Foundations and Applications of Statistics simultaneously emphasizes both the foundational and the computational aspects of modern statistics. Engaging and accessible, this book is useful to undergraduate students with a wide range of backgrounds and career goals. The exposition immediately begins with statistics, presenting concepts and results from probability along the way. Hypothesis testing is introduced very early, and the motivation for several probability distributions comes from p-value computations. Pruim develops the students' practical statistical reasoning through explicit examples and through numerical and graphical summaries of data that allow intuitive inferences before introducing the formal machinery. The topics have been selected to reflect the current practice in statistics, where computation is an indispensible tool. In this vein, the statistical computing environment $\mathsf{R}$ is used throughout the text and is integral to the exposition. Attention is paid to developing students' mathematical and computational skills as well as their statistical reasoning. Linear models, such as regression and ANOVA, are treated with explicit reference to the underlying linear algebra, which is motivated geometrically. Foundations and Applications of Statistics discusses both the mathematical theory underlying statistics and practical applications that make it a powerful tool across disciplines. The book contains ample material for a two-semester course in undergraduate probability and statistics. A one-semester course based on the book will cover hypothesis testing and confidence intervals for the most common situations. In the second edition, the $\mathsf{R}$ code has been updated throughout to take advantage of new $\mathsf{R}$ packages and to illustrate better coding style. New sections have been added covering bootstrap methods, multinomial and multivariate normal distributions, the delta method, numerical methods for Bayesian inference, and nonlinear least squares. Also, the use of matrix algebra has been expanded, but remains optional, providing instructors with more options regarding the amount of linear algebra required.
Author(s): Randall Pruim
Series: Pure and Applied Undergraduate Texts 28
Edition: 2
Publisher: American Mathematical Society
Year: 2018
Language: English
Pages: 842
Cover......Page 1
Title page......Page 2
Contents......Page 4
Preface to the Second Edition......Page 8
Acknowledgments......Page 9
Preface to the First Edition......Page 12
What Is Statistics?......Page 18
Chapter 1. Data......Page 22
1.1. Data Frames......Page 23
1.2. Graphical and Numerical Summaries Data......Page 26
1.3. Summary......Page 47
Exercises......Page 49
Chapter 2. Probability and Random Variables......Page 54
2.1. Introduction to Probability......Page 55
2.2. Additional Probability Rules and Counting Methods......Page 60
2.3. Discrete Distributions......Page 78
2.4. Hypothesis Tests and p-Values......Page 87
2.5. Mean and Variance of a Discrete Random Variable......Page 95
2.6. Joint Distributions......Page 103
2.7. Other Discrete Distributions......Page 113
2.8. Summary......Page 128
Exercises......Page 134
3.1. pdfs and cdfs......Page 152
3.2. Mean and Variance......Page 166
3.3. Higher Moments......Page 168
3.4. Other Continuous Distributions......Page 176
3.5. Kernel Density Estimation......Page 188
3.6. Quantile-Quantile Plots......Page 194
3.7. Exponential Families......Page 201
3.8. Joint Distributions......Page 204
3.9. Multivariate Normal Distributions......Page 216
3.10. Summary......Page 231
Exercises......Page 235
4.1. Statistical Models......Page 246
4.2. Fitting Models by the Method of Moments......Page 248
4.3. Estimators and Sampling Distributions......Page 255
4.4. Limit Theorems......Page 266
4.5. Inference for the Mean (Variance Known)......Page 274
4.6. Estimating Variance......Page 283
4.7. Inference for the Mean (Variance Unknown)......Page 289
4.8. Confidence Intervals for a Proportion......Page 300
4.9. Paired Tests......Page 304
4.10. Developing New Hypothesis Tests......Page 309
4.11. The Bootstrap......Page 322
4.12. The Delta Method......Page 338
4.13. Summary......Page 348
Exercises......Page 352
5.1. Maximum Likelihood Estimators......Page 368
5.2. Numerical Maximum Likelihood Methods......Page 377
5.3. Likelihood Ratio Tests in One-Parameter Models......Page 390
5.4. Confidence Intervals in One-Parameter Models......Page 400
5.5. Inference in Models with Multiple Parameters......Page 407
5.6. Goodness of Fit Testing......Page 411
5.7. Inference for Two-Way Tables......Page 426
5.8. Rating and Ranking Based on Pairwise Comparisons......Page 436
5.9. Bayesian Inference......Page 445
5.10. Summary......Page 462
Exercises......Page 466
Chapter 6. Introduction to Linear Models......Page 476
6.1. The Linear Model Framework......Page 477
6.2. Parameter Estimation for Linear Models......Page 483
6.3. Simple Linear Regression......Page 486
6.4. Inference for Simple Linear Regression......Page 501
6.5. Regression Diagnostics......Page 515
6.6. Transformations in Linear Regression......Page 526
6.7. Categorical Predictors......Page 534
6.8. Categorical Response (Logistic Regression)......Page 543
6.9. Simulating Linear Models to Check Robustness......Page 555
6.10. Summary......Page 559
Exercises......Page 563
7.1. The Multiple Quantitative Predictors......Page 574
7.2. Assessing the Quality of a Model......Page 596
7.3. One-Way ANOVA......Page 620
7.4. Two-Way ANOVA......Page 655
7.5. Model Selection......Page 669
7.6. More Examples......Page 677
7.7. Permutation Tests......Page 689
7.8. Non-linear Least Squares......Page 693
7.9. Summary......Page 701
Exercises......Page 705
A.1. Getting Up and Running......Page 718
A.2. Getting Data into \R......Page 727
A.3. Saving Data......Page 733
A.4. Transforming Data with dplyr and tidyr......Page 734
A.5. Primary \R Data Structures......Page 746
A.6. Functions in \R......Page 756
A.7. ggformula Graphics......Page 759
Exercises......Page 766
Appendix B. Some Mathematical Preliminaries......Page 770
B.1. Sets......Page 771
B.2. Functions......Page 773
B.3. Sums and Products......Page 774
Exercises......Page 776
C.1. Vectors, Spans, and Bases......Page 780
C.2. Dot Products and Projections......Page 784
C.3. Orthonormal Bases......Page 791
C.4. Matrices......Page 792
Exercises......Page 800
Hints, Answers, and Solutions to Selected Exercises......Page 804
Bibliography......Page 820
Index to R Functions, Packages, and Data Sets......Page 828
Index......Page 834
Back Cover......Page 842
