Table Of Content Chance/Rossman, 2018 ISCAM III
Acknowledgements
We would like to thank the following reviewers for their continued feedback and suggestions. They have
greatly improved the quality of these materials. We would like to especially thank Julie Clark of Hollins
University for her gracious and extensive attention to detail.
Doug Andrews, Wittenberg University; Julie Clark, Hollins University; David Cruz-Uribe,
Trinity College; Jo Hardin, Pomona College; Allen Hibbard, Central College; Sharon Lane-
Getaz, St. Olaf University; Tom Linton, Central College; Mark Mills, Central College; Paul
Roback, St. Olaf College; Soma Roy, Cal Poly – San Luis Obispo; Michael Stob, Calvin College
Cover Design: Danna Joselovitch, DannaJDesigns
Initial Solutions to Investigations: Chelsea Lofland, based on classroom notes
Initial Exercise Compilation and Solutions: Julie Clark
Copy-editing Assistance: Virginia Burroughs
Companion Website: www.rossmanchance.com/iscam3/
Online Resources include solutions to investigations, exercises, and additional investigations.
Also see Resources for Instructors.
Copyright August 2018 Beth Chance and Allan Rossman
San Luis Obispo, California
ALL RIGHTS RESERVED. No part of this work covered by the copyright hereon may be produced or
used in any form or by any means – graphic, electronic, or mechanical, including photocopying,
recording, taping, Web distribution, information storage retrieval systems, or in any other manner –
without the written permission of the authors.
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Chance/Rossman, 2018 ISCAM III Table of Contents
Table of Contents
To The Student ............................................................................................................................................ 3
Investigation A: Traffic Fatalities And Federal Speed Limits .................................................................... 4
Investigation B: Random Babies .............................................................................................................. 12
Chapter 1: Analyzing One Categorical Variable ...................................................................................... 19
Section 1: Analyzing A Process Probability ................................................................................. 20
Section 2: Normal Approximations For Sample Proportions ....................................................... 62
Section 3: Sampling From A Finite Population ............................................................................ 91
Chapter 2: Analyzing Quantitative Data ................................................................................................. 132
Section 1: Descriptive Statistics.................................................................................................. 133
Section 2: Inference For Population Mean ................................................................................. 148
Section 3: Inference For Other Statistics .................................................................................... 167
Chapter 3: Comparing Two Proportions ................................................................................................. 182
Section 1: Comparing Two Population Proportions ................................................................... 183
Section 2: Types Of Studies ........................................................................................................ 199
Section 3: Comparing Two Treatment Probabilities .................................................................. 208
Section 4: Other Statistics ........................................................................................................... 224
Chapter 4: Comparisons With Quantitative Variables ............................................................................ 252
Section 1: Comparing Groups – Quantitative Reponse .............................................................. 253
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Section 4: Matched Pairs Designs............................................................................................... 288 et
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Chapter 5: Comparing Several Populations, Exploring Relationships ................................................... 317 d
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Section 1: Two Categorical Variables ........................................................................................ 318 s,
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Section 2: Comparing Several Population Means ...................................................................... 338 ati
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Section 3: Relationships Between Quantitative Variables.......................................................... 352 pli
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Section 4: Inference For Regression ........................................................................................... 381 s,
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Index ..................................................................................................................................................... 424 ce
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Chance/Rossman, 2018 ISCAM III To the Student
TO THE STUDENT
Statistics is a mathematical science.
Although this is a very short sentence, perhaps a self-evident one, and certainly one of the shortest that
you will find in this book, we want to draw your attention to several things about it:
We use the singular “is” and not the plural “are.” It is certainly grammatically correct and more
common usage to say “statistics are...”, but that use of the term refers to statistics as numerical values.
In this sentence we mean statistics as a field of study, one that has its own concepts and techniques,
and one that can be exciting to study and practice.
We use “mathematical” as an adjective. Statistics certainly makes use of much mathematics, but it is a
separate discipline and not a branch of mathematics. Many, perhaps most, of the concepts and
methods in statistics are mathematical in nature, but there are also many that do not involve
mathematics. You will see an example of this early in the book as you study the difference between
observational studies and controlled experiments. You will find that even in cases where the
mathematical aspects of two situations may be identical, the scope of one’s conclusions depends
crucially on how the data were collected, a statistical rather than a mathematical consideration.
We use the noun “science.” Statistics is the science of gaining insight from data. Data are (notice the
plural here) pieces of information (often but not always numerical) gathered on people or objects or
processes. The science of statistics involves all aspects of inquiry about data. Well-designed studies
begin with a research question or hypothesis, devise a plan for collecting data to address that issue,
proceed to gather the data and analyze them, and then often make inferences about how the findings
generalize beyond the particular group being studied. Statistics concerns itself with all phases of this
process and therefore encompasses the scientific method.
In these materials, our goal is to introduce you to this practice of statistics, to help you think about the
applications of statistics and to study the mathematical underpinnings of the statistical methods. Most of
all, we hope you will find fun and engaging examples. Statistics is a vitally important subject, and also fun
to study and practice, largely because it brings you into contact with all kinds of interesting questions. You
will analyze data from medical studies, legal cases, psychology experiments, sociological studies, and many
other contexts. To paraphrase the late statistician John Tukey, “the best thing about statistics is that it
allows you to play in everyone else’s backyard.” You never know what you might learn in a statistics class!
One of the first features you will notice about these materials is that you will play the active role of
investigator. You will read about an actual study and consider the research question, and then we will lead
you to discover and apply the appropriate tools for carrying out the analysis. A primary reason for the
investigative nature of these materials is that we strongly believe that you will better understand and retain
the concepts if you build your own knowledge and are engaged in the context. Be sure to also pay attention
to the Study Conclusions to see how to effectively convey statistical information and the Practice Problems
for testing your understanding. Though you will only scratch the surface of the statistical methods used in
practice, you will learn fundamental concepts (such as variability, randomness, confidence, and
significance) that are an integral part of many statistical analyses. A distinct emphasis will be the focus on
how the data are collected and how this determines the scope of conclusions that you can draw from the
data.
3
Chance/Rossman, 2018 ISCAM III Investigation A
Investigation A: Traffic Fatalities and Federal Speed Limits
These first two investigations give you a very brief introduction to some big ideas for the course. Some of you
will have seen some of these ideas before and can use the investigations to refresh your memory. Some of the
ideas may be new and you will see them again in later chapters. For now, try to focus on the bigger picture of
analyzing data and drawing appropriate conclusions.
Increases and decreases in motor vehicle fatalities can be monitored to help assess impacts of changes in
policy. For example, in 1974, the US Federal government passed the National Maximum Speed Law,
capping the speed limit on all interstate roads to 55mph. At the time, the goal was to improve fuel
consumption in response to the 1973 oil embargo. However, when Congress repealed this law in 1995,
allowing states to raise the legal speed limit, many wondered if that would also lead to an increase in
accidents and fatalities. Frieman, Hedeker, and Richter (2009) took data from the Fatality Analysis
Reporting System (FARS) to explore this issue, focusing in particular on the repeal of federal speed
limit controls on road fatalities.
(a) To the right is a portion of Wikipedia’s “List of motor vehicle deaths in
the U.S. by year” data (through 2013) on the number of total motor vehicle
deaths in the United States per year. Notice the data are given in a data
table: each row represents a different year and the second column is the
number of deaths in the U.S. that year. Suggest a research question that
you could investigate with the full dataset.
(b) From the data provided here, do you see any trend or pattern to how the
number of deaths per year is changing? Does this pattern make sense?
What other information would you like to consider?
A time plot is a natural way to examine how a variable changes over time,
but not about how other variables are also changing over the same time
period. In particular, there are a lot more drivers on the roads now than in
1899, which is going to impact the number of deaths, so we really should
take that into account. Typically, data like this would be scaled in some
way, such as dividing by number of vehicle miles traveled by all cars that
year or by the population size of the United States that year.
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Chance/Rossman, 2018 ISCAM III Investigation A
Below is a time plot of fatalities per 100,000 population using Wikipedia’s data from 1899 to 2013.
(c) Write a few sentences summarizing how this new variable was calculated and what you learn from
the time plot of these data.
(d) In 1973 (before the speed limit was capped at 55 mph nationwide), the data report 25.507 fatalities
per 100,000 people, compared to 21.134 in 1974 (after the national limit). Did the rate of fatalities
decrease between these two years? If so, by how much? Does that appear to be a “large” difference?
How are you deciding?
Absolute differences, like you calculated in (d), can be difficult to evaluate, especially if you don’t
consider the magnitude of values they come from. You might respond differently to a box of M&Ms
coming up 4 candies short, than to a difference of 4 fatalities per 100,000 people (about 10,000 more
deaths across the U.S. in one year). One way to take into account the magnitude of the data values you
are comparing is to compute the percentage change:
percentage change = (current rate – previous rate) × 100%
previous rate
(e) Calculate and interpret this value. In particular, is this a positive or negative value; what does that
imply?
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Chance/Rossman, 2018 ISCAM III Investigation A
Below is a time plot of the year-to-year percentage change, starting in 1900.
(f) Why doesn’t this time plot start in 1899?
(g) Write a few sentences describing what you learn from this graph. Are there any unusual features?
Can you conjecture an explanation for any of these features?
From this graph, we notice that there is variability in how these values change from year to year. Much
of statistical investigation is measuring, accounting for, and trying to explain variability. In particular,
we still can’t judge whether the percentage change you computed between 1973 and 1974 is particularly
large until we compare it to the percentage changes for the other years.
Below is a dotplot displaying the distribution of the percentage change values for this data set.
(h) Explain what each dot represents. What information is revealed by this plot that was less clear in the
time plot? How might you use this graph to measure how unusual the 1973/1974 percentage change is?
What information is lost in this graph compared to the time plot?
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Chance/Rossman, 2018 ISCAM III Investigation A
When looking at a distribution of a single variable like this, we are often interested in three key features:
Center: What would you consider a “typical” value in the distribution?
Variability: How clustered together or consistent are the observations?
Shape: Are some values more common than others? Are the values symmetric about the center?
Are there any unusual observations that don’t follow the overall pattern?
In describing these features, it is often helpful to summarize a characteristic with a single number. To
summarize the center of the distribution, we often report the mean or the median. To summarize the
variability of the distribution, we often report the standard deviation or the interquartile range. For now
we will focus on the mean and standard deviation, a very common pairing.
Definitions: If there are n such values and we refer to them as x , x , …, x ,
1 2 n
The mean is the average of all numerical values in n
x
your data set. i
mean x = i1
n
The median is a value such that 50% of the data
median position: (n+1)/2
lies below and 50% of the data lies above that value
standard deviation s =
The standard deviation measures the size of a
(x x)2
“typical” deviation of the data values from the i
mean. n1
Quick Check: Which set of values will have the largest standard deviation?
(a) 1, 3, 5, 7, 9 (b) 1, 1, 5, 9, 9 (c) 5, 5, 5, 5, 5
(i) Open the TrafficFatalitiesYears.txt file and copy the PercentChange column into the Descriptive
Statistics applet:
Select and highlight all the data values in the text file and copy to the clipboard (e.g., ctrl-C or right
click and select copy).
In the applet, keep the Includes header box checked. (Note that the header must consist of just one
word.)
Press the Clear button and then click inside the Sample data window. Then paste the data from the
clipboard (e.g., ctrl-V or right click and select paste).
You can delete the * (representing the percentage change for the initial year) or the applet will warn
you that it is ignoring that value but allow you to proceed.
Press Use Data.
What do you see?
Because the data set doesn’t consist of a lot of identical numerical values, the dotplot is not very
informative, looking somewhat flat. We could round the data values to the nearest integer or we can
“bin” the observations together a bit: group similar values into non-overlapping categories and then
display the count of observations in each category. The dotplot shown before (h) is binned.
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Chance/Rossman, 2018 ISCAM III Investigation A
(j) In the applet, select the radio button for Histogram to have the applet bin the observations together
and create a histogram. The height of each bar counts the number of observations that fall (uniquely) in
each bin. How does the histogram representation compare to the original dotplot? Which display do you
prefer? What is a potential downside to a histogram with not very many bins? [Hint: You can adjust the
number of bins below the graph.]
(k) Now check the boxes for Actual in the Mean row and in the Std dev row to the left of the graph.
Report these values and summarize what they tell you.
(l) We can use this information to help us measure how “large” the percentage change was after the
repeal of the federal speed limit. Where does the percentage change you calculated in (e) fall in this
distribution? (Give a pretty detailed description.) Would you consider it an atypical value?
(m) The percentage change between 1994 and 1995 (before and after the federal speed limit was
repealed) was 1.74%. Where does that value fall in the above distribution? Does it appear unusually
large?
Of course, one issue with this analysis is that after the repeal of the federal speed limit different states
chose to implement changes to the speed limits in different years, so focusing on the 1994-95 change is
too simplistic. One way to account for this source of variability is by focusing on an individual state.
The CAFatalities.xls file contains the number of fatalities and population size (in thousands) for 1993-
2012 from www-fars.nhtsa.dot.gov/Trends/TrendsGeneral.aspx.
(n) What was (roughly) the California population in 1993?
(o) Use Excel to compute the fatality rates per thousand residents (e.g., In cell E2, enter =C2/D2, then
put the cursor back over that cell and double click on the lower right corner to “fill down” the column.
Remember to name the column!). What was the fatality rate in 1993? (Include “measurement units” with
your answer.)
(p) Now compute the percentage change in fatality rates (e.g., In cell F3, enter =(E3-E2)/E2*100
and fill down). What was the percentage change from 1993 to 1994? Was it positive or negative?
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Chance/Rossman, 2018 ISCAM III Investigation A
(q) Use technology to create a dotplot or histogram of the percentage change in fatality rates for
California during this time frame, and calculate the mean and standard deviation. Compare these results
to what you found above for the entire United States.
(r) In particular, is the standard deviation in (q) larger or smaller than for the U.S. data? Give some
explanations why the standard deviation has changed this way. [Hint: Focus on issues other than the
sample size (the number of years of data), because the number of observations is not generally an
indicator of how large the “typical deviation from the mean” will be. In fact, if we consider the smaller
population size in CA, we might expect there to be more variability in the rate of accidents per year…]
(s) On Jan. 7, 1996, California allowed rural speed limits to increase from 65 to 70 mph and urban roads
to increase from 55 to 65 mph. What is the percentage change in the fatality rate for California between
1995 and 1996? Is this a particularly large value in the distribution for California? [Be sure to justify
your answer.]
(t) Suppose we did find a large increase or decrease in a state’s fatality rate before and after the change
in speed limit laws. Would it be reasonable to conclude that the change in speed limit laws caused the
change in fatality rate? Why or why not? What else should we consider?
Study Conclusions
Many researchers are interested in whether changes in speed limits have had an impact on the
number of traffic fatalities. In this investigation, we focused on the number of fatalities per
100,000 in the United States. (You could also look at other rates like fatalities per 100 million
miles traveled.) We see an increasing and then overall decreasing trend in the fatality rate over
time, though the rates appear to be leveling off after about 1950. Looking at percentage change,
the second largest decrease (more than one standard deviation from the mean) occurred in 1976,
after the imposition of a federal maximum speed limit. We did not see a similar increase when
the federal maximum speed limit was repealed, though not all the states changed their speed
limits at this time. A follow-up study could examine whether the fatality rates differed between
states that did and did not increase speed limits (see Example 4.2).
9
