Mathematical Statistics with Resampling and R E-Book

Laura M. Chihara

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 793

Veröffentlichungsjahr: 2022

Beschreibung

Mathematical Statistics with Resampling and R

This thoroughly updated third edition combines the latest software applications with the benefits of modern resampling techniques

Resampling helps students understand the meaning of sampling distributions, sampling variability, P-values, hypothesis tests, and confidence intervals. The third edition of Mathematical Statistics with Resampling and R combines modern resampling techniques and mathematical statistics. This book is classroom-tested to ensure an accessible presentation, and uses the powerful and flexible computer language R for data analysis.

This book introduces permutation tests and bootstrap methods to motivate classical inference methods, as well as to be utilized as useful tools in their own right when classical methods are inaccurate or unavailable. The book strikes a balance between simulation, computing, theory, data, and applications.

Throughout the book, new and updated case studies representing a diverse range of subjects, such as flight delays, birth weights of babies, U.S. demographics, views on sociological issues, and problems at Google and Instacart, illustrate the relevance of mathematical statistics to real-world applications.

Changes and additions to the third edition include:

• New and updated case studies that incorporate contemporary subjects like COVID-19
• Several new sections, including introductory material on causal models and regression methods for causal modeling in practice
• Modern terminology distinguishing statistical discernibility and practical importance
• New exercises and examples, data sets, and R code, using dplyr and ggplot2
• A complete instructor’s solutions manual
• A new github site that contains code, data sets, additional topics, and instructor resources

Mathematical Statistics with Resampling and R is an ideal textbook for undergraduate and graduate students in mathematical statistics courses, as well as practitioners and researchers looking to expand their toolkit of resampling and classical techniques.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Dedication

Preface

Note

1 Data and Case Studies

1.1 Case Study: Flight Delays

1.2 Case Study: Birth Weights of Babies

1.3 Case Study: Verizon Repair Times

1.4 Case Study: Iowa Recidivism

1.5 Sampling

1.6 Parameters and Statistics

1.7 Case Study: General Social Survey

1.8 Sample Surveys

1.9 Case Study: Beer and Hot Wings

1.10 Case Study: Black Spruce Seedlings

1.11 Studies

1.12 Google Interview Question: Mobile Ads Optimization

Exercises

Notes

2 Exploratory Data Analysis

2.1 Basic Plots

2.2 Numeric Summaries

2.3 Boxplots

2.4 Quantiles and Normal Quantile Plots

2.5 Empirical Cumulative Distribution Functions

2.6 Scatter Plots

2.7 Skewness and Kurtosis

Exercises

3 Introduction to Hypothesis Testing: Permutation Tests

3.1 Introduction to Hypothesis Testing

3.2 Hypotheses

3.3 Permutation Tests

3.4 Matched Pairs

3.5 Cause and Effect

Exercises

Notes

4 Sampling Distributions

4.1 Sampling Distributions

4.2 Calculating Sampling Distributions

4.3 The Central Limit Theorem

Exercises

5 Introduction to Confidence Intervals: The Bootstrap

5.1 Introduction to the Bootstrap

5.2 The Plug‐in Principle

5.3 Bootstrap Percentile Intervals

5.4 Two Sample Bootstrap

5.5 Other Statistics

5.6 Bias

5.7 Monte Carlo Sampling

5.8 Accuracy of Bootstrap Distributions

5.9 How Many Bootstrap Samples Are Needed?

Exercises

Notes

6 Estimation

6.1 Maximum Likelihood Estimation

6.2 Method of Moments

6.3 Properties of Estimators

6.4 Statistical Practice

Exercises

Notes

7 More Confidence Intervals

7.1 Confidence Intervals for Means

7.2 Confidence Intervals Using Pivots

7.3 One‐Sided Confidence Intervals

7.4 Confidence Intervals for Proportions

7.5 Bootstrap Confidence Intervals

7.6 Confidence Interval Properties

7.7 The Delta Method*

Exercises

Notes

8 More Hypothesis Testing

8.1 Hypothesis Tests for Means and Proportions: One Population

8.2 Bootstrap

Tests

8.3 Hypothesis Tests for Means and Proportions: Two Populations

8.4 Type I and Type II Errors

8.5 Interpreting Test Results

8.6 Likelihood Ratio Tests

8.7 Statistical Practice

Exercises

Notes

9 Regression

9.1 Covariance

9.2 Correlation

9.3 Least Squares Regression

9.4 The Simple Linear Model

9.5 Resampling Correlation and Regression

9.6 Logistic Regression

Exercises

Notes

10 Categorical Data

10.1 Independence in Contingency Tables

10.2 Permutation Test of Independence

10.3 Chi‐Square Test of Independence

10.4 Chi‐Square Test of Homogeneity

10.5 Goodness‐of‐Fit Tests

10.6 Chi‐Square and the Likelihood Ratio*

Exercises

Notes

11 Bayesian Methods

11.1 Bayes Theorem

11.2 Binomial Data: Discrete Prior Distributions

11.3 Binomial Data: Continuous Prior Distributions

11.4 Continuous Data

11.5 Sequential Data

Exercises

Notes

12 One‐Way ANOVA

12.1 Comparing Three or More Populations

Exercises

Notes

13 Additional Topics

13.1 Smoothed Bootstrap

13.2 Parametric Bootstrap

13.3 Stratified Sampling

13.4 Control Variates and Casual Modeling

13.5 Computational Issues in Bayesian Analysis

13.6 Monte Carlo Integration

13.7 Importance Sampling

13.8 The EM Algorithm

Exercises

Notes

Appendix A: Review of Probability

A.1 Basic Probability

A.2 Mean and Variance

A.3 Marginal and Conditional Distributions

A.4 The Normal Distribution

A.5 The Mean of a Sample of Random Variables

A.6 Sums of Normal Random Variables

A.7 The Law of Averages

A.8 Higher Moments and the Moment Generating Function

Appendix B: Probability Distributions

B.1 The Bernoulli and Binomial Distributions

B.2 The Multinomial Distribution

B.3 The Geometric Distribution

B.4 The Negative Binomial Distribution

B.5 The Hypergeometric Distribution

B.6 The Poisson Distribution

B.7 The Uniform Distribution

B.8 The Exponential Distribution

B.9 The Gamma Distribution

B.10 The Chi‐Square Distribution

B.11 The Student's

Distribution

B.12 The Beta Distribution

B.13 The

Distribution

Exercises

Appendix C: Distributions Quick Reference

Problem Solutions

Bibliography

Index

End User License Agreement

List of Tables

Chapter 1

Table 1.1 Partial view of

FlightDelays

data.

Table 1.2 Variables in data set

FlightDelays

.

Table 1.3 Variables in data set

NCBirths2004

.

Table 1.4 Variables in data set

Verizon

.

Table 1.5 Variables in data set Iowa

Recidivism

.

Table 1.6 Variables in data set

GSS2018

.

Table 1.7 Variables in data set

Beerwings

.

Table 1.8 Variables in data set

Spruce

.

Table 1.9 Variables in data set

MobileAds

.

Table 1.10 Partial view of

MobileAds

data.

Chapter 2

Table 2.1 Counts for the

Carrier

variable.

Table 2.2 Counts of Delayed Flights grouped by carrier.

Table 2.3 Distribution of length of flight delays for United Airlines.

Table 2.4 A set of 21 data values.

Chapter 3

Table 3.1 All possible partitions of {30, 25, 20, 18, 21, 22} into two sets...

Table 3.2 Hot wings consumption.

Table 3.3 Partial view of

Beerwings

data set.

Table 3.4 Partial view of diving scores in file

Diving2017

.

Chapter 5

Table 5.1 Summary of center and spread for the normal distribution example....

Table 5.2 Summary of center and spread for the gamma distribution example....

Chapter 6

Table 6.1 Sample of wind speeds (m/s) from Carleton College turbine.

Chapter 8

Table 8.1 Decisions by a jury in a murder trial.

Table 8.2 Cumulative probabilities for

.

Chapter 9

Table 9.1 Partial view of

Spruce

data.

Table 9.2 Bushmeat: local supply of fish per capita and biomass of 41 speci...

Table 9.3 Part of the data on driver fatalities in Pennsylvania.

Chapter 10

Table 10.1 Counts of death penalty opinions grouped by degree.

Table 10.2 Expected counts of death penalty opinions grouped by degree.

Table 10.3 Contingency table after permuting

DeathPenalty

column.

Table 10.4 Observed counts.

Table 10.5 General

contingency table.

Table 10.6 Counts of candy preferences.

Table 10.7 Counts of home runs (in 162 games).

Table 10.8 Probabilities for Poisson distribution with

, and expected coun...

Table 10.9 Chi‐square test for Poisson goodness‐of‐fit to home run data.

Table 10.10 Distribution of wind speed data into intervals.

Chapter 11

Table 11.1 Calculations for Example 11.2.

Chapter 12

Table 12.1 Summary statistics for birth weights.

Table 12.2 Observations drawn from the

populations.

Table 12.3 ANOVA table.

Appendix B

Table B.1 Table for hypergeometric distribution.

List of Illustrations

Chapter 1

Figure 1.1 Density plot for CPR error.

Chapter 2

Figure 2.1 Bar chart of

Carrier

variable.

Figure 2.2 Histogram of lengths of flight delays for United Airlines. The di...

Figure 2.3 Histogram of average January temperatures in Washington state (18...

Figure 2.4 Example of a dot plot.

Figure 2.5 Boxplot for Table 2.4.

Figure 2.6 Distribution of lengths of the flight delays for United Airlines ...

Figure 2.7 Distribution of birth weights for North Carolina babies.

Figure 2.8 Density for

with

.

Figure 2.9 (a) Example of normal quantile plot for data in Example 2.11. (b)...

Figure 2.10 Normal quantile plot for a random sample of the flight delay tim...

Figure 2.11 Normal quantile plot for average January temperatures in Washing...

Figure 2.12 (a) Example of sample with symmetric, bell‐shaped distribution w...

Figure 2.13 (a) Empirical cumulative distribution function for the data 3, 6...

Figure 2.14 Ecdf's for male and female beer consumption. The vertical line i...

Figure 2.15 A scatter plot of

Beer

against

Hotwings

.

Figure 2.16 Examples of scatter plots.

Figure 2.17 Examples of skewness and kurtosis for four distributions, includ...

Figure 2.18 Empirical cdf for a data set,

.

Chapter 3

Figure 3.1 Empirical cumulative distribution function of the null distributi...

Figure 3.2 Number of hot wings consumed, by gender.

Figure 3.3 Permutation distribution of the difference in means, male 

 femal...

Figure 3.4 Normal quantile plots of the ILEC and CLEC data.

Figure 3.5 Permutation distribution of difference of means (ILEC 

 CLEC) for...

Figure 3.6 Repair times for Verizon data. (a) Permutation distribution for d...

Figure 3.7 Repair times for Verizon data. (a) Difference in proportion of re...

Figure 3.8 A diagram to indicate that X causes Y.

Figure 3.9 Confounding variable C affecting both treatment X and outcome Y....

Figure 3.10 Possible confounders in a study of time on Instagram and mental ...

Figure 3.11 Both treatment X and outcome Y cause the collider B.

Figure 3.12 Outcome on green die is not independent of red die if conditioni...

Chapter 4

Figure 4.1 Distribution of

after

sets of

tosses.

Figure 4.2 Distribution of

after

sets of

tosses.

Figure 4.3 Dot plot for sampling distribution of sample means.

Figure 4.4 Density for the exponential distribution with

.

Figure 4.5 (a) Histogram and (b) quantile‐normal plots for simulated samplin...

Figure 4.6 Simulated sampling distribution of the maximum of a sample.

Figure 4.7 Simulated sampling distribution of the sum of two Poissons.

Figure 4.8 Density for the gamma distribution

and

.

Figure 4.9 (a) Histogram and (b) quantile‐normal plots for the sampling dist...

Figure 4.10 Binomial distribution with

and

, with CLT approximation, and

Chapter 5

Figure 5.1 Bootstrap distribution of means for the North Carolina birth weig...

Figure 5.2 (a) The population distribution,

. (b) The theoretical sampling ...

Figure 5.3 Sampling and bootstrap distribution from a gamma distribution. (a...

Figure 5.4 Diagram of the process of creating a sampling distribution. Many ...

Figure 5.5 Diagram of the process of creating a bootstrap distribution. This...

Figure 5.6 (a) Histogram and (b) normal quantile plot(b) QQ plot of arsenic ...

Figure 5.7 (a) Histogram and (b) normal quantile plot of the bootstrap distr...

Figure 5.8 (a) Histogram and (b) normal quantile plot of the bootstrap distr...

Figure 5.9 (a) Histogram and (b) normal quantile plot of the permutation dis...

Figure 5.10 (a) Histogram and (b) quantile normal plots of the bootstrap dis...

Figure 5.11 Bootstrap distribution for the sample mean of the Verizon CLEC d...

Figure 5.12 Bootstrap distribution for the difference in means, ILEC–CLEC. T...

Figure 5.13 Bootstrap distribution for the difference in 25% trimmed means f...

Figure 5.14 Bootstrap distribution for the ratio of means.

Figure 5.15 Bootstrap distribution of relative risk.

Figure 5.16 Bootstrapped proportions of the high blood pressure group agains...

Figure 5.17 Bootstrap distribution for the mean,

. The left column shows th...

Figure 5.18 Bootstrap distributions for the mean,

. The left column shows t...

Figure 5.19 Bootstrap distributions for the median,

. The left column shows...

Figure 5.20 Bootstrap distributions for the mean,

, exponential population....

Chapter 6

Figure 6.1 Likelihood for

, after five heads and three tails.

Figure 6.2 Estimate of area under curve by rectangle.

Figure 6.3 Likelihood for

.

Figure 6.4 (a) Histogram of wind speeds (m/s) with the pdf for the Weibull d...

Figure 6.5 Sampling distribution of estimators for

. (a) Distribution of

....

Figure 6.6 Mean square error against

,

.

Figure 6.7 (a) Running mean for Cauchy. The

value is the running mean from...

Chapter 7

Figure 7.1 Random confidence intervals,

,

. Notice that several miss the m...

Figure 7.2 Standard normal density with shaded area

.

Figure 7.3 Normal quantile plot for sampling distribution of

.

Figure 7.4 Density for standard normal and students

distributions with 4, ...

Figure 7.5 Normal quantile plot of weights of baby girls.

Figure 7.6 Quantile–quantile plot of

 statistics versus a

 distribution wi...

Figure 7.7 Comparison of the

empirical cumulative distribution function

s (

ec

...

Figure 7.8 Quantile–quantile plot of

 statistics versus a

 distribution wh...

Figure 7.9 Weights of boy and girl babies born in Texas in 2004.

Figure 7.10 Density for

. Shaded region represents an area of

.

Figure 7.11 Sampling distributions for a location parameter. (a) Sampling di...

Figure 7.12 Sampling distributions for a scale parameter. (a) Sampling distr...

Figure 7.13 (a) Plot of

against

. (b) A normal quantile plot of the boots...

Figure 7.14 How often confidence intervals miss on each side. Confidence int...

Figure 7.15 Delta method.

: sample means

from 200 samples of size

from ...

Chapter 8

Figure 8.1 Distribution of weights of babies born to nonsmoking and smoking ...

Figure 8.2 Critical region, critical value

, and a test statistic

for whi...

Figure 8.3 Distributions of the test statistic under the null and alternativ...

Figure 8.4 Distributions under the null and alternative hypotheses. Shaded r...

Figure 8.5 Plot of power against different alternatives

for

and

when h...

Figure 8.6 Correspondence between hypothesis tests and confidence intervals....

Figure 8.7 Likelihood ratio for Cauchy,

,

, for a single observation. The ...

Figure 8.8 Google Mobile Ads Optimization, distribution of return on investm...

Figure 8.9 Google Mobile Ads Optimization, (a) Positive values on the

‐axis...

Figure 8.10 Return on investment (ROI) is more variable when the size of the...

Chapter 9

Figure 9.1 Change in diameter against change in height of seedlings over a 5...

Figure 9.2 Two scatter plots of two numeric variables.

Figure 9.3 Change in diameter against change in height of seedlings over a 5...

Figure 9.4 Examples of correlation. (a) Correlation −0.9, (b) Correlation −0...

Figure 9.5 “Best fit” line.

Figure 9.6 (a) The relationship between the regression slope,

and

with p...

Figure 9.7 Heights of parents and children. The data are

jittered

 – a small ...

Figure 9.8 The partitioning of the variability. The horizontal line is the m...

Figure 9.9 Residuals are the (signed) lengths of the line segments drawn fro...

Figure 9.10 Examples of residual plots. (a) A good straight line fit. (b) Th...

Figure 9.11 Residual plot for the Spruce data.

Figure 9.12 Residual plot for the Spruce data, with a scatter plot smooth in...

Figure 9.13 (a) Scatter plot of fat content against sugar content in ice cre...

Figure 9.14 Scores of the 24 finalists in the 2010 Olympics men's figure ska...

Figure 9.15 Linear regression conditions: each

is normal with mean

and c...

Figure 9.16 Plot of pointwise confidence and prediction intervals for the Ol...

Figure 9.17 Bootstrap distributions of (a) correlations and (b) slopes for s...

Figure 9.18 Distribution of bootstrapped free program scores when the short ...

Figure 9.19 Bushmeat data; (a) fish per capita and (b) biomass of 41 species...

Figure 9.20 Scatter plot of percent change in biomass against fish supply wi...

Figure 9.21 (a) Regression lines from 40 bootstrap samples of the bushmeat d...

Figure 9.22 Plot of a typical logistic curve, Equation (9.20).

Figure 9.23 (a) Plot of estimated probability of alcohol being involved agai...

Figure 9.24 (a) Histogram and (b) normal quantile plot for

, the logistic r...

Figure 9.25 (a) Histogram and (b) normal quantile plot for probability of in...

Figure 9.26 Residuals plot for regression models.

Chapter 10

Figure 10.1 Null distribution for chi‐square statistic for death penalty opi...

Figure 10.2 Densities for the chi‐square distribution.

Chapter 11

Figure 11.1 Prior distribution and posterior distributions after three games...

Figure 11.2 Venn diagrams with box areas showing joint probabilities for

a...

Figure 11.3

with different

's and

's.

Figure 11.4 Pdf for

, resulting from a uniform prior and 110 successes in 2...

Figure 11.5 Pdfs for the prior

and posterior

for Bayesian B.

Figure 11.6 Comparison of pdfs for the priors of Bayesians A and B, as well ...

Figure 11.7 Prior

and posterior

distributions of

. The observed mean

...

Figure 11.8 Posterior distribution for artificial Google Analytics Content E...

Chapter 12

Figure 12.1 (a) Distribution of the birth weights of boys born in Illinois i...

Figure 12.2 Distribution of the resistivity measurements across the five ins...

Chapter 13

Figure 13.1 Ordinary bootstrap distributions for the median and mean,

. (a)...

Figure 13.2 Kernel density estimates. (a) A kernel density estimate using th...

Figure 13.3 Smoothed bootstrap distributions for the median and the mean,

....

Figure 13.4 Empirical cumulative distribution function of wind speeds (m/s),...

Figure 13.5 Bootstrap distributions for 10% quantile of wind speed. (a) The ...

Figure 13.6 Bootstrap distributions for fitted Weibull distribution estimate...

Figure 13.7 A/B Trial, linear relationships between outcome metric and covar...

Figure 13.8 Logistic relationships between outcome metric and covariate. The...

Figure 13.9 Prior and posterior densities for Pew survey example.

Figure 13.10 Importance sampling for the European option example, where

is...

Figure 13.11 Importance sampling for the Twitter account example,

and

, w...

Figure 13.12 Weighted ecdf from importance sampling for the Twitter account ...

Figure 13.13 Weighted ecdf from importance sampling for the Twitter account ...

Appendix B

Figure B.1 Densities of gamma distribution.

Figure B.2 Densities for the chi‐square distribution.

Figure B.3 Densities for the

distribution.

Figure B.4 Densities for the Beta distribution (

,

).

Figure B.5

distribution with (

,

) degrees of freedom.

Guide

Cover

Table of Contents

Title Page

Copyright

Dedication

Preface

Begin Reading

Appendix A: Review of Probability

Appendix B: Probability Distributions

Appendix C: Distributions Quick Reference

Problem Solutions

Bibliography

Index

End User License Agreement

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Mathematical Statistics with Resampling and R

Third Edition

Laura M. ChiharaCarleton College, Northfield, MN, US

Tim C. HesterbergInstacart, Seattle, WA, US

This third edition first published 2022© 2022 John Wiley & Sons, Inc.

Edition History2e: 2018, Wiley; 1e: 2011, Wiley

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, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.

The right of Laura M. Chihara and Tim C. Hesterberg to be identified as the authors of this work has been asserted in accordance with law.

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Library of Congress Cataloging‐in‐Publication Data applied forHardback ISBN: 9781119874034

Cover image: Courtesy of Carleton CollegeCover design by Wiley

The world seldom notices who teachers are; but civilization depends on what they do.– Lindley Stiles

To:Theodore S. Chihara

To:Bev Hesterberg

Preface

Mathematical Statistics with Resampling and R is a one term undergraduate statistics textbook aimed at sophomores or juniors who have taken a course in probability (at the level of, for instance, Ross (2009), Ghahramani (2004), or Scheaffer and Young (2010)) but may not have had any previous exposure to statistics.

What sets this book apart from other mathematical statistics texts is the use of modern resampling techniques – permutation tests and bootstrapping. We begin with permutation tests and bootstrap methods before introducing classical inference methods. Resampling helps students understand the meaning of sampling distributions, sampling variability, ‐values, hypothesis tests, and confidence intervals. We are inspired by the textbooks of Wardrop (1995) and Chance and Rossman (2005), two innovative introductory statistics books which also take a non‐traditional approach in the sequencing of topics.

We believe the time is ripe for this book. Many faculty have learned resampling and simulation‐based methods in graduate school and/or use them in their own work, and are eager to incorporate these ideas into a mathematical statistics course. Students and faculty today have access to computers that are powerful enough to perform resampling quickly.

A major topic of debate about the Mathematical Statistics course is how much theory to introduce. We want mathematically talented students to get excited about statistics, so we try to strike a balance between theory, computing, and applications. We feel that it is important to demonstrate some rigor in developing some of the statistical ideas presented here, but that mathematical theory should not dominate the text. To keep the size of the text reasonable, we omit some topics such as sufficiency and Fisher information (though we plan to make some omitted topics available as supplements on the text web page https://github.com/lchihara/MathStatsResamplingR).

We have compiled the definitions and theorems of the important probability distributions into an appendix (see Appendix B). Instructors who want to prove results on distributional theory can refer to that chapter. Instructors who wish to skip the theory can continue without interrupting the flow of the statistical discussion.

Incorporating resampling and bootstrapping methods requires that students use statistical software. We use R or RStudio because they are freely available (www.r-project.org or http://rstudio.com), powerful, flexible, and a valuable tool in future careers. One of us worked at Google where there was an explosion in the use of R, with more and more non‐statisticians learning R (the statisticians already know it). We realize that the learning curve for R is high, but believe that the time invested in mastering R is worth the effort. We have written some basic materials on R that are available on the website for this text. We recommend that instructors work through the introductory worksheet with the students on the first or second day of the term, in a computer lab if possible.

For the third edition, we decided to incorporate the packages in Hadley Wickham's tidyverse, including ggplot2. And though some R packages exist that implement some of the bootstrap and permutation algorithms that we teach, we felt that students understand and internalize the concepts better if they are required to write the code themselves. We do provide R scripts or R Markdown files with code on our website, and we may include alternate coding using some of the many R packages available.

Statistical computing is necessary in statistical practice and for people working with data in a wide variety of fields. There is an explosion of data – more and more data – and new computational methods are continuously being developed to handle this explosion. Statistics is an exciting field, dare we even say sexy?1

Third Edition: The issue of ‐values has generated much discussion in the statistics community (Wasserstein and Lazar, 2016; Wasserstein et al., 2019), and has even made it into the popular press (Resnick, 2019). In light of this controversy, one of the major changes we have made in this edition is to move away from the term “statistically significant” in favor of “statistically discernible.” We discuss this in Section 8.5 (Interpreting Test Results).

As noted above, we also updated the R code, using the ggplot2 package for plots instead of base R. We've added sections on cause‐and‐effect, control variates, expanded on stratification (Chapter 13), moved the section on the delta method from Chapter 13 to Chapter 7, updated the General Social Survey data set, included more examples, and developed more exercises. Through‐out the text, we have updated, clarified or made small changes to the exposition.

Pathways: This textbook contains more than enough material for a one term undergraduate course. We have written the textbook in such a way that instructors can choose to skip or gloss over some sections if they wish to emphasize others. In some instances, we have labeled a section or subsection with an asterisk (*) to denote it as optional. For classes comprised primarily of students who have no statistics background, a possible sequence includes Chapters 1–10. For courses focused more on applications, instructors could omit, for example, Sections 7.2 (Confidence Intervals Using Pivots) and 8.6 (Likelihood Ratio Tests). For classes in which students come in with an introductory statistics course background, instructors could have students read the first two chapters on their own, beginning the course at Chapter 3. In this case, instructors may wish to spend more time on theory, Bayesian methods, the delta method in Chapter 7, or the topics in Chapter 13, including the parametric bootstrap, stratified sampling and importance sampling. These topics could also be assigned as final projects for an undergraduate course or a senior capstone thesis.

Acknowledgments: This textbook could not have been completed without the assistance of many colleagues and students. In particular, for the first edition, we would like to thank Professor Katherine St. Clair of Carleton College who bravely class‐tested an early (very!) rough draft in her Introduction to Statistical Inference class during Winter 2010. In addition, Professor Julie Legler of St. Olaf College adopted the manuscript in her Statistical Theory class for Fall 2010. Both instructors and students provided valuable feedback that improved the exposition and content of this textbook. For the second edition, we would like to thank Professors Elaine Newman (Sonoma State University), Eric Nordmoe (Kalamazoo College), Nick Horton (Amherst College), and Carleton College faculty Katie St. Clair, Andy Poppick and Adam Loy for their helpful comments. We thank Ed Lee of Google for the Mobile Ads data and explanation. For the third edition, we would like to thank Jeff Witmer (Oberlin) and Miles Ott (Johnson & Johnson) for insightful comments; Ruhan Zhang of Instacart for the ad evaluation example in Exercise 13.14; and Rachel Zhang, Nick Cooley, and Jeff Moulton of Instacart for the human evaluation framework Exercise 13.15. Finally, we thank Professor Albert Y. Kim (Smith College) who compiled the data sets into an R package (resampledata).

We would also like to thank Siyuan (Ernest) Liu and Chen (Daisy) Sun, two Carleton College students, for solving many of the exercises in the first edition and writing up the solutions with.

Finally, we are also grateful for the valuable assistance provided by the staff at Wiley, including Skyler Van Valkenburgh, Judit Anbu Hena, Isabella Proietti, and Kalli Schutea. For the first and second editions, we were helped by Jon Gurstelle, Amudhapriya Sivamurthy, Kshitija Iyer, Vishnu Narayanan, Kathleen Pagliaro, Steve Quigley, Sanchari Sill, Dean Gonzalez, and Jackie Palmieri.

Additional Resources: We will place additional materials including R scripts, data sets, tutorials, and errata at our github site

https://github.com/lchihara/MathStatsResamplingR.

Northfield MN

Laura M. Chihara

Seattle WA

Tim C. Hesterberg

April 2022

Note

1

Try googling “statistics sexy profession.”

1Data and Case Studies

Statistics is the art and science of collecting and analyzing data and understanding the nature of variability. Mathematics, especially probability, governs the underlying theory, but statistics is driven by applications to real problems.

In this chapter, we introduce several data sets that we will encounter throughout the text in the examples and exercises. These data sets are available in the R package resampledata3 or at the textbook website https://github.com/lchihara/MathStatsResamplingR.

1.1 Case Study: Flight Delays

If you have ever traveled by air, you probably have experienced the frustration of flight delays. The Bureau of Transportation Statistics maintains data on all aspects of air travel, including flight delays at departure and arrival.1

LaGuardia Airport (LGA) is one of three major airports that serves the New York City metropolitan area. In 2008, over 23 million passengers and over 375 000 planes flew in or out of LGA. United Airlines and American Airlines are two major airlines that schedule services at LGA. The data set FlightDelays contains information on all 4029 departures of these two airlines from LGA during May and June 2009 (Tables 1.1 and 1.2).

Table 1.1 Partial view of FlightDelays data.

Flight

Carrier

FlightNo

Destination

DepartTime

Day

1

UA

403

DEN

4–8 a.m.

Friday

2

UA

405

DEN

8–noon 

Friday

3

UA

409

DEN

4–8 p.m.

Friday

4

UA

511

ORD

8–noon 

Friday

Table 1.2 Variables in data set FlightDelays.

Variable

Description

Carrier

UA = United Airlines, AA = American Airlines

FlightNo

Flight number

Destination

Airport code

DepartTime

Scheduled departure time in 4 h intervals

Day

Day of week

Month

May or June

Delay

Minutes flight delayed (negative indicates early departure)

Delayed30

Departure delayed more than 30 min?

FlightLength

Length of time of flight (minutes)

Each row of the data set is an observation. Each column represents a variable – some characteristic that is obtained for each observation. For instance, on the first observation listed, the flight was a United Airlines plane, flight number 403, destined for Denver, and departing on Friday between 4 and 8 a.m. This data set consists of 4029 observations and 9 variables.

Questions we might ask include the following: Are flight delay times different between the two airlines? Are flight delay times different depending on the day of the week? Are flights scheduled in the morning less likely to be delayed by more than 15 min?

1.2 Case Study: Birth Weights of Babies

The birth weight of a baby is of interest to health officials since many studies have shown possible links between this weight and conditions in later life, such as obesity or diabetes. Researchers look for possible relationships between the birth weight of a baby and the age of the mother or whether or not she smoked cigarettes or drank alcohol during her pregnancy. The Centers for Disease Control and Prevention (CDC) maintains a database on all babies born in a given year,2 incorporating data provided by the US Department of Health and Human Services, the National Center for Health Statistics, and the Division of Vital Statistics. We will investigate different samples taken from the CDC's database of births.

One data set that we will investigate consists of a random sample of 1009 babies born in North Carolina during 2004 (Table 1.3). The babies in the sample had a gestation period of at least 37 weeks and were single births (i.e. not a twin or triplet).

Table 1.3 Variables in data set NCBirths2004.

Variable

Description

MothersAge

Mother's age

Smoker

Mother smoker or non‐smoker

Gender

Gender of baby

Weight

Weight at birth (grams)

Gestation

Gestation time (weeks)

In addition, we will also investigate a data set, Girls2004, consisting of a random sample of 40 baby girls born in Alaska and 40 baby girls born in Wyoming. These babies also had a gestation period of at least 37 weeks and were single births.

The data set TXBirths2004 contains a random sample of 1587 babies born in Texas in 2004. In this case, the sample was not restricted to single births, nor to a gestation period of at least 37 weeks. The numeric variable Number indicates whether the baby was a single birth, or one of a twin, triplet, and so on. The variable Multiple is a factor variable indicating whether or not the baby was a multiple birth.

1.3 Case Study: Verizon Repair Times

Verizon is the primary local telephone company (incumbent local exchange carrier (ILEC)) for a large area of the Eastern United States. As such, it is responsible for providing repair service for the customers of other telephone companies known as competing local exchange carriers (CLECs) in this region. Verizon is subject to fines if the repair times (the time it takes to fix a problem) for CLEC customers are substantially worse than those for Verizon customers.

The data set Verizon contains a sample of repair times for 1664 ILEC and 23 CLEC customers (Table 1.4). The mean repair times are 8.4 h for ILEC customers and 16.5 h for CLEC customers. Could a difference this large be easily explained by chance?

Table 1.4 Variables in data set Verizon.

Variable

Description

Time

Repair times (in hours)

Group

ILEC or CLEC

1.4 Case Study: Iowa Recidivism

When a person is released from prison, will he or she relapse into criminal behavior and be sent back? The state of Iowa tracks offenders over a 3‐year period, and records the number of days until recidivism for those who are readmitted to prison. The Department of Corrections uses this recidivism data to determine whether or not their strategies for preventing offenders from relapsing into criminal behavior are effective.

The data set Recidivism contains all offenders convicted of either a misdemeanor or felony who were released from an Iowa prison during the 2010 fiscal year (ending in June) (Table 1.5). There were 17 022 people released in that period, of whom 5386 were sent back to prison in the following 3 years (through the end of the 2013 fiscal year).3

The recidivism rate for those under the age of 25 years was 36.5% compared to 30.6% for those 25 years or older. Does this indicate a real difference in the behavior of those in these age groups, or could this be explained by chance variability?

Table 1.5 Variables in data set Iowa Recidivism.

Variable

Description

Gender

F, M

Age

Age at release: Under 25, 25–34, 35–44, 45–54, 55 and Older

Age25

Under 25, Over 25 (binary)

Offense

Original conviction: Felony or Misdemeanor

Recid

Recidivate? No, Yes

Type

New (crime), No Recidivism, Tech (technical violation,

such as a parole violation)

Days

Number of days to recidivism; NA if no recidivism

1.5 Sampling

In analyzing data, we need to determine whether the data represent a population or a sample. A population represents all the individual cases, whether they are babies, fish, cars, or coin flips. The data from the flight delays case study in Section 1.1 are all the flight departures of United Airlines and American Airlines out of LGA in May and June 2009; thus, this data set represents the population of all such flights. On the other hand, the North Carolina data set contains only a subset of 1009 births from over 100 000 births in North Carolina in 2004. In this case, we will want to know how representative statistics computed from this sample are for the entire population of North Carolina babies born in 2004.

Populations may be finite, such as births in 2004, or infinite, such as coin flips or births next year.

Throughout this book, we will talk about drawing random samples from a population. We will use capital letters (e.g. , , , and so on) to denote random variables and lower‐case letters (e.g. , , , and so on) to denote actual values or data.

There are many kinds of random samples. Strictly speaking, a “random sample” is any sample obtained using a random procedure. However, we use random sample to mean a sample of independent and identically distributed (i.i.d.) observations from the population, if the population is infinite.

For instance, suppose you toss a fair coin 20 times and consider each head a “success.” Then your sample consists of the random variables each a Bernoulli random variable with success probability 1/2. We use the notation , .

If the population of interest is finite , we can choose a random sample as follows: Label balls with the numbers and place them in an urn. Draw a ball at random, record its value and then replace the ball. Draw another ball at random, record its value, and replace. Continue until you have a sample . This is sampling with replacement. For instance, if and , then there are different samples of size 2 (where order matters). (Note: By “order matters” we do not imply that order matters in practice, rather we mean that we keep track of the order of the elements when enumerating samples. For instance, the set is different from .)

However, in most real situations, for example, in conducting surveys, we do not want to have the same person polled twice. So we would sample without replacement, in which case, we will not have independence. For instance, if you wish to draw a sample of size from a population of people, then the probability of any one person being selected is . However, after having chosen that first person, the probability of any one of the remaining people being chosen is now .

In cases where populations are very large compared to the sample size, calculations under sampling without replacement are reasonably approximated by calculations under sampling with replacement.

Example 1.1 Consider a population of 1000 people, 350 of whom are smokers, and the rest are nonsmokers. If you select 10 people at random but with replacement, then the probability that 4 are smokers is . If you select without replacement, then the probability is .

1.6 Parameters and Statistics

When discussing numeric information, we will want to distinguish between populations and samples.

Definition 1.1 A parameter is a (numerical) characteristic of a population or of a probability distribution.

A statistic is a (numerical) characteristic of data.

Any function of a parameter is also a parameter; any function of a statistic is also a statistic. When the statistic is computed from a random sample, it is itself random, and hence is a random variable.

Example 1.2 and are parameters of the normal distribution with pdf .

The variance and signal‐to‐noise ratio are also parameters.

Example 1.3 If are a random sample, then the mean is a statistic.

Example 1.4 Consider the population of all babies born in the United States in 2017. Let denote the average weight of all these babies. Then is a parameter. The average weight of a sample of 2500 babies born in that year is a statistic.

Example 1.5 If we consider the population of all adults in the United States today, the proportion who approve of the president's job performance is a parameter. The fraction who approve in any given sample is a statistic.

Example 1.6 The average weight of 1009 babies in the North Carolina case study in Section 1.2 is 3448.26 g. This average is a statistic.

Example 1.7 If we survey 1000 adults and find that intend to vote in the next presidential election, then is a statistic – it estimates the parameter , the proportion of all adults who intend to vote in the next election.

1.7 Case Study: General Social Survey

The General Social Survey (GSS) is a major survey that has tracked American demographics, characteristics, and views on social and cultural issues since the 1970s. It is conducted by the National Opinion Research Center (NORC) at the University of Chicago. Trained interviewers meet face to face with the adults chosen for the survey and question them for about 90 min in their homes.

The GSS case study includes the responses of 2348 participants selected in 2018 to a subset of the questions asked, listed in Table 1.6 (Smith et al., 2014). For example, one of the questions (Courts) asked whether the respondent thinks that the courts in their area deal too harshly or not harshly enough with criminals.

Table 1.6 Variables in data set GSS2018.

Variable

Description

Region

Interview location

GenderNow

Current gender

Age

Age of respondent (

Note

: 89 = 89 or older)

Marital

Marital status

Degree

Highest level of education

Employed

Respondent employed? (Yes = full/part‐time; No = unemployed, retired, in school, etc.)

Income

Respondents income

Polviews

Respondents political views

Pres16

Whom did you vote for in the 2016 presidential election?

DeathPenalty

Death penalty for murder?

Courts

Courts deal harshly with criminals?

Attend

Attendance at religious services

Postlife

Believe in life after death?

Happy

General happiness

Satfin

Satisfied with financial situation?

Energy

Amount of spending on alternative energy

One of the core variables that has been included in all GSS surveys since its inception is Sex with the values (Female/Male) coded by the interviewer. Starting in 2018, GSS added a new question, “What is your current gender?”, to reflect recent research on the nature of gender identity which indicates that gender is not binary.4 Definitions of terms and concepts are always changing over time so it is important to consider the context when analyzing data.

We will analyze the GSS data to investigate questions such as the following: Is there a relationship between a person's marital status and whom they voted for in the 2016 presidential election? Are people who live in certain regions happier? Are there educational differences in support for the death penalty? These data can be obtained with the GSS Data Explorer.5

1.8 Sample Surveys

“Who do you plan to vote for in the next presidential election?” “Would you purchase our product again in the future?” “Do you smoke cigarettes? If yes, how old were you when you first started?” Questions such as these are typical of sample surveys. Researchers want to know something about a population of individuals, whether they are registered voters, online shoppers, or American teenagers, but to poll every individual in the population – that is, to take a census – is impractical and costly. Thus, researchers will settle for a sample from the target population. But if, say, of those in your sample of 1000 adults intend to vote for candidate Wong in the next election, how close is this to the actual percentage who will vote for Wong? How can we be sure that this sample is truly representative of the population of all voters? We will learn techniques for statistical inference, drawing a conclusion about a population based on information about a sample.

When conducting a survey, researchers will start with a sampling frame  – a list from which the researchers will choose their sample. For example, to survey all students at a college, the campus directory listing could be a sampling frame. For pre‐election surveys, many polling organizations use a sampling frame of registered voters. Note that the choice of sampling frame could introduce the problem of undercoverage: omitting people from the target population in the survey. For instance, young people were missed in many pre‐election surveys during the 2008 Obama–McCain presidential race because they had not yet registered to vote.

Once the researchers have a sampling frame, they will then draw a random sample from this frame. Researchers will use some type of probability (scientific) sampling scheme, that is, a scheme that gives everybody in the population a positive chance of being selected. For example, to obtain a sample of size 10 from a population of 100 individuals, write each person's name on a slip of paper, put the slips of paper into a basket, and then draw out 10 slips of paper. Nowadays, statistical software is used to draw random samples from a sampling frame.

Another basic survey design uses stratified sampling: The population is divided into nonoverlapping strata, and then random samples are drawn from each stratum. The idea is to group individuals who are similar in some characteristic into homogeneous groups, thus reducing variability. For instance, in a survey of university students, a researcher might divide the students by class: first year, sophomores, juniors, seniors, and graduate students. A market analyst for an electronics store might choose to stratify customers based on income levels.

In cluster sampling, the population is divided into nonoverlapping clusters, and then a random sample of clusters is drawn. Every person in a chosen cluster is then interviewed for the survey. An airport wanting to conduct a customer satisfaction survey might use a sampling frame of all flights scheduled to depart from the airport on a certain day. A random sample of flights (clusters) is chosen, and then all passengers on these flights are surveyed. A modification of this design might involve sampling in stages: For instance, the analysts might first choose a random sample of flights, and then from each flight choose a random sample of passengers.

The GSS uses a more complex sampling scheme in which the sampling frame is a list of counties and county equivalents (standard metropolitan statistical areas) in the United States. These counties are stratified by region, age, and race. Once a sample of counties is obtained, a sample of block groups and enumeration districts is selected, stratifying these by race and income. The next stage is to randomly select blocks and then interview a specific number of men and women who live within these blocks.

Indeed, all major polling organizations such as Gallup or Roper as well as the GSS use a multistage sampling design. In this book, we use the GSS data or polling results for examples as if the survey design used simple random sampling. Calculations for more complex sampling scheme are beyond the scope of this book, and we refer the interested reader to Lohr (1991) for details.

1.9 Case Study: Beer and Hot Wings

Carleton student Nicki Catchpole conducted a study of hot wings and beer consumption at the Williams Bar in the Uptown area of Minneapolis (N. Catchpole, private communication). She asked patrons at the bar to record their consumption of hot wings and beer over the course of several hours. She wanted to know if people who ate more hot wings would then drink more beer. In addition, she asked each person their gender to investigate whether or not gender had an impact on hot wings or beer consumption.

Table 1.7 Variables in data set Beerwings.

Variable

Description

Gender

Male or female

Beer

Ounces of beer consumed

Hotwings

Number of hot wings eaten

The data for this study are in Beerwings (Table 1.7). There are 30 observations and 3 variables.

1.10 Case Study: Black Spruce Seedlings

Black spruce (Picea mariana) is a species of a slow‐growing coniferous tree found across the northern part of North America. It is commonly found on wet organic soils. In a study conducted in the 1990s, a biologist interested in factors affecting the growth of the black spruce planted its seedlings on sites located in boreal peatlands in northern Manitoba, Canada (Camill et al., 2010).

The data set Spruce contains a part of the data from the study (Table 1.8). Seventy‐two black spruce seedlings were planted in four plots under varying conditions (fertilizer–no fertilizer, competition–no competition), and their heights and diameters were measured over the course of 5 years.

Table 1.8 Variables in data set Spruce.

Variable

Description

Tree

Tree number

Competition

C (competition), CR (competition removed)

Fertilizer

F (fertilized), NF (not fertilized)

Height0

Height (cm) of seedling at planting

Height5

Height (cm) of seedling at year 5

Diameter0

Diameter (cm) of seedling at planting

Diameter5

Diameter (cm) of seedling at year 5

Ht.change

Change (cm) in height

Di.change

Change (cm) in diameter

The researcher wanted to see whether the addition of fertilizer or the removal of competition from other plants (by weeding) affected the growth of these seedlings.

1.11 Studies

Researchers carry out studies to understand the conditions and causes of certain outcomes: Does smoking cause lung cancer? Do teenagers who smoke marijuana tend to move on to harder drugs? Do males eat more hot wings than females? Do black spruce seedlings grow taller in fertilized plots?

The beer and hot wings case study in Section 1.9 is an example of an observational study, a study in which researchers observe participants but do not influence the outcome. In this case, the student just recorded the gender and the number of hot wings eaten and beer consumed as reported by these patrons in the Williams bar.

Example 1.8 The first Nurses' Health Study is a major observational study funded by the National Institutes of Health. Over 120 000 registered female nurses who were married, between the ages of 33 and 55 years, and lived in the 11 most populous states (all in 1976) have been responding every 2 years to written questions about their health and lifestyle, including smoking habits, hormone use, and menopause status. Many results on women's health have come out of this study, such as finding an association between taking estrogen after menopause and lowering the risk of heart disease, and determining that for nonsmokers there is no link between taking birth control pills and developing heart disease.

Because this is an observational study, no cause and effect conclusions can be drawn. For instance, we cannot state that taking estrogen after menopause will cause a lowering of the risk for heart disease. In an observational study, there may be many unrecorded or hidden factors that impact the outcomes. Also, because the participants in this study are registered nurses, we need to be careful about making inferences about the general female population. Nurses are more educated and more aware of health issues than the average person.

On the other hand, the black spruce case study in Section 1.10 was an experiment. In an experiment, researchers will manipulate the environment in some way to observe the response of the objects of interest (people, mice, ball bearings, etc.). When the objects of interest in an experiment are people, we refer to them as subjects; otherwise, we call them experimental units. In this case, the biologist randomly assigned the experimental units – the seedlings – to plots subject to four treatments: fertilization with competition, fertilization without competition, no fertilization with competition, and no fertilization with no competition. He then recorded their height over a period of several years.

A key feature in this experiment was the random assignment of the seedlings to the treatments. The idea is to spread out the effects of unknown or uncontrollable factors that might introduce unwanted variability into the results. For instance, if the biologist had planted all the seedlings obtained from one particular nursery in the fertilized, no competition plot and subsequently recorded that these seedlings grew the least, then he would not be able to discern whether this was due to this particular treatment or due to some possible problem with seedlings from this nursery. With random assignment of treatments, the seedlings from this particular nursery would usually be spread out over the four treatments. Thus, the differences between the treatment groups should be due to the treatments (or chance).

Example 1.9 Knee osteoarthritis (OA