Probability and Statictics for Science and Engineering with Examples in R

337 Pages·010.065 MB·English

by  Hongshik Ahn

A
H
N
PROBABILITY STATISTICS
AND
WITH EXAMPLES IN R
FOR SCIENCE & ENGINEERING
Probability and Statistics for Science and Engineering with Examples in R teaches students how to
use R software to obtain summary statistics, calculate probabilities and quantiles, find confidence
P
intervals, and conduct statistical testing.
R
O
B
The first chapter introduces methods for describing statistics. Over the course of the subsequent
A
eight chapters students will learn about probability, discrete and continuous distributions, multiple
B
random variables, point estimation and testing, and inferences based on one and two samples. The I
L
book features a comprehensive table for each type of test to help students choose appropriate
I
T
statistical tests and confidence intervals. PROBABILITY STATISTICS
Y

A AND
Based on years of classroom experience and extensively class-tested, Probability and Statistics for ND

S
Science and Engineering with Examples in R is designed for one-semester courses in probability
T
and statistics, and specifically for students in the natural sciences or engineering. The material is A FOR SCIENCE & ENGINEERING WITH EXAMPLES IN R
also suitable for business and economics students who have studied calculus. T
I
S
E D I T I O N BY HONGSHIK AHN
T
Hongshik Ahn holds a Ph.D. in statistics from the University of Wisconsin, Madison. Dr. Ahn is
I
currently a professor in the Department of Applied Mathematics and Statistics at Stony Brook C
S
University. He worked at National Center for Toxicological Research, FDA before joining Stony Brook
University. Recently he served as the vice president of SUNY Korea. His research interests include
tree-structured regression and classification, bioinformatics, generalized linear modeling, and risk
assessment. Dr. Ahn has been working on NIH grants on various biostatistical and medical
researches. He has been published in three book chapters and over 60 peer-reviewed journals. Dr.
Ahn also published a book entitled Mathematical Analysis of Genesis, from Shinil Books.
c
o
g
n
e
ll
a
cognella® ACADEMIC ®
PUBLISHING SKU 81544-1C GPUBLISHINACADEMIC
FIRST EDITION
PROBABILITY
AND STATISTICS
FOR SCIENCE AND ENGINEERING
W I T H E X A M P L E S I N R
BY HONGSHIK AHN
Bassim Hamadeh, CEO and Publisher
Kassie Graves, Director of Acquisitions
Jamie Giganti, Senior Managing Editor
Miguel Macias, Senior Graphic Designer
John Remington, Senior Field Acquisitions Editor
Monika Dziamka, Project Editor
Brian Fahey, Senior Licensing Specialist
Christian Berk, Associate Editor
Copyright © 2017 by Cognella, Inc. All rights reserved. No part of this publication may be reprinted, reproduced,
transmitted, or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented,
including photocopying, microfilming, and recording, or in any information retrieval system without the written permis-
sion of Cognella, Inc.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for
identification and explanation without intent to infringe.
Cover image copyright © by Depositphotos / ikatod.
Printed in the United States of America
ISBN: 978-1-5165-1398-7 (pb) / 978-1-5165-1399-4 (br)
Contents
Preface ix
CH. 1 Describing Data 1
1. Display of Data by Graphs and Tables 1
2. Measures of Central Tendency 11
3. Measures of Variation 16
Summary of Chapter 1 27
Exercises 28
CH. 2 Probability 39
1. Sample Spaces and Events 39
2. Counting 43
3. Relative Frequency (Equally Likely Outcomes) 48
4. Probability 48
iii
iv COnTEnTS
5. Conditional Probability 54
6. Independence 56
7. Bayes’ Theorem 58
Summary of Chapter 2 60
Exercises 62
CH. 3 Discrete Distributions 71
1. Random Variables 71
2. Probability Distribution 72
3. The Mean and Variance of Discrete
Random Variables 77
4. The Binomial Distribution 81
5. The Hypergeometric Distribution 85
6. The Poisson Distribution 89
7. The Geometric Distribution 95
8. Chebyshev’s Inequality 97
9. The Multinomial Distribution 98
Summary of Chapter 3 99
Exercises 101
COnTEnTS v
CH. 4 Continuous Distributions 111
1. Probability Density 111
2. The Uniform Distribution 113
3. The Exponential Distribution 116
4. The Cumulative Distribution Function 117
5. Expectations 123
6. The normal Distribution 124
7. The Gamma Distribution 137
8. The Beta Distribution 139
Summary of Chapter 4 141
Exercises 144
CH. 5 Multiple Random Variables 153
1. Discrete Distributions 153
2. Continuous Distributions 156
3. Independent Random Variables 159
4. Conditional Distributions 160
5. Expectations 162

Summary of Chapter 5 169

Exercises 170
vi COnTEnTS
CH. 6 Sampling Distributions 177
1. Populations and Samples 177
2. Distribution of the Sample Mean When σ Is Known 179
3. Central Limit Theorem 182
4. Distribution of the Sample Mean for a normal
Population Whenσ Is Unknown 184
5. Sampling Distribution of the Variance 188
Summary of Chapter 6 192
Exercises 193
CH. 7 Introduction to Point Estimation and Testing 199
1. Point Estimation 199
2. Tests of Hypotheses 201
Summary of Chapter 7 208
Exercises 209
CH. 8 Inferences Based on One Sample 213
1. Inferences Concerning a Mean 213
2. Inferences Concerning a Population Proportion 228
3. Inferences Concerning a Variance 233
Summary of Chapter 8 236
Exercises 239
COnTEnTS vii
CH. 9 Inferences Based on Two Samples 249
1. Inferences Concerning Two Means 249
2. Inferences Concerning Two Population Proportions 267
3. Inferences Concerning Two Variances 272

Summary of Chapter 9 278

Exercises 280
Appendix 291
Answers to Selected Exercise Problems 317
Index 329
Preface
This book is designed for a one-semester course in probability and statistics, specifically for students
in the natural science or engineering. It is also suitable for business and economics students with a
calculus background. The text is based on my past teachings over the course of many years at Stony
Brook University. Most existing textbooks contain topics to fulfill at least one whole year of instruc-
tion, and therefore they are excessive for a one-semester course in probability and statistics. The
purpose of this book is to cover just the necessary topics for a one-semester course, thus reducing
the typical volume of a textbook and lowering the financial burden on students.
This book provides examples of how to use the R software to obtain summary statistics, calculate
probabilities and quantiles, find confidence intervals, and conduct statistical testing. In addition to
using distribution tables, students can calculate probabilities of various distributions using their
smartphones or computers. Since R is available under an open-source license, everyone can down-
load it and use it free of charge.
This book is organized as follows: Chapter 1 covers descriptive statistics, Chapters 2 through
Chapter 5 cover probability and distributions, Chapter 6 provides concepts about sampling, and
Chapters 7 through 9 cover estimations and hypothesis testing. Hypothesis testing can be over-
whelming for students, due to the inundating formulas needed for numerous cases. To help students
understand and identify the correct formula to use, a comprehensive table for each type of test is
provided. Students can easily follow these tables to choose the appropriate confidence intervals
and statistical tests. A summary is given at the end of each chapter. Distribution tables for various
distributions are provided in the appendix.
I wish to thank Dr. David Saltz for contributing some examples and exercise problems; Chelsea
Kennedy for proofreading the preliminary edition and checking the solutions to the exercise
problems; Yan Yu for proofreading the preliminary edition; Jerson Cochancela for proofreading the
manuscript, contributing an exercise problem, and providing helpful suggestions; and Mingshen
Chen for checking the solutions to the exercise problems. I also thank Cognella for inviting me to
write this book.
Hongshik Ahn
Professor
Department of Applied Mathematics and Statistics
Stony Brook University
1
Describing
Data
1. Display of Data by Graphs and Tables
There are various ways to describe data. In this section, we study how to organize and describe a set
of data using graphs and tables.
A. FREQUENCY DISTRIBUTIONS
A frequency distribution is a table that displays the frequency of observations in each interval in a
sample. To build a frequency distribution, we need the following steps.
Basic Steps
1. Find the minimum and the maximum values in the data set.
2. Determine class intervals: intervals or cells of equal length that cover the range between the
minimum and the maximum without overlapping
e.g., minimum 0, maximum 100: [0, 10), [10, 20), …, [90, 100]
3. Find frequency: the number of observations in the data that belong to each class interval. Let’s
denote the frequencies as f , f , .
1 2
4. Find relative frequency:
Class frequency

Total number of observations
The relative frequencies are denoted as f n, f n,  if the total sample size is n.
1 2
EXAMPLE 1.1 Midterm scores of an introductory statistics class of 20 students are given below.
69 84 52 93 81 74 89 85 88 63 87 64 67 72 74 55 82 91 68 77
1