SCHAUM'S OUTLINE OF THEORY AND PROBLEMS OF BEGINNING

380 Pages·2006·8.93 MB·English

SCHAUM’S OUTLINE OF
THEORY AND PROBLEMS
OF
BEGINNING STATISTICS
LARRY J. STEPHENS, Ph.D.
Professor of Mathematics
University of Nebrasku at Oriialin
SCHAUM’S OUTLINE SERIES
McGRAW-HILL
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To My Mother and Father, Rosie, and Johnie Stephens
LARRY J. STEPHENS is Professor of Mathematics at the University of Nebraska at
Omaha. He received his bachelor’s degree from Memphis State University in Mathematics,
his master’s degree from the University of Arizona in Mathematics, and his Ph.D. degree
from Oklahoma State University in Statistics. Professor Stephens has over 40 publications in
professional journals. He has over 25 years of experience teaching Statistics. He has taught at
the University of Arizona, Christian Brothers College, Gonzaga University, Oklahoma State
University, the University of Nebraska at Kearney, and the University of Nebraska at Omaha.
He has published numerous computerized test banks to accompany elementary Statistics texts.
He has worked for NASA, Livermore Radiation Laboratory, and Los Alamos Laboratory.
Since 1989, Dr. Stephens has consulted with and conducted Statistics seminars for the
engineering group at 3M, Valley, Nebraska plant.
Schaurn’s Outline of Theory and Problems of
BEGlNNlNG STATISTICS
Copyright 0 1998 by the McGraw-Hill Companies, Inc. All rights reserved. Printed in the United
States of Atnerica. Except as permitted under the Copyright Act of 1976, no part of this publication
tna he reproduced or distributed in any form or by any means, or stored in a data base or retrieval
system, without the prior written permission of the publisher.
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ISBN 0-07-06 1259-5
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Library of Congress Cataloging-in-PublicationD ata
Stephens. Larry J.
Schaum’s outline of theory and problems of beginning statistics /
Larry J. Stephens.
p. cm. - (Schaurn’s outline series)
Includes index.
ISBN 0-07-06 1259-5 (pbk.)
I. Mathematical statistics-Outlines, syllabi, etc.
2. Mathematical statistics-Problems, exercises, etc. I. Title.
11. Series.
QA276.19.S74 1998
519.5’0764~21 97-45979
CIP
AC
Preface
Statistics is a required course for undergraduate college students in a number of majors. Students in the
following disciplines are often required to take a course in beginning statistics: allied health careers, biology,
business, computer science, criminal justice, decision science, engineering, education, geography, geology,
information science, nursing, nutrition, medicine, pharmacy, psychology, and public administration. This
outline is intended to assist these students in the understanding of Statistics. The outline may be used as a
supplement to textbooks used in these courses or a text for the course itself.
The author has taught such courses for over 25 years and understands the difficulty students encounter with
statistics. I have included examples from a wide variety of current areas of application in order to motivate an
interest in learning statistics. As we leave the twentieth century and enter the twenty-first century, an
understanding of statistics is essential in understanding new technology, world affairs, and the ever-expanding
volume of knowledge. Statistical concepts are encountered in television and radio broadcasting, as well as in
magazines and newspapers. Modern newspapers, such as USA Today, are full of statistical information. The
sports section is filled with descriptive statistics concerning players and teams performance. The money section
of USA Today contains descriptive statistics concerning stocks and mutual funds. The life section of USA Today
often contains summaries of research studies in medicine. An understanding of statistics is helpful in evaluating
these research summaries.
The nature of the beginning statistics course has changed drastically in the past 30 or so years. This change
is due to the technical advances in computing. Prior to the 1960s statistical computing was usually performed
on mechanical calculators. These were large cumbersome computing devices (compared to today’s hand-held
calculators) that performed arithmetic by moving mechanical parts. Computers and computer software were no
comparison to today’s computers and software. The number of statistical packages available today numbers in
the hundreds. The burden of statistical computing has been reduced to simply entering your data into a data file
and then giving the correct command to perform the statistical method of interest.
One of the most widely used statistical packages in academia as well as industrial settings is the package
called Minitab (Minitab Inc., 3081 Enterprise Drive, State College, PA 16801-3008). I wish to thank Minitab
Inc. for granting me permission to include Minitab output, including graphics, throughout the text. Most
modern Statistics textbooks include computer software as part of the text. I have chosen to include Minitab
because it is widely used and is very friendly. Once a student learns the various data file structures needed to
use Minitab, and the structure of the commands and subcommands, this knowledge is readily transferable to
other statistical software.
The outline contains all the topics, and more, covered in a beginning statistics course. The only
mathematical prerequisite needed for the material found in the outline is arithmetic and some basic algebra. I
wish to thank my wife, Lana, for her understanding during the preparation of the book. I wish to thank my
friend Stanley Wileman for all the computer help he has given me during the preparation of the book. I wish to
thank Dr. Edwin C. Hackleman of Delta Software, Inc. for his timely assistance as compositor of the final
camera-ready manuscript. Finally, I wish to thank the staff at McGraw-Hill for their cooperation and
helpfulness.
LARRYJ . STEPHENS
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Contents
Chapter 1 INTRODUCTION ................................................................................. 1
Statistics. Descriptive Statistics. Inferential Statistics: Population and
Sample. Variable, Observation. and Data Set. Quantitative Variable:
Discrete and Continuous Variable. Qualitative Variable. Nominal,
Ordinal, Interval, and Ratio Levels of Measurement. Summation Notation.
Computers and Statistics.
Chapter 2 ORGANIZING DATA .......................................................................... 14
Raw Data. Frequency Distribution for Qualitative Data. Relative
Frequency of a Category. Percentage. Bar Graph. Pie Chart.
Frequency Distribution for Quantitative Data. Class Limits, Class
Boundaries, Class Marks, and Class Width. Single-Valued Classes.
Histograms. Cumulative Frequency Distributions. Cumulative Relative
Frequency Distributions. Ogives. Stem-and-Leaf Displays.
Chapter 3 DESCRIPTIVE MEASURES .............................................................. 40
Measures of Central Tendency. Mean, Median, and Mode for Ungrouped
Data. Measures of Dispersion. Range, Variance, and Standard Deviation
for Ungrouped Data. Measures of Central Tendency and Dispersion for
Grouped Data. Chebyshev’s Theorem. Empirical Rule. Coefficient of
Variation. Z Scores. Measures of Position: Percentiles, Deciles, and
Quartiles. Interquartile Range. Box-and-Whisker Plot.
Chapter 4 PROBABILITY ..................................................................................... 63
Experiment, Outcomes, and Sample Space. Tree Diagrams and the
Counting Rule. Events, Simple Events, and Compound Events. Probability.
Classical, Relative Frequency and Subjective Probability Definitions.
Marginal and Conditional Probabilities. Mutually Exclusive Events.
Dependent and Independent Events. Complementary Events.
Multiplication Rule for the Intersection of Events. Addition Rule for the
Union of Events. Bayes’ Theorem. Permutations and Combinations. Using
Permutations and Combinations to Solve Probability Problems.
Chapter 5 DISCRETE RANDOM VARIABLES ................................................ 89
Random Variable. Discrete Random Variable. Continuous Random
Variable. Probability Distribution. Mean of a Discrete Random Variable.
Standard Deviation of a Discrete Random Variable. Binomial Random
Variable. Binomial Probability Formula. Tables of the Binomial
Distribution. Mean and Standard Deviation of a Binomial Random
Variable. Poisson Random Variable. Poisson Probability Formula.
Hypergeome tric Random Variable. Hypergeometric Probability Formula.
V
vi CONTENTS
Chapter 6 CONTINUOUS RANDOM VARIABLES AND THEIR
PROBABILITY DISTRIBUTIONS. ................................................... 113
Uniform Probability Distribution. Mean and Standard Deviation for the
Uniform Probability Distribution. Normal Probability Distribution.
Standard Normal Distribution. Standardizing a Normal Distribution.
Applications of the Normal Distribution. Determining the z and x Values
When an Area under the Normal Curve is Known. Normal Approximation
to the Binomial Distribution. Exponential Probability Distribution.
Probabilities for the Exponential Probability Distribution.
Chapter 7 SAMPLING DISTRIBUTIONS .......................................................... 140
Simple Random Sampling. Using Random Number Tables. Using the
Computer to Obtain a Simple Random Sample. Systematic Random
Sampling. Cluster Sampling. Stratified Sampling. Sampling Distribution
of the Sampling Mean. Sampling Error. Mean and Standard Deviation of the
Sample Mean. Shape of the Sampling Distribution of the Sample Mean
and the Central Limit Theorem. Applications of the Sampling Distribution
of the Sample Mean. Sampling Distribution of the Sample Proportion.
Mean and Standard Deviation of the Sample Proportion. Shape of the
Sampling Distribution of the Sample Proportion and the Central Limit
Theorem. Applications of the Sampling Distribution of the Sample Proportion.
Chapter 8 ESTIMATION AND SAMPLE SIZE DETERMINATION:
ONE POPULATION ........................................................................... 166
Point Estimate. Interval Estimate. Confidence Interval for the Population
Mean: Large Samples. Maximum Error of Estimate for the Population
Mean. The t Distribution. Confidence Interval for the Population Mean:
Small Samples. Confidence Interval for the Population Proportion: Large
Samples. Determining the Sample Size for the Estimation of the
Population Mean. Determining the Sample Size for the Estimation of the
Population Proportion.
Chapter 9 TESTS OF HYPOTHESIS: ONE POPULATION ............................ 185
Null Hypothesis and Alternative Hypothesis. Test Statistic, Critical
Values, Rejection and Nonrejection Regions.Type I and Type I1 Errors.
Hypothesis Tests about a Population Mean: Large Samples. Calculating
Type I1 Errors. P Values. Hypothesis Tests about a Population Mean:
Small Samples. Hypothesis Tests about a Population Proportion: Large
Samples.
CONTENTS vii
Chapter 10 INFERENCES FOR TWO POPULATIONS ..................................... 211
x,
Sampling Distribution of - for Large Independent Samples.
Estimation of p1- p2 Using Large Independent Samples. Testing
Hypothesis about pI- p, Using Large Independent Samples.
-x2
Sampling Distribution of XI for Small Independent Samples
from Normal Populations with Equal (but unknown) Standard
Deviations. Estimation of p1- p, Using Small Independent Samples froin
Normal Populations with Equal (but unknown) Standard Deviations.
Testing Hypothesis about p, - p, Using Small Independent
Samples from Normal Populations with Equal (but Unknown) Standard
x2
Deviations. Sampling Distribution of XI- for Small Independent Samples
from Normal Populations with Unequal (and Unknown) Standard Deviations.
Estimation of p, - p2 Using Small Independent Samples from Normal
Populations with Unequal (and Unknown) Standard Deviations. Testing
Hypothesis about p1- p2 Using Small Independent Samples from Normal
Populations with Unequal (and Unknown) Standard Deviations.
a
Sampling Distribution of for Normally Distributed Differences
Computed from Dependent Samples. Estimation of pd Using Normally
Distributed Differences Computed from Dependent Samples. Testing
Hypothesis about pd Using Normally Distributed Differences Computed
from Dependent Samples. Sampling Distribution of PI- p2 for Large Independent
Samples. Estimation of PI- P2 Using Large Independent Samples. Testing
Hypothesis about PI- P2 Using Large Independent Samplcs.
Chapter 11 CHI-SQUARE PROCEDURES. .......................................................... 249
Chi-square Distribution. Chi-square Tables. Goodness-of-Fit Test.
Observed and Expected Frequencies. Sampling Distribution of the
Goodness-of-Fit Test Statistic. Chi-square Independence Test.
Sampling Distribution of the Test Statistic for the Chi-square
Independence Test. Sampling Distribution of the Sample Variance.
Inferences Concerning the Population Variance.
Chapter 12 ANALYSIS OF VARIANCE (ANOVA) ............................................. 272
F Distribution. F Table. Logic Behind a One-way ANOVA. Sum of
Squares, Mean Squares, and Degrees of Freedom for a One-way ANOVA.
Sampling Distribution for the One-way ANOVA Test Statistic. Building
One-way ANOVA Tables and Testing the Equality of Means. Logic
Behind a Two-way ANOVA. Sum of Squares, Mean Squares, and Degrees
of Freedom for a Two-way ANOVA. Building Two-way ANOVA Tables.
Sampling Distributions for the Two-way ANOVA. Testing Hypothesis
Concerning Main Effects and Interaction.
... -~
CONTENTS
Vlll
Chapter 13 REGRESSION AND CORRELATION. ............................................. 309
Straight Lines. Linear Regression Model. Least Squares Line. Error Sum
of Squares. Standard Deviation of Errors. Total Sum of Squares.
Regression Sum of Squares. Coefficient of Determination. hiean, Standard
Deviaticjii, and Sampling Distribution of the Slope of the Estimated Regression
Equation. Inferences Concerning the Slope of the Population Regression Line.
Estimation and Prediction in Linear Regression. Linear Correlation
Coefficient. Inference Concerning the Population Correlation Coefficient.
Chapter 14 NONPARAMETRIC STATISTICS... ................................................. 334
NonparametricMethods. Sign Test. Wilcoxon Signed-Ranks Test for Two
Dependent Samples. Wilcoxon Rank-Sum Test for Two Independent Samples.
Kruskal-Wall6 Test. Rank Correlation. Runs Test for Randomness.
APPENDIX
1 Binomial Probabilities ....................................................................................................................... 359
2 Areas under the Standard Normal Curve from 0 to 2 ....................................................................... 364
~~-
3 Area in the Right Tail under the t Distribution Curve ....................................................................... 365
~
...
4 Area in the Right Tail under the Chi-square Distribution Curve ....................................................... 366
~
5 Area in the Right Tail under the F Distribution Curve ...................................................................... 367
~
INDEX .................................................................................................................................................... 369
-
Chapter 1
Introduction
STATISTICS
Statistics is a discipline of study dealing with the collection, analysis, interpretation, and
presentation of data. Statistical methodology is utilized by pollsters who sample our opinions
concerning topics ranging from art to zoology. Statistical methodology is also utilized by business
and industry to help control the quality of goods and services that they produce. Social scientists and
psychologists use statistical methodology to study our behaviors. Because of its broad range of
applicability, a course in statistics is required of majors in disciplines such as sociology, psychology,
criminal justice, nursing, exercise science, pharmacy, education, and many others. To accommodate
this diverse group of users, examples and problems in this outline are chosen from many different
sources.
DESCRIPTIVE STATISTICS
The use of graphs, charts, and tables and the calculation of various statistical measures to
organize and summarize information is called descriptive statistics. Descriptive statistics help to
reduce our information to a manageable size and put it into focus.
EXAMPLE 1.1 The compilation of batting average, runs batted in, runs scored, and number of home runs for
each player, as well as earned run average, wordlost percentage, number of saves, etc., for each pitcher from the
official score sheets for major league baseball players is an example of descriptive statistics. These statistical
measures allow us to compare players, determine whether a player is having an “off year” or “good year,” etc.
EXAMPLE 1.2 The publication entitled Crime in the United States published by the Federal Bureau of
Investigation gives summary information concerning various crimes for the United States. The statistical
measures given in this publication are also examples of descriptive statistics and they are useful to individuals in
law enforcement.
INFERENTIAL STATISTICS: POPULATION AND SAMPLE
The complete collection of individuals, items, or data under consideration in a statistical study
is referred to as the pupulatiurz. The portion of the population selected for analysis is called the
sample. Inferential statistics consists of techniques for reaching conclusions about a population
based upon information contained in a sample.
EXAMPLE 1.3 The results of polls are widely reported by both the written and the electronic media. The
techniques of inferential statistics are widely utilized by pollsters. Table 1.1 gives several examples of
populations and samples encountered in polls reported by the media. The methods of inferential statistics are
used to make inferences about the populations based upon the results found in the samples and to give an
indication about the reliability of these inferences. The results of a poll of 600 registered voters might be
reported as follows: Forty percent of the voters approve of the president’s economic policies. The margin of
error for the survey is 4%. The survey indicates that an estimated 40% of all registered voters approve of the
economic policies, but it might be as low as 36% or as high as 44%.
1