Understanding Statistical Concepts Using S-plus

Randall E. Schumacker, Allen Akers

• Publisher: Psychology Press
• Year: 2001
• ISBN:
  9781135657031
• Edition: 1
• Topic: Psychology
• Subtopic: History & Theory in Psychology
• Index: Psychology

Written as a supplemental text for an introductory or intermediate statistics course, this book is organized along the lines of many popular statistics texts. The chapters provide a good conceptual understanding of basic statistics and include exercises that use S-PLUS simulation programs. Each chapter lists a set of objectives and a summary.

The book offers a rich insight into how probability has shaped statistical procedures in the behavioral sciences, as well as a brief history behind the creation of various statistics. Computational skills are kept to a minimum by including S-PLUS programs that run the exercises in the chapters. Students are not required to master the writing of S-PLUS programs, but explanations of how the programs work and program output are included in each chapter. S-PLUS is an advanced statistical package that has an extensive library of functions, which offer flexibility in writing customized routines.

The S-PLUS functions provide the capability of programming object and dialog windows, which are commonly used in Windows software applications. The S-PLUS program also contains pull-down menus for the statistical analysis of data.

A ZIP file containing programs that work in S-PLUS 6.2 for use with this book is available for download from http://www.psypress.com/resources/9780805836233.zip - please note that these scripts will only run in S-PLUS 6.2 and not later versions due to changes in the programming language syntax.

I
INTRODUCTION AND STATISTICAL THEORY

CHAPTER 1
STATISTICAL THEORY

CHAPTER OBJECTIVES

• To see how the computer can be used to generate random samples of data.

• To understand the difference between a population and a sample.

• To observe that different samples may lead to different sample estimates of population parameters.

• To see that most estimates from samples contain some error of estimation.

• To find a relationship between the size of a sample and the accuracy of the estimate.

The field of statistics uses numerical information obtained from samples to draw inferences about populations. A population is a well-defined set of individuals, events, or objects. A sample is a selection of individuals, events, or objects taken from a well-defined population. A sample is generally taken from a population with each individual, event, or object being independent and having an equally likely chance of selection. The sample average is an example of a random sample estimate of a population value, i.e., population mean. Population characteristics or parameters are inferred from sample estimates, which are called statistics. Examples of population parameters are population proportion, population mean, and population correlation. For example, a student wants to estimate the proportion of teachers in the state who are in favor of year-round school. The student might make the estimate on the basis of information received from a random sample of 500 teachers in the population comprised of all teachers in the state. In another example, a biologist wants to estimate the proportion of tree seeds that will germinate. The biologist plants 1,000 tree seeds and uses the germination rate to establish the rate for all seeds. In marketing research, the proportion of 1,000 randomly sampled consumers who buy one product rather than another helps advertising executives determine product appeal.

Because a sample is only a part of the population, how can the sample estimate accurately reflect the population characteristic? There is an expectation that the sample estimate will be close to the population value if the sample is representative of the population. The difference between the sample estimate and the population value is called sample error. In a random sample, all objects have an equal chance of being selected from the population. If the sample is reasonably large, this equally likely chance of any individual, event, or object being selected makes it likely that the random sample will represent the population well. Most statistics are based upon this concept of random sampling from a well-defined population. Sampling error, or the error in using a sample estimate as a population estimate does occur. In future chapters, you will learn that several random sample estimates can be averaged to better approximate a population value, although sampling error is still present.

In this chapter, you will use S-PLUS computer software and a STATISTICS program to simulate the sampling of data from a population. You are to determine what proportion of a certain large population of people favor stricter penalties. A random number generator will determine the responses of the people in the sample. A random number generator uses an initial start number to begin data selection, and then uses the computer to generate other numbers at random. You will use the results of these simulated random samples to draw conclusions about the population.

You will be asked to run the S-PLUS STATISTICS program five times. The program can be run more times if you like. Each time you run the program, the true population proportion will be different. Consequently, each time you will be simulating a sample data set from a different popul...

Table of contents

• TABLE OF CONTENTS
• PREFACE
• ACKNOWLEDGEMENTS
• I INTRODUCTION AND STATISTICAL THEORY
• II PROBABILITY AND PROBABILITY DISTRIBUTIONS
• III MONTE CARLO AND STATISTICAL DISTRIBUTIONS
• IV SAMPLING AND INFERENCE
• V HYPOTHESIS TESTING IN RESEARCH
• VI REPLICABILITY OF FINDINGS
• GLOSSARY OF TERMS
• ABOUT THE AUTHORS
• APPENDIX
• ANSWERS TO CHAPTER EXERCISES
• AUTHOR INDEX
• SUBJECT INDEX