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Part Ⅰ Descriptive Statistics3

Unit 1 Statistics3

1.1 What is Statistics？4

1.1.1 Meanings of Statistics4

1.1.2 Definition of Statistics5

1.1.3 Types of Statistics6

1.1.4 Applications of Statistics6

1.2 The language of Statistics9

1.2.1 Population and Sample9

1.2.2 Kinds of Variables11

1.3 Measurability and Variability14

1.4 Data Collection16

1.4.1 The Data Collection Process17

1.4.2 Sampling Frame and Elements18

1.5 Single-Stage Methods21

1.5.1 Simple Random Sample21

1.5.2 Systematic Sample22

1.6 Multistage Methods25

1.7 Types of Statistical Study27

1.8 The Process of a Statistical Study31

Glossary34

Reading English Materials35

Passage 1．What is Statistics？35

Passage 2．From Data to Foresight35

Problems36

Unit 2 Descriptive Analysis of Single-Variable Data40

2.1 Graphs，Pareto Diagrams，and Stem-and-Leaf Displays41

2.1.1 Qualitative Data41

2.1.2 Quantitative Data43

2.2 Frequency Distributions and Histograms47

2.2.1 Frequency Distribution47

2.2.2 Histograms51

2.2.3 Cumulative Frequency Distribution and Ogives53

2.3 Measures of Central Tendency55

2.3.1 Finding the Mean55

2.3.2 Finding the Median56

2.3.3 Finding the Mode57

2.3.4 Finding the Midrange58

2.4 Measures of Dispersion60

2.4.1 Sample Standard Deviation62

2.5 Measures of Position64

2.5.1 Quartiles64

2.5.2 Percentiles64

2.5.3 Other Measures of Position66

2.6 Interpreting and Understanding Standard Deviation70

2.6.1 The Empirical Rule and Testing for Normality70

2.6.2 Chebyshev's Theorem72

Glossary74

Problems75

Unit 3 Descriptive Analysis of Bivariate Data79

3.1 Bivariate Data80

3.1.1 Two Qualitative Variables80

3.1.2 One Qualitative and One Quantitative Variable82

3.1.3 Two Quantitative Variables83

3.2 Linear Correlation85

3.2.1 Calculating the Linear Correlation Coefficient,r86

3.2.2 Causation and Lurking Variables89

3.3 Linear Regression91

3.3.1 Line of Best Fit92

3.3.2 Making Predictions97

Reading English Materials99

Passage 1．The First Regression99

Passage 2．Simpson's Paradox99

Problems100

Unit 4 Introduction to Probability104

4.1 Sample Spaces,Events and Sets105

4.1.1 Introduction105

4.1.2 Sample Spaces105

4.1.3 Events106

4.1.4 Set Theory108

4.2 Probability Axioms and Simple Counting Problems109

4.2.1 Probability Axioms and Simple Properties109

4.2.2 Interpretations of Probability111

4.2.3 Classical Probability112

4.2.4 The Multiplication Principle113

4.3 Permutations and Combinations115

4.3.1 Introduction115

4.3.2 Pemutations116

4.3.3 Combinations118

4.3.4 The Difference Between Permutations and Combinations120

4.4 Conditional Probability and the Multiplication Rule122

4.4.1 Conditional Probability122

4.4.2 The Multiplication Rule123

4.5 Independent Events，Partitions and Bayes Theorem124

4.5.1 Independence124

4.5.2 Partitions125

4.5.3 Law of Total Probability126

4.5.4 Bayes Theorem126

4.5.5 Bayes Theorem for Partitions127

Reading English Materials130

Passage 1．Probability and Odds130

Passage 2．The Relationship between Odds and Probability130

Passage 3．How the Odds Change across the Range of the Probability131

Problems132

Unit 5 Discrete Probability Models134

5.1 Introduction,Mass Functions and Distribution Functions135

5.1.1 Introduction135

5.1.2 Probability Mass Functions（PMFs）136

5.1.3 Cumulative Distribution Functions（CDFs）137

5.2 Expectation and Variance for Discrete Random Quantities138

5.2.1 Expectation138

5.2.2 Variance139

5.3 Properties of Expectation and Variance140

5.3.1 Expectation of a Function of a Random Quantity140

5.3.2 Expectation of a Linear Transformation140

5.3.3 Expectation of the Sum of Two Random Quantities141

5.3.4 Expectation of an Independent Product141

5.3.5 Variance of an Independent Sum142

5.4 The Binomial Distribution142

5.4.1 Introduction142

5.4.2 Bemoulli Random Quantities143

5.4.3 The Binomial Distribution143

5.4.4 Expectation and Variance of a Binomial Random Quantity145

5.5 The Geometric Distribution146

5.5.1 PMF146

5.5.2 CDF147

5.5.3 Useful Series in Probability148

5.5.4 Expectation and Variance of Geometric Random Quantities148

5.6 The Poisson Distribution149

5.6.1 Poisson as the Limit of a Binomial149

5.6.2 PMF150

5.6.3 Expectation and Variance of Poisson151

5.6.4 Sum of Poisson Random Quantities152

5.6.5 The Poisson Process152

Reading English Materials154

Passage 1．The Founder of Modern Statistics—Karl Pearson154

Passage 2．The Relations of Several Discrete Probability Models154

Problems155

Unit 6 Discrete Probability Models158

6.1 Introduction，PDF and CDF159

6.1.1 Introduction159

6.1.2 The Probability Density Function159

6.1.3 The Distribution Function160

6.1.4 Median and Quartiles161

6.2 Properties of Continuous Random Quantities161

6.2.1 Expectation and variance of continuous random quantities161

6.2.2 PDF and CDF of a Linear Transformation162

6.3 The Uniform Distribution163

6.4 The Exponential Distribution165

6.4.1 Definition and Properties165

6.4.2 Relationship with the Poisson Process166

6.4.3 The Memoryless Property167

6.5 The Normal Distribution168

6.5.1 Definition168

6.5.2 Properties168

6.6 The Standard Normal Distribution169

6.6.1 Properties of the Standard Normal Distribution170

6.6.2 Finding Area to The Right of z=0171

6.6.3 Finding Areain The Right Tail of a Normal Curve171

6.6.4 Finding Area to the Left of a Positive z Value172

6.6.5 Finding Area from a Negative z to z=0172

6.6.6 Finding Area in the Left Tail of a Normal Curve172

6.6.7 Finding Area from A Negative z to a Positive z172

6.6.8 Finding Area Between two z Values of the Same Sign173

6.6.9 Finding z-Scores Associated with a Percentile173

6.6.10 Finding z-scores that Bound an Area174

6.7 Applications of Normal Distributions175

6.7.1 Probabilities and Normal Curves175

6.7.2 Using the Normal Curve and z176

6.8 Specific z-score178

6.8.1 Visual Interpretation of z（a）179

6.8.2 Determining Corresponding z Values for z（a）179

6.8.3 Determining z-scores for Bounded Areas180

6.9 Normal Approximation of Binomial and Poisson181

6.9.1 Normal Approximation of the Binomial181

6.9.2 Normal Approximation of the Poisson182

Problems182

Unit 7 Sampling Distributions and CLT187

7.1 Sampling Distributions188

7.1.1 Forming a Sampling Distribution of Means188

7.1.2 Creating a Sampling Distribution of Sample Means189

7.2 The Sampling Distribution of Sample Means192

7.2.1 Central Limit Theorem193

7.2.2 Constructing a Sampling Distribution of Sample Means194

7.3 Application of the Sampling Distribution of Sample Means199

7.3.1 Converting？Information into z-scores199

7.3.2 Distribution of？and Increasing Individual Sample Size200

7.4 Advanced Central Limit Theorem202

7.4.1 Central Limit Theorem（Sample Mean）203

7.4.2 Central Limit Theorem（Sample Sum）203

Problems207

Part Ⅱ Inferential Statistics210

Unit 8 Introduction to Statistical Inferences210

8.1 Point Estimation and Interval Estimation211

8.1.1 Point Estimate211

8.1.2 Interval Estimate212

8.2 Estimation of Meanμ（σKnown）214

8.2.1 The Principle of Constructing a Confidence Interval214

8.2.2 Applications216

8.2.3 Sample Size and Confidence Interval217

8.3 Introduction to Hypothesis Testing220

8.3.1 Null Hypothesis and Alternative Hypothesis220

8.3.2 Four Possible Outcomes in a Hypothesis Test222

8.4 Formulating the Statistical Null and Alternative Hypotheses226

8.4.1 Writing Null and Alternative Hypothesis in One-Tailed Situation226

8.4.2 Writing Null and Alternative Hypothesis in Two-Tailed Situation227

8.5 Hypothesis Test of Meanμ（σKnown）：A Probability-Value Approach228

8.5.1 One-Tailed Hypothesis Test Using the p-Value Approach229

8.5.2 Two-Tailed Hypothesis Test Using the p-Value Approach233

8.5.3 Evaluating the p-Value Approach234

8.6 Hypothesis Test of Meanμ（σKnown）：A Classical Approach235

8.6.1 One-Tailed Hypothesis Test Using the Classical Approach236

8.6.2 Two-Tailed Hypothesis Test Using the Classical Approach239

Problems241

Unit 9 Inferences Involving One Population246

9.1 Inferences about the Meanμ（σUnknown）247

9.1.1 Using the t-Distribution Table249

9.1.2 Confidence Interval Procedure251

9.1.3 Hypothesis-Testing Procedure252

9.2 Inferences about the Binomial Probability of Success258

9.2.1 Confidence Interval Procedure259

9.2.2 Determining Sample Size261

9.2.3 Hypothesis-Testing Procedure263

9.3 Inferences about the Variance and Standard Deviation268

9.3.1 Critical Values of Chi-Square269

9.3.2 Hypothesis-Testing Procedure270

Problems279

Unit 10 Inferences Involving Two Populations284

10.1 Dependent and Independent Samples285

10.2 Inferences Concerning the Mean Difference Using Two Dependent Samples287

10.2.1 Procedures and Assumptions for Inferences Involving Paired Data287

10.2.2 Confidence Interval Procedure288

10.2.3 Hypothesis-Testing Procedure290

10.3 Inferences Conceming the Difference between Means Using Two Independent Samples294

10.3.1 Confidence Interval Procedure295

10.3.2 Hypothesis-Testing Procedure297

10.4 Inferences Concerning the Difference between Proportions301

10.4.1 Confidence Interval Procedure303

10.4.2 Hypothesis-Testing Procedure304

10.5 Inferences Concerning the Ratio ofVariances Using Two Independent Samples308

10.5.1 Writing for the Equality of Variances308

10.5.2 Using the F-Distribution309

10.5.3 One-Tailed Hypothesis Test for the Equality of Variances310

10.5.4 Critical F-Values for One-and Two-Tailed Tests313

Problems315

Unit 11 An Introduction to Simple Regression321

11.1 Regression as a Best Fitting Line322

11.1.1 Regression as a Best Fitting Line322

11.1.2 Errors and Residuals324

11.2 Interpreting OLS Estimates326

11.3 Fitted Values and R2：Measuring the Fit of a Regression Model328

11.4 Nonlinearity in Regression331

Reading English Materials335

Problems336

Part Ⅲ Statistical Methods and Data Science339

Unit 12 Statistics and Data Science339

12.1 Statistics and Data Science（Ⅰ）340

12.1.1 What is Data Science340

12.1.2 Statistics and Data Science340

12.2 Statistics and Data Science（Ⅱ）343

12.2.1 Statistics as Part of Data Science343

12.2.2 The Modern Statistical Analysis Process344

12.2.3 Statistician and Data Scientist345

12.3 Statistical Thinking348

12.3.1 What is Statistical Thinking348

12.3.2 The Two Cultures of Statistical Modeling348

12.3.3 A New Research Community350

12.4 Distinguishing Analytics，Business Intelligence，Data Science352

12.4.1 Analytics352

12.4.2 Business Intelligence355

12.4.3 Data Science356

Reading English Materials359

Problems361

Commonly Used Statistical Terms362

Appendix A Commonly Used Statistical Tables367

Appendix B Summary of Univariate Descriptive Statistics and Graphs for the Four Level of Measurement379

Appendix C Order of Magnitude of Data380

References381