** **I Introduction to Statistics 1

**1.1 **Introduction 1

What Is Statistics? 1

** ****1.2 **Why Sample? 8

Why Sample? 8

** ****1.3 **Sampling Techniques 10

Random or Probability Sampling Techniques 11

** ****1.4 **Uses of Statistics 16

** ****1.5 **Misuses of Statistics 18

Misleading Graphs 18

Non-Representative Samples 20

Inappropriate Comparisons 21

The Omission of Variation About an Average 21

** ****1.6 **Overview and Summary 21

** **

2 Organizing and Presenting Data 31

**2.1 **Introduction 31

** ****2.2 **Classifications of Data 33

Categorical and Numerical Variables 34

Continuous and Discrete Data 34

** ****2.3 **Exploring Data Using the Stem-and-Leaf Display 36

Back-to-Back Stem-and-Leaf Display 40

** ****2.4 **Frequency Distribution Tables 41

Frequency Distribution Table for Categorical Data 41

Frequency Distribution Table for Numerical Data 42

Relative Frequency Distributions 47

Cumulative Frequency Distributions 49

** ****2.5 **Graphs: Bar Graph, Histogram, Frequency Polygon and Ogive 52

Bar Graph 52

Histogram 54

Frequency Polygon and Ogive 61

** ****2.6 **Specialty Graphs: Pie Chart and Pictograph 65

** ****2.7 **Identifying Shapes and Interpreting Graphs 66

Shapes of Distributions 66

Interpreting Graphs 69

** ****2.8 **Using Technology 71

Graphing Calculator Display of a Histogram, Frequency Polygon and Ogive 73

** **** **

** **3 Numerical Techniques for Describing Data 85

**3.1 **Measures of Central Tendency 86

The Mean 86

Sample Mean 87

Population Mean 88

Other Calculations Using Σ 89

The Median 92

The Mode 95

The Relationship of the Mean, Median and Mode 98

The Mode and Its Location within the Distribution Shapes 99

The Median and Its Location within the Distribution Shapes 99

The Mean and Its Location within the Distribution Shapes 99

Symmetric Bell-Shaped Distribution 99

Skewed to the Left Distribution 100

Skewed to the Right Distribution 100

Comparing the Mean, the Median and the Mode 101

** ****3.2 **Measures of Variability 103

The Range 103

The Variance and Standard Deviation of a Sample 105

Interpretation of the Standard Deviation 108

The Variance and Standard Deviation of a Population 110

Using the Sample Standard Deviation to Estimate the Population Standard

Deviation 114

** ****3.3 **Applications of the Standard Deviation 115

Chebyshev’s Theorem 115

Empirical Rule 117

Using the Range to Obtain an Estimate of the Standard Deviation 118

** ****3.4 **Measures of Relative Standing 122

z Score 123

Detecting Outliers Using z Scores 126

Converting z Scores to Raw Scores 126

Percentile Rank and Percentiles 128

Deciles and Quartiles 130

** ****3.5 **Box-and-Whisker Plot: An Exploratory Data Analysis Technique 133

** ****3.6 **Using Technology 143

** **

4 Linear Correlation and Regression Analysis 169

**4.1 **Introduction 169

** ****4.2 **The Scatter Diagram 170

** ****4.3 **The Coefficient of Linear Correlation 177

Some Cautions Regarding the Interpretation of Correlation Results 181

** ****4.4 **More on the Relationship Between the Correlation Coefficient

and the Scatter Diagram 181

** ****4.5 **The Coefficient of Determination 182

** ****4.6 **Linear Regression Analysis 183

Interpolation versus Extrapolation 188

Regression Analysis Concept Revisited 190

** ****4.7 **Using Technology 193

** **

5 Probability 205

**5.1 **Introduction 205

The Birthday Problem 205

Chance! Chance! Chance! 206

** ****5.2 **Some Terms Used in Probability 206

Using a Tree Diagram to Construct a Sample Space 208

** ****5.3 **Permutations and Combinations 213

Permutation 213

Combination 219

Methods of Selection 223

Explaining the Difference Between the Idea of a Permutation

and a Combination 224

** ****5.4 **Probability 225

Alternate Approaches to Assigning a Probability 226

Another Approach to Defining Probability 231

Relative Frequency Concept of Probability or Posteriori Probability 231

Another Approach to Defining Probability 233

Subjective or Personal Probability 233

** ****5.5 **Fundamental Rules and Relationships of Probability 235

Probability Problems Using Permutations and Combinations 247

** ****5.6 **Conditional Probability 251

** ****5.7 **Using Technology 260

** **

6 Random Variables and Discrete Probability

Distributions 277

**6.1 **Introduction 277

** ****6.2 **Random Variables 278

** ****6.3 **Probability Distribution of a Discrete Random Variable 282

** ****6.4 **Mean and Standard Deviation of a Discrete Random Variable 288

The Mean Value of a Discrete Random Variable 288

The Variance and Standard Deviation of a Discrete Random Variable 292

** ****6.5 **Binomial Probability Distribution 299

Binomial Probability Formula 302

An Application of the Binomial Distribution: Acceptance Sampling 314

** ****6.6 **The Poisson Distribution 318

Shape of the Poisson Distribution 323

** ****6.7 **Using Technology 324

** **7 Continuous Probability Distributions

and the Normal Distribution 341

**7.1 **Introduction 341

** ****7.2 **Continuous Probability Distributions 341

** ****7.3 **The Normal Distribution 346

** ****7.4 **Properties of a Normal Distribution 348

The Standard Normal Curve 350

** ****7.5 **Using the Normal Curve Area Table 353

Finding a z-score Knowing the Proportion of Area to the Left 366

** ****7.6 **Applications of the Normal Distribution 371

** ****7.7 **Percentiles and Applications of Percentiles 376

** ****7.8 **Probability Applications 387

** ****7.9 **The Normal Approximation to the Binomial Distribution 392

** ****7.10 **Using Technology 404

** **

8 Sampling and Sampling Distributions 421

**8.1 **The Sampling Distribution of the Mean 421

** ****8.2 **The Mean and Standard Deviation of the Sampling Distribution

of the Mean 429

Mean of the Sampling Distribution of the Mean 429

Standard Deviation of the Sampling Distribution of the Mean 429

Interpretation of the Standard Error of the Mean 432

** ****8.3 **The Finite Correction Factor 433

** ****8.4 **The Shape of the Sampling Distribution of the Mean 436

Sampling from a Normal Population 436

Sampling from a Non-Normal Population 439

The Central Limit Theorem 439

** ****8.5 **Calculating Probabilities Using the Sampling Distribution of the Mean 444

** ****8.6 **The Effect of Sample Size on the Standard Error of the Mean 450

** ****8.7 **The Sampling Distribution of the Proportion 452

Sampling Error of the Proportion 461

Interpretation of the Standard Error of the Proportion 462

Shape of the Sampling Distribution of the Proportion 462

Calculating Probabilities Using the Sampling Distribution

of the Proportion 464

** ****8.8 **Using Technology 468

** **

9 Estimation 481

**9.1 **Introduction 481

** ****9.2 **Point Estimate of the Population Mean and the Population Proportion 482

** ****9.3 **Interval Estimation 484

** ****9.4 **Interval Estimation: Confidence Intervals for the Population Mean 485

Constructing a Confidence Interval for a Population Mean: When the Population

Standard Deviation Is Unknown 493

The t Distribution 494

** ****9.5 **Interval Estimation: Confidence Intervals for the Population Proportion 500

** ****9.6 **Determining Sample Size and the Margin of Error 504

Sample Size for Estimating a Population Mean, μ 504

Sample Size for Estimating a Population Proportion, p 508

Summary of Confidence Intervals 513

** ****9.7 **Using Technology 513

** **

10 Introduction to Hypothesis Testing 523

**10.1 **Introduction 523

** ****10.2 **Hypothesis Testing 523

Null and Alternative Hypotheses 524

** ****10.3 **The Development of a Decision Rule 532

** ****10.4 **p-Values for Hypothesis Testing 547

Procedure to Calculate the p-Value of a Hypothesis Test 550

** ****10.5 **Using Technology 552

** **

11 Hypothesis Testing Involving One Population 563

**11.1 **Introduction 563

** ****11.2 **Hypothesis Testing Involving a Population Proportion 563

Hypothesis Testing Procedure Involving a Population Proportion 565

** ****11.3 **Hypothesis Testing Involving a Population Mean: Population Standard Deviation

Known 574

** ****11.4 **The t Distribution 579

Using Table III: Critical Value for the t Distribution 580

** ****11.5 **Hypothesis Testing Involving a Population Mean: Population Standard Deviation

Unknown 582

** ****11.6 **p-Value Approach to Hypothesis Testing Using the TI-84 Plus Calculator 591

** ****11.7 **Using Technology 598

** **

12 Hypothesis Testing Involving Two Population Proportions

Using Independent Samples 609

**12.1 **Introduction to Hypothesis Tests Involving a Difference Between Two Population

Proportions Using Independent Samples 609

** ****12.2 **The Sampling Distribution of the Difference Between Two Proportions 610

** ****12.3 **Hypothesis Testing Involving Two Population Proportions Using Large Samples 619

Hypothesis Testing Procedure Involving the Difference Between the Proportions

of Two Populations for Large Samples 619

** ****12.4 **Hypothesis Testing Involving Two Population Proportions Comparing

Treatment and Control Groups 630

** ****12.5 **p-Value Approach to Hypothesis Testing Involving Two Population Proportions

Using the TI-84 Calculator 636

** ****12.6 **Two Population Hypothesis Testing Summaries Using Independent Samples 640

** ****12.7 **Using Technology 642

** **13 Hypothesis Test Involving Two Population Means

Using Independent Samples 657

**13.1 **Introduction 657

** ****13.2 **The Sampling Distribution of the Difference Between Two Means 657

** ****13.3 **Hypothesis Testing Involving Two Population Means and Unknown Population

Standard Deviations 660

Two Sample t Test 660

** ****13.4 **Hypothesis Tests Comparing Treatment and Control Groups 666

** ****13.5 **p-Value Approach to Hypothesis Testing Involving Two Population Means

Using the TI-84 Plus Calculator 672

** ****13.6 **Using Technology 677

** **

14 Chi-Square 687

**14.1 **Introduction 687

** ****14.2 **Properties of the Chi-Square Distribution 689

** ****14.3 **Chi-Square Hypothesis Test of Independence 690

** ****14.4 **Assumptions Underlying the Chi-Square Test 700

** ****14.5 **Test of Goodness-of-Fit 700

** ****14.6 **p-Value Approach to Chi-Square Hypothesis Test of Independence

Using the TI-84 Plus Calculator 707

** ****14.7 **Using Technology 712

** **

15 Inferences for Correlation and Regression 723

**15.1 **Introduction 723

** ****15.2 **Testing the Significance of the Correlation Coefficient 725

Procedure to Test the Significance of the Population Correlation

Coefficient, 725

** ****15.3 **Assumptions for Linear Regression Analysis 733

** ****15.4 **p-Value Approach to Testing the Significance of the Correlation Coefficient

Using the TI-84 Calculator 733

** ****15.5 **Introduction to Multiple Regression 738

** ****15.6 **Using Technology 745

** **

**Answer Section **757

** ****Appendices **791

A: Databases 792

B: Chapter Formulas 797

C: Summary of Hypothesis Tests 803

D: Statistical Tables 805

** ****Index **811

** **16 The F-Distribution and An Introduction

to Analysis of Variance (ANOVA)

17 NonParametric Statistics

** **