculate how long it will take to earn that back
through reduced electricity costs. Use information from your local area for this, and state
your assumptions clearly.
(a) Estimate the area of the roof where the panels will be installed. Solar panels work
best when placed on south-facing roofs, so work that into your calculation.
(b) Do some research about options for installing solar panels. One approach is to sign a
long-term contract with a solar services provider to purchase the power that the panels
generate. Or a homeowner could apply for a grant to cover some of the cost. The federal
government also allows homeowners to take a tax credit of up to 30% of the cost of
installing panels. There are other options and it’s your job to look into them.
(c) Write a proposal for the customer that outlines the different options. For each option,
calculate the up-front costs and the long-term costs (or benefits).
(d) What would you recommend? Write a clear recommendation, justifying your answer
with the data you have found and calculated.
Exercise 2.9.60. [Section 2.8][Goal 2.4] [Goal 2.1] Fire and Ice
In 2013, the Northwest camps of Wayne County Community College District installed a
thermal energy cooling system on its campus. This system consists of large water-filled tanks.
The water in the tanks is frozen at night, when electricity is less expensive, and used in
the cooling system during the day. The college installed its tanks underground in a space
between two buildings, measuring approximately 60 by 90 feet.
(a) Standard cooling tanks (from the Calmac corporation, for example, at http://www.
calmac.com/products/icebankc_specs.asp are 89 inches wide, 91 inches long and 69.5
inches tall. Use this information to determine the maximum number of tanks that the
college could install in that space. Make a sketch of the layout of the installation.
(b) Another type of tank from the same company is cylindrical, measuring 48 inches tall
and with a diameter of 73.75 inches. How many tanks of this style could be installed
into the space? Make a sketch of the layout of the installation.
47
c 2014 Ethan Bolker, Maura Mast
2.9. COMMENTS ON THE EXERCISES
Exercise 2.9.61. [U][Section 2.1][Goal 2.1] Back in the saddle again
Bradley Wiggins won the 2012 Tour de France, spending spent 87 hours 34 minutes 47
seconds in the saddle to travel 3,497 km. 1
(a) What was his average speed, in km/hour?
(b) What was his average speed, in mi/hour?
(c) Did he travel as far as the distance from Maine to Florida? If not, how far would he
have gone on that route?
(d) Did he travel as far as the distance from California to New York? If not, how far
would he have gone on that route?
Exercise 2.9.62. [U] To Match Walton Heirs’ Fortune, You’d Need to Work at Walmart
for 7 Million Years
The businessman Sam Walton founded the WalMart company in 1962. Since then the
company has expanded globally and earns hundreds of billions of dollars in sales each year.
Sam Walton died in 1992 and at that time was among the richest people in the United States.
Just how rich are the Waltons? According to the latest edition of the Forbes
400, released yesterday, the six wealthiest heirs to the Walmart empire are together
worth a staggering $115 billion.
...
The average Walmart worker earns just $8.81 an hour. At that wage, the
union-backed Making Change at Walmart campaign calculates that a Walmart
worker would need:
• 7 million years to earn as much wealth as the Walton family has (presuming
the worker doesn’t spend anything)
• 170,000 years to earn as much money as the Walton family receives annually
in Walmart dividends
• 1 year to earn as much money as the Walton family earns in Walmart
dividends every three minutes
2
(a) Verify Making Change at Walmart’s calculations.
1
Statistics from The New York Times, quoting Reuter’s, July 23, 2012, http://www.nytimes.com/
reuters/2012/07/23/sports/cycling/23reuters-cycling-britain-wiggins.html
2
http://www.motherjones.com/mojo/2012/09/sam-waltons-fortune-walmart-employees-7-million-years
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2.9. COMMENTS ON THE EXERCISES
(b) What is the average annual salary of a Walmart worker?
Exercise 2.9.63. [U] The 2012 Nobel Prize in Physics
Serge Haroche and David J. Wineland won that prize for experimental work in quantum
physics. One of their experiments built an optical clock.
The precision of an optical clock is better than one part in 1017 which means
that if one had started to measure time at the beginning of the universe in the
Big Bang about 14 billion years ago, the optical clock would only have been off
by about five seconds today. 3
Check the arithmetic in this paragraph: does one part in 1017 work out to five seconds in 14
billion years?
Exercise 2.9.64. [S] Keeping wikipedia solvent.
This appeal appeared December 2012 on the Wikipedia website:
Dear Wikipedia readers: We are the small non-profit that runs the #5 website
in the world. We have only 150 staff but serve 450 million users, and have costs
like any other top site. To protect our independence, we’ll never run ads. We
take no government funds. We run on donations. We just need 0.3% of readers
to donate an average of about $30. We’re not there yet. Please help us forget
fundraising and get back to Wikipedia.
If everyone reading this gave the price of a cup of coffee, our fundraiser would
be done within an hour. If Wikipedia is useful to you, take one minute to keep it
online another year by donating whatever you can today.
(a) Use the data in the first paragraph to estimate Wikipedia’s fundraising goal. Is your
answer believable?
(b) Estimate the percentage of the fundraising goal that goes toward staff salaries.
(c) Use the data in the second paragraph to estimate the number of users who visit
Wikipedia in an hour. (The average cost of a cup coffee on campus is $2.00.) For a little
bit of extra credit, spend no more than five minutes on the internet to see if you can
confirm your estimate
3
http://www.nobelprize.org/nobel_prizes/physics/laureates/2012/
popular-physicsprize2012.pdf
49
c 2014 Ethan Bolker, Maura Mast
2.9. COMMENTS ON THE EXERCISES
(a) Use the data in the first paragraph to estimate Wikipedia’s fundraising goal. Is your
answer believable?
0.3% × 450 million users ×
$30
= $40, 500, 000 ≈ $40 million.
user
That sounds reasonable to me.
(b) Estimate the percentage of the fundraising goal that goes toward staff salaries.
I suspect that the 150 employees running this important web site are pretty highly trained
in technical matters, so I will estimate their average annual salary as $100K. Then it will
take $15,000,000 to cover salaries. That’s $15, 000, 000/$400, 000, 000 = 0.375 ≈ 38% or
about one third of their fundraising goal
If you estimate the average annual salary as just $50K then the fraction is closer to 20%,
or one fifth.
Any answer in this range makes sense to me.
The rest of the money is for “costs like any other top site” – things like computers and
office space.
(c) Use the data in the second paragraph to estimate the number of users who visit
Wikipedia in an hour. (The average cost of a cup coffee on campus is $2.00.) For a little
bit of extra credit, spend no more than five minutes on the internet to see if you can
confirm your estimate
It would take 20 million $2 contributions to raise $40,000,000, so that’s the number of
hourly visits if their estimate is right.
According to http://meta.wikimedia.org/wiki/Wikimedia_in_figures_-_Wikipedia
In June 2007 a one-time measurement was made of number of visitors to
all Wikimedia projects combined . . . [resulted in] an average view rate of more
than 2500 Wikimedia pages per second, 24 hours per day.
That converts to a rate of 9 million pages per hour, which is in the same ballpark as my
estimate.
Exercise 2.9.65. [U][Goal 2.4] [Goal 1.2] [Section 2.5] Claim your money, please!
One of the authors received this in email:
I am thompson Cole the newly appointed United Nations Inspection Agent in
JFK Airport New York. During our Investigation, we discovered An abandoned
shipment on your name through a Diplomat from London. which was transferred
to our facility here in JF Kennedy Airport and when scanned it revealed an
c 2014 Ethan Bolker, Maura Mast
50
2.9. COMMENTS ON THE EXERCISES
undisclosed sum of money in a Metal Trunk Box weighing approximately 55kg
each.
I beleived each of the boxes will contain more that $4M or above in each
and the consignment is still left in storage house till today through a registered
shipping Company, Courier Dispatch Service Limited a division of Tran guard
LTD. The Consignment are two metal box with weight of about 55kg each (Internal
dimension: W61 x H156 x D73 (cm). Effective capacity: 680 L.)Approximately.
Mr. thompson Cole goes on to ask for identification, and offers to “ bring it by myself to
avoid any more trouble. But we will share it 70% for you and 30% for me. But you have to
assure me of my 30%.”
(a) Does a box with the dimensions specified contain approximately 680 liters?
(b) What denominations (bills or coins) would you expect to find inside to make a total
of $4M?
(c) Does a weight of 55kg seem reasonable for that box full of money?
(d) What things about the email make you suspicious?
Exercise 2.9.66. [U] Cleaning up Everest.
Nepal’s government announced on March 3 that it would
require every
climber returning from the summit of Mount Everest to bring back at least 18
pounds of garbage, the first concerted effort to eliminate the estimated 50 tons
of trash that has been left on the mountain over the past six decades. The waste
includes empty oxygen bottles, torn tents and discarded food containers. 4
(a) How many people would have to bring back 18 pounds of garbage to eliminate that 50
tons?
(b) Estimate How many years would it take to bring back the garbage if everyone who
made it to the top did what he or she should.
(c) The quotation says only people who reach the summit would be required to bring back
garbage. Answer the previous question if everyone on each expedition had to do that.
[See the back of the book for a hint.] Use the web to find out enough about the history
of Mount Everest expeditions to estimate the average number of climbers and the average
number of people who actually reach the summit each year.
4
http://www.nytimes.com/2014/03/28/world/asia/nepal-considering-local-guide-requirement-for-peaks.
html
51
c 2014 Ethan Bolker, Maura Mast
2.9. COMMENTS ON THE EXERCISES
Exercise 2.9.67. [U][N] Digital photography
The relative increase in detail resulting from an increase in resolution is better
compared by looking at the number of pixels across (or down) the picture, rather
than the total number of pixels in the picture area. For example, a sensor of
2560 × 1600 sensor elements is described as “4 megapixels” (2560 × 1600 =
4, 096, 000). Increasing to 3200 × 2048 increases the pixels in the picture to
6,553,600 (6.5 megapixels), a factor of 1.6, but the pixels per cm in the picture (at
the same image size) increases by only 1.25 times. A measure of the comparative
increase in linear resolution is the square root of the increase in area resolution,
i.e., megapixels in the entire image. 5
2.9
Exercise 2.9.68. [S][Section 2.6][Goal 2.1] [Goal 2.2] [Goal 2.5] The flight of the honeybee
An article in the April 28, 2009 edition of The New York Times quotes Dr. Anna Dornhaus:
A honeybee is less than an inch long. If it flies 20 kilometers that equals
787,400 body lengths. If we say a human is two meters tall, then in human terms,
that would be like traveling 394 kilometers and back, possibly several times a day.
6
(a) Use the figures she provided to determine the body length of a honeybee. Is she correct
in stating that it is “less than an inch”?
(b) Is Dr. Dornhaus’ estimate of “394 kilometers and back” as the human equivalent of
the honeybee’s 20 kilometers reasonable?
(a) The Google calculator tells me that 20 kilometers = 787,401.575 inches. If each body
length were less than an inch then 787,400 of them would be less than 20 kilometers (by
just a little bit), so she’s not quite right.
She might have arrived at her answer by first converting kilometers to miles: 20 kilometers
= 12.4274238 miles. If she rounded before converting to inches she’d get 12.427 miles =
787,374.72 inches. That would be consistent with her assertion.
She should probably have said “just about an inch” instead of “less than an inch.”
5
6
http://en.wikipedia.org/wiki/Digital_photography
http://www.nytimes.com/2009/04/28/science/28prof.html?scp=1&sq=dornhaus%20&st=cse
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2.9. COMMENTS ON THE EXERCISES
(b) 394 kilometers is 394,000 meters, which is 394,000 body lengths per round trip for a 2
meter tall person. That’s just about half the number of body lengths flown by the bee.
So “possibly several times a day” should be “twice a day.”
Exercise 2.9.69. [S][Section 2.1][Goal 2.1] Cheers from (no) peanut gallery
The Boston Globe reported on June 12, 2011 that
At a typical game at Fenway, spectators peel 3,000 bags of peanuts, haphazardly
scattering a half-ton worth of shells. 7
(a) How many ounces of shells are there in each bag of peanuts?
(b) About what fraction of the weight of a bag of peanuts are the shells?
(c) Are the numbers in this quotation reasonable?
(a) How many ounces of shells are there in each bag of peanuts?
1
2
ton
2000 pounds 16 ounces
ounces
ounces
×
×
= 5.333 . . .
≈5
3000 bags
ton
pound
bag
bag
(b) About what fraction of the weight of a bag of peanuts are the shells?
A bag of peanuts probably weighs eight ounces. I think it’s reasonable that the weight is
more than half shells.
(c) Are the numbers in this quotation reasonable?
Fenway Park seats about 30,000 people. If ten percent of them bought peanuts that
would be 3,000 bags. I can believe that estimate.
Exercise 2.9.70. [U][N] Cleaning the sand.
From The New York Times, January 18, 2013, reporting on cleaning the beach sand in the
Rockaways after hurricane Sandy:
The Army Corps of Engineers, which is overseeing the operation on behalf of
the city’s parks department, says that in the six weeks leading up to New Year’s
Day, the crews in Queens had filtered 94,000 cubic yards of sand. That is enough,
the agency estimated, to fill a football field to a depth four feet higher than the
7
http://www.boston.com/lifestyle/health/articles/2011/06/12/with_ban_allergic_red_sox_
fans_cheer_breathe/
53
c 2014 Ethan Bolker, Maura Mast
2.9. COMMENTS ON THE EXERCISES
goal posts – or 44 feet – said Robert Schneider, an engineer technician for the
Corps.
. . . The items plucked from the sand have been part of the 10,800 truckloads
of debris that have rumbled out of the city to landfills in upstate New York and
in Pennsylvania. Strung end to end, the tractor-trailers would extend from New
York City to Albany. 8
8
http://www.nytimes.com/2013/01/10/nyregion/after-hurricane-sandy-cleaning-up-sand-and-returning-ithtml
c 2014 Ethan Bolker, Maura Mast
54
Chapter 3
Percentages, Sales Tax and Discounts
3.1
Don’t drive and text
In principle, the content of this chapter is a review of material on percentages students
learned in high school. In fact the review is necessary. Moreover, the presentation is a little
more sophisticated, and, we hope, both more useful and more interesting, than what they’ve
seen already. So even for those who remember it, there’s value added here.
3.2
Red Sox ticket prices
3.3
Patterns in percentage calculations
3.4
The one-plus trick
Speaking mathematically, we’d rather have the relative change be the fraction new/old,
paralleling the definition of absolute change as new-old. But speaking practically, for the
target student audience it’s better to think of the relative change as the difference (new-old)-1.
But note that when computing exponential growth later in the book we’ll want to go back to
new/old.
It’s hard to persuade students to learn the “multiply by 1+change” trick. But it’s well worth
the effort, which will pay big dividends later when looking at inflation, interest rates and
exponential growth. And it helps wake up students who might otherwise see this material as
55
3.5. TWO RAISES IN A ROW
just boring review of things they once knew and think they still know.
3.5
Two raises in a row
3.6
Sales tax
3.7
20 percent off
3.8
Successive discounts
3.9
Percentage points
3.10
Comments on the Exercises
.5 The Gulf oil spill in the summer of 2010 generated lots of data along with the oil. If it had
happened during the semester we might have used it daily.
.26 This exercise is worth spending class time on.
.30 Much to our surprise, we found that about a third of the class didn’t know what “wholesale
price” meant when we first assigned this ptoblem on markups. Now there’s a hint in the back
of the book.
.40 This problem comparing private label to branded goods turned out to be more interesting
than we thought. There are several decreases (negative values) to deal with. The answer in
the solutions manual shows how the 1+ trick lets you find the relative increase without first
finding the absolute increase.
.45 This exercise is an advance look at the mathematics of compound interest.
ch semester we try to find a current news story that students might respond to with a letter
to the editor, or an online comment at the appropriate web site. We assign a draft letter or
comment as an exercise, discuss the results in class, and offer extra credit for submitted or
published comments.
c 2014 Ethan Bolker, Maura Mast
56
3.10. COMMENTS ON THE EXERCISES
Extra (ideas for) Exercises
3.10
Exercise 3.10.65. [U] On September 27, 2011, PCMag reported that
Visits to Google+ jumped 1269 percent during the week of September 17,
while total U.S. site visits were up from 1.1 million to almost 15 million during
that same time period, according to Hitwise. . . .
Market leader Facebook is still miles ahead with almost 1.8 billion U.S. visits,
followed by YouTube with 530 million and Twitter with 33 million. 1
(a) Verify that the numbers in the first paragraph are consistent.
(b) Compare Google+ visits with Facebook visits, or Google+ visits with YouTube or
Twitter visits.
(c) Paul Allen calls himself the Google+ unofficial statistician. He is also the founder
of Ancestry.com. His post on December 27, 2011 contains a lot of Google+ data
https://plus.google.com/+PaulAllen/posts/ZcPA5ztMZaj. In that post, he said
that he expects the growth to continue to accelerate. Was he correct? Be careful – he’s
saying more than that the growth will continue. He is saying that it will continue to
accelerate.
Exercise 3.10.66. [S][Goal 3.4] Living Without a Cellphone
2
On September 28, 2012 The Wall Street Journal reported that
In the second quarter, the number of cellphone subscribers on contract plans
rose just 0.5% from the year before, to 217 million, according to UBS AG. The
number of prepaid customers grew about 11% to 74 million.
(a) What percentage of the people using cellphones have prepaid service now?
(b) How many subscribers had contract plans in the second quarter of 2011 (“the year
before”)?
(c) How many prepaid customers were there in the second quarter of 2011?
(d) How did the percentage of people using a prepaid service rather than a contract plan
change in the last year?
1
2
http://www.pcmag.com/article2/0,2817,2393640,00.asp
Suggested exercise (Lea Ferone, October 2012)
57
c 2014 Ethan Bolker, Maura Mast
3.10. COMMENTS ON THE EXERCISES
(a) What percentage of the people using cellphones have prepaid service now?
The percentage is
74
= 0.254295533 ≈ 25%.
74 + 217
It makes no sense to keep more than two significant digits.
(b) How many subscribers had contract plans in the second quarter of 2011 (“the year
before”)?
This is a place where the 1+ trick comes in handy. Since the current number is 0.5%
more than last year’s number,
1.005 × 217 = last year’s number
so
217
= 215.920398.
1.005
Almost all those digits are nonsense. The answer is about 216 million.
last year’s number =
(c) How many prepaid customers were there in the second quarter of 2011?
I used the same method and calculated 74/1.11 = 66.6666667 ≈ 67 million.
(d) What percentage of the people using cellphones had prepaid service last year?
The percentage of people using a prepaid service last year was
67
= 0.236749117 ≈ 24%
67 + 216
(e) How did the percentage of people using a prepaid service rather than a contract plan
change in the last year?
The percentage increased by about one percentage point from 24% to 25%. Not much at
all.
Exercise 3.10.67. [W][R][S] Doublespeak
In his Letter from Rangoon in the August 25, 2008 New Yorker, George Packer wrote:
The minister of planning . . . gave a long speech that attempted to rebut
Petrie’s remarks, using the U.N.’s own statistics. “Some of it was really funny,”
Petrie recalls. “He said, for example, ’The U.N. states that a third of the children
under five are malnourished. That’s absolutely not true. It’s 31.2 per cent. The
U.N. states that three-quarters of an average family’s income is used on food.
That’s actually not true. It’s 68.7 per cent.’ He was using our statistics to say
there was no poverty – that everything was fine.” 3
3
http://www.newyorker.com/reporting/2008/08/25/080825fa_fact_packer
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58
3.10. COMMENTS ON THE EXERCISES
(a) Where is Rangoon?
(b) The quote describes how Myanmar’s minister of planning tried to rebut the conclusions
of a U.N. report. Do you think he succeeded?
(c) Why did Petrie think the attempt was “really funny”?
(d) Estimate the number of malnourished children in Burma in 2008.
(e) (optional) What is doublespeak? What’s the origin of the term? Is the minister of
planning’s response doublespeak?
(a) Where is Rangoon?
Rangoon, now known as Yangon, is the largest city in Myanmar (formerly known as
Burma). It used to be the capital.
(b) The quote describes how Myanmar’s minister of planning tried to rebut the conclusions
of a U.N. report. Do you think he succeeded?
No, I don’t think so. The report is describing how much poverty there is in Myanmar.
All the minister can do to contradict the report is to criticize the perfectly reasonable
approximations it uses to describe the situation in words. 31.2 percent rounds to one
third, 68.7 percent is nearly three quarters.
(c) Why did Petrie think the attempt was “really funny”?
The minister was nitpicking the numbers but refusing to acknowledge what they meant.
(d) Estimate the number of malnourished children in Burma in 2008.
According to http://www.indexmundi.com/burma/population.html the population of
Burma in 2008 was 47,758,180, or approximately 48 million. The site says the numbers
come from estimates made by the US Census Bureau. I see no reason to doubt this figure.
Since Myanmar is a very poor country, and poor countries tend to have a disproportionate
fraction of small children, I’ll estimate that 20% of the population, or about 10 million,
are children. That would mean about three and a half million undernourished children.
(e) (optional) What is doublespeak? What’s the origin of the term? Is the minister of
planning’s response doublespeak?
According to http://dictionary.reference.com/browse/doublespeak, doublespeak
is “evasive, ambiguous language that is intended to deceive or confuse.” I think the
minister’s comment qualifies.
They say the source of the word might be an analogy to “doublethink”. That cool
word, meaning “the act of simultaneously accepting two mutually contradictory beliefs
as correct” (according to Wikipedia, which in this case is trustworthy) was invented by
George Orwell in his novel “1984”.
59
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3.10. COMMENTS ON THE EXERCISES
Exercise 3.10.68. [C][U] Colleges and the numbers game
On February 4, 2008 Ralph Whitehead Jr. wrote in The Boston Globe that
The share of the nation’s 18-year-olds who are from households where no
adult holds a four-year degree is 60 percent. If Princeton looked like America,
its first-generation number would be 60 [percent], not 11. Its number is about
one-fifth of a representative number. Blacks make up 12 percent of America. If
Princeton’s enrollment represented one-fifth of black America, the black share of
its students would be under 2 percent. If it did the same for women, the female
share of its students would be under 10 percent. 4
We find these numbers quite confusing. Can you figure out what Whitehead is trying to say?
This exercise might make an interesting class discussion. Assign it first, then build a class
around student solutions.
Exercise 3.10.69. [U][Goal 1.2] [Goal 1.4] [Goal 3.3] Writing Your Dissertation in Fifteen
Minutes a Day
Author Joan Bolker’s book with that title was published in 1998. The hundred thousandth
copy was bought in 2011.
Approximately what percentage of the graduate students writing dissertations in those years
bought the book?
Exercise 3.10.70. [U][N] ADHD
The web page http://www.cdc.gov/ncbddd/adhd/data.html/ from the Centers for Disease
Control (CDC) reports this data on Attention-Deficit / Hyperactivity Disorder (ADHD):
The American Psychiatric Association states in the Diagnostic and Statistical
Manual of Mental Disorders (DSM-IV-TR) that 3%-7% of school-aged children
have ADHD. However, studies have estimated higher rates in community samples.
Parents report that approximately 9.5% or 5.4 million children 4-17 years of
age have ever been diagnosed with ADHD, as of 2007.
The percentage of children with a parent-reported ADHD diagnosis increased
by 22% between 2003 and 2007.
Rates of ADHD diagnosis increased an average of 3% per year from 1997 to
2006 and an average of 5.5% per year from 2003 to 2007.
4
http://pqasb.pqarchiver.com/boston-sub/access/1423321841.html?FMT=FT
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60
3.10. COMMENTS ON THE EXERCISES
Boys (13.2%) were more likely than girls (5.6%) to have ever been diagnosed
with ADHD.
(a) Make sense of these statistics.
(b) Read and comment on Bronwen Hruska’s opinion piece Raising the Ritalin Generation
at http://www.nytimes.com/2012/08/19/opinion/sunday/raising-the-ritalin-generation.
html
Exercise 3.10.71. [N] Food security
http://www.ers.usda.gov/media/884525/err141.pdf
Exercise 3.10.72. [N][G] Poor Smokers in New York State Spend 25% of Income on
Cigarettes, Study Finds
The article at http://www.nytimes.com/2012/09/20/nyregion/poor-smokers-in-new-york-state-spe
html
reports on a study at
http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0043838#s1
The discussion at Andrew Gelman’s blog (http://andrewgelman.com/2012/09/poor-smokers-in-new-yo
#comments is a gem, dissecting the assumptions behind the headline and the study.
Exercise 3.10.73. [N] How many women work outside the home?
This comment appeared in The Times on November 25, 2013 in response to the Opinionator
blog post How Can We Jump-Start the Struggle for Gender Equality?
HKGuy
NYC
If surveys have measured 46% of the labor force as female, even aside from
the fact that is a statistically negligible percentage, that means only 4% of women
are stay-at-home moms and full-time homemakers. I find that astonishing. 5
Exercise 3.10.74. [U]
Organic foods account for 4.2 percent of retail food sales, according to the U.S.
Department of Agriculture. It certifies products as organic if they meet certain
requirements including being produced without synthetic pesticides or fertilizers,
or routine use of antibiotics or growth hormones.
5
opinionator.blogs.nytimes.com/2013/11/23/how-can-we-jump-start-the-struggle-for-gender-equality
61
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3.10. COMMENTS ON THE EXERCISES
Consumers can pay a lot more for some organic products but demand is rising:
Organic foods accounted for $31.4 billion sales last year, according to a recent
Obama administration report. That’s up from $3.6 billion in 1997. 6
(a) Use these figures to find total retail food sales “last year” – 2011.
(b) Estimate total annual retail food sales as a Fermi problem.
(c) Are your answers in the same ballpark?
Interesting followup question/discussion:
by jhbeck23 September 5, 2012 9:41 AM EDT:
Tom Philpott at MotherJones (dot-com) sees this study rather differently.
The study’s view reported above that “Organic produce had a 30 percent lower
risk of containing detectable pesticide levels” was reached by subtracting a 5%
chance with organic from a 35% chance with non-organic. Wouldn’t you usually
say that the chance was seven times higher with non-organic? Or that the chance
was 1 in 20 for organic, one in three for non-organic? The substraction is not
wrong, but it seems to be stated so as to minimize the difference.
A critique of original Stanford study confirms this confusion between percentages and
percentage points: http://www.tfrec.wsu.edu/pdfs/P2566.pdf
Exercise 3.10.75. [S] Hubway
Figure 3.1 accompanied an article in The Boston Globe about the Hubway bicycle exchange
program. According to the article:
A typical station with bikes costs about $50,000 and requires $10,000 in
annual subsidies, though operations are expected to be covered fully by user
fees and advertising as Hubway matures. Currently, there are 1,003 bicycles
distributed across the Hubway’s 105 stations. Bike users can pay a one-time fee
or buy an annual membership for $85.
(a) What was the percentage increase in annual use (trips) of Hubway bicycles from 2011
to 2012?
6
http://www.cbsnews.com/8301-204_162-57505328/organic-food-hardly-healthier-study-suggests/
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62
3.10. COMMENTS ON THE EXERCISES
Figure 3.1: Hubway
(b) Revise your answer to (a) to take into account the fact that Hubway opened for
business in July 2011, so it was available for only half of 2011.
(c) On average, how many trips did each active annual member take in 2012?
(d) On average, how many bikes are available at each station?
(e) Compare the revenue from annual subscriptions to the cost of maintaining the Hubway
stations.
(a) What was the percentage increase in annual use (trips) of Hubway bicycles from 2011
to 2012?
533, 257 trips in 2012
= 3.74770361729 ≈ 375%.
142, 289 trips in 2011
That’s a 275% increase.
(b) Revise your answer to (a) to take into account the fact that Hubway opened for
business in July 2011, so it was available for only half of 2011.
I will double the number of 2011 trips to get an estimate of what they would have been
for a whole year. Then the computation is
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c 2014 Ethan Bolker, Maura Mast
3.10. COMMENTS ON THE EXERCISES
533, 257 trips in 2012
= 1.87385180864 ≈ 187%.
2 × 142, 289 trips in 2011
That’s an 87% increase. This answer makes more sense than the previous one.
(c) On average, how many trips did each active annual member take in 2012?
365, 257 trips by active members
trips
≈ 46
.
7, 885 active members
active member
I can believe that – it’s about two per week for six months of bikeable weather.
(d) On average, how many bikes are available at each station?
There are about 1000 bikes and about 100 stations, so about 10 bikes per station on
average.
(e) Compare the revenue from annual subscriptions to the cost of maintaining the Hubway
stations.
Revenue from annual subscriptions is
$85
× 11, 130 subscribers ≈ $950, 000.
subscriber
It costs $10,000 to maintain each of the 105 stations, for a total maintenance cost of
about $1 million. So the subscriptions pay for about 95% of the maintenance.
A more precise calculation gives
$85
subscriber
× 11, 130 subscribers
105 stations ×
$10,000
station
= 0.901 ≈ 90%.
3.10
Exercise 3.10.76. [S][Section 3.1] [Goal 3.3] Gasoline in Europe
In The New York Times on July 20, 2011 reader Robin Foor commented on the article
Bipartisan Plan for Budget Deal Buoys President:
European nations use a value added tax and the gasoline tax to generate
revenue. The tax on a gallon of gas in Germany is more than $3. The purpose is
to raise revenue and discourage the use of cars.
The US per gallon tax on gasoline is 18.4 cents per gallon, which it has been
since 1993. It is a per unit tax, not a percentage of value tax. . . .
c 2014 Ethan Bolker, Maura Mast
64
3.10. COMMENTS ON THE EXERCISES
In 1993 gasoline cost $1.16 per gallon. The 18.4 cents per gallon tax was
15.9 percent of the value of a gallon of gas. Today gas is about $3.70 a gallon.
Equivalent tax at 15.9 percent is 58.7 cents per gallon. The increased tax of 40.3
per gallon would fund fixing the road. 7
(a) Check the percentage calculations in this quote.
(b) Correct the writing errors in the last sentence.
(c) Verify as many of the historical quantitative claims as you can.
(a) Check the percentage calculations in this quote.
0.159 × 1.16 = 0.18444 so the 18.4 cents per gallon claim is right.
0.159 × 3.70 = 0.5883 rounds to 58.8 cents per gallon. I don’t know how the Times came
up with 58.7 cents per gallon.
(b) Correct the writing errors in the last sentence.
The last sentence is missing the “cents” in cents per gallon and an “s” at the end of road.
What’s worse is that it mixes up the new value of the tax with the increase in the tax. It
should read
The increase in tax to 58.7 cents per gallon would fund fixing the roads.
or
The increase in tax of of 40.3 cents per gallon would fund fixing the road.
(c) Verify as many of the historical quantitative claims as you can.
Der Spiegel, a well known German magazine, reported in its English language web page8
on April 5. 2012, that
Tax authorities apply a set tax rate to every liter (quarter gallon) of fuel –
47.40 cents for diesel fuel, for example, and 65.45 cents for gas
I assume that’s U.S. cents, not hundredths of a Euro. Since four times 65.45 cents is
more than three dollars, the Times is right.
Many web sites confirm the 18.4 cents per gallon tax. (For example, http://taxfoundation.
org/article/federal-gasoline-excise-tax-rate-1932-2008.)
7
http://community.nytimes.com/comments/www.nytimes.com/2011/07/20/us/politics/20fiscal.
html
8
http://www.spiegel.de/international/germany/high-german-fuel-taxes-have-been-a-bonanza-for-governme
html
65
c 2014 Ethan Bolker, Maura Mast
3.10. COMMENTS ON THE EXERCISES
Many web sites confirm the total 1993 cost of (approximately) $1.16 per gallon. The graph
at http://flowingdata.com/2008/08/08/watch-the-rise-of-gasoline-retails-prices-1993-2
is cool.
Exercise 3.10.77. [S][Section 3.3][Goal 3.1] [Goal 3.3]
service cut
Top official favors fare hike over
On Sunday, February 19, 2012 The Boston Globe reported that
. . . [The MBTA] provides 1.3 million rides each weekday
...
The proposed increases - a one-way subway ride, now $1.70 with the reloadable
CharlieCard and $2 with paper CharlieTicket, could rise to as much as $2.40 and
$3, respectively - would be the largest by percentage since 1949,
...
the advisory group said fares should rise no more than 25 percent, acknowledging the need for more money but wary of pricing people off the system. Such
a fare increase would generate about $80 million . . . 9
(a) Compare the proposed increases in CharlieCard and CharlieTicket fares, in absolute
and relative terms.
(b) Estimate how much extra revenue would be generated in a year from weekday rides if
CharlieCard and CharlieTicket fares increased by just 25%.
(c) Compare your answer to the assertion in the quotation that a 25% increase in fares
would yield $80 million in new revenue.
(d) What was the percent increase in T fares in 1949?
(a) Compare the proposed fares to the current fares in absolute and relative terms.
The CharlieCard fare will go up $0.70 (the absolute change). Since $2.40/$1.70 =
1.41176471 that’s about a 40% increase.
The corresponding increases for the CharlieTicket fares are $1 (absolute change) and
50% (relative change).
(b) Estimate how much extra revenue would be generated in a year from weekday rides if
CharlieCard and CharlieTicket fares increased by just 25%, as proposed in the last part
of the quotation.
9
http://bostonglobe.com/metro/2012/02/19/top-official-favors-fare-hike-over-service-cut/
D08UlBCew6682ddN4FLFBL/story.html
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66
3.10. COMMENTS ON THE EXERCISES
There are 52 weeks per year, 5 weekdays per week, so 260 days on which extra fare is
collected. There are 1.3 million riders each weekday, so 260 × 1.3 = 338 million weekday
rides.
Now I need to know how much more money per ride would come from a 25% increase in
the fare. For CharlieCards that would be 0.25 × $1.70 = $0.43. For CharlieTickets it
would be 0.25 × $2.00 = $0.50. If I assume about half the riders use Cards and the other
half use Tickets I can estimate about $0.48 extra per ride. Then then increase in revenue
would be
$
≈ $162 million.
338 million rides × 0.48
ride
(c) Compare your answer to the assertion in the quotation that a 25% increase in fares
would yield $80 million in new revenue.
If my calculation is correct, that 25% increase would collect about double the $80 million
claimed. I wonder why.
(d) What was the percent increase in T fares in 1949?
In 1949, the MBTA proposed increasing the charge for customers exiting trains above
ground. The increase would be 5 cents. It’s not clear how this would increase fares, as
apparently fares were very complicated back then. However, this particular fare increase
is famous for the folk song “M.T.A.” (or “Charlie on the M.T.A.” as most people refer to
it) about a fictional subway rider named Charlie who was doomed to ride forever because
he didn’t have that nickel to get off of the T. The MBTA 10 has a short description.
When the MBTA introduced plastic fare cards they named them “Charlie cards”.
If the song is correct, then the previous fare was 10 cents. It increased to 15 cents total,
which is a 50% hike.
The Kingston Trio made the song famous in 1959.
Exercise 3.10.78. [S][Section 3.3] [Goal 3.1] [Goal 3.3] Natural Born Drillers
On March 16, 2012 Paul Krugman wrote in The New York Times:
And this tells us that giving the oil companies carte blanche isn’t a serious
jobs program. Put it this way: Employment in oil and gas extraction has risen
more than 50 percent since the middle of the last decade, but that amounts to
only 70,000 jobs, around one-twentieth of 1 percent of total U.S. employment. So
the idea that drill, baby, drill can cure our jobs deficit is basically a joke. 11
(a) Use the data in the quotation to estimate total U. S. employment.
10
11
http://www.mbta.com/about_the_mbta/history/?id=19582
www.nytimes.com/2012/03/16/opinion/krugman-natural-born-drillers.htm
67
c 2014 Ethan Bolker, Maura Mast
3.10. COMMENTS ON THE EXERCISES
(b) Use the data here (not a web search!) to estimate oil and gas industry employment in
2005
Think carefully first about what the 70,000 oil industry jobs represents. There are several
possibilities. Write them down; then decide which you think is right, and why.
(c) Use the web to check your estimate.
(a) Use the data in the quotation to estimate total U. S. employment.
(b) Use the data here (not a web search!) to estimate oil and gas industry employment in
2005
Think carefully first about what the 70,000 oil industry jobs represents. There are several
possibilities. Write them down; then decide which you think is right, and why.
The quotation says “risen more than 50 percent since the middle of the last decade, but
that amounts to only 70,000 jobs”. I can’t tell whether the 70,000 is the amount that
employment rose or the number of jobs after the increase.
(c) Use the web to check your estimate.
Exercise 3.10.79. [S][Section 3.4][Goal 3.3] [Goal 3.4] At Sleepaway Camp, Math Is Main
Sport
On July 27, 2011 The New York Times reported from a math camp for middle school kids
who like mathematics:
In a Bard classroom one afternoon, it seemed for a moment that Arturo
Portnoy had stumped everyone. Dr. Portnoy, a math professor visiting from
the University of Puerto Rico, posed this question: “The length of a rectangle
is increased by 10 percent and the width is decreased by 10 percent. What
percentage of the old area is the new area?” 12
What’s the answer to Portnoy’s question?
The crucial mathematics is 1.1 × 0.9 = 0.99 – the new area is 99% of the original.
If you’re uncomfortable with just the 1+ trick, work it out for a square that’s 10 inches on
each side, with an area of 100 square inches.
Exercise 3.10.80. [U][R][Section 3.7][Goal 3.2] [Goal 3.3] [Goal 3.6] National Grid plans to
downsize by 1,200 US jobs
On January 31, 2011 Erin Ailworth reported in The Boston Globe that
12
http://www.nytimes.com/2011/07/28/nyregion/a-sleepaway-camp-for-low-income-ny-math-whizzes.
html
c 2014 Ethan Bolker, Maura Mast
68
3.10. COMMENTS ON THE EXERCISES
Electric and gas supplier National Grid will downsize its US workforce by
1,200 jobs, or about 7 percent, in a restructuring that the London-based utility
company says will help it save $200 million.
It was not immediately clear how many of the lost jobs – focused at management
levels – would be in Massachusetts, where National Grid employs roughly 5,000.
The company currently has nearly 18,000 total workers in the US. 13
(a) Use the information in the first paragraph to find the size of National Grid’s U.S.
workforce.
(b) Is the 18,000 figure in the second paragraph consistent with your answer to the previous
part of the problem?
(c) How much money will the company save per worker fired?
(d) What percentage of National Grid’s U.S. employees are in Massachusetts?
(e) If the firings in Massachusetts occur at the 7% rate how many jobs will be lost there?
Exercise 3.10.81. [S][Section 3.2][Goal 3.3] ‘The Scream’ Is Auctioned for a Record $119.9
Million
On May 12, 2012 The New York Times reported on another sale at that auction: “Picasso’s
’Femme Assise Dans un Fauteuil,’ a 1941 portrait of Dora Maar, the artist’s muse and lover,
posed in a chair . . . went for $26 million, or $29.2 million with fees.”
Earlier, the article described how Sotheby’s computes fees: “Final prices include the buyer’s
commission to Sotheby’s: 25 percent of the first $50,000; 20 percent of the next $50,000 to $1
million and 12 percent of the rest.” 14
Check that the Times’ explanation about how the commission was computed does in fact
match the data in the quotation about fees.
Sotheby’s buyer’s commission on a $26 million sale is
25% of $50,000
20% of $950,000
12% of $25 million
$12,500
$190,000
$3,000.000
for a total of $3,202,500. When I add that to the $26 million sale price I do get the Times’
$29.2 million, plus an extra $2,500. I guess $2,500 is just chump change in this game.
13
http://www.boston.com/business/ticker/2011/01/national_grid_p_1.html
http://www.nytimes.com/2012/05/03/arts/design/the-scream-sells-for-nearly-120-million-at-sothebys-a
html
14
69
c 2014 Ethan Bolker, Maura Mast
3.10. COMMENTS ON THE EXERCISES
Exercise 3.10.82. [S] [Section 3.3] [Goal 3.1] [Goal 3.2] Small business loans
Figure 3.2 appeared in The Boston Globe on June 12, 2010.
Figure 3.2: Small Business Loans
(a) Compute the absolute and relative changes in small business lending for Massachusetts
and for the United States between 2009 and 2010.
(b) Did small business lending change more in Massachusetts than in the United States?
Explain how you could justify answering this question either “yes” or “no”.
(c) How did the proportion of small business lending in Massachusetts compare to that in
the United States change between 2009 and 2010?
(d) Compare the average size of a small business loan in Massachusetts in 2009 to the
nationwide average. Did Massachusetts companies get more (per loan) or less?
(a) Compute the absolute and relative changes in small business lending for Massachusetts
and for the United States between 2009 and 2010.
Small business lending for Massachusetts increased from $61.8 million to $142.3 million.
That’s an absolute change of $142.3 − $61.8 = $80.5 million, and a relative change of
80.5/61.8 = 1.302589 ≈ 130 percent. The number more than doubled.
Note that the relative change is a percent, not some number of dollars.
The corresponding changes for the United States are $9.8 − $5.0 = $4.8 billion (absolute)
and 4.8/5.0 = 0.96 = 96% (relative).
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70
3.10. COMMENTS ON THE EXERCISES
(b) Did small business lending change more in Massachusetts than in the United States?
Explain how you could justify answering this question either “yes” or “no”.
Small business lending increased more in the United States in absolute terms (which is no
surprise since Massachusetts is only a small part of the United States) but that lending
increased much more in Massachusetts in relative terms: 300% rather than just 96%.
(c) How did the proportion of small business lending in Massachusetts compare to that in
the United States change between 2009 and 2010?
In 2009 the proportion of small business lending in Massachusetts was
$61.8 million
= 0.01236 ≈ 1.2%
$5 billion
while in 2010 it was
$142.3 million
== 0.0145204082 ≈ 1.45%.
$9.8 billion
So the percentage coming to Massachusetts increased by 0.25 percentage points. That
was an increase of 0.25/1.2 = 0.208333333 or almost 21%.
(d) Compare the average size of a small business loan in Massachusetts in 2009 to the
nationwide average. Did Massachusetts companies get more (per loan) or less?
Not yet done. Not hard.
Exercise 3.10.83. [S][Section 3.1] [Goal 3.3] The prison population.
From The New York Times, May 31, 2011:
Every year America spends close to $66 billion to keep people behind bars.
But almost 500,000 of the 2.3 million prisoners arent convicts; rather, they are
accused individuals awaiting trial. 15
(a) What percentage of the population is behind bars?
(b) How does that percentage compare to the percentages in other countries?
(c) How much does it cost to incarcerate a prisoner for a year?
(d) Can you verify $66 billion and 2.3 million figures in this quotation?
15
http://www.nytimes.com/2011/05/31/opinion/31baradaran.html
71
c 2014 Ethan Bolker, Maura Mast
3.10. COMMENTS ON THE EXERCISES
(a) What percentage of the population is behind bars?
I want two significant digits for the U.S. population, so my usual estimate of 300 million
won’t do. I looked up the population on September 24, 2012 at http://www.census.
gov/main/www/popclock.html and found 314,445,535.
2.3 million prisoners
= 0.0073 = 0.73%.
314 million people
(b) How does that percentage compare to the percentages in other countries?
I googled “prison population by country” and found the wikipedia page at http:
//en.wikipedia.org/wiki/List_of_countries_by_incarceration_rate. There the
United States is at the top of the list. The rate of 730 inmates per 100,000 people
confirms my answer to the previous question.
Wikipedia got all its numbers from the International Centre for Prison Studies (http:
//www.prisonstudies.org/); I never quote wikipedia without looking to see where they
got their information.
(c) How much does it cost to incarcerate a prisoner for a year?
$66 × 109
$
= 29, 000
6
2.3 × 10 prisoners
prisoner
(rounded to two significant digits).
(d) Can you verify $66 billion and 2.3 million figures in this quotation?
The wikipedia article above verifies the number of prisoners.
On April 22, 2012 CBS News reported on a study from “the non-partisan Vera Institute
of Justice” that said (in part) that “Our epidemic of incarceration costs us taxpayers
$63.4 billion a year.” That’s in the same ballpark as the $66 billion figure above. I didn’t
check further, but could have if I’d needed to.
c 2014 Ethan Bolker, Maura Mast
72
Chapter 4
Inflation
4.1
Red Sox ticket prices
Computing the effective increase after taking inflation into account is subtle. You can’t just
subtract the percentages. We return to this question in the section below addressing what
a raise is worth. It’s probably best to finesse the question here by making only qualitative
statements – e.g. “faster than inflation.”
4.2
Inflation is a rate
It’s difficult to keep the printed text up to date on current events. When you teach this
section, study inflation from last year to this year. Then the students have two separate
treatments to learn from – yours in class and the one in the book.
4.3
More than 100%
4.4
The consumer price index
The inflation calculator gives $215.40 for 1983 and $206.48 for 1984. The average is
(214.40 ∗ 206.48 ∗ 222.32)1/3 = 214.302433
but you don’t want to teach that!
73
4.5. HOW MUCH IS YOUR RAISE WORTH?
4.5
How much is your raise worth?
4.6
The minimum wage
4.7
Comments on the Exercises
22 This would make an interesting spreadsheet exercise when we get to spreadsheet calculations
and graphing. Then you can ask if there is any year in which the minimum wage went up
fast enough to account for inflation.
Extra (ideas for) Exercises
4.7
Exercise 4.7.27. [N][G][Section 4.6] [Goal 4.2] [Goal 1.1] [Goal 1.2]
SUPPORT OF A $10.50 U.S. MINIMUM WAGE
ECONOMISTS IN
At http://www.peri.umass.edu/fileadmin/pdf/resources/Minimum_Wage_petition_website.
pdf you can read a July 2013 open letter from dozens of economists advocating an increase in
the federal minimum wage. (There’s a local copy at http://www.cs.umb.edu/~eb/qrbook/
Instructor/EconomistsMinimumWagePetition.pdf .)
The essay suggests lots of good quantitative reasoning questions: check their arithmetic,
verify their estimates, put their conclusions in a personal context, . . .
Here’s one particularly interesting paragraph.
On average, even fast food restaurants, which employ a disproportionate
share of minimum wage workers, are likely to see their overall business costs
increase by only about 2.7 percent from a rise today to a $10.50 federal minimum
wage. That means, for example, that McDonalds could cover fully half of the cost
increase by raising the price of a Big Mac, on average, from $4.00 to $4.05.
Exercise 4.7.28. [S][Section 4.2][Goal 4.2] [Goal 4.5] Middle income family spends $235,000
to raise baby.
On June 23, 2012, you could read on the Dallas Morning News website that
c 2014 Ethan Bolker, Maura Mast
74
4.7. COMMENTS ON THE EXERCISES
[In 1960] the cost of raising a child was just over $25,000 for middle-income
families. That would be $191,720 today when adjusted for inflation. 1
(a) Use the inflation calculator to check this inflation assertion.
(b) Has the cost of raising a child increased faster or slower than inflation? Does that
question even make sense?
[See the back of the book for a hint.] You will have to think about whether $190K was
enough to raise a child in 2012.
(a) Use the inflation calculator to check this inflation assertion.
The inflation calculator tells me $193,913.85 for the 2012 value of $25,000 in 1960. That’s
a little larger than the $191,720 reported in the newspaper. That article appeared in
June, so maybe I should check the 2011 inflation result. OK, that’s $189,982.26, which is
a little low. Perhaps if I dug deeper I could find the midyear data and see whether it
matches the article, but my time is worth more than it would take me to do that.
(b) Has the cost of raising a child increased faster or slower than inflation? Does that
question even make sense?
There’s really no way to answer this question without data about what expenses were
counted in the 1960 report. Even with that data I’d have to quantify the changes in what
one had to buy to raise a kid. And I’d have to wonder about whether the $25,000 figure
was itself adjusted for inflation during the 18 years that child was considered a child.
On the whole, I think this kind of article is a headline grabbing waste of time and energy.
Exercise 4.7.29. [N][U][Section 4.5] [Goal 4.1] [Goal 4.2] [Goal 4.4] COLA
The website http://www.fedsmith.com featured an article in October 2011 about the
projected 2012 cost of living increase for federal retirees. It noted that
For the first time in three years, there will be an increase in the cost of living
adjustment for federal retirees and those receiving Social Security checks. The
amount of the increase: 3.6% for CSRS retirees and Social Security recipients. . . .
The last COLA increase was 5.8% in 2009 (based on 2008 inflation figures). The
two years without any increase in the cost of living had not happened since the
automatic increase formula for Social Security was established in 1975. Prior to
these two years without any increase, the lowest annual adjustment was 1.3% in
1
http://www.dallasnews.com/business/headlines/20120623-middle-income-family-spends-235000-to-raise-ba
ece
75
c 2014 Ethan Bolker, Maura Mast
4.7. COMMENTS ON THE EXERCISES
1998. The reason for the lack of an increase the past two years was because, as
reported by the federal government for this purpose, there was no inflation. 2
Need questions here. This may be too complicated because the COLA does not seem to
be equal to the inflation rate. The idea of zero inflation and no cost of living raise is an
interesting one.
Exercise 4.7.30. [N][Goal 3.3] [Goal 2.1] [Goal 4.2]
World War II Shipwreck
48 Tons of Silver Recovered From
On July 18, 2012 ABC News reported that
An American company has made what is being called the heaviest and deepest
recovery of precious metals from a shipwreck.
The recovery is being made under a contract awarded by the U.K. government,
which will keep 20 percent of the cargo’s value, estimated to be in the tens of
millions of dollars. The Gairsoppa became U.K. property after the government
paid the owners of the ship an insurance sum of 325,000 in 1941. Records indicate
the silver was valued at 600,000 in 1941.
...
The Tampa, Fla.-based Odyssey Marine Exploration, Inc. announced Wednesday that it had recovered 48 tons of silver [and] that about 240 tons of silver may
be recovered by the end of the operation.
3
Possible questions:
(a) some routine percentage calculations
(b) value of silver then and now? did it match inflation rate?
(c) currency conversion
Exercise 4.7.31. [U][N][W][Goal 4.2] [Goal 4.4] Apple Becomes the Most Valuable Public
Company Ever, With an Asterisk
From The New York Times blogs, August 20, 2012:
2
http://www.fedsmith.com/article/3150/36-percent-cola-2012.html
http://abcnews.go.com/International/record-setting-silver-recovery-made-world-war-ii/
story?id=16806534#.UAgLMZHCbTo
3
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76
4.7. COMMENTS ON THE EXERCISES
With a surge in its share price, Apple broke the record for the biggest market
capitalization, $620.58 billion, set by Microsoft on Dec. 30, 1999, . . . That record
was for an intraday market value – a company’s total outstanding shares multiplied
by its stock price at some point during the trading day. On Monday, Apple had
an intraday market value of $623.14 billion when its shares traded at $664.74.
To exceed Microsoft’s record for market value at the close of a trading day,
Apple shares needed to close at $657.50 or higher, . . . which they did. The stock
closed at $665.15.
Another analyst [noted] that Microsoft’s 1999 market value is still far higher
than Apple’s when adjusted for inflation. The Microsoft of late 1999 would be
worth $850 billion in today’s dollars . . . The Microsoft of August 2012 is worth
$257 billion.
To beat Microsoft’s inflation-adjusted market value, Apple needs to close at
$910 . . . . Perhaps not this week.
Exercise 4.7.32. [S] Running to keep up with inflation
(a) Use the data in Exercise 3.10.2 (page 72) to compare the growth in the number of
finishers in the New York marathon to the inflation rate for the corresponding years.
(b) Why is this a ridiculous question?
(a) Use the data in Exercise 3.10.2 (page 72) to compare the growth in the number of
finishers in the New York marathon to the inflation rate for the corresponding years.
46, 795 runners
≈ 3.54
13, 203 runners
so the 2011 number is about three and a half times the 1981 number. It’s a 254% increase.
The CPI inflation calculator told me it would take $247 in 2011 to buy what cost $100 in
1981. That’s an increase of 147%. That’s about about 100 percentage points less than
the increase in the number of runners.
(b) Why is this a ridiculous question?
It makes no sense to compare the increase in popularity of a race with the increase in
the dollar cost of living.
Exercise 4.7.33. [U][Section 4.2][Goal 4.2] Charging for Debit Cards Is Robbery
On October 7, 2011 Lloyd Constantine wrote an Op-Ed piece in The New York Times that
said (in part)
77
c 2014 Ethan Bolker, Maura Mast
4.7. COMMENTS ON THE EXERCISES
Debit cards were developed by banks as a replacement for paper checks. When
a consumer pays with a debit card instead of a check, the bank saves money. In
the 1980s, Visa calculated the savings at 55 cents to $1.60 per check. The savings
is much higher today. 4
(a) If we assume that the cost of processing checks has grown with inflation we should
calculate the cost “55 cents to $1.60 . . . [in] the 1980s” in 2011 dollars. Do that.
(b) Estimate the number of debit card transactions per day in the United States.
(c) How much do banks earn on debit card transactions because they save the cost of
processing checks?
(d) How should the “$6.6 billion cost” in Exercise 3.10.35 (page 82) be revised to take
into account your answer to the previous part of this exercise?
4.7
Exercise 4.7.34. [U][C][Section 4.2][Goal 4.2] [Goal 4.5] Making Good on Pledge, Gates
Outlines Military Cuts
In the article with that headline in The New York Times on August 9, 2010 you can read
that
Mr. Gates is calling for the Pentagon’s budget to keep growing in the long
run at 1 percent a year after inflation, plus the costs of the war. It has averaged
an inflation-adjusted growth rate of 7 percent a year over the last decade (nearly
12 percent a year without adjusting for inflation), including the costs of the wars.
So far, Mr. Obama has asked Congress for an increase in total spending next
year of 2.2 percent, to $708 billion – 6.1 percent higher than the peak under the
Bush administration. 5
(a) Make sense of these numbers.
Some readers were confused, and said so:
Unless I missed something in this article, there are no cuts at all if the overall
budget is increasing 1% above inflation. This is a shell game.
(b) What is a shell game?
4
5
www.nytimes.com/2011/10/07/opinion/debit-card-fees-are-robbery.html
http://www.nytimes.com/2010/08/10/us/10gates.htm
c 2014 Ethan Bolker, Maura Mast
78
Chapter 5
Average Values
5.1
Average test score
Decide whether you want to point out that the units of the average are the same as the
units of the things you are averaging, since the weights are dimensionless. Sometimes we do,
sometimes we don’t. The observation may confuse students while they are working to master
the new concept.
5.2
Grade point average
Students find this section on grade point average computation compelling – many often say
they had no idea how it was done and are delighted to have found out. It makes the weighted
average concept clear in a context that really matters.
5.3
Improving averages
We find that many students solve this equation by writing some version of
90 × 2.8 + 30 G
= 3.0 = 360 = 108 = 3.6.
120
You can of course see what they’re thinking, and the answer is the right value of G. Their
prose (if that’s what you can call it) conflates “=” meaning “is the same number as” with
“=” meaning “is the same equation as”. We constantly ask for “more words” and can’t seem
to get them. If you know how, please let us know.
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5.4. THE CONSUMER PRICE INDEX
One of the readers of an early version of Common Sense Mathematics suggested that we
start the solution with guess-and-check and finish with algebra, to reinforce the value of a
viable understood strategy compared to one that may be murky and must be remembered.
Consider that when you lecture on the material.
It’s worth taking a little time to practice the guess-and-adjust-your-guess strategy. It may
not be as efficient as algebra in situations like this, but it’s much more generic and much less
arcane. Students can understand and appreciate it and might even remember it.
5.4
The consumer price index
5.5
New car prices fall . . .
5.6
An averaging paradox
This paradox is a well known phenomenon. See Chapter 6, A Small Paradox, in Is Mathematics
Necessary?, Underwood Dudley (ed.), Mathematical Association of America, 2008, and further
references there.
5.7
Comments on the Exercises
Extra (ideas for) Exercises
5.7
Exercise 5.7.19. [U][N][Section 5.1][Goal 5.1] Improving Reading Skills Should be a state
priority
On June 29, 2010 the Cape Cod Times reported that “Almost half of Massachusetts thirdgraders are not proficient readers.”
Last year, 65 percent of low-income third grade students scored below proficient
on the MCAS reading test. And overall, the percentage of third-graders receiving below-proficient scores has hovered around 40 percent over the last decade.
http://www.capecodonline.com/apps/pbcs.dll/article?AID=/20100629/OPINION/6290319
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5.7. COMMENTS ON THE EXERCISES
There’s a weighted average hidden here. If we knew either the percentage of low-income third
graders in the population or the percentage of non-low-income students who scored below
proficient we could find the other percentage.
Exercise 5.7.20. [U][C][Section 5.5][Goal 5.1] Average Boston homeowner can expect $220
tax increase
The property tax bill for the average home in Boston is increasing by $220 in
the current fiscal year, the city announced yesterday.
...
Ronald W. Rakow, the city’s commissioner of assessing, said in a telephone
interview that the bill for the average single-family residence will come to $3,155
in fiscal 2011, which began in July, up from $2,935 in fiscal 2010. He said the
rate is “still very competitive, compared to the statewide average” of $4,390.
“Due to the current economic climate, reductions in the [city’s] other revenue
sources, most notably state aid, left the [city] with little choice but to increase
the property tax levy by the maximum allowed under Proposition 2 1/2,” the
state law capping property tax rate increases at 2.5 percent without an override,
Mayor Thomas M. Menino’s office said in a statement yesterday.
Menino’s office said the tax rate for residential properties in fiscal 2011 is
$12.79 per $1,000 of assessed value, up from $11.88.
The rate for commercial properties is $31.04 per $1,000 of assessed value, up
from $29.38 in the last fiscal year.
...
The tax rate for homes in the city was $11.12 per $1,000 of assessed value in
fiscal 2006, $10.99 in fiscal 2007, $10.97 in fiscal 2008, and $10.63 in fiscal 2009,
records show. 1
Figure 5.1 accompanied the article.
Lots more questions to ask
(a) Verify that the rate increases for residential property for the years reported were
constent with Proposition 2 1/2?
(b) Is the rate increase for commercial property consistent with Proposition 2 1/2 ?
(c) What does “average homeowner” mean? Median, mean or mode?
(d) To do: write a question about total assessment and tax income - probably a weighted
average.
1
http://www.boston.com/news/local/massachusetts/articles/2010/12/18/average_boston_
homeowner_can_expect_220_tax_increase/
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5.7. COMMENTS ON THE EXERCISES
Figure 5.1: Boston Real Estate Assessments
5.7
Exercise 5.7.21. [S][Section 5.1][Goal 5.1] Texting teens.
The chart in Figure 5.2 appeared in The Boston Globe on April 15, 2012.
Figure 5.2: Teen Texting
(a) Estimate from the chart the mode, median, and mean number of text messages sent
by teenagers each day.
(b) In total, approximately how many text messages are sent by the 23 million American
teens each day?
(c) The percentages don’t add up to 100%. Why might that have happened?
(d) If you asked a random teenager how many text messages she sent yesterday what are
the chances (what is the probability) that it was more than 50? More than 100? More
than 25?
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5.7. COMMENTS ON THE EXERCISES
(e) What percent of teenagers text more than the median amount?
(a) Estimate from the chart the mode, median, and mean number of text messages sent
by teenagers each day.
The mode is the tallest column: more teenagers text 101-200 messages a day than any
other category. It’s also reasonable to say that the mode is about 150 messages per day.
Adding up the percentages for the first three columns comes to 2% + 22% + 28% = 52%.
That means just over half the teenagers text up to 50 messages per day. That’s a good
estimate for the median.
I will compute the mean as a weighted average, using the midpoints of the ranges for
each range.
0.02 × 0 + 0.22 × 5 + 0.28 × 30 + 0.16 × 75 + 0.31 × 150 = 68,
so the mean number of daily teen text messages is about 70.
I could also have computed using the exact midpoints of the ranges. The arithmetic is
uglier:
0.02 × 0 + 0.22 × 5.5 + 0.28 × 30.5 + 0.16 × 75.5 + 0.31 × 150.5 = 68.705,
which is still about 70. The extra precision isn’t worth the effort.
(b) In total, approximately how many text messages are sent by the 23 million American
teens each day?
Since I have the mean, 70 messages per teenager on average, all I have to do is multiply
by 23 million to get about 1.6 billion messages per day.
(c) The percentages don’t add up to 100%. Why might that have happened?
The percentages sum to 99%. The missing one percent is probably roundoff error.
(d) If you asked a random teenager how many text messages she sent yesterday what are
the chances (what is the probability) that it was more than 50? More than 100? More
than 25?
There’s a 47% chance that a random teenager texts more than 50 messages in a day and
a 31% chance that s/he texts more than 100.
For 25 messages I have to use a fraction of the 11-51 message column. Since 25 is about
one third of the way from 11 to 50 I will suppose that about two thirds or 18% of the
teens in that category text more than 25 messages a day. Adding that 18% to the 47%
for the two top columns gives me an estimate of 65% for the chances that a teenager
texts more than 25 times a day.
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5.7. COMMENTS ON THE EXERCISES
(e) What percent of teenagers text more than the median amount?
By definition that’s 50%.
Exercise 5.7.22. [S][Section 5.3][Goal 5.1] [Goal 5.2] Some UMass unions agree to wage
freeze
The Boston Globe reported on February 25, 2009 in its New England in Brief column that
Faculty members at the Boston and Amherst campuses of the University of
Massachusetts have agreed to a three-year contract that freezes their wages for the
current academic year. . . . The new contract raises salaries by 1.5 percent in the
coming academic year and 3.5 percent the following year. . . . At UMass-Amherst,
full professors earn an average of $117,000 a year. Faculty members earn an
average of $90,000, according to university data.
(a) What was the average Amherst full professor’s salary in the 2009-2010 academic year?
(b) What is the percentage increase in faculty salaries over the life of the contract?
(c) There was no raise in 2009, so the salaries then were the same as they were the year
before. What is the buying power of the a faculty member’s $90K salary in 2009, in 2008
dollars?
(d) Suppose 30% of the faculty at Amherst were full professors. What was the average
salary of the other 70%?
(a) What was the average Amherst full professor’s salary in the 2009-2010 academic year?
Using the 1+ trick is the fastest way to do the problem: in 2010 the average UMassAmherst full professor salary was 1.015 × $117, 000 = $118, 755.
You can also find the answer by figuring out the raise and then adding it to the base
salary.
(b) What is the percentage increase in faculty salaries over the life of the contract?
To compute the effect of successive raises of 1.5% and 3.5% I multiplied 1.015 × 1.035 =
1.050525. The result corresponds to a raise of 5.0525%. I know I couldn’t just add the
two percentage raises to get 5%, but the difference in this case is pretty small: just over
$50 – a dollar a week – on a salary of $100K.
(c) There was no raise in 2009, so the salaries then were the same as they were the year
before. What is the buying power of the a faculty member’s $90K salary in 2009, in 2008
dollars?
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5.7. COMMENTS ON THE EXERCISES
The inflation calculator tells me $90K in 2009 was worth $90,321 in 2008 dollars. So
although there was no raise, professors were actually a little better off because the
economic meltdown that year led to a little deflation.
(d) Suppose 30% of the faculty at Amherst were full professors. What was the average
salary of the other 70%?
The $90K average is a weighted average of the salaries of the full professors (weight 30%)
and that of the rest of the faculty (weight 70%). So I need to solve the equation
0.7x + 0.3 × $117K = $90K
for x. The answer is about $78,400.
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5.7. COMMENTS ON THE EXERCISES
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86
Chapter 6
Income Distribution – Excel, Charts
and Statistics
6.1
Salaries at Wing Aero
We find students come to our course with a wide range of technology experience. They can
all manage word processing, web searching and email. Some have used Excel. Those for
whom it is new find this chapter hard going. Often pairing experienced and inexperienced
students in the lab classroom works well.
6.2
What if ?
6.3
Using software
6.4
Median
6.5
Bar charts
When teaching Excel in a classroom we strongly recommend drawing bar charts first by hand
on the board, and asking students to do the same on paper on homework and exams. That
strategy helps later when they accidentally draw a line chart where a scatter plot is called for.
87
6.6. PIE CHARTS
Excel can use separate vertical scales to plot two data series. We think it’s more useful and
more interesting to teach this ad hoc solution.
6.6
Pie charts
6.7
Histograms
6.8
Mean, median, mode
Th hand drawn figure showing when ( mode ¡ median ¡ mean ) may seem unprofessional, but
in fact we think it’s useful. It’s more like what a student could produce than a fancy graphic
would be. We should probably have more pictures like this.
6.9
Computing averages from histograms
We think the material in this section is among the most valuable in the text. Working through
it and the exercises that reinforce it take lots of time well spent.
We find this a particularly valuable section – it forces students to come to grips with the real
meanings of each kind of average. Just memorizing definitions suffices for small data sets,
but with grouped data the mean is a weighted average and real understanding is required.
We recommend thinking about mode, median and mean in that order.
You will need to take time here to explain both the miracle – Excel guessed that updating
references was what was wanted – and its value in saving time and typing. It takes getting
used to.
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88
6.10. PERCENTILES
6.10
Percentiles
6.11
The bell curve
6.12
Margin of error
This is one of those places where it’s hard to find the right compromise between appropriate
simplifications and actual untruths about what the concepts mean and the numbers say. It
might well take a whole class period to expand on the discussion here. You have to decide
whether that’s how you want to spend class time.
6.13
Bimodal data
6.14
Comments on the Exercises
ere’s an important and subtle distinction here between weights as a percentage of revenue
and weights as a percentage of trips. Worth a discussion in class if you have time and a good
enough class.
.15 This exercise may be worth assigning for the Excel practice, and for reinforcing the
computations of various averages from summarized data. But the conclusions aren’t very
striking.
.23 Your students will probably not think this topic is useful or interesting but it might be
fun if introduced in class.
Extra (ideas for) Exercises
6.14
Exercise 6.14.33. [S][Section 6.9][Goal 6.6] [Goal 6.7] [Goal 6.8] Faculty salaries
The chart in Figure 6.1 shows the salary distribution of tenured faculty at the University of
Texas Houston. Vertical scale represents the number of faculty, the horizontal axis shows
salary ranges in increments of $20,000.
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6.14. COMMENTS ON THE EXERCISES
Figure 6.1: Faculty Salaries
(a) Recreate this data in Excel as both a table and a histogram.
(b) What is the modal salary for UTH faculty?
(c) What is the median salary for UTH faculty?
(d) What is the mean salary for UTH faculty?
(a) Recreate this data in Excel as both a table and a histogram.
(b) What is the modal salary for UTH faculty?
The mode is the highest bar – about $60-$80K.
(c) What is the median salary for UTH faculty?
The halfway point to the 203 faculty members (I added up the heights of the bars in
Excel) occurs partway through the second bar, so the median is about $70K.
(d) What is the mean salary for UTH faculty?
The mean salary is about $81K – computed in Excel as a weighted average. The
spreadsheet shows mean 81.03448276
Exercise 6.14.34. [U][Section 6.4][Goal 6.1] Doublethink.
Where The One Percent Live: The 15 Richest Counties In America
Living in Arlington isn’t cheap, so you’d better be making at least the median
household income to live in this county just outside Washington, D.C. 1
1
http://www.businessinsider.com/where-the-one-percent-live-the-15-richest-counties-in-america-2012-2
5th-richest-arlington-county-va-11
c 2014 Ethan Bolker, Maura Mast
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6.14. COMMENTS ON THE EXERCISES
How does this statement contradict itself?
2
Exercise 6.14.35. [U][Section 6.8][Goal 6.3] The article on new car and truck prices that
we studied in Section 5.5 first asserts that
. . . the average price of a new vehicle in the second quarter [of 2008] fell 2.3
percent from a year earlier to $25,632 . . .
and later
The result is the average new vehicle now costs less than 40 percent of an
average household’s median annual income, the analysts said, whereas from 1991
to 2007, it would cost more than half of the median income. 3
Verify as much of this last assertion as you can.
Exercise 6.14.36. [U][A][Section 6.5][Goal 6.4] [Goal 6.5] [Goal 6.6] Enrollments
The final enrollment report for the past year at an unnamed small college provided the
following information about students: 450 students freshmen; 421 students sophomores; 400
students juniors and 511 seniors.
(a) Create an Excel spreadsheet containing this data. Label the columns appropriately.
(b) Ask Excel to calculate the total number of students enrolled during the past year.
Label this result.
(c) Create a properly labeled bar chart of the student data.
(d) A corrected enrollment report noted that there were 419 juniors. Make that adjustment
in your spreadsheet and check that the other information (total number of students, bar
chart) is correctly updated.
(e) Using this new information, ask Excel to calculate the percentage of students who
are freshmen, sophomores, juniors and seniors. Copy and paste so that you type as few
formulas as possible.
(f) Create a new bar chart displaying the percentages.
(g) Convert your bar chart to a pie chart.
2
Thanks to David Kung for this exercise.
http://www.boston.com/business/articles/2008/09/05/new_car_prices_fall_at_fastest_
rate_ever
3
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6.14. COMMENTS ON THE EXERCISES
(h) Arrange the data with percentages and the final bar and pie charts so that they print
on one page.
Exercise 6.14.37. [U][A][Section 6.10][Goal 6.9] SAT exam percentiles
A student received this notification on his college entrance exam:
English Language Arts: 77th percentile
Mathematics:
88th percentile
Explain these this report in everyday language.
[See the back of the book for a hint.] Your answer might begin “More than three quarters of
the students taking this test . . . ”
Exercise 6.14.38. [U][Section 6.8][Goal 6.1] Comparing the states
You can do this exercise using Excel, or with properly documented research. (Your instructor
may specify one method or the other.)
Find the mean, median and mode for the populations of the 50 states.
Display the answers to the previous question on a properly labeled histogram. Discuss your
findings – is the distribution skewed?
Redo parts (a) and (b) for the areas of the states.
Redo parts (a) and (b) for the population densities (people per square mile).
Exercise 6.14.39. [N][Section 6.12][Goal 6.10] Jellybean margin of error
http://andrewgelman.com/2011/08/that_xkcd_carto/
Exercise 6.14.40. [U][Section 6.9][Goal 6.6] [Goal 5.1] [Goal 6.7] [Goal 6.8] Reputation on
stack exchange
Stackexchange (http://stackexchange.com) is a network of online question and answer
websites. Users who post questions and provide answers earn reputation based on community
feedback. Table 6.2 shows the number of users with reputations in certain ranges on January 6,
2013 for all stackexchange sites and for the particular site http://tex.stackexchange.com/
where the authors have asked and answered questions about the TEXsoftware used to prepare
the manuscript for Common Sense Mathematics. 4
4
Data sources:
http://stackexchange.com/leagues/1/week/stackoverflow/2012-12-30?sort=
reputationchange&page=1,
http://stackexchange.com/leagues/29/week/tex/2012-12-30?sort=
reputationchange
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6.14. COMMENTS ON THE EXERCISES
Estimate the mode, median and mean for each distribution. This is subtle in several ways.
The bucket sizes vary. Data at the top and bottom end of the range are very scarce. Ask
about sensitivity to the assumptions made there about the actual means for the top and
bottom categories.
Reputation Users
100,000+
2
50,000+
14
25,000+
24
10,000+
60
5,000+
91
3,000+
132
2,000+
179
1,000+
281
500+
395
200+
591
1+
18,417
TEX– LATEX
Reputation
Users
100,000+
97
50,000+
297
25,000+
938
10,000+
3,249
5,000+
6,874
3,000+
11,150
2,000+
15,650
1,000+
24,867
500+
34,857
200+
45,107
1+
1,478,007
All Sites
Table 6.2: Stackexchange reputation
Exercise 6.14.41. [U][Section 6.6][Goal 6.6] Deceptive pie charts
Build the pie chart from Section 6.6.
(a) Experiment with the pie chart features to make it look like the managers’ salaries are
the largest. You can’t actually change the data to do this – you need to use the 3D and
other pie chart features to make it look like the managers’ salaries are large.
(b) Play around with different types of charts in Excel. Find a chart type and an effect
(3D, most likely) that really distorts the data.
Exercise 6.14.42. [U][C][Section 6.7] [Goal 6.6] [Goal 6.7] River lengths
There should be a good Benford’s law exercise somewhere, maybe here. But spending time on
this peculiar phenomenon is probably not worth the time it takes away from other more useful
topics.
At http://en.wikipedia.org/wiki/List_of_rivers_by_length Wikipedia offers a chart
of 163 major rivers, organized by length.
(a) Construct a bar chart with nine categories, using the first digit of the length of the
river to determine the category.
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6.14. COMMENTS ON THE EXERCISES
You might expect all the bars to be the same height, since there are nine possible starting
digits. But they’re not. In fact, there are no rivers longer than 6000 km.
The fact that for the short rivers there are more whose length begins with small digits is
an instance of Benford’s Law . You can look up more about it if you’re curious.
(b) Use Excel to create new columns with river lengths measured in yards, in feet and in
inches. Draw each of those bar charts. Discuss what you see.
Exercise 6.14.43. [U][Section 6.9][Goal 6.1] [Goal 6.8] When Does Old Age Begin?
On June 29, 2009 The Chattanooga Times Free Press reported that
Americans differ on when old age begins. On average, they say 68. People
under age 30 believe it begins at 60, while those 65 and older push the threshold
to 74. Of all those surveyed, most said they wanted to live to 89. 5
The survey that provided statistics is at http://pewresearch.org/pubs/1269/aging-survey-expectatio
src=prc-latest&proj=peoplepress where you can find the graph in Figure 6.3.
The “average” of 68 is clearly a mean computed as weighted average. Try to verify it. You
may need to estimate some statistics – in particular, the fraction of respondents in each age
category.
Can you estimate the median age at which people think old age begins?
Figure 6.3: When does old age begin?
6.14
Exercise 6.14.44. [U][C][Section 6.5][Goal 6.6] [Goal 6.1] Median wages and incomes.
The Boston Globe printed Figures 6.4 and 6.5 on January 1, 2011.
5
http://www.timesfreepress.com/news/2009/jun/29/study-finds-widening-generation-gap-us/
?print
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6.14. COMMENTS ON THE EXERCISES
Figure 6.4: Median Wages
Figure 6.5: Wage and Income
(a) Reproduce Figure 6.4 in Excel.
(b) Compare changes in median wage with changes in median income.
(c) Are these data consistent with other reports of median income?
Exercise 6.14.45. [C][Section 6.8][Goal 6.1] The Lahey Clinic
Figure 6.6 appeared in an advertisement for the Lahey Clinic in The Boston Globe on March
6, 2009.
Our first reaction to the ad was “That can’t be true. There are thousands of hospitals, so
more than nine of them must be above average.” Then we thought about it some more.
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6.14. COMMENTS ON THE EXERCISES
Figure 6.6: Lahey Clinic Advertisement
(a) Use what you learned in the previous exercise to show that the assertion in the ad
isn’t impossible, although it is unlikely.
(b) Invent some reasonable numbers for hospital heart attack survival rates that would
make the ad correct.
(c) The ad included a reference to http://knowyournumbers.lahey.org/ for more
information. There we found the source http://www.usatoday.com/news/health/
2008-08-21-hospitals-standouts_N.htm (Note date).
Describe how the author of the ad reached the conclusion she wanted.
Note that the table provides information only about the exceptional hospitals. It does
not tell you how many unexceptional ones that are not reported on here. Try to find out
more about the missing data.
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Chapter 7
Electricity Bills and Income Taxes –
Linear Functions
7.1
Rates
In a course devoted more to real life applications than to modeling, you can move directly to
the section on taxes. There’s no explicit need there for the slope-intercept description of a
linear model.
7.2
Reading your electricity bill
7.3
Linear functions
We usually treat the slope and intercept as given data, since that’s how they appear in the
world. Computing the slope as ∆y/∆x belongs in an algebra class that’s leading to calculus,
but not here.
7.4
Linear functions in Excel
Since the entries in the Tamworth Electricity Bill table are out of order, Excel has drawn
some of the segments in the graph twice. You can see that if you look carefully. You might
want to point this out, or not.
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7.5. COMPARING ELECTRICITY BILLS
7.5
Comparing electricity bills
7.6
Energy and power
We’re undecided about how much time to spend on power and energy. It takes a lot of
teaching time (and energy) to convey the distinction convincingly in class. Perhaps those
resources are better spent on other parts of the curriculum. But the topic is important and
ultimately interesting, because students care about climate change and alternative power
sources. We have included problems that explore the issue further.
If you do choose to spend more time on the difference between power and energy, and the
confusion because the name “watt-hour” contains a unit of time, consider discussing the
light-year, which is a measure of distance, not time. So “light-years ago” is never right. The
knot – one nautical mile per hour – is also a rate, like power, that doesn’t mention time.
7.7
Taxes: sales, social security, income
In the tax rate brouhaha in the 2008 election we recall reading a story about a dentist who
complained that he would need to be careful not to let his income exceed $250,000 – where
candidate Obama drew no-new-taxes line – lest his overall tax rate increase. If you find the
story let us know and we’ll turn it into an exercise.
7.8
Comments on the Exercises
ponential decay might be a better model. Consider returning to this exercise when you reach
that topic later in the text.
20 The article on the quarry water cooling system also asserts that
The system ... cost only about $700,000 more than a traditional cooling
system, meaning Biogen Idec should get a return on its investment in eight to 10
years.
Discussing payback time might be interesting – or too difficult.
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7.8. COMMENTS ON THE EXERCISES
Figure 7.1: Federal Tax Rate by Income, 2007
Extra (ideas for) Exercises
7.8
Exercise 7.8.37. [N] The more you earn the less you pay
Figure 7.1 shows the effective Federal tax rate by household income for the year 2007. The
rate is lower for the wealthier households because much of their income is taxed as capital
gains rather than as ordinary income.
Exercise 7.8.38. [U][N][Section 7.6] [Goal 7.1] [Goal 7.7] Atop TV Sets, a Power Drain
That Runs Nonstop
The New York Times reported on June 26, 2011 that
Those little boxes that usher cable signals and digital recording capacity into
televisions have become the single largest electricity drain in many American
homes, with some typical home entertainment configurations eating more power
than a new refrigerator and even some central air-conditioning systems.
A new study has found that some home entertainment systems eat more
energy than refrigerators or central air-conditioning systems.
There are 160 million so-called set-top boxes in the United States, one for
every two people, and that number is rising. Many homes now have one or more
basic cable boxes as well as add-on DVRs, or digital video recorders, which use
40 percent more power than the set-top box.
One high-definition DVR and one high-definition cable box use an average of
446 kilowatt hours a year, about 10 percent more than a 21-cubic-foot energyefficient refrigerator, a recent study found.
These set-top boxes are energy hogs mostly because their drives, tuners and
other components are generally running full tilt, or nearly so, 24 hours a day,
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7.8. COMMENTS ON THE EXERCISES
even when not in active use. The recent study, by the Natural Resources Defense
Council, concluded that the boxes consumed $3 billion in electricity per year in
the United States – and that 66 percent of that power is wasted when no one is
watching and shows are not being recorded. That is more power than the state of
Maryland uses over 12 months 1
The graphic in Figure 7.2 accompanied the article.
Figure 7.2: Energy Use Comparisons
Hiawatha Bray’s Stop power leaks; smile at savings 2 from The Boston Globe on January
15, 2009 refers you to http://standby.lbl.gov/standby.html, a site maintained by the
Lawrence Berkeley National Laboratory, at which you can find out lots of useful stuff about
standby power consumption. This reference to the Hiawatha Bray article is the first of
several possible exercises on the same theme – if you combine them and assign them and let
us know what happened we’ll rewrite them.
Exercise 7.8.39. [U][Section 7.6][Goal 7.1] [Goal 7.7] Not so deep sleep
Many electronic devices use a small amount of electricity even when they are turned off. A
tv, for example, is always in standby mode unless it is unplugged, and so is drawing a small
amount of electricity. Similarly, computers, modems, cell phones and other devices draw
electricity even when turned off. This is known as “leaking electricity”. An article in 1998
estimated that this accounts for about 45 billion kilowatt-hours of electricity consumed in
the United States each year, or about 5% of the total electricity use by individuals. (see
http://www.aceee.org/pubs/a981.htm)
1
http://www.nytimes.com/2011/06/26/us/26cable.htm
http://www.boston.com/business/technology/articles/2009/01/15/stop_power_leaks_smile_
at_savings/
2
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7.8. COMMENTS ON THE EXERCISES
(a) Choose one electric item in your home and research how much electricity it uses when
it is off. You may be able to find this in the documentation for the device, or on the web.
Make an estimate if you can’t find the exact specifications.
(b) Suppose you unplugged this device at night, when you are done using it. Estimate
how much electricity this will save and how much money you will save in a year.
Exercise 7.8.40. [N] Romney Says His Effective Tax Rate Is ’Probably’ 15%
An Associated Press article in the Houston Chronicle on January 18, 2012 about the tax
returns of then Presidential candidate Mitt Romney said the following:
Speaking to reporters after a campaign stop in South Carolina, Romney
said most of his income comes from investments, not regular wages and salary.
The tax rate on investment income is 15 percent, much lower than the 35 percent
rate applied to wages for those in the highest tax bracket.
“What’s the effective rate I’ve been paying? It’s probably closer to the 15
percent rate than anything,” Romney said. 3
(a) The median net income of an American family in 2012 was about $50,000. What is
the effective tax rate for someone with that income?
(b) How does Romney’s tax rate of “probably” 15% compare to the “average” American?
Exercise 7.8.41. [U][C][N] Does virtual save energy?
MOST PEOPLE take for granted the Earth-friendly nature of electronic
communication. Paperless, ink-free, no shipping supplies, no gas for transportation:
the environmental benefits of virtual communication are obvious. But the reality is
more complicated, at least according to a growing number of concerned technology
experts and scientists. Vast stockpiles of digital data waste energy, too.
Everyday emails aren’t to blame. But large photo and video attachments,
cluttered inboxes, and massive email forwards may be. Some analysts estimate
that emailing a 4.7-megabyte attachment – the equivalent of four large digital
photos – can use as much energy as it takes to boil about 17 kettles of water. The
problem is magnified when large emails are forwarded to many people and left in
inboxes undeleted. As long as emails remain in your inbox, the data they create
is physically stored somewhere.
And that’s where the problems arise: The total amount of digital storage
worldwide is approaching 1 zettabyte, or 1 million times the contents of the
3
http://www.houstonchronicle.com/news/politics/article/GOP-front-runner-Romney-backpedals-on-releasin
php
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Earth’s largest library. Currently, that information is archived on equipment with
a mass equivalent to 20 percent of Manhattan. Global data storage is expected
to reach 35 zettabytes by 2020, which means more equipment, land, and energy.
The information industry already accounts for approximately 2 percent of global
carbon dioxide emissions. That’s the same amount as the airline industry blasts
into the atmosphere. Coupled with the rapid increase in stored data, it’s an
unsustainable scenario.
Technology firms must create systems that store data with less energy, and
governments should provide incentives for them to do so. Just as important,
consumers must demand products that save energy, and use websites like Flickr
and MediaFire that allow them to share large files without emailing. Better still,
they could consider keeping some of those embarrassing photos and home videos
to themselves. 4
Exercise 7.8.42. [U][N] The Gas Is Greener
Robert Bryce’s op-ed in The New York Times on June 7, 2011 (http://www.nytimes.com/
2011/06/08/opinion/08bryce.html) has lots of interesting numbers about the costs in steel
and land area for solar and wind electricity generation.
Exercise 7.8.43. [U][N] Every little bit counts.
On March 5, 2012 The Boston Globe reported on the Ocean Renewable Power Company’s
plans to install tidal powered generators in Maine:
The first unit capable of powering 20 to 25 homes will be hooked up to the
grid this summer, and four more units will be installed next year at a total cost
of $21 million . . .
Eventually, Ocean Renewable hopes to install more units to bring its electrical
output to 4 megawatts. 5
Exercise 7.8.44. [N] Power, Pollution and the Internet
On September 23, 2012 The New York Times printed an article with that headline. It’s full
of numbers that make for interesting explorations. More will be forthcoming:
This is the first article in a series about the physical structures that make up
the cloud, and their impact on our environment.
4
http://www.boston.com/bostonglobe/editorial_opinion/editorials/articles/2010/09/07/
dont_forward_those_photos/
5
www.bostonglobe.com/business/2012/03/05/maine-company-ready-install-tidal-power-unit/
daJ3ivfrUNUnHejoOt2lqJ/story.html
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7.8. COMMENTS ON THE EXERCISES
Nationwide, data centers used about 76 billion kilowatt-hours in 2010, or
roughly 2 percent of all electricity used in the country that year,
Here’s one of the comments (many are very interesting).
This article is weakened by hysterical reporting that relies on meaningless
unscaled scare statistics rather than a balanced presentation of the issues.
Buried deep in the article, one finds a more meaningful statistic: data centers
use only 2 percent of the electricity used in the country, hardly a huge figure,
given their importance in today’s economy.
You also fail to consider the cost of alternatives. How much does it cost to
sort and deliver a first class letter? To print and distribute a paperback book?
To answer a phone bank call? To drive to a movie theater? It is likely that the
data center is saving energy, not costing it.
No doubt energy efficiency can be improved, and the Times is to be lauded for
pointing that out. But no one who has ever worked with this kind of technology
can suppose that facilities of this kind can be built without backup generators,
batteries, and air conditioning: so crucial are these servers to today’s economy
that when a major service goes down for an hour, it is front page news. And in
the absence of context, the implication that data centers are disproportionately
responsible for wasting energy is questionable indeed.
Exercise 7.8.45. [N][Section 7.6][Goal 7.7] Graph gives calculations for force of a bomb
Figure 7.3: Energy and Power from an Atomic Bomb
This graph is good for discussing the difference between energy and power (the power energy
curve is the accumulation of the power curve) but as evidence about the Iranian nuclear
program it’s quite suspect – see http://www.guardian.co.uk/commentisfree/2012/nov/
28/ap-iran-nuclear-bomb
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Exercise 7.8.46. [S][Section 3.1][Goal 3.3] [Goal ??] Trucks test brakes.
Figure 7.4 shows some road signs you might see at the top of a hill on a highway, warning
that a steep grade lies ahead.
Figure 7.4: Steep grade
(a) If the grade is two miles long and the grade is 10%, how many feet lower is the bottom
of the hill than the top?
(b) What could a grade of 100% mean?
(a) If the grade is two miles long and the grade is 10%, how many feet lower is the bottom
of the hill than the top?
Two miles is about 10,000 feet. 10% of that is 1,000 feet. It’s how much lower you’ll be
at the end of the grade. I’ve never seen a 10% grade on a highway!
(b) What could a grade of 100% mean?
Fortunately the question asks what it could mean, not what it does mean.
I can interpret it as a change of one foot in height for every foot of travel on the road, or
as one foot of travel for every foot of horizontal progress, or as the edge of a cliff.
If this book had a chapter on geometry this discussion would go there.
Exercise 7.8.47. [U][Section 7.5] [Goal 7.2] [Goal 7.3]
meaning to “disappearing ink”
Today’s inkjet printers give new
On July 5, 2013, an article in the Jefferson City, MO News Tribune noted that
Printers are getting less expensive all the time but
keeping them fully
stocked with ink seems to be getting more expensive. No sooner do you load
new ink cartridges than you get messages warning you that the ink is running
low. 6
6
http://www.newstribune.com/news/2013/jul/05/todays-inkjet-printers-give-new-meaning-disappeari/
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7.8. COMMENTS ON THE EXERCISES
The linear equation that describes this situation is
total cost = (cost of printer) + (ink cost per page) × (pages printed) .
(a) Estimate the slope and intercept for this equation.
(b) What fraction of the cost is the initial purchase price?
[See the back of the book for a hint.] You may want to think in terms of hundreds of pages
rather than pages.
The answer to the second question depends on how many pages you print over the lifetime of
the printer.
Exercise 7.8.48. [S][A][Section 7.5][Goal 7.2] [Goal 7.3] CFLs
Consumers are being encouraged to replace ordinary light bulbs with compact fluorescent
bulbs. (CFLs). Soon they will be required to.7 A CFL uses less energy than an ordinary
incandescent bulb that produces the same amount of light, but it costs more to buy. This
table provides data with which you can compare the two.
bulb
ordinary
CFL
initial cost
$2.00
$9.00
power
100 watts
25 watts
Suppose electricity costs $0.20 per kwh.
You can use pencil and paper, a calculator, or Excel to do the arithmetic.
(a) Write a linear equation with which you can calculate the total cost C of using the
ordinary bulb for H hours.
(b) What is the slope of that equation (with its units)?
(c) What is the intercept of that equation (with its units)?
(d) How much would it cost to buy the ordinary bulb and use it for 1000 hours?
(e) Write a linear equation with which you can calculate the total cost C of using the
CFL for H hours.
7
This exercise should be updated to discuss LED bulbs too, and to use real rather than invented numbers.
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7.8. COMMENTS ON THE EXERCISES
(f) How much would it cost to buy the CFL and use it for 1000 hours?
(g) How long would you have to use the CFL to make it worth having paid the higher
purchase price?
Are the five numbers given in this exercise reasonable?
What does “incandescent” mean? Why are incandescent light bulbs called that?
(a) Write a linear equation with which you can calculate the total cost C of using the
ordinary bulb for H hours.
C = 2.00 + 0.1 × 0.20H
(b) What is the slope of that equation (with its units)?
The slope is 0.02 $/hour, or 2 cents/hour.
(c) What is the intercept of that equation (with its units)?
The intercept is $2.00.
(d) How much would it cost to buy the ordinary bulb and use it for 1000 hours?
C = $2.00 + 0.02
$
× 1000 hours = $22.00.
hour
(e) Write a linear equation with which you can calculate the total cost C of using the
CFL for H hours.
C = 9.00 + 0.025 × 0.20H
(f) How much would it cost to buy the CFL and use it for 1000 hours?
C = $9.00 + 0.005
$
× 1000 hours = $14.00.
hour
(g) How long would you have to use the CFL to make it worth having paid the higher
purchase price?
From my previous work, the answer will be less than 1000 hours. I can find the exact
time by algebra or by trial and error. It turns out to be 467 or about 500 hours. That
makes sense – the CFL costs a penny and a half less per hour to run, which means it will
take about 500 hours to save the $7 difference in initial cost.
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106
7.8. COMMENTS ON THE EXERCISES
(h) Are the five numbers given in this exercise reasonable?
Waiting for student input.
(i) What does “incandescent” mean? Why are incandescent light bulbs called that?
“Incandescent” means “giving off light because it’s hot”. That’s just how ordinary old
fashioned light bulbs work. There’s a thin filament (wire) inside that heats up and glows.
7.8
Exercise 7.8.49. [U][S][Section 7.6][Goal 7.7]
puters Alter Science
Digging Deeper, Seeing Farther: Supercom-
In The New York Times on April 25, 2011 you could read that
“The killer application is collaboration; that is what people want,” Dr. DeFanti
said. “You can save so much energy by not flying to London that it will run a
rack of computers for a year.” 8
Estimate the energy costs of flying to London and running a rack of computers for a year to
see if they are of the same order of magnitude.
[See the back of the book for a hint.] If you’re a physicist you can make these estimates with
your common knowledge. If you’re not, you can put together reliable information from the
web. Try searching for the energy cost of flying an airplane and the energy cost of running a
computer.
From http://www.inference.phy.cam.ac.uk/withouthotair/c5/page_35.
shtml:
Imagine that you make one intercontinental trip per year by plane. How much
energy does that cost?
A Boeing 747-400 with 240 000 litres of fuel carries 416 passengers about 8,800
miles (14,000 km). And fuel’s calorific value is 10 kWh per litre. (We learned
that in Chapter 3.) So the energy cost of one full-distance roundtrip on such a
plane, if divided equally among the passengers, is
2 × 240, 000 litre
× 10 kWh/litre ≈ 12, 000 kWh per passenger.
416 passengers
If you make one such trip per year, then your average energy consumption per
day is
12, 000 kWh
≈ 33 kWh/day.
365 days
8
http://www.nytimes.com/2011/04/26/science/26planetarium.htm
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7.8. COMMENTS ON THE EXERCISES
The round trip airline distance from Boston to London is about 6,500 miles, so that trip will
cost somewhat less than the trip above, figure 24 kwh/day.
At (say) 100 watts to run a computer (that’s the right order of magnitude) you use about
2,400 watt-hours or 2.4 kwh in a day. So 24 kwh will power ten computers for a day. That’s
a pretty small rack, but the order of magnitude is right.
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Chapter 8
Climate Change – Linear Models
8.1
Climate change
The students may all be interested in this topic, so they will want to think about it. The
consensus among climate scientists is that it’s real and anthropogenic, but the real science is
complex.
We do teach how to find regression lines (using Excel). But you can’t draw reliable conclusions
from simple regressions like the ones here. So treat this material respectfully and cautiously.
Our approach stresses skepticism throughout. Rather than teaching regression as a tool they
can use, we treat it as a tool often misused.
This part of this chapter, like the start of the last one, is written as an Excel tutorial. If
possible, students should follow along, checking the steps using Excel as they read or as you
lecture.
If you are working on this section in a classroom which allows you to project a spreadsheet
onto a screen you can reach you can eyeball the regression line with a yardstick.
This error surprised us when it occurred during a class we hadn’t prepared carefully. That
turned out to be useful – the students saw their teacher seeing that a number made no sense,
then looking for an explanation.
This is an important point – one worth emphasizing whenever it comes up. Encourage your
students to use more digits in any intermediate calculation – in particular, when working
with the trendline equation. Even better, encourage them to think about the numbers and
the graph. That’s the best approach.
When we reviewed this chapter, we ended up arguing about the validity of a prediction 10
109
8.2. THE GREENHOUSE EFFECT
years into the future. Students who know a bit about extrapolation may raise that issue;
others may ask why we bothered to predict to 2010 if we already had the data. Here’s part
of our dialogue:
Maura: Philosophically, I’m more comfortable with a prediction into the next year as opposed
to a prediction 10 years in the future. That’s the other reason why I’d like to include the
data through 2010 in the graph.
Ethan: The temperature data is so erratic that any prediction is likely to be wrong. I picked
10 years because we have actual data that far out so I could use a part of the data as an
experiment. One year wouldn’t be good visually.
8.2
The greenhouse effect
8.3
How good is the linear model?
We have deliberately omitted any discussion of the correlation coefficient R. We found when
we taught that material from an early draft of Common Sense Mathematics we used up a
lot of class time on material that did not meet our “what should students remember ten
years from now?” criterion for inclusion. We think that thinking qualitatively about R2 is
sufficient.
8.4
Regression nonsense
Do spend some time teaching the students to scrape data using cut and paste. That and
the fact that Excel can read .csv files will save them grief in this course and whenever
they use Excel. Tell them “csv” stands for “comma separated values”. Point them to
http://en.wikipedia.org/wiki/Comma-separated_values.
8.5
Comments on the Exercises
1 This is an interesting exercise to work in class.
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8.5. COMMENTS ON THE EXERCISES
Extra (ideas for) Exercises
8.5
Exercise 8.5.20. [U][Section 8.1][Goal 8.1] White House Opens Door to Big Donors, and
Lobbyists Slip In
On April 15 2012 The New York Timespublished Figure 8.1
Figure 8.1: Odds of an invitation to the White House
Fit a linear trendline to this data to predict the size of donation that would guarantee an
invitation to visit the White House.
You can do this with a ruler and get a good-enough approximate answer. No need to put the
data into Excel.
Exercise 8.5.21. 2% per degree Celsius . . . the magic number for how worker productivity
responds to warm/hot temperatures
http://andrewgelman.com/2012/09/persistently-reduced-labor-productivity-may-be-one-of-th
Exercise 8.5.22. [U][C][Section 8.1][Goal 8.1] Polarization
Note: The data need work before we can ask the students to deal with them.
Figure 8.2 appeared in The Boston Globeon November 6, 2010. 1 We extracted the
numerical data from the graph; you can find it at http://www.cs.umb.edu/~eb/qrbook/
Instructor/polarization.csv .
1
http://www.boston.com/news/nation/washington/articles/2010/11/06/election_opens_up_a_
gaping_divide/
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8.5. COMMENTS ON THE EXERCISES
Figure 8.2: Income Disparity and Political Polarization
(a) Find the trendline modeling a linear relationship between the income share of the top
1 percent of the population and the political Polarization index.
(b) Find the trendline modeling a linear relationship between the income share of the
top 1 percent of the population in a year and the political Polarization index four years
earlier.
Exercise 8.5.23. [U][C] Do the math on overrides
Barry Bluestone and Anna Gartsman wrote an op-ed with that headline in The Boston
Globe on June 4, 2010. Implicit in what they propose are several linear dependencies among
statistics describing towns in Massachusetts:
We decided to test this theory by simulating the impact on home values of a
change in school spending due to a Prop. 2 override, controlling for other factors.
We obtained data on housing values in 2005, the change in housing values between
2005 and 2010, and two measures of perceived school quality: school-wide SAT
scores and per pupil expenditures. We found complete data for 176 of the 351
cities and towns in the Commonwealth.
According to our analysis, which controlled for initial home value in 2005, a
municipality with SAT scores and per pupil spending levels 20 percent higher than
average experienced a 24 percent increase in nominal home value between 2005
and 2010. In contrast, a municipality with SAT scores and per pupil spending 20
percent below average experienced a loss in home value of 11 percent.
So, how much difference would the passage of the Hull override have potentially
meant for home values in that community? Hull’s 2005 average home value of
$366,343 was near the mean for the communities in our study. The average SAT
score in the Hull public schools was 961 compared with an average of 1047 for
all the study communities. Average per pupil expenditure in Hull was $11,491,
some $1,500 higher than average. Based on our home value model, the predicted
increase in home values in Hull between 2005 and 2010 was 3.85 percent.
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8.5. COMMENTS ON THE EXERCISES
Now what would likely have happened to the average home value in Hull if
the recent proposed $1.9 million override had been passed back in 2005? This
tax increase would have cost the average homeowner in Hull $506 per year. Over
five years, it would have totaled $2,530. However, that tax increase would have
resulted in an additional $1,442 spent per pupil. This increase would result in a
predicted increase in home value of 6.57 percent rather than the increase of 3.85
percent. The difference between the two predicted values results in an average
increase in home value in Hull of $9,970. 2
These figures are probably the result of a regression study. Identify the slopes of the regression
lines involved, and verify the predictions.
Exercise 8.5.24. [U][C] Say “Cheese”
Figure 8.3 appeared in The New York Times on November 7, 2010
3
4
.
Figure 8.3: Cheese Consumption
The graphs suggest a linear model. Explore.
Exercise 8.5.25. [N] Climate changes
The Economist published Figure 8.4 on May 12, 2010, along with the following paragraph:
How global surface temperature, ocean heat and atmospheric CO2 levels have
risen since 1960.
THE record of atmospheric carbon-dioxide levels started by the late Dave
Keeling of the Scripps Institute of Oceanography is one of the most crucial of
2
http://www.boston.com/bostonglobe/editorial_opinion/oped/articles/2010/06/24/do_the_
math_on_overrides/
3
http://www.nytimes.com/interactive/2010/11/07/health/nutrition/fat.html
4
http://www.nytimes.com/2010/11/07/us/07fat.html?_r=1&hp
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8.5. COMMENTS ON THE EXERCISES
the data sets dealing with global warming. When the measurements started in
1959 the annual average level was 315 parts per million, and it has gone up every
year since. To begin with it went up by roughly one part per million per year.
Now it is more like two parts per million per year. The figure for 2011 is 391.6.
More carbon dioxide in the atmosphere means a stronger greenhouse effect, and
various measurements speak to this. Global surface temperature records show a
warming over the same period, though because of fluctuations in the climate, air
pollution, volcanic eruptions and other confounding factors the rise is nothing
like as smooth. A steadier rise can be seen in the heat content of the oceans,
measured in terms of the energy stored, rather than the temperature.
Figure 8.4: Climate Change
Exercise 8.5.26. [U][Section 8.2] 2013 Carbon Dioxide
On September 10, 2014 The Associated Press reported that
The heat-trapping gas blamed for the largest share of global warming rose
to worldwide concentrations of 396 parts per million last year, the biggest
year-to-year change in three decades, the World Meteorological Organization
said in an annual report.
Thats an increase of 2.9 parts per million from the previous year and is 42
percent higher than before the Industrial Age, when levels were about 280 parts
per million.
Based on the current rate, the world’s carbon dioxide pollution level is expected
to cross the 400 parts per million threshold by 2016, said organization Secretary
General Michel Jarraud. That is way beyond the 350 amount that some scientists
and environmental groups promote as a safe level and that was last seen in 1987.
5
5
http://hosted.ap.org/dynamic/stories/E/EU_UNITED_NATIONS_GLOBAL_WARMING?SITE=AP&
SECTION=HOME&TEMPLATE=DEFAULT&CTIME=2014-09-09-15-29-18
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8.5. COMMENTS ON THE EXERCISES
How might the data in this quotation change the discussion in Section 8.2?
Exercise 8.5.27. [N] Start with a graph
This is a placeholder for Exercises as suggested by a Cengage reviewer:
I’d like to see more homework problems here that begin from a graph and
trend line, rather than beginning from a data set. Given that it is so easy to
mislead with graphs, this would help students to develop those “defensive reading”
skills that appear to be one of the goals of this chapter.
Exercise 8.5.28. [U][Goal 8.3] Web search correlations.
Visit the Google correlation site http://www.google.com/trends/correlate/, choose a
search term you are interested in and find out what other search terms its correlated with.
Write about what you discover. Is causation involved?
Exercise 8.5.29. [U][N] Heart attack risk
At the website www.cardiosource.org/en/Science-And-Quality/Practice-Guidelines-and-Quality2013-Prevention-Guideline-Tools.aspxi you can download a spreadsheet with which to
predict your risk of a heart attack. You fill in some values (like your age, blood pressure and
cholesterol count) and the spreadsheet tells you your risk.
The formulas it uses are hidden, but you can figure out something about them by experimenting.
For example, try filling in all the fields, then vary the total cholesterol count while keeping all
the other values the same. Record the results in another spreadsheet, and produce a graph
showing how risk depends on that variable. Is it linear? Approximately linear?
Do the same for some of the other variables.
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Chapter 9
Compound Interest – Exponential
Growth
9.1
Money earns money
9.2
Using Excel to explore exponential growth
9.3
Depreciation
9.4
Doubling times and half lives
Of course you don’t need to rely on experiments to know that the doubling time is independent
of the initial value. It’s very easy to prove with a little bit of algebra. But this is a book
about quantitative reasoning, not about algebra. For its intended audience the experiments
are more convincing than the more formal mathematics many people find mysterious.
If we were teaching algebra and not quantitative reasoning we might use a negative exponent
to write (1/2)10 as 2−10 . But we’re not, so we don’t, so we avoid the time it would take to
remind students about working with negative exponents.
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9.5. EXPONENTIAL MODELS
9.5
Exponential models
The material in this section on bacterial growth is at the edge of what we think students in a
quantitative reasoning course need. It deals with real data, not the artificial doubling time
problems in most books. Carrying through the discussion in sufficient detail to allow them to
solve similar problems would take time better spent on other topics. But if there’s time in
the syllabus there are ideas here worth exploring. They tie together all the themes of the
chapter.
9.6
Comments on the Exercises
Extra (ideas for) Exercises
9.6
Exercise 9.6.36. [S] So many words!
On December 1, 2012 R. Alexander Bentley and Micahel J. O’Brien wrote in The New York
Times that
[F]or the last 300 years, the number of words published annually grew exponentially by about 3 percent per year. From about 20 million words for 1700, the
annual word count grew to several trillion for 2000. 1
(a) Check the authors’ arithmetic.
(b) If the growth continues at the same rate how many words will be published in the
year 3000?
(c) How much confidence do you have in your prediction?
(a) Check the authors’ arithmetic.
The coolest way to do this is with the rule of 70. A 3% annual increase has a doubling
time of 70/3 ≈ 25 years. The question asks about 300 years of growth, which would be
12 doublings.
1
www.nytimes.com/2012/12/02/opinion/sunday/science-and-buzzwords.htm
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9.6. COMMENTS ON THE EXERCISES
But to get from 20 million to 20 trillion you must multiply by 106 . Since 210 ≈ 1, 000,
that’s 20 doublings. It would take 19 doublings to get to 10 trillion, and 16 to get to 1
trillion. So 12 is not enough.
You can, of course, solve this problem the boring way with some arithmetic:
(20 million) × (1.03)300 ≈ 140, 000 million = 140 billion.
That’s at least an order of magnitude short of “several trillion”.
To get to 2 trillion you’d need 19 doublings in 300 years. That’s a doubling time of about
16 years. Then the rule of 70 says you’d have to have had an annual growth rate of about
4.3%.
(b) If the growth continues at the same rate how many words will be published in the
year 3000?
Another 100 years at 4.3% would be about 6 more doublings, so each trillion words would
grow to 26 = 64 trillion.
(c) How much confidence do you have in your prediction?
Not much!
Exercise 9.6.37. [S] World population
According to a Harvard School of Public Health press release The world’s population has
grown slowly for most of human history. It took until 1800 for the population to hit 1
billion. However, in the past half-century, population jumped from 3 to 7 billion. In 2011,
approximately 135 million people will be born and 57 million will die, a net increase of 78
million people.
(a) By what percent did world population increase in 2011?
(b) Write the equation for a mathematical model for world population growth for years
since 2011 if the annual net increase seen in 2011 remains constant for the remainder of
the century. (Use 7 billion as the 2011 population.)
(c) Write the equation for a mathematical model for world population growth for years
since 2011 if the annual percentage increase seen in 2011 remains constant for the
remainder of the century. (Use 7 billion as the 2011 population.)
(d) Construct an Excel spreadsheet predicting world population for the years through
2100 using both models. Graph both predictions on the same chart.
(e) The table below gives the United Nations’ high estimate for world population growth
during the remainder of this century.
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Year
2011
2025
2050
2100
Population
7 billion
8.5 billion
10.6 billion
15.8 billion
Which of your models most closely matches the UN high estimate?
(a) By what percent did world population increase in 2011?
The percentage increase was
78 × 106
= 0.01114285714 ≈ 1.1%.
7 × 109
(b) Write the equation for a mathematical model for world population growth for years
since 2011 if the annual net increase seen in 2011 remains constant for the remainder of
the century. (Use 7 billion as the 2011 population.)
Let Y be the number of years since 2011 and P the world population, in billions. Then
the equation is
P = 7 + 0.078Y .
(c) Write the equation for a mathematical model for world population growth for years
since 2011 if the annual percentage increase seen in 2011 remains constant for the
remainder of the century. (Use 7 billion as the 2011 population.)
With the same variables as above, the exponential equation is
P = 7 × (1.011)Y .
(d) Construct an Excel spreadsheet predicting world population for the years through
2100 using both models. Graph both predictions on the same chart.
(e) The table below gives the United Nations’ high estimate for world population growth
during the remainder of this century.
Which of your models most closely matches the UN high estimate?
I’ve added the predictions to the table. Numbers are in billions. The last two columns
show the relative errors in the predictions.
Year
2011
2025
2050
2100
UN
7.0
8.5
10.6
15.8
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linear
7.00
8.19
10.32
14.57
exp
7.00
8.17
10.78
18.77
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lin/UN
1.00
0.96
0.97
0.92
exp/UN
1.00
0.99
1.02
1.19
9.6. COMMENTS ON THE EXERCISES
The linear and exponential predictions are both pretty close for the first half of the
21st century. By 2100 the linear prediction is 8% lower than the UN’s, the exponential
prediction 19% higher.
Exercise 9.6.38. [N] Making it into the Hall of Fame.
On January 12, 2013, Nate Silver wrote in his blog at The New York Times that
Based on an analysis of Hall of Fame voting between 1967 and 2011, I found
that the increase in a player’s vote total is typically proportional to his percentage
from the previous year. In his second year on the ballot, for example, the typical
player’s vote share increases by a multiple of about 1.1.
Thus, a player who received 10 percent of the vote in his first year would be
expected to receive about 11 percent on his second try, while a player who got 50
percent of the vote would go up to 55 percent. 2
Explain why this is exponential growth. Look up the original. Make some projections.
Exercise 9.6.39. [U][Section 9.2][Goal 9.1] Computing gets cheaper
The web site http://www.freeby50.com/2009/04/cost-of-computers-over-time.html
offers Figure 9.1 along with the text
The Bureau of Labor Statistics tracks consumer price data. Computer
equipment is one of the price categories that they track.
As you can see the price of computers has dropped drastically over the years.
From 1999 to 2003 the index dropped over 20% every year. Lately the rate of
price drop has slowed but the prices for computers are still dropping. The index
has decreased at a rate of 11-12% annually for the last 3 years.
Figure 9.1: The Cost of Computers
(a) Reproduce this chart in Excel.
2
http://fivethirtyeight.blogs.nytimes.com/2013/01/12/in-cooperstown-a-crowded-waiting-room/
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(b) Why do you think this chart was created using Excel?
(c) Check the writer’s claim about the percentage decreases.
(d) What happens when you adjust the data to take inflation into account?
Exercise 9.6.40. [N] The Richter scale
Exercise 9.6.41. [U] [N] Massive Increase in Ethanol Production
Figure 9.2 graphs the annual U. S. production of ethanol in the years since 1980. It’s from http:
//www.graphoftheweek.org/2012/02/massive-increase-in-ethanol-production.html
where you can read this:
In 2010, the United States produced 13.2 billion gallons of ethanol. That
sounds like a lot until you compare it with the amount of imported oil (180.8
billion gallons for the same year). In other words, for each gallon of ethanol
produced locally, the U.S. imports nearly 14 gallons of crude oil.
Questions:
1) Will production of ethanol continue to increase exponentially?
2) By how much can we reduce our dependence on foreign oil, in practical
terms?
3) How has this affected agriculture in the United States?
Figure 9.2: Ethanol Use
Exercise 9.6.42. [U][N] The 1% More Savings Calculator
The interactive web based calculator provided by The New York Times at http://www.
nytimes.com/interactive/2010/03/24/your-money/one-pct-more-calculator.html suggests many questions about savings, compound interest and inflation.
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9.6. COMMENTS ON THE EXERCISES
Exercise 9.6.43. [N] Chicken or beef?
http://www.ers.usda.gov/amberwaves/april06/findings/chicken.htm
Exercise 9.6.44. [U][N] London ready to complete Olympic triple play
On July 22, 2012 The Boston Globe wrote abut the third Olympic games to be hosted by
London:
The Games have grown geometrically during the past 104 years – from 2,023
athletes representing 22 countries competing in 109 events in 24 sports in 1908 to
4,064 athletes, 59 countries, 136 events, and 19 sports in 1948 to 10,500 athletes,
204 countries, 302 events, and 37 sports in 2012. 3
Note: “geometric” is a synonym for “exponential”.
Possible questions: find exponential regression curves for the numbers of athletes, countries
and events.
Exercise 9.6.45. [N] Monk parakeets
Figure 9.3 shows the bird population as a function of time.
(We have a better picture, the raw data, and a reference.)
Figure 9.3: Monk Parakeets
Exercise 9.6.46. [U][N][Section 9.3] [Goal 9.1] Radioactive waste.
The web site http://www.nirs.org/factsheets/hlwfcst.htm we quoted earlier offers much
more information about radioactive waste.
The majority of high-level radioactive waste is the fuel from the hot core
of commercial nuclear power plants. This irradiated fuel is the most intensely
3
http://www.bostonglobe.com/sports/2012/07/22/london-ready-complete-olympic-triple-play/
iaAdNViKdg53AyrkHMORwN/story.html
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9.6. COMMENTS ON THE EXERCISES
radioactive material on the planet, and unshielded exposure gives lethal radiation
doses. It accounts for 95% of the radioactivity generated in the last 50 years
from all sources, including nuclear weapons production. Uranium is processed
into fuel rods and loaded into nuclear power reactors where it undergoes the
nuclear fission reaction. This increases the radioactivity due to the formation
of intensely radioactive elements known as fission products, such as cesium and
strontium, resulting from the physical splitting of uranium-235 atoms. Heavier
elements, known as transuranics, are also formed – including plutonium. Each
1000 megawatt nuclear power reactor annually produces about 500 pounds of
plutonium, and about 30 metric tons of high-level waste in the form of irradiated
fuel. After several years, when removed from the reactor core, the fuel is about
one million times more radioactive than when it was loaded. This irradiated fuel
is currently stored at the reactor sites.
Ask and answer some interesting Fermi problems suggested by the data in this paragraph.
You could consider what it says about a nuclear power plant near you.
We could construct the Fermi problems based on this radioactive waste data ourselves, and
ask the students to solve them. But by this time in the course we hope they can start from
the numbers and create their own.
Exercise 9.6.47. [U][N][Section 9.5] [Goal 9.5]
accepted
Cash, credit card, and . . . smartphones
Figure 9.4 appeared in The Boston Globe on November 2, 2011 along with the quote
Mobile payments are just a sliver of retail sales. They accounted for $3 billion
in sales in 2010, or 1 percent of e-commerce transactions, but they are expected
to double this year and reach $31 billion by 2016, according to Forrester Research
of Cambridge.
We’ve entered the data in a spreadsheet at http://www.cs.umb.edu/~eb/qrbook/Instructor/
MobileCommerce.xlsx .
The percentage on mobile devices is projected to grow linearly at one percentage point per
year. The amount of mobile payment is growing faster – because total is e-commerce is
growing faster.
Compute total e-commerce, fit an exponential trendline.
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9.6. COMMENTS ON THE EXERCISES
Figure 9.4: Mobile Commerce
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9.6. COMMENTS ON THE EXERCISES
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Chapter 10
Borrowing and Saving
10.1
Credit card interest
10.2
Can you afford a mortgage?
10.3
Saving for college or retirement
10.4
Annual and effective percentage rates
10.5
Instantaneous compounding
If you want to take this just a little bit further you can tell the class that the doubling time
for continuous compounding is ln 2/r ≈ 0.6931/r, hence the rule of 70.
10.6
Comments on the Exercises
Extra (ideas for) Exercises
10.6
Exercise 10.6.15. [N][U] Debit Card Trap
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10.6. COMMENTS ON THE EXERCISES
On August 20, 2009 The New York Times editorialized that
A study by the Center for Responsible Lending, a nonpartisan research
and policy group, describes what it calls the “overdraft domino effect.” One
college student whose bank records were analyzed by the center made seven small
purchases including coffee and school supplies that totaled $16.55 and was hit
with overdraft fees that totaled $245.
Some bankers claim the system benefits debit card users, allowing them to
keep spending when they are out of money. But interest rate calculations tell a
different story. Credit card companies, for example, were rightly criticized when
some drove up interest rates to 30 percent or more. According to a 2008 study by
the F.D.I.C., overdraft fees for debit cards can carry an annualized interest rate
that exceeds 3,500 percent. 1
.
We haven’t made up any questions yet to go with this interesting quote.
Exercise 10.6.16. [S][Section 10.4][Goal 10.1] [Goal 10.2] [Goal 10.4] Regulating the credit
card industry.
The Boston Globe reported on May 13, 2009 that the Senate might consider cap on card
interest rates. There you can read
The Senate legislation would require lenders to apply credit card payments to
balances with the highest interest rates first.
and
Senator Bernie Sanders, a Vermont independent, will offer an amendment
limiting all lenders to 15 percent interest rates, he said in a written statement.
That’s the same limit that applies to cards offered by credit unions, according to
the statement. 2
Suppose a credit card user has a balance of $100 at 24% for purchases and $1000 at 0% for a
debt she transferred from another credit card. She makes no new purchases, and pays off her
loan at the rate of $100 per month.
1
http://www.nytimes.com/2009/08/20/opinion/20thu1.html
http://www.boston.com/business/personalfinance/articles/2009/05/13/senate_might_
consider_cap_on_card_interest_rates/
2
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10.6. COMMENTS ON THE EXERCISES
(a) When will her loan be paid off and how much interest will she have paid under the
2009 rules?
(b) Under the new rules, which are now law?
(c) If Sanders’ amendment had passed? (It didn’t.)
(a) When will her loan be paid off and how much interest will she have paid under the
2009 rules?
Under the 2009 rules it will take her 10 months to pay off the $1,000 transfer. During
that time her $100 balance will have been accruing interest at the rate of 2% per month.
Then she will owe $100(1.02)10 = $121.90. (I could have found the same answer with
the spreadsheet.) Then she will pay that off in two months, so she’ll have one month’s
interest on the unpaid balance of $21.90: another 0.02 × $21.90 = $0.44. Her total interest
payments will be $21.90 + $0.44 = $22.34
(b) Under the new rules she will pay off the high interest part of her bill in the first month,
with no interest charge, and then the rest in 10 months, again with no interest charge.
(c) If Sanders’ amendment had passed her interest rate would be capped at 15% annually,
so if she pays off the zero interest balance first her unpaid balance will become 100 × (1 +
0.15/12)10 = 113.23. The last interest charged will be just (0.15/12) × 13.23 = 0.17 for a
total interest payment of $113.40.
She’ll pay no interest if she pays off the purchases first.
Exercise 10.6.17. [N][Section 10.1][Goal 10.1] Reward cards
In The Boston Globe on December 18, 2009, Candice Choi wrote about credit card reward
programs:
http://www.boston.com/business/personalfinance/articles/2009/12/18/rewards_cards_
may_be_a_bit_less_rewarding_after_you_consider_the_higher_fees/.
Exercise 10.6.18. [U][N][Section 10.4][Goal 7.6] [Goal 10.4] IRS gives taxpayers more time
to file, not to pay
On April 16, 2012 The Boston Globe carried an Associated Press story with that headline
and this information:
If you file your tax return on time or get an extension, but fail to pay, the
IRS will charge you interest on unpaid taxes. That rate currently works out to
about 3.25 percent and is compounded daily.
The IRS also will charge you a late payment penalty of one-half of 1 percent
of any tax not paid by April 17. That translates to a $25 penalty if you owe
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10.6. COMMENTS ON THE EXERCISES
$5,000. It is charged each month or part of a month the tax goes unpaid, up to
25 percent, or $1,250 on that $5,000.
That interest rate can jump to 1 percent, however, if the tax bill hasn’t been
paid within 10 days after the IRS issues a notice of intent to levy. But if you
work out a payment plan with the IRS, it will reduce the rate to one-quarter of 1
percent.
The last thing you want to do is miss the tax-filing deadline and not ask for
an extension. That triggers a penalty of 5 percent of the tax owed for each month
the tax return is late, for up to five months. It’s best not to try to calculate what
you might owe in penalties and interest; the IRS will send a bill if you underpay.
3
Work out the total owed the government on a $5,000 tax bill under various late payment
scenarios.
In each case compare the penalty and the interest. Which contributes more?
10.6
3
http://www.bostonglobe.com/business/2012/04/16/irs-gives-taxpayers-more-time-file-not-pay/
ZqlllJRYJvGjCVkHIRHNVK/story.html
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Chapter 11
Probability – Counting, Betting,
Insurance
robability is hard, often counterintuitive. We deal with it in three chapters. This one is
about the basic quantitative notion, focusing first on the easy cases coming from games of
chance, but not spending significant time on the combinatorics. In the next chapter we take
on repeated independent events, the bell curve, and rare events. In the one after that we
take on conditional probability, but without formulas. Throughout the discussion we often
find that there are ideas about probability that should be thought about but that don’t fit
nicely into simple numerical examples, either real or imagined.
Our choice of “invented” instead of “discovered” mathematics in the chapter introduction is
deliberate. You might want to discuss that philosophical question in class.
11.1
Equally likely
11.2
Odds
Consider not even mentioning the formulas for converting from odds to probabilities and
back lest the students latch on to them as more important than they are.
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11.3. RAFFLES
11.3
Raffles
11.4
State lotteries
11.5
The house advantage
11.6
One time events
11.7
Insurance
11.8
Sometimes the numbers don’t help at all
11.9
Comments on the Exercises
.3 Ben Bolker suggests analyzing this hiring dilemma using a payoff matrix, with utilities
associated with each state (awful, ok, great). Then we could compute an expected value
for each action (hire known, hire unknown) in terms of the various probability and payoff
assumptions. This would be cool in Excel.
.11 This Exercise is worth class time. You could ask your students to update it with current
data about your state.
Extra (ideas for) Exercises
11.9
Exercise 11.9.17. [U][N] Playing the Odds on Saving
On January 15, 2014 Tina Rosenberg blogged at The New York Times that
Lotteries aren’t usually considered part of the solution to our savings crisis.
They’re usually cited as a big part of the problem. Lotteries offer the worst odds
in legal gambling – about 55 percent of what people pay for tickets is paid out in
prizes. Yet we spend an average of $540 per household on lottery tickets every
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11.9. COMMENTS ON THE EXERCISES
year – about $100 more than we spend on milk or beer. That is disproportionately
spent by African-Americans, who spend five times as much on lottery tickets per
person than whites, and the very poor. People with a household income of less
than $10,000 a year who play the lottery spend $597 a year on tickets. 1
Read further for the savings strategy . . .
Exercise 11.9.18. [N] Statistical Probability That Mitt Romney’s New Twitter Followers
Are Just Normal Users: 0%
From the Atlantic:
http://www.theatlantic.com/technology/archive/2012/07/statistical-probability-that-mitt260539/
See the facebook exercise in the chapter on Averages.
11.9
1
http://opinionator.blogs.nytimes.com/2014/01/15/playing-the-odds-on-saving/
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Chapter 12
Break the Bank – Independent Events
12.1
A coin and a die
We’ve deliberately avoided the technical term “sample space”. We do present the idea, but
don’t think combinatorial computations that require formal reasoning belong in a course at
this level.
12.2
Repeated coin flips
12.3
Double your bet?
12.4
Cancer clusters
12.5
The hundred year storm
Of course
0.99100 = 0.366032341 ≈
but you don’t want to go there with this class.
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1
= 0.367879441
e
12.6. IMPROBABLE THINGS HAPPEN ALL THE TIME
12.6
Improbable things happen all the time
12.7
Comments on the Exercises
Extra (ideas for) Exercises
12.7
Exercise 12.7.20. [N] What are the chances of six double-yolkers?
http://www.bbc.co.uk/news/magazine-16118149
Exercise 12.7.21. [N] Gladwell and success
From Andrew Gelman’s blog: http://andrewgelman.com/2013/10/11/gladwell-vs-chabris-david-vsHeres another example. A few years ago, I criticized the following passage
from Gladwell:
Its one thing to argue that being an outsider can be strategically useful. But
Andrew Carnegie went farther. He believed that poverty provided a better preparation for success than wealth did; that, at root, compensating for disadvantage
was more useful, developmentally, than capitalizing on advantage.
I argued that Gladwell was making a statistical fallacy:
At some level, theres got to be some truth to this: you learn things from the
school of hard knocks that youll never learn in the Ivy League, and so forth. But
. . . there are so many more poor people than rich people out there. Isnt this
just a story about a denominator? Heres my hypothesis:
Pr (success — privileged background) ¿¿ Pr (success — humble background)
# people with privileged background ¡¡ # of people with humble background
Multiply these together, and you might find that many extremely successful
people have humble backgrounds, but it does not mean that being an outsider is
actually an advantage.
The comments on that post are worth reading!
Exercise 12.7.22. [U][N][Section 12.1][Goal 12.1] What’s the problem with stories and the
story with probability?
This material appears in both http://www.irishtimes.com/newspaper/finance/2012/
0808/1224321714580.html and http://www.ft.com/cms/s/0/bc86b5dc-dfcb-11e1-9bb7-00144feab49
html#axzz239alNGMW
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12.7. COMMENTS ON THE EXERCISES
The mark of a first-rate intelligence is the ability to hold conflicting ideas in
the mind at the same time and still function
LINDA IS single, outspoken and deeply engaged with social issues. Which of
the following is more likely? That Linda is a bank manager or that Linda is a
bank manager who is an active feminist?
This is one of the best-known problems in behavioural economics. Many
people say the second option is more likely. Yet, the standard response goes, this
cannot be. The rules of probability tell us the probability that both A and B
are true cannot exceed the probability that either A or B is true. It is less likely
that someone is a female Jamaican Olympic gold medallist than that a person is
female, or that a person is Jamaican, or that a person is a gold medallist. Yet
even people trained in probability make a mistake with the Linda problem.
Or is it a mistake? Little introspection is required to understand what is
going on. Respondents do not interpret the question as one about probability.
They think it is a question about believability. The description of Linda that
ends with the statement Linda is a bank manager is designed to be incongruous.
The addition of who is an active feminist begins to restore coherence. The story
becomes more believable, even if less probable. No one gets the Jamaican problem
wrong because every part of the story makes sense.
We do not often, or easily, think in terms of probabilities, because there are
not many situations in which this style of thinking is useful. Probability theory is
a marvellous tool for games of chance, such as spinning a roulette wheel. The
structure of the problem is comprehensively defined by the rules of the game.
The set of outcomes is well defined and bounded, and we will soon know which
outcome has occurred.
An answer/comment at http://www.ft.com/intl/cms/s/0/7b3c497e-e21a-11e1-b3ff-00144feab49a.
html#axzz239alNGMW
(“When storytelling leads to an unhappy ending”, August 8). It is that,
regardless of the question actually asked, we think that the question concerns
conditional probabilities.
Which is the more likely: that Linda is single and outspoken given that Linda
is a bank manager; or that Linda is single and outspoken given that she is a bank
manager who is an active feminist?
Here our intuition is correct: the latter probability is higher.
Robert Simons, Eudoxus Systems, Leighton Buzzard, Beds, UK
Exercise 12.7.23. [N] Mathematician plots pregnancy probability
Figure 12.1 shows . . .
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12.7. COMMENTS ON THE EXERCISES
Figure 12.1: Odds of conceiving . . .
Exercise 12.7.24. [U][N] Is Failure to Predict a Crime?
This oped from The New York Times has an interesting discussion of the difficulty working
with small probabilities for rare events.
urlhttp://www.nytimes.com/2012/10/27/opinion/a-failed-earthquake-prediction-a-crime.html
Exercise 12.7.25. [U] Russian roulette
(a) What is Russian roulette?
(b) What is the probability of surviving 1, 2, 3, 4, 5 or 6 rounds?
12.7
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Chapter 13
How Good Is That Test?
This chapter focuses on two way contingency tables in order to discuss several important
common logical pitfalls dealing with everyday probabilities. We think that approach makes
more sense, and is easier to remember and apply, than an explicit treatment of dependent
events and Bayes’ theorem. That’s too technical for our goals in this quantitative reasoning
text, and so better left for a full course in probability and statistics. In fact, many of the
examples in this chapter employ qualitative rather than quantitative reasoning.
You can even skip the first two sections and the vocabulary of dependent events and start
with the section on screening for rare diseases.
If you want to go further into the analysis of dependence (perhaps leading to Bayes’ theorem)
consider two way tables as the entry point. Independence corresponds to tables whose rows
(and hence columns) are proportional. Those are the only ones that can be modeled using
areas of parts of a square, as in the last chapter.
Causation corresponds to tables with a 0 in one quadrant.
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13.1. UMASS BOSTON ENROLLMENT
13.1
UMass Boston enrollment
13.2
False positives and false negatives
13.3
Screening for a rare disease
13.4
Trisomy 18
13.5
The prosecutor’s fallacy
13.6
The boy who cried “Wolf ”
13.7
Comments on the Exercises
Extra (ideas for) Exercises
13.7
Exercise 13.7.16. [U][N][Section 13.2][Goal 13.1]
Testing in NYC STD Clinics
False Positive Oral Fluid Rapid HIV
Read the City of New York DEPARTMENT OF HEALTH AND MENTAL HYGIENE Health
Advisory #20 at http://www.nyc.gov/html/doh/downloads/pdf/cd/08md20.pdf.
Build the two way contingency tables based on the data there, and discuss the consequences
of the data.
Exercise 13.7.17. [U][C][N] Missile defense.
Theodore A. Postel wrote in The Boston Globe on April 15, 2008 that
THE HOUSE Subcommittee on National Security and Foreign Affairs will
hold a long-overdue oversight hearing tomorrow on the prospects for national
missile defense. The most basic question that needs to be addressed is the inability
of the national missile defense to tell the difference between simple warheads and
decoys.
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13.7. COMMENTS ON THE EXERCISES
...
The issue of the effectiveness of decoys against the missile defense is easy to
understand. The national missile defense is designed to destroy warheads by
hitting them with infrared homing Kill Vehicles while the warheads are in the
near vacuum of space. Since there is no air-drag in space, a warhead weighing
thousands of pounds and a balloon weighing almost nothing will travel together.
Warheads could be placed inside balloons, and many balloons could be deployed
along with the warheads. . . . Since there would be no way for the Kill Vehicle to
know which balloons contain warheads, the chances of actually hitting a warhead
would be minuscule. 1
Not yet sure how to phrase a question about this. Is there a natural two way contingency table?
Exercise 13.7.18. [N] Asian carp eDNA testing can lead to false positives, study finds
The technical team developed a strategy to test for false positives.
Water
samples were collected in April from a metro lake that served as a negative control
(very little chance Asian carp could be present). Twenty samples were sent to
the Corps of Engineers laboratory in Vicksburg and 20 were sent to the private
contractor that did the 2011 analysis. All of the samples from the Corps of
Engineers lab tested negative, while one sample from the private contractor tested
positive for silver carp. This sample was tested again and the positive was
verified.
There is a high likelihood this is a false positive which creates uncertainty
about previous results.
The percentage of positives in the 2012 samples was
much lower than previous samples suggesting there may have been a mix of real
and false positive samples in 2011. This does not minimize eDNA testing as
an important tool for detecting Asian carp, but it does emphasize the need to
determine the source of false positives and to review and modify sampling and
analytical procedures. In addition, we have collected live Asian carp from the St.
Croix and Mississippi Rivers which are definitive evidence these fish are present
and pose a threat to Minnesota. 2
The posting has no numbers, so it’s hard to write questions.
Exercise 13.7.19. [N] Shaky Foundations for the New Mammogram Economy
From Bloomberg news, on August 1, 2012:
1
http://www.boston.com/bostonglobe/editorial_opinion/oped/articles/2008/04/15/
troubling_questions_about_missile_defense/
2
http://blogs.twincities.com/outdoors/2012/07/27/asian-carp-edna-testing-can-lead-to-false-positives-
141
c 2014 Ethan Bolker, Maura Mast
13.7. COMMENTS ON THE EXERCISES
Part of the problem is that, on a mammogram, a noncancerous abnormality
can look very much like cancer. . . . This causes 10 percent to 15 percent of
screened women in the U.S. to be recalled for more evaluation. Most (95 percent)
screening-detected abnormalities are ultimately found to be noncancerous. An
American woman who is regularly screened during her 40s has a 61 percent chance
of getting a false positive result.
...
Now, for every $100 spent on screening, an additional $30 to $33 is spent to
evaluate false positive findings. In the Medicare population, the workup of false
positive mammogram results is estimated to total $250 million a year. 3
Also see the report High Rate of False-Positives with Annual Mammogram from UCSF:
For the false-positive study, the researchers found that after a decade of
annual screening, a majority of women will receive at least one false-positive
result, and 7 to 9 percent will receive a false-positive biopsy recommendation. 4
Exercise 13.7.20. [U] Testing for prostate cancer
Figure 13.1 from the Department of Family Medicine at Virginia Commonwealth University
illustrates the possible outcomes of a PSA screening test for prostate cancer.
Figure 13.1: Prostate cancer screening test results
This quotation spells out some of the arguments for and against the test:
There are possible advantages to having a PSA test.
3
http://www.bloomberg.com/news/2012-08-01/shaky-foundations-for-the-new-mammogram-economy.
html
4
http://www.ucsf.edu/news/2011/10/10778/high-rate-false-positives-annual-mammogram
c 2014 Ethan Bolker, Maura Mast
142
13.7. COMMENTS ON THE EXERCISES
1. A normal PSA test may reassure you.
2. A PSA test may find prostate cancer early before it has spread.
3. Treatment of prostate cancer in early stages may help some men to avoid
problems from cancer.
4. Treatment of prostate cancer in early stages may help some men live longer.
There are possible disadvantages to having a PSA test.
1. A normal PSA test may miss some prostate cancers.
2. A false positive PSA test may cause unnecessary anxiety.
3. A false positive PSA test may cause an unneeded prostate biopsy.
4. You may find out that you have prostate cancer, but it may be a cancer that
would never cause you any problems.
5. Treatment of prostate cancer may cause you harm. Difficulties with getting
erections or problems with controlling your bladder or bowels are some
potential harms.
(a) What does “PSA” stand for?
(b) Construct the two way contingency table based on this data.
(c) If your screening test is positive, what is the probability that you have prostate cancer?
(d) If you have prostate cancer, what is the probability that this screening test will detect
it?
The figure shows that the overall incidence of prostate cancer is 10%. That is probably an
invented statistic, to make the arguments easier to understand. The reality is complex. Here’s
a start on it, from the American Cancer Society:
What are the key statistics about prostate cancer?
Other than skin cancer, prostate cancer is the most common cancer in American
men. The latest American Cancer Society estimates for prostate cancer in the
United States are for 2012:
About 241,740 new cases of prostate cancer will be diagnosed
About 28,170 men will die of prostate cancer
About 1 man in 6 will be diagnosed with prostate cancer during his lifetime.
Prostate cancer occurs mainly in older men. Nearly two thirds are diagnosed
in men aged 65 or older, and it is rare before age 40. The average age at the time
of diagnosis is about 67.
143
c 2014 Ethan Bolker, Maura Mast
13.7. COMMENTS ON THE EXERCISES
Prostate cancer is the second leading cause of cancer death in American men,
behind only lung cancer. About 1 man in 36 will die of prostate cancer.
Prostate cancer can be a serious disease, but most men diagnosed with prostate
cancer do not die from it. In fact, more than 2.5 million men in the United States
who have been diagnosed with prostate cancer at some point are still alive today.
5
13.7
5
http://www.cancer.org/Cancer/ProstateCancer/DetailedGuide/prostate-cancer-key-statistics
c 2014 Ethan Bolker, Maura Mast
144
Index
ADHD, 60
All Things Considered, 44
Attention-Deficit / Hyperactivity Disorder,
60
HIV testing, 140
honeybee, 52
Hubway, 62
Krugman, Paul, 67
Benford’s Law, 94
Bolker, Benjamin, 132
Bray, Hiawatha, 42, 100
Lahey Clinic, 95
likely, 113
marathon, 77
MBTA, 37
missile defense, 140
MobileCommerce.xlsx, 124
Mother Goose and Grimm, 38
carbon dioxide, 114
carbon footprint, 113
CDC, 60
Centers for Disease Control, 60
CFL, 105
Charlie Card, 37
Cheerios, 33
chump change, 69
Clarke, Arthur C., 42
COLA, 75
compact fluorescent bulb, 105
New York Times, The, 31, 33, 36, 42, 48,
53, 64, 67–69, 71, 76–78, 99, 102,
107, 111, 113, 118, 121, 122, 132,
138
optical clock, 49
debit card, 77
peanuts, 53
Picasso, 69
pinochle, 30
plastic bags, 34
Plutonium, 123
polarization.csv, 111
Postel, Theodore, 140
prison population, 71
probably, 101, 113
Proposition 2 1/2, 81
PSA, 142
EconomistsMinimumWagePetition.pdf, 74
Everest, 51
Facebook, 57
Fenway Park, 53
five second rule, 39
Google, 36
Google correlate, 115
Hall of Fame, 121
heart attack risk, 115
radioactive waste, 123
145
13.7. COMMENTS ON THE EXERCISES
reward program, 129
Richter scale, 122
Royko, Mike, 30
Russian roulette, 138
Snapple facts, 36
Sotheby, 69
spreadsheet link, 74, 111, 124
stack exchange, 92
STD testing, 140
Sanders, Bernie, 128
shell game, 78
Silver, Nate, 121
small business loans, 70
weighted average, 22
wikipedia, 49
c 2014 Ethan Bolker, Maura Mast
zillion, 30
146