For a one-semester course in Mathematical Statistics.
This innovative new introduction to Mathematical Statistics covers the important concept of estimation at a point much earlier than other texts (Chapter 2). Thought-provoking pedagogical aids help students test their understanding and relate concepts to everyday life. Ideal for courses that offer a little less probability than usual, this book requires one year of calculus as a prerequisite.
1.1 Basic Concepts
1.2 Methods of Enumeration
1.3 Conditional Probability
1.4 Independent Events
1.5 Bayes's Theorem
Chapter One Comments
2. Discrete Distributions
2.1 Discrete Probability Distributions
2.3 Special Discrete Distributions
2.5 Linear Functions of Independent Random Variables
2.6 Multivariate Discrete Distributions
Chapter Two Comments
3. Continuous Distributions
3.1 Descriptive Statistics and EDA
3.2 Continuous Probability Distributions
3.3 Special Continuous Distributions
3.4 The Normal Distribution
3.5 Estimation in the Continuous Case
3.6 The Central Limit Theorem
3.7 Approximations for Discrete Distributions
Chapter Three Comemnts
4. Applications of Statistical Inference
4.1 Summary of Necessary Theoretical Results
4.2 Confidence Intervals Using X2 F,and T
4.3 Confidence Intervals and Tests of Hypotheses
4.4 Basic Tests Concerning One Parameter
4.5 Tests of the Equality of Two Parameters
4.6 Simple Linear Regression
4.7 More on Linear Regression
4.8 One-Factor Analysis of Variance
4.9 Distribution-Free Confidence and Tolerance Intervals
4.10 Chi-Square Goodness of Fit Tests
4.11 Contingency Tables
Chapter Four Comments
5. Computer Oriented Techniques
5.1 Computation of Statistics
5.2 Computer Algebra Systems
Chapter Five Comments
6. Some Sampling Distribution Theory
6.1 Moment-Generation Function Technique
6.2 M.G.F of Linear Functions
6.3 Limiting Moment-Generating Functions
6.4 Use of Order Statistics in Non-regular Cases
Chapter Six Comments
Almost all of the content in Chapters 1-3 and Sections 4.1-4.6 will likely be covered in a one-semester course, allowing the Instructor to select topics from Sections 4.7-4.11 and Chapters 5 and 6 to complete the course.
• Content and Organization:
– Chapter 1: With a little algebra of sets and a standard course in calculus as the mathematical background, some basic probability is presented in this chapter.
– Chapter 2: Presents certain discrete distributions. Included are the topics of expectations, maximum likelihood estimation, as well as expectations and variances of linear functions, in particular those of the sample mean; and this makes it possible to introduce confidence intervals for means of distributions. Also included is a section on multivariate discrete distributions. (Early introduction of estimation is unique to this text.)
– Chapter 3: Focuses on the continuous case and corresponding estimation problems.
– Chapter 4: Includes some statistical inferences. Tests of statistical hypotheses and confidence intervals are tied together throughout. Material on linear regression is included. The section on distribution-free confidence intervals for percentiles also includes the topic of tolerance intervals. The last two sections concern chi-square tests.
– Chapter 5: Provides computer applications. This chapter is devoted to a discussion of some uses of the computer both for data analysis and also for theoretical solutions such as simulation and bootstrapping. This is illustrated using Minitab for data analysis and Maple for theoretical solutions, simulations, and bootstrapping. (Other computer packages and Computer Algebra Systems could be used.) More than 100 probability and statistics procedures have been written for Maple. These are stored as stat.m along with some additional supplementary procedures stored as text files in “Maple Examples” that is available in an online web page: www.prenhall.com/statistics. Several statistical applications of Maple are included here.
– Chapter 6: Introduces the moment-generating function. This allows the student to see how important theoretical results are proved. The Central Limit Theorem is explained and used in Chapters 3-6 with a proof of it given in this chapter. The use of order statistics in non-regular cases closes this chapter.
• End pages – include summaries of the most important aspects of discrete distributions, continuous distributions, confidence intervals, and tests of hypotheses.
• End-of-chapter Comments – Brief summaries provide some interesting aspects of probability and statistics.
• Abundant Exercises – The average number of exercises is 11 per section with many exercises having several parts, giving instructors a wide selection for each assignment.
• Wide range of fields used for examples, exercises, and applications:
– Includes biology, education, economics, engineering, environmental studies, exercise science, health science, manufacturing, opinion polls, psychology, sociology, and sports.
– In particular, the reader might be interested in those concerning insurance, Pap smear tests, estimating the number of whales in an ocean, fitting models, filling 12 ounce containers, environmental issues, and results in certain sporting events.
Elliot A. Tanis: Tanis has written 30 articles in probability and statistics, many illustrating applications using the computer. He has authored or co-authored four books in probability and statistics. These include “Probability & Statistics Explorations with MAPLE,” 2nd edition, with Zaven Karian in 1999 and “Probability and Statistical Inference,” 7th edition, with Robert V. Hogg in 2006. He was Chairperson (1976-77) and Governor (1989-92) of the Michigan Section of the Mathematical Association of America. He was a winner of the Hope’s Outstanding Professor Educator (H.O.P.E.) award in 1989 and received the award for Distinguished College or University Teaching of Mathematics, Michigan Section, MAA, in 1992. Tanis became Professor Emeritus of Mathematics at Hope College in 2000 after teaching there 35 years.
Robert V. Hogg: Hogg has written over 70 research articles and coauthored five books, including “Introduction of Mathematical Statistics,” 6th edition, with J. W. McKean and A.T. Craig, and “Probability and Statistical Inference,” 7th edition, with E.A. Tanis. He was President of the American Statistical Association in 1988 and was given the Founders and Noether awards of that society. Earlier he received an Outstanding Teacher award of the Mathematical Association of America and, in 2007, will receive the Carver Medal from the Institute of Mathematical Statistics. Hogg became Professor Emeritus of Statistics at the University of Iowa in 2001 after teaching there 51 years.