# Solution Manual for Introductory Statistics: Exploring the World Through Data, 3rd Edition

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INSTRUCTORโS
SOLUTIONS MANUAL
I NTRODUCTORY S TATISTICS :
E XPLORING THE W ORD T HROUGH D ATA
THIRD EDITION
Robert Gould
University of California, Los Angeles
Rebecca Wong
West Valley College
Colleen Ryan
Moorpark Community College
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
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Reproduced by Pearson from electronic files supplied by the author.
Copyright ยฉ 2020 by Pearson Education, Inc. 221 River Street, Hoboken, NJ 07030. All rights reserved.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or
by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the
publisher. Printed in the United States of America.
ISBN-13: 978-0-13-519061-6
ISBN-10: 0-13-519061-4
CONTENTS
Chapter 1: Introduction to Data
Section 1.2: Classifying and Storing Data ……………………………………………………………1
Section 1.3: Investigating Data …………………………………………………………………………..3
Section 1.4: Organizing Categorical Data ……………………………………………………………3
Section 1.5: Collecting Data to Understand Causality……………………………………………6
Chapter Review Exercises …………………………………………………………………………………7
Chapter 2: Picturing Variation with Graphs
Section 2.1: Visualizing Variation in Numerical Data
and Section 2.2: Summarizing Important Features of a Numerical Distribution ….9
Section 2.3: Visualizing Variation in Categorical Variables
and Section 2.4: Summarizing Categorical Distributions ………………………………..14
Section 2.5: Interpreting Graphs ……………………………………………………………………….15
Chapter Review Exercises ……………………………………………………………………………….16
Chapter 3: Numerical Summaries of Center and Variation
Section 3.1: Summaries for Symmetric Distributions ………………………………………….19
Section 3.2: Whatโs Unusual? The Empirical Rule and z-Scores …………………………..22
Section 3.3: Summaries for Skewed Distributions ………………………………………………23
Section 3.4: Comparing Measures of Center ………………………………………………………24
Section 3.5: Using Boxplots for Displaying Summaries ………………………………………27
Chapter Review Exercises ……………………………………………………………………………….28
Chapter 4: Regression Analysis: Exploring Associations
between Variables
Section 4.1: Visualizing Variability with a Scatterplot ………………………………………..31
Section 4.2: Measuring Strength of Association with Correlation …………………………31
Section 4.3: Modeling Linear Trends ………………………………………………………………..32
Section 4.4: Evaluating the Linear Model ………………………………………………………….37
Chapter Review Exercises ……………………………………………………………………………….40
Chapter 5: Modeling Variation with Probability
Section 5.1: What Is Randomness? ……………………………………………………………………49
Section 5.2: Finding Theoretical Probabilities…………………………………………………….49
Section 5.3: Associations in Categorical Variables ……………………………………………..54
Section 5.4: Finding Empirical and Simulated Probabilities …………………………………56
Chapter Review Exercises ……………………………………………………………………………….58
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iii
Chapter 6: Modeling Random Events: The Normal and Binomial Models
Section 6.1: Probability Distributions Are Models of Random Experiments …………..65
Section 6.2: The Normal Model………………………………………………………………………..67
Section 6.3: The Binomial Model (Optional) ……………………………………………………..79
Chapter Review Exercises ……………………………………………………………………………….81
Chapter 7: Survey Sampling and Inference
Section 7.1: Learning about the World through Surveys ………………………………………85
Section 7.2: Measuring the Quality of a Survey ………………………………………………….86
Section 7.3: The Central Limit Theorem for Sample Proportions ………………………….88
Section 7.4: Estimating the Population Proportion with Confidence Intervals ………..90
Section 7.5: Comparing Two Population Proportions with Confidence………………….94
Chapter Review Exercises ……………………………………………………………………………….97
Chapter 8: Hypothesis Testing for Population Proportions
Section 8.1: The Essential Ingredients of Hypothesis Testing …………………………….101
Section 8.2: Hypothesis Testing in Four Steps ………………………………………………….102
Section 8.3: Hypothesis Tests in Detail ……………………………………………………………107
Section 8.4: Comparing Proportions from Two Populations ……………………………….108
Chapter Review Exercises ……………………………………………………………………………..112
Chapter 9: Inferring Population Means
Section 9.1: Sample Means of Random Samples ………………………………………………121
Section 9.2: The Central Limit Theorem for Sample Means ……………………………….122
Section 9.3: Answering Questions about the Mean of a Population……………………..123
Section 9.4: Hypothesis Testing for Means ………………………………………………………125
Section 9.5: Comparing Two Population Means ……………………………………………….131
Chapter Review Exercises ……………………………………………………………………………..138
Chapter 10: Associations between Categorical Variables
Section 10.1: The Basic Ingredients for Testing with Categorical Variables …………147
Section 10.2: The Chi-Square Test for Goodness of Fit ……………………………………..149
Section 10.3: Chi-Square Tests for Associations between
Categorical Variables ……………………………………………………………………………….153
Section 10.4: Hypothesis Tests When Sample Sizes Are Small…………………………..160
Chapter Review Exercises ……………………………………………………………………………..165
Chapter 11: Multiple Comparisons and Analysis of Variance
Section 11.1: Multiple Comparisons………………………………………………………………..173
Section 11.2: The Analysis of Variance …………………………………………………………..175
Section 11.3: The ANOVA Test ……………………………………………………………………..176
Section 11.4: Post-Hoc Procedures ………………………………………………………………….180
Chapter Review Exercises ……………………………………………………………………………..184
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Chapter 12: Experimental Design: Controlling Variation
Section 12.1: Variation Out of Control …………………………………………………………….187
Section 12.2: Controlling Variation in Surveys …………………………………………………192
Section 12.3: Reading Research Papers ……………………………………………………………192
Chapter 13: Inference without Normality
Section 13.1: Transforming Data …………………………………………………………………….197
Section 13.2: The Sign Test for Paired Data……………………………………………………..199
Section 13.3: Mann-Whitney Test for Two Independent Groups…………………………201
Section 13.4: Randomization Tests………………………………………………………………….203
Chapter Review Exercises ……………………………………………………………………………..204
Chapter 14: Inference for Regression
Section 14.1: The Linear Regression Model……………………………………………………..209
Section 14.2: Using the Linear Model ……………………………………………………………..210
Section 14.3: Predicting Values and Estimating Means ……………………………………..212
Chapter Review Exercises ……………………………………………………………………………..213
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v
Chapter 1: Introduction to Data
Section 1.2: Classifying and Storing Data
1.1
There are eight variables: โFemaleโ, โCommute Distanceโ, โHair Colorโ, โRing Sizeโ, โHeightโ, โNumber of
Auntsโ, โCollege Units Acquiredโ, and โLiving Situationโ.
1.2
There are eleven observations.
1.3
a.
Living situation is categorical.
b.
Commute distance is numerical.
c.
Number of aunts is numerical.
a.
Ring size is numerical.
b.
Hair color is categorical.
c.
Height is numerical.
1.4
1.5
Answers will vary but could include such things as number of friends on Facebook or foot length. Donโt copy
these answers.
1.6
Answers will vary but could include such things as class standing (โFreshmanโ, โSophomoreโ, โJuniorโ, or
โSeniorโ) or favorite color. Donโt copy these answers.
1.7
0 = male, 1 = female. The sum represents the total number females in the data set.
1.8
There would be seven 1โs and four 0โs.
1.9
Female is categorical with two categories. The 1โs represent females, and the 0โs represent males. If you
added the numbers, you would get the number of females, so it makes sense here.
1.10 a.
Freshman
0
1
1
0
1
1
0
1
1
0
0
b.
numerical
c.
categorical
1.11 a.
The data is stacked.
b.
1 = male, 0 = female.
c.
Male
1916
183
836
95
512
Female
9802
153
1221
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Introductory Statistics: Exploring the World Through Data, 3rd edition
1.12 a.
The data is unstacked.
b.
Labels for columns will vary.
Gender
Age
1
29
1
23
1
30
1
32
1
25
0
24
0
24
0
32
0
35
0
23
c.
Gender is categorical; Age is numerical
1.13 a.
Stacked and coded:
Calories
90
310
500
500
600
90
150
600
500
550
Sweet
1
1
1
1
1
1
0
0
0
0
The second column could be labeled โSaltyโ with the 1โs being 0โs and the 0โs being 1โs.
b.
Unstacked:
Sweet
90
310
500
500
600
90
1.14 a.
Salty
150
600
500
550
Stacked and coded:
Cost
Male
10
1
15
1
15
1
25
1
12
1
8
0
30
0
15
0
15
0
The second column could be labeled โFemaleโ with the 1โs being 0โs and the 0โs being 1โs.
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Chapter 1: Introduction to Data
b.
Unstacked:
Male
Female
10
15
15
25
12
8
30
15
15
Section 1.3: Investigating Data
1.15 Yes. Use College Units Acquired and Living Situation.
1.16 Yes. Use Female and Height.
1.17 No. Data on number of hours of study per week are not included in the table.
1.18 Yes. Use Ring Size and Height.
1.19 a.
Yes. Use Date.
b.
No. data on temperature are not included in the table.
c.
Yes. Use Fatal and Species of Shark.
d.
Yes. Use Location.
1.20 Use Time and Activity.
Section 1.4: Organizing Categorical Data
1.21 a.
33/40 = 82.5%
b.
32/45 = 71.1%
c.
33/65 = 50.8%
d.
82.5% of 250 = 206
1.22 a.
4/27 = 14.8%
b.
14/27 = 51.9%
c.
4/18 = 22.2%
d.
14.8% of 600 = 89 men
1.23 a.
15/38 = 39.5% of the class were male.
b.
0.64(234) = 149.994, so 150 men are in the class.
c.
0.40(x) = 20, so 20/0.40 = 50 total students in the class.
1.24 a.
0.35(346) = 121 male nurses.
b.
66/178 = 37.1% female engineers.
c.
0.65(x) = 169 so 169/0.65 = 260 lawyers in the firm.
1.25 The frequency of women 6, the proportion is 6/11, and the percentage is 54.5%.
1.26 The frequency is 8, the proportion is 8/11, and the percentage is 72.7%.
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Introductory Statistics: Exploring the World Through Data, 3rd edition
1.27 a. and b.
Dorm
Commuter
Total
c.
4/6 = 66.7%
d.
4/7 = 57.1%
e.
7/11 = 63.6%
f.
66.7% of 70 = 47
Men
Women
3
2
5
4
2
6
Total
7
4
11
Men
3
2
0
5
Women
5
0
1
6
Total
8
2
1
11
1.28 a. and b.
Brown
Black
Blonde
Total
c.
5/6 = 83.3%
d.
5/8 = 62.5%
e.
8/11 = 72.7%
f.
83.3% of 60 = 50
1.29 1.26(x) = 160328 so 160328/1.26 = 127,244 personal care aids in 2014
1.30 .1295(x) = 3480000 so 3480000/.1295 = $26,872,587.87 total candy sales
1.31
State
California
New York
Illinois
Louisiana
Mississippi
Prison
136,088
52518
48278
30030
18793
Rank
Prison
1
2
3
4
5
Population
39,144,818
19,795,791
12,859,995
4,670,724
2,992,333
Population
(thousands)
39145
19796
12860
4671
29922
Prison per 1000
3.48
2.65
3.75
6.43
6.28
California has the highest prison population. Louisiana has the highest rate of imprisonment.
The two answers are different because the state populations are different.
1.32 a.
Miami: 4,919,000/2891 = 1701
Detroit: 3,903,000/3267 = 1195
Atlanta: 3,500,000/5083 = 689
Seattle: 2,712,000/1768 = 1534
Baltimore: 2,076,000/1768 = 1174
Ranks: 1- Miami, 2- Seattle, 3- Detroit, 4- Baltimore, 5- Atlanta
b.
Atlanta
c.
Miami
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Rank
Rate
4
5
3
1
2
Chapter 1: Introduction to Data
5
1.33
Year
1990
2000
%Uncovered
34, 719
๏ฝ 13.9%
249, 778
36,586
๏ฝ 13.1%
279, 282
29758
๏ฝ 9.4%
316574
The percentage of uninsured people have been declining.
2015
1.34
Year
% Subscribers
103.6
๏ฝ 90.3%
2012
114.7
103.3
๏ฝ 90.5%
2013
114.1
103.7
๏ฝ 89.6%
2014
115.7
100.2
๏ฝ 86.0%
2015
116.5
97.8
๏ฝ 84.0%
2016
116.4
The percentage of cable subscribers rose slightly between 2012 and 2013 but has declined each year since then.
1.35
%Older
Population
54.8
๏ฝ 16.4%
2020
334
70.0
๏ฝ 19.6%
2030
358
81.2
๏ฝ 21.4%
2040
380
88.5
๏ฝ 22.1%
2050
400
The percentage of older population is projected to increase.
Year
1.36
%Older
Population
4.0
๏ฝ 48.8%
2000
8.2
3.6
๏ฝ 47.4%
2005
7.6
3.6
๏ฝ 52.9%
2010
6.8
3.2
๏ฝ 46.4%
2014
6.9
The rate has fluctuating over this period, decreasing, then increasing, and then decreasing again.
Year
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Introductory Statistics: Exploring the World Through Data, 3rd edition
1.37 We donโt know the percentage of female students in the two classes. The larger number of women at 8a.m.
may just result from a larger number of students at 8 a.m., which may be because the class can accommodate
more students because perhaps it is in a large lecture hall.
1.38 No, we need to know the population of each city so we can compare the rates.
Section 1.5 Collecting Data to Understand Causality
1.39 Observational study.
1.40 Controlled experiment.
1.41 Controlled experiment.
1.42 Controlled experiment.
1.43 Controlled experiment.
1.44 Observational study.
1.45 Anecdotal evidence are stories about individual cases. No cause-and effect conclusions can be drawn from
anecdotal evidence.
1.46 These testimonials are anecdotal evidence. There is no control group and no comparison. No cause-and-effect
conclusions can be drawn from anecdotal evidence.
1.47 This was an observational study, and from it you cannot conclude that the tutoring raises the grades. Possible
confounders (answers may vary): 1. It may be the more highly motivated who attend the tutoring, and this
motivation is what causes the grades to go up. 2. It could be that those with more time attend the tutoring, and
it is the increased time studying that causes the grades to go up.
1.48 a.
If the doctor decides on the treatment, you could have bias.
b.
To remove this bias, randomly assign the patients to the different treatments.
c.
If the doctor knows which treatment a patient had, that might influence his opinion about the
effectiveness of the treatment.
d.
To remove that bias, make the experiment double-blind. The talk-therapy-only patients should get a
placebo, and no patients should know whether they have a placebo or antidepressant. In addition, the
doctor should not know who took the antidepressants and who did not.
1.49 a.
The sample size of this study is not large (40). The study was a controlled experiment and used random
assignment. It was not double-blind since researchers new what group each participant was in.
b.
The sample size of the study was small, so we should not conclude that physical activity while learning
caused higher performance.
1.50 This is an observational study because researchers did not determine who received PCV7 and who did not.
You cannot conclude causation from an observational study. We must assume that it is possible that there
were confounding factors (such as other advances in medicine) that had a good effect on the rate of
pneumonia.
1.51 a.
Controlled experiment. Researchers used random assignment of subjects to treatment or control groups.
b.
Yes. The experiment had a large sample size, was controlled, randomized, and double-blind; and used a
placebo.
1.52 a.
Observational study. There was no random assignment to treatment/control groups. The subjects kept a
food diary and had their blood drawn.
b.
We cannot make a cause-and-effect conclusion since this was an observational study.
1.53 No, this was not a controlled experiment. There was no random assignment to treatment/control groups and no
use of a placebo.
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Chapter 1: Introduction to Data
7
1.54 No. There was no control group and no comparison. From observation of 12 children it is not possible to
come to a conclusion that the vaccine causes autism. It may simply be that autism is usually noticed at the
same age the vaccine is given.
1.55 a.
Intervention remission: 11/33 = 33.3%; Control remission: 3 /34 = 8.8%
b.
Controlled experiment. There was random assignment to treatment/control groups.
c.
While this study did use random assignment to treatment/control groups, the sample size was fairly small
(67 total) and there was no blinding in the experimental design. The difference in remission may indicate
that the diet approach is promising and further research in this area is needed.
1.56 Ask whether there was random assignment to groups. Without random assignment there could be bias, and we
cannot infer causation.
1.57 No. This is an observational study.
1.58 This is likely a conclusion from observational studies since it would not be ethical to randomly assign a
subject to a group that drank large quantities of sugary drinks. Since this was likely based on observational
studies, we cannot conclude drinking sugary beverages causes lower brain volume.
Chapter Review Exercises
1.59 a.
61/98 = 62.2%
b.
37/82 = 45.1%
c.
Yes, this was a controlled experiment with random assignment. The difference in percentage of homes
adopting smoking restrictions indicates the intervention may have been effective.
1.60 No. Cause-and-effect conclusions cannot be drawn from observational studies.
1.61 a.
Gender (categorical) and whether students had received a speeding ticket (categorical)
b.
c.
1.62 a.
Male
Female
Yes
6
5
No
4
10
Men: 6/10=60%; Women: 5/15 = 33.3%; a greater percentage of men reported receiving a speeding
ticket.
Gender (categorical) and whether students had driven over 100 mph (categorical).
b.
c.
Male
Female
Yes
6
5
No
3
10
Men: 6/9 = 66.7%; Women: 5/15 = 33.3%; a greater percentage of men reported driving over 100 mph.
1.63 Answers will vary. Students should not copy the words they see in these answers. Randomly divide the group
in half, using a coin flip for each woman: Heads she gets the vitamin D, and tails she gets the placebo (or vice
versa). Make sure that neither the women themselves nor any of the people who come in contact with them
know whether they got the treatment or the placebo (โdouble-blindโ). Over a given length of time (such as
three years), note which women had broken bones and which did not. Compare the percentage of women with
broken bones in the vitamin D group with the percentage of women with broken bones in the placebo group.
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Introductory Statistics: Exploring the World Through Data, 3rd edition
1.64 Answers will vary. Students should not copy the words they see here. Randomly divide the group in half,
using a coin flip for each person: Heads they get Coumadin, and tails they get aspirin (or vice versa). Make
sure that neither the subjects nor any of the people who come in contact with them know which treatment they
received (โdouble-blindโ). Over a given length of time (such as three years), note which people had second
strokes and which did not. Compare the percentage of people with second strokes in the Coumadin group with
the percentage of people with second strokes in the aspirin group. There is no need for a placebo because we
are comparing two treatments. However, it would be acceptable to have three groups, one of which received a
placebo.
1.65 a.
The treatment variable is mindful yoga participation. The response variable is alcohol use.
b.
Controlled experiment (random assignment to treatment/control groups).
c.
No, since the sample size was fairly small; however, the difference in outcomes for treatment/control
groups may indicate that further research into the use of mindful yoga may be warranted.
1.66 a.
The treatment variable was neurofeedback; the response variable is ADHD symptoms.
b.
Controlled experiment (random assignment to treatment/control groups).
c.
No because there were no significant differences in outcomes between any of the groups.
1.67 No. There was no control group and no random assignment to treatment or control groups.
1.68 a.
Long course antibiotics: 39/238 = 16.4%; short course antibiotics: 77/229 = 33.6%.
The longer course recipients did better.
b.
10 days 5 days
Failure
39
77
Success 199
152
c.
Controlled experiment (random assignment to treatment/control groups).
d.
Yes. This was a controlled, randomized experiment with a large sample size.
1.69 a.
LD: 8% tumors; LL: 28% tumors A greater percentage of the 24 hours of light developed tumors.
b.
A controlled experiment. You can tell by the random assignment.
c.
Yes, we can conclude cause and effect because it was a controlled experiment, and random assignment
will balance out potential confounding variables.
1.70 a.
b.
43/53, or about 81.1%, of the males who were assigned to Scared Straight we rearrested. 37/55, or 67.3%,
of those receiving no treatment were rearrested So the group from Scared Straight had a higher arrest rate.
No, Scared Straight does not cause a lower arrest rate because the arrest rate was higher.
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Chapter 2: Picturing Variation with Graphs
Section 2.1: Visualizing Variation in Numerical Data
and Section 2.2: Summarizing Important Features of a Numerical Distribution
2.1
2.2
a.
4 people had resting pulse rates more than 100.
b.
4
๏ฝ 3.2% of the people had resting pulse rates of more than 100.
125
a.
8 people have glucose readings above 120 mg/dl.
b.
8 ๏ฝ 6.1% of these people have glucose readings above 120 mg/dl.
5
1
2
3
4
๏ฝ 0.20
๏ฝ 0.04,
๏ฝ 0.08,
๏ฝ 0.12,
๏ฝ 0.16,
25
25
25
25
25
2.3
New vertical axis labels:
2.4
a.
The bin width is 100.
b.
The histogram is bimodal because two bins have a much higher relative frequency than the others.
c.
About 19% (combine 6% and 13%). Due to the scale on the graph, any answer between 18% to 20% is
acceptable.
2.5
Yes, since only about 7% of the pulse rates were higher than 90 bpm. Conclusion might vary, but students
must mention that 7% of pulse rates were higher than 90 bpm.
2.6
No, because on roughly half of the days the post office served more than 250 customers, so 250 would not be
unusual.
2.7
a.
Both cereals have similar center values (about 110 calories). The spread of the dotplots differ.
b.
Cereal from manufacturer K tend to have more variation.
a.
Both distributions have more than one mode. The center for the coins from the United States is much
larger than the center for other countries. The spreads are similar.
b.
Coins in the United States tend to weigh more, as we conclude because the center of the distribution is
higher for the United States coins.
2.8
2.9
Roughly bell shaped. The lower bound is 0, the mean will be a number probably below 9, but a few students
might have slept quite a bit (up to 12 hours?) which creates a right-skew.
2.10 Roughly right-skewed (most students with no tickets, very few with many tickets).
2.11 It would be bimodal because the men and women tend to have different heights and therefore different arm
spans.
2.12 It might be bimodal because private colleges and public colleges tend to differ in amount of tuition.
2.13 About 75 beats per minute.
2.14 About 500 Calories.
2.15 The BMI for both groups are right skewed. For the men it is maybe bimodal (hard to tell). The typical values
for the men and women are similar although the value for the men appears just a little bit larger than the
typical value for the women. The womenโs values are more spread out.
2.16 a.
b.
Both distributions are right skewed. They have similar typical values.
The menโs distribution is more spread out and has a greater percentage of values that are considered high.
So, the womenโs levels are somewhat better.
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Introductory Statistics: Exploring the World Through Data, 3rd edition
2.17 a.
The distribution is multimodal with modes at 12 years (high school), 14 years (junior college), 16 years
(bachelorโs degree), and 18 years (possible masterโs degree). It is also left-skewed with numbers as low
as 0.
b.
Estimate: 300 + 50 + 100 + 40 + 50, or about 500 to 600, had 16 or more years.
c.
Between
2.18 a.
500
600
, or about 25%, and
, or about 30%, have a bachelorโs degree or higher. This is very
2018
2018
similar to the 27% given.
The distribution is right-skewed.
b.
About 2 or 3.
c.
Between 80 and 100.
d.
80
100
๏ฝ 4% or
๏ฝ 5%
2000
2000
2.19 Ford typically has higher monthly costs (the center is near 250 dollars compared with 225 for BMW) and
more variation in monthly costs.
2.20 Both makes have similar typical mpg (around 23 mpg). BMW has more variation in mpg (more horizontal
spread in the data).
2.21 1.
The assessed values of homes would tend to be lower with a few higher values: This is histogram B.
2.
The number of bedrooms in the houses would be slightly skewed right: This is histogram A.
3.
The height of house (in stories) for a region would be that allows up to 3 stories would be histogram C.
2.22 1.
The consumption of coffee by a person would be skewed right with many people who do not drink coffee
and a few who drink a lot: This is histogram A.
2.
The maximum speed driven in a car would be roughly symmetrical with a few students who drive very
fast: This is histogram C.
3.
The number of times a college student had breakfast would skew left with students who rarely eat
breakfast: This is histogram B.
2.23 1.
The heights of students would be bimodal and roughly symmetrical: This is histogram B.
2.
The number of hours of sleep would be unimodal and roughly symmetrical, with any outliers more likely
being fewer hours of sleep: This is histogram A.
3.
The number of accidents would be left skewed, with most student being involved in no or a few
accidents: This is histogram C.
2.24 1.
The SAT scores would be unimodal and roughly symmetrical: This is histogram C.
2.
The weights of men and women would be bimodal and roughly symmetrical, but with more variation that
SAT scores: This is histogram A.
3.
The ages of students would be left skewed, with most student being younger: This is histogram B.
2.25 Students should display a pair of dotplots or histograms. One graph for Hockey and one for Soccer. The
hockey team tends to be heavier than the soccer team (the typical hockey player weighs about 202 pounds
while the typical soccer player weighs about 170 pounds). The soccer team has more variation in weights than
the hockey team because there is more horizontal spread in the data. Statistical Question (answers may vary):
Are hockey players heavier than soccer players? Which type of athlete has the most variability in weight?
Copyright ยฉ 2020 Pearson Education, Inc.
Chapter 2: Picturing Variation with Graphs
11
2.26 (Answers may vary). Which type of apartment tends to cost more? See histograms.
Studio apartments tend to be less expensive and have more variation in price than do one-bedrooms.
2.27 See histogram. The shape will depend on the binning used. The histogram is bimodal with modes at about $30
and about $90.
Output for Exercise 27
12
Frequency
10
8
6
4
2
0
30
60
90
Cost (dollars)
120
2.28 See histogram. The shape will depend on the binning used. The 800 score could be an outlier or not, and the
graph could appear left-skewed or not.
Output for Exercise 28
14
12
Frequency
10
8
6
4
2
0
480
560
640
720
SAT Score
800
Copyright ยฉ 2020 Pearson Education, Inc.
12
Introductory Statistics: Exploring the World Through Data, 3rd edition
2.29 See histogram. The histogram is right-skewed. The typical value is around 12 (between 10 and 15) years, and
there are three outliers: Asian elephant (40 years), African elephant (35 years), and hippo (41 years). Humans
(75 years) would be way off to the right; they live much longer than other mammals.
Output for Exercise 29
14
Frequency
12
10
8
6
4
2
0
5
10 15 20 25 30 35 40
Average Longevity (years)
2.30 The histogram is right-skewed and also bimodal (at least with this grouping). The modes are at about 80 days
and 240 days. The typical value is about 240 days (between 160 and 320 days). There are two outliers at more
than 600 days, the Asian elephant and the African elephant. Humans (266 days) would be near the middle of
the graph.
Output for Exercise 30
12
Frequency
10
8
6
4
2
0
80 160 240 320 400 480 560 640
Gestation Period (days)
2.31 Both graphs are multimodal and right-skewed. The Democrats have a higher typical value, as shown by the
fact that the center is roughly around 35 or 40%, while the center value for the Republicans is closer to 20 to
30%. Also note the much larger proportion of Democrats who think the rate should be 50% or higher. The
distribution for the Democrats appears more spread out because the Democrats have more people responding
with both lower and higher percentages.
Output for Exercise 31
Democrat
Republican
0
15
30
45
60
75
90
Ideal Maxim um Tax Rate (percentage)
E ach sy mbol represents up to 2 observ ations.
Copyright ยฉ 2020 Pearson Education, Inc.
Chapter 2: Picturing Variation with Graphs
13
2.32 Both distributions are right-skewed. A large outlier did represent a cat lover, but typically, cat lovers and dog
lovers both seem to have about 2 pets, although there are a whole lot of dog lovers with one dog.
Output for Exercise 32
C at
Dog
0
2
4
6
8
10
Num ber of Pets
12
14
E ach sy mbol represents up to 2 observ ations.
2.33 The distribution appears left-skewed because of the low-end outlier at about $20,000 (Brigham Young
University).
Output for Exercise 33
14
12
Frequency
10
8
6
4
2
0
20 25 30 35 40 45 50 55
Tuition (thousands of dollars)
2.34 The histogram is strongly right-skewed, with outliers.
Output for Exercise 34
70
Frequency
60
50
40
30
20
10
0
0
40
80 120 160 200 240 280
Text Messages Sent in One Day
Copyright ยฉ 2020 Pearson Education, Inc.
14
Introductory Statistics: Exploring the World Through Data, 3rd edition
2.35 With this grouping the distribution appears bimodal with modes at about 110 and 150 calories. (With fewerโ
that is, widerโbins, it may not appear bimodal.) There is a low-end outlier at about 70 calories. There is a bit
of left skew.
Output for Exercise 35
25
Frequency
20
15
10
5
0
80 100 120 140 160 180 200
Calories in 12 Ounces of Beer
2.36 The distribution is left-skewed primarily because of the outliers at about 0% alcohol.
Output for Exercise 36
30
Frequency
25
20
15
10
5
0
0
1
2
3
4
5
6
Percent Alcohol in Beer
Section 2.3: Visualizing Variation in Categorical Variables
and Section 2.4: Summarizing Categorical Distributions
2.37 No, the largest category is Wrong to Right, which suggests that changes tend to make the answers more likely
to be right.
2.38 a.
About 7.5 million.
b.
About 5 million.
c.
No, overweight and obesity do not result in the highest rate. That is from high blood pressure.
d.
This is a Pareto chart.
2.39 A bar graph with the least variability would be one in which most children favored one particular flavor (like
chocolate). A bar graph with most variability, would be one in which children were roughly equally split in
their preference. with 1/3 choosing vanilla, 1/3 chocolate, 1/3 strawberry.
2.40 Least amount of variability would be one where most of the applicants had the same education goal (like
transfer). Most variability would be one where applicants were roughly equally divided among the five
choices.
Copyright ยฉ 2020 Pearson Education, Inc.

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