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- Introduction to Real World Statistics: With Step-By-Step SPSS Instructions

Edward T. Vieira, Jr.

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March 9, 2017
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*Introduction to Real World Statistics* provides students with the basic concepts and practices of applied statistics, including data management and preparation; an introduction to the concept of probability; data screening and descriptive statistics; various inferential analysis techniques; and a series of exercises that are designed to integrate core statistical concepts. The author’s systematic approach, which assumes no prior knowledge of the subject,** **equips student practitioners with a fundamental understanding of applied statistics that can be deployed across a wide variety of disciplines and professions.

Notable features include:

- short, digestible chapters that build and integrate statistical skills with real-world applications, demonstrating the flexible usage of statistics for evidence-based decision-making
- statistical procedures presented in a practical context with less emphasis on technical jargon
- early chapters that build a foundation before presenting statistical procedures
- SPSS step-by-step detailed instructions designed to reinforce student understanding
- real world exercises complete with answers
- chapter PowerPoints and test banks for instructors.

Preface

Why Read This Book?

Notable Features

Assumes No Prior Knowledge of Statistics

Short Digestible Chapters That Build and Integrate Real World Statistical Skills

An Alternative to the Traditional Hypothesis Testing Approach

Interdisciplinary Applications

SPSS Step-by-Step Detailed Instructions with Screenshots

Chapter PowerPoints and Test Bank

A Systematic Approach to Teaching Statistics

Book Organization

Acknowledgments

PART I: GETTING STARTED

1 Introduction to Real World Statistics

Learning Objectives

1.1 What Is Statistics?

Sample Data vs. Census Data

1.2 Reification

1.3 Naïve Science: The Deception of Common Sense

Real World Snapshot

1.4 Importance of Statistics

Statistical Assumptions

Summary of Key Concepts

Introductory Applied Exercises

2 Statistics: Descriptive, Correlation, and Inferential

Learning Objectives

2.1 Introduction to Descriptive, Correlation, and Inferential Statistics

2.2 Descriptive Statistics

Measures of Variation

2.3 Correlation Statistics

2.4 Inferential Statistics

Real World Snapshot

2.5 Descriptive, Correlation, and Inferential Statistics

Summary of Key Concepts

Descriptive, Correlation, and Inferential Statistics Applied Exercises

3 Data and Types of Variables

Learning Objectives

3.1 Introduction to Variables

3.2 Kinds of Variables

3.3 Variables by Type of Data

Categorical Data

Binary-Level Data (Variable)

Nominal-Level Data (Variable)

Ordinal-Level Data (Variable)

Numeric Data

Ratio-Level Data (Variable)

Interval-Level Data (Variable)

Scale Response Formatted Variables: A Special Case

Appropriate Analysis for Variable (Data) Type

Real World Snapshot

3.4 Variables by Influence

Independent Variables (Predictors)

Dependent Variables (Outcomes)

Control Variables

Interaction Variables

Summary of Key Concepts

Variables Applied Exercises

4 SPSS Statistics Data Management Basics: Preparing Data for Analysis

Learning Objectives

4.1 Introduction to SPSS and Data, Output, and Syntax Files

4.2 Setting up the Data File

4.3 Key SPSS Data Management Tools

4.4 Opening SPSS

4.5 Formatting the Variables’ Data

Name

Type

Width

Decimals

Label

Values

Missing

Column

Align

Measure

Role

4.6 The SPSS Data File

Data Access

Manual Entry

Opening an Existing SPSS Data File

Opening Other Formatted Spreadsheet Data Files

Text Files

Cut and Paste

Saving the Data File

Saving Data in SPSS

Saving Data in Other Spreadsheet Formats

4.7 The SPSS Output File

Creating a New SPSS Output File

Opening an Existing SPSS Output File

Displaying the Full P-Value in the Output File

Saving an SPSS Output File

Saving Output for the SPSS Output Viewer

Saving SPSS Output in Another Format

4.8 A Brief Review of the Syntax File

4.9 Creating a Codebook

Creating a Codebook from Scratch

Real World Snapshot

The SPSS Codebook

Summary of Key Concepts

Data Management Applied Exercises

PART II: SAMPLING CONSIDERATIONS

5 Sampling Strategies

Learning Objectives

5.1 Introduction to the Sampling Process

5.2 Probability Sampling

Random vs. Representative Sampling

Random Sampling and the Shape of Data Distribution

Simple Random Sampling

Systematic Random Sampling

Cluster (Random) Sampling

Stratified Random Sampling

Real World Snapshot

5.3 Nonprobability Sampling

Convenience Sampling

Expert Sampling

Quota Sampling

Snowball Sampling

Summary of Key Concepts

Sampling Applied Exercises

6 Sample Size

Learning Objectives

6.1 Introduction to Sample Size

Numeric Data

Central Limit Theorem

Categorical Data

Other Considerations

6.2 Power Analysis and Sample Size

Finite Population Correction

Informed Power Analysis

Real World Snapshot

Comparison of Unequal Sample Sizes

Data Assumptions

6.3 Examples Using SPSS: Step-by-Step Instructions

Example 6.1: Means: One-sample t-test that mean = specific value

Interpretation

Example 6.2: Means: Paired t-test that mean = 0

Interpretation

Example 6.3: Proportions: One-sample test that proportion = ".50"

Interpretation

Example 6.4: Proportions: 2 × 2 for independent samples (chi-square or Fisher’s exact test)

Interpretation

Example 6.5: Correlations: One-sample test that correlation = 0

Interpretation

Example 6.6: ANOVA: One-way analysis of variance

Interpretation

Example 6.7: Regression: One set of predictors

Interpretation

Example 6.8: Clustering

Interpretation

Summary of Key Concepts

Sample Size Applied Exercises

7 Sources and Types of Statistical Error

Learning Objectives

7.1 Introduction to Sources of Statistical Error

Real World Snapshot

7.2 Sampling Error

Sampling Random Error

Sampling Systematic Error

7.3 Nonsampling Error

Nonsampling Random Error

Nonsampling Systematic Error

Summary of Key Concepts

Statistical Error Applied Exercises

8 Missing Data

Learning Objectives

8.1 Introduction to Missing Data

8.2 Missing Value (Data) Analysis

Real World Snapshot

8.3 Methods for Replacing Missing Values

Listwise (Casewise)

Pairwise

Series Mean

Mean of Nearby Points

Median of Nearby Points

Linear Interpolation

Linear Trend at Point

8.4 New Data Replacement Methods

Expectation Maximization

Multiple Imputation

8.5 Examples Using SPSS: Step-by-Step Instructions

Example 8.1: The MCAR Case

Interpretation

Write-Up

Example 8.2: The NMAR Case

Interpretation

Write-Up

Summary of Key Concepts

Missing Data Applied Exercises

PART III: DATA SCREENING, DESCRIBING, AND PROBABILITIES

9 Describing Categorical Variables

Learning Objectives

9.1 Introduction to Describing Categorical Variables

Real World Snapshot

9.2 Charting Categorical Variables

Pie Chart

Dichotomous (Two-Category) Pie Chart

Example 9.1: A Pie Chart with Two Categories

Describing and Reporting

Multiple Category Pie Chart

Example 9.2: A Pie Chart with More Than Two Categories

Describing and Reporting

Bar Chart

Example 9.3: A Bar Chart with More Than Two Categories

Describing and Reporting

9.3 Categorical Variable Tables

Single Variable Tables

Example 9.4: A Single Categorical Variable with Two Categories Table

Describing and Reporting

Example 9.5: A Single Categorical Variable with More Than Two Categories Table

Describing and Reporting

Multiple Variable Tables

Two Variables

Example 9.6: Two Categorical Variables Each with Two or More Categories Table

Describing and Reporting

Three Variables

Example 9.7: Three Categorical Variables Each with Two or More Categories Table

Describing and Reporting

Summary of Concepts

Describing Categorical Variables Applied Exercises

10 Basic Probabilities for Categorical Variables

Learning Objectives

10.1 Introduction to Basic Probability

10.2 Assumptions

Real World Snapshot

10.3 Simple (Marginal) Probability

10.4 Joint Probability

10.5 Conditional Probability

10.6 Tables

10.7 Multiplication Rule in Probability

10.8 Addition Rule in Probability

Summary of Key Concepts

Categorical Data Probability Applied Exercises

11 The Concepts of Data Distribution, Probability Values, and Significance Testing

Learning Objectives

11.1 Introduction to Data Distribution and Probability

11.2 Numerical Data Distribution

Standard Deviation and the Normal Distribution

Real World Snapshot

Z-Distribution and Z-Scores

T-Distribution

Probability Based on the Normal Distribution

Probability Value (P-Values)

Level of Significance (Alpha) and Significance Testing

11.3 Categorical Data Distribution

The Chi-Square Significance Test

Degrees of Freedom

Two Types of Expected Observations

Probability (P-Values) Based on the Chi-Square Distribution

11.4 Confidence Intervals

11.5 Conclusion

Summary of Key Concepts

Distribution and Significance Testing Applied Exercises

12 Numeric Variables: Data Screening and Removing Outliers

Learning Objectives

12.1 Introduction to Numeric Data Screening and Removing Outliers

Real World Snapshot

12.2 Measuring Central Tendency

Mean

Median

Mode

Coefficient of Skewness

12.3 Measuring Dispersion

Range

Variance

Standard Deviation

Coefficient of Variation

Coefficient of Kurtosis

12.4 Screening Data: Identifying and Removing Outliers

Outliers

Visual Assessment

Statistical Measures

Methods for Identifying and Removing Outliers

Simple Outlier Removal

Standard Deviation Rule

Trimming or Truncating

Winsorizing

Outlier Labeling Rule

Data Removal and Analysis

Data Screening and the Removal of Outliers Assumptions

12.5 Examples Using SPSS: Step-by-Step Instructions

Example 12.1: Simple Outlier Removal

SPSS Output Interpretation

Example 12.2: Outlier Labeling Rule Removal

SPSS Output Interpretation

12.6 Other Remedies for Non-Normal Data Distribution

Summary of Key Concepts

Numeric Data Screening and Removing Outliers Applied Exercises

PART IV: STATISTICAL ANALYSIS

Categorical Variables

13 Chi-Square Goodness of Fit Test: Comparing Counts in a Single Variable with Two or More Categories

Learning Objectives

13.1 Introduction to the Chi-Square Goodness of Fit Test

13.2 Calculating and Understanding the Chi-Square Statistic

Real World Snapshot

13.3 Data Assumptions

13.4 Examples Using SPSS: Step-by-Step Instructions

Example 13.1: Equal Expected Counts: The Significant Case

SPSS Output Interpretation

Data Screening

Chi-Square Goodness of Fit Test Analysis

Reporting Results

Write-Up

Example 13.2: Equal Expected Counts: The Nonsignificant Case

SPSS Output Interpretation

Data Screening

Chi-Square Goodness of Fit Test Analysis

Reporting Results

Write-Up

Example 13.3: Specified Expected Counts: Both Cases

SPSS Output Interpretation

Data Screening

Chi-Square Goodness of Fit Analysis

Reporting Significant Results

Write-Up

Reporting Nonsignificant Results

Write-Up

Summary of Key Concepts

Chi-Square Goodness of Fit Test Applied Exercises

14 Chi-Square Test of Independence: Comparing Counts between Two Variables Each with Two or More Categories

Learning Objectives

14.1 Introduction to the Chi-Square Test of Independence

14.2 Calculating and Understanding the Chi-Square Test of Independence

14.3 Data Assumptions

Real World Snapshot

14.4 Examples Using SPSS: Step-by-Step Instructions

Example 14.1: A 2 × 2 Chi-Square Test of Independence: The Significant Case

SPSS Output Interpretation

Data Screening

Chi-Square Test of Independence Analysis

Reporting Results

Write-Up

Example 14.2: A 2 × 2 Chi-Square Test of Independence: The Nonsignificant Case

SPSS Output Interpretation

Data Screening

Chi-Square Test of Independence Analysis

Reporting Results

Write-Up

Example 14.3: A 3 × 5 Chi-Square Test of Independence: Both Cases

SPSS Output Interpretation

Data Screening

Chi-Square Test of Independence Analysis

Reporting Significant Results

Write-Up

Reporting Nonsignificant Results

Write-Up

Summary of Key Concepts

Chi-Square Test of Independence Applied Exercises

15 Chi-Square Test of the Same Sample: Comparing Counts of the Same Sample Measured Twice Using a Categorical Variable

Learning Objectives

15.1 Introduction to the Same Sample Measured Twice Using a Categorical Variable

15.2 Data Assumptions

Real World Snapshot

15.3 Examples Using SPSS: Step-by-Step Instructions

Example 15.1: A Crosstabs 2 × 2 Repeated Measures McNemar Test: The Significant Case

SPSS Output Interpretation

Data Screening

Chi-Square Test for Repeated Counts Analysis

Reporting Results

Write-Up

Example 15.2: A Crosstabs 2 × 2 Repeated Measures McNemar Test: The Nonsignificant Case

SPSS Output Interpretation

Data Screening

Chi-Square Test for Repeated Counts Analysis

Reporting Results

Write-Up

Example 15.3: A Crosstabs 4 × 2 Repeated Measures McNemar-Bowker Test: Both Cases

SPSS Output Interpretation

Data Screening

Chi-Square Test for Repeated Counts Analysis

Reporting Significant Results

Write-Up

Reporting Nonsignificant Results

Write-Up

Summary of Key Concepts

Chi-Square Test of Two Related Samples Measured Twice Applied Exercises

Numeric Variables

16 T-Test: Comparing a Single Sample Mean to a Specific Value

Learning Objectives

16.1 Introduction to the Single Sample T-Test

16.2 Confidence Interval for a Single Sample T-Test

Real World Snapshot

16.3 Data Assumptions

16.4 Examples Using SPSS: Step-by-Step Instructions

Example 16.1: Single Sample T-Tests: The Significant Case

SPSS Output Interpretation

Data Screening

*Single Sample *T*-Test Analysis *

Reporting Results

Write-Up

Example 16.2: Single Sample T-Tests: The Nonsignificant Case

SPSS Output Interpretation

Data Screening

*Single Sample *T*-Test Analysis *

Reporting Results

Write-Up

Summary of Key Concepts

16.5 Single Sample T-Test Applied Exercises

17 T-Test: Comparing Two Independent Samples’ Variable Means

Learning Objectives

17.1 Introduction to the Two Independent Samples T-Test

17.2 Equality of Variance

Real World Snapshot

Pooled or Separate Two Independent Samples T-Test

17.3 Data Assumptions

17.4 Examples Using SPSS: Step-by-Step Instructions

Example 17.1: Two Independent Samples T-Tests: The Significant Case

SPSS Output Interpretation

Data Screening

*Two Independent Samples *T*-Test Analysis *

Reporting Results

Write-Up

Example 17.2: Two Independent Samples T-Tests: The Nonsignificant Case

SPSS Output Interpretation

Data Screening

*Two Independent Samples *T*-Test Analysis *

Reporting Results

Write-Up

Summary of Key Concepts

Two Independent Samples T-Test Applied Exercises

18 Analysis of Variance (ANOVA): Comparing More Than Two Independent Samples’ Means to Test for Differences among Them by One Type of Classification

Learning Objectives

18.1 Introduction to One-Way ANOVA

18.2 Variance

Real World Snapshot

18.3 Data Assumptions

18.4 Strategies for Addressing Violations of Assumptions

18.5 Examples Using SPSS: Step-by-Step Instructions

Example 18.1: ANOVA F-Test: The Significant Case

SPSS Output Interpretation

Data Screening

ANOVA (Analysis)

Reporting Results

Write-Up

Example 18.2: ANOVA F-Test: The Nonsignificant Case

SPSS Output Interpretation

Data Screening

ANOVA (Analysis)

Reporting Results

Write-Up

Summary of Key Concepts

One-Way ANOVA F-Test Applied Exercises

19 Paired T-Test: Comparing the Means of the Same Sample Measured Twice Using a Numeric Variable

Learning Objectives

19.1 Introduction to the Paired-Sample T-Test

19.2 Paired T-Test Calculations

Real World Snapshot

19.3 Data Assumptions

19.4 Examples Using SPSS: Step-by-Step Instructions

Example 19.1: Paired-Sample T-Tests: The Significant Case

SPSS Output Interpretation

Data Screening

*Paired-Sample *T*-Test Analysis *

Reporting Results

Write-Up

Example 19.2: Single Sample T-Tests: The Nonsignificant Case

SPSS Output Interpretation

Data Screening

*Paired-Sample *T*-Test Analysis *

Reporting Results

Write-Up

Summary of Key Concepts

Paired-Samples T-Test Applied Exercises

20 General Linear Model Repeated Measures: Comparing Means of the Same Sample Measured More Than Twice Using a Numeric Variable

Learning Objectives

20.1 Introduction to General Linear Model Repeated Measures

Real World Snapshot

20.2 Data Assumptions

20.3 Strategies for Addressing Violations of Assumptions

20.4 Examples Using SPSS: Step-by-Step Instructions

Example 20.1: General Linear Model Repeated Measures: The Significant Case

SPSS Output Interpretation

Data Screening

General Linear Model Repeated Measures Analysis

Reporting Results

Write-Up

Example 20.2: General Linear Model Repeated Measures: The Nonsignificant Case

SPSS Output Interpretation

Data Screening

General Linear Model Repeated Measures Analysis

Reporting Results

Write-Up

Summary of Key Concepts

General Linear Model Repeated Measures Applied Exercises

21 Correlation Analysis: Looking for an Association between Two Variables

Learning Objectives

21.1 Introduction to Pearson, Spearman, and Partial Bivariate Correlations

Explained Variance (r2)

21.2 Strength and Directionality of Correlations

Correlation Strength

Correlation Directionality

Linear Correlation Strength and Directionality Together

21.3 Calculating a Correlation for Numeric Data

Real World Snapshot

21.4 Types of Correlations

21.5 General Data Assumptions

21.6 Examples Using SPSS: Step-by-Step Instructions

Example 21.1: Pearson Correlation: The Significant Case

SPSS Output Interpretation

Data Screening

Pearson Correlation Analysis

Reporting Results

Write-Up

Example 21.2: Pearson Correlation: The Nonsignificant Case

SPSS Output Interpretation

Data Screening

Pearson Correlation Analysis

Reporting Results

Write-Up

Example 21.3: Spearman Correlation: Both Cases

SPSS Output Interpretation

Data Screening

Spearman Correlation Analysis

Reporting Significant Results

Write-Up

Reporting Nonsignificant Results

Write-Up

Example 21.4: Partial Correlation: Both Cases

SPSS Output Interpretation

Data Screening

Partial Correlation Analysis

Reporting Nonsignificant Results

Write-Up

Reporting Significant Results

Write-Up

Summary of Key Concepts

Correlation Analysis Applied Exercises

22 Single Linear Regression

Learning Objectives

22.1 Introduction to Single Linear Regression

Prediction vs. Cause and Effect

22.2 Prediction Model

Applying the Prediction Model

Standardized Regression Coefficients

Real World Snapshot

22.3 Data Assumptions

Testing Data Assumptions

22.4 Examples Using SPSS: Step-by-Step Instructions

Example 22.1: Single Linear Regression: The Significant Case

SPSS Output Interpretation

Data Screening

Single Linear Regression Analysis

Reporting Results

Write-Up

Example 22.2: Single Linear Regression: The Nonsignificant Case

SPSS Output Interpretation

Data Screening

Single Linear Regression Analysis

Reporting Results

Write-Up

Summary of Key Concepts

Single Linear Regression Applied Exercises

23 Multiple Linear Regression

Learning Objectives

23.1 Introduction to Multiple Linear Regression

23.2 Prediction Model

R-Square and Adjusted R-Square

Real World Snapshot

23.3 Data Assumptions

23.4 Examples Using SPSS: Step-by-Step Instructions

Example 23.1: Multiple Linear Regression: The Significant Case

SPSS Output Interpretation

Data Screening

Multiple Linear Regression Analysis

Reporting Results

Write-Up

Example 23.2: Multiple Linear Regression: The Nonsignificant Case

SPSS Output Interpretation

Data Screening

Multiple Linear Regression Analysis

Reporting Results

Write-Up

Summary of Key Concepts

Multiple Linear Regression Applied Exercises

APPENDICES

Appendix A: Glossary

Appendix B: Chapter Statistical Exercise Solutions

B.1 Chapter 1

B.2 Chapter 2

B.3 Chapter 3

B.4 Chapter 4

B.5 Chapter 5

B.6 Chapter 6

B.7 Chapter 7

B.8 Chapter 8

B.9 Chapter 9

B.10 Chapter 10

B.11 Chapter 11

B.12 Chapter 12

B.13 Chapter 13

B.14 Chapter 14

B.15 Chapter 15

B.16 Chapter 16

B.17 Chapter 17

B.18 Chapter 18

B.19 Chapter 19

B.20 Chapter 20

B.21 Chapter 21

B.22 Chapter 22

B.23 Chapter 23

Appendix C: Case Studies and Solutions

C.1 Case Study Questions

Financial Attributes

Gift Shop Customers

Health Issues

Moving Services

Sample Size Matters

Violent Crime Recidivism

What Motivates Students to Perform

Psychological Effects of the Workplace

C.2 Case Study Solutions

Financial Attributes

Gift Shop Customers

Health Issues

Moving Services

Sample Size Matters

Violent Crime Recidivism

What Motivates Students to Perform

Psychological Effects of the Workplace

Appendix D: Research Goal and Objectives

D.1 Research Goal

E.2 Research Objectives

Developing and Testing Research Statements

Developing and Answering Research Questions

D.3 The Interconnected Parts of Research Goals and Objectives

Appendix E: Types of Research Design

E.1 Introduction to Research Designs

E.2 Survey or Self-Report Research Design

Person-to-Person Administered Survey

Self-Administered Survey

E.3 Experimental Research Design

Cause and Effect Relationship

Lab Experiment

Field Experiment

Manipulation Check

External Influences

Managing the Effects of Unaccounted for Extraneous Variables

The Experimental Design Process Model

E.4 Observational Research Design

Personal Observation

Mechanical Observation

Content Analysis

E.5 Other Research Designs

Single Time vs. Repeated Measures Designs

Cross-Sectional Design

Longitudinal Design

Mixed Research Designs

Appendix F: Comparing Counts of the Same Sample Measured More Than

Twice Using a Categorical Variable

F.1 A Categorical Variable Measured More Than Twice Using the Same Sample

F.2 Data Assumptions

Appendix G: More on Linear Regression

G.1 Introduction to Other Tools in Regression Analysis

G.2 The Influence of Outliers on Linear Regression Results

G.3 Linear Regression Methods

Stepwise

Hierarchical

G.4 Dummy Coding

G.5 Interaction Terms (Variables)

G.6 Residual Analysis

G.7 Multicollinearity

Appendix H: Statistics Flow Chart

References

Index

Professor of Marketing & Statistics, School of Business, Simmons University

Boston, Massachusetts, USA

**Edward T. Vieira, Jr. **is an Associate Professor, Research Director, and member of the Institutional Review Board at Simmons College, Boston, Massachusetts, USA. He earned his M.B.A from Bryant University and Ph.D. from the University of Connecticut. Currently, Dr. Vieira serves on the editorial boards of seven peer-reviewed journals providing statistical and methodological expertise. He has over 30 years of management, research, and consulting experience in areas such as marketing research, community outreach focus groups, organizational research, and education evaluation research.

*This book serves students being introduced to quantitative research as well as research professionals seeking to add to their statistical analysis and quantitative reasoning skills. Its emphasis on providing the reasons and prerequisites for using a statistical procedure make it valuable as a book-shelf reference as well as a textbook. Integration of relevant exercises to be carried out with SPSS enhances understanding of general statistical concepts with a body of hands-on experience, and that combination results in a very valuable skill set that will serve the reader well for years.*

**James H. Watt, Professor Emeritus, University of Connecticut**

*Professor Vieira combines a straightforward approach to applied statistics with the most accessible statistical package, SPSS. His non-technical, clear, and concise writing style makes* Introduction to Real World Statistics *a valuable handbook and reference for students and practitioners as well as an effective text for introductory statistics courses.*

**John Lowe, Associate Dean for Undergraduate Programs, Simmons College**

*This is a textbook that attempts to bridge areas of content that are traditionally addressed independently: a the conceptual knowledge of technical information; b. the real world application of such technical knowledge and c. a leading software tool that assists a researcher in applying the conceptual knowledge of technical information to a real world context. Vieira seems to have used the feedback he received from his students over several decades to successfully build a bridge across these three content areas. This textbook and its approach are a welcome addition for making the process of learning statistics in the social sciences a lot smoother and a lot more relevant to students. *** **

**Michael G. Elasmar, Ph.D., Associate Professor and Director of the Marketing Communication Research graduate program, Boston University**

*Dr. Vieira’s book is comprehensive, clear, and has great examples to illustrate the concepts. I particularly liked the section on Sampling. I envision that this book would be suitable for a variety of audiences and levels.*

**Clayton W. Barrows, University of New Hampshire**

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