Statistics II For Dummies

Deborah J. Rumsey, PhD, is a Statistics Education Specialist and Associated Professor in the Department of Statistics at The Ohio State University. She is also a Fellow of the American Statistical Association and has received the Presidential Teaching Award from Kansas State University. Dr. Rumsey has published numerous papers and given many professional presentations on the subject of statistics education.

Introduction 1

About This Book 1

Foolish Assumptions 3

Icons Used in This Book 3

Beyond the Book 4

Where to Go from Here 4

Part 1: Tackling Data Analysis and Model-Building Basics 7

Chapter 1: Beyond Number Crunching: The Art and Science of Data Analysis 9

Data Analysis: Looking before You Crunch 9

Nothing (not even a straight line) lasts forever 10

Data snooping isn’t cool 11

No (data) fishing allowed 12

Getting the Big Picture: An Overview of Stats II 13

Population parameter 13

Sample statistic 13

Confidence interval 14

Hypothesis test 14

Analysis of variance (ANOVA) 15

Multiple comparisons 15

Interaction effects 16

Correlation 16

Linear regression 17

Chi-square tests 18

Chapter 2: Finding the Right Analysis for the Job 21

Categorical versus Quantitative Variables 22

Statistics for Categorical Variables 23

Estimating a proportion 23

Comparing proportions 24

Looking for relationships between categorical variables 25

Building models to make predictions 26

Statistics for Quantitative Variables 27

Making estimates 27

Making comparisons 28

Exploring relationships 28

Predicting y using x 30

Avoiding Bias 31

Measuring Precision with Margin of Error 33

Knowing Your Limitations 35

Chapter 3: Having the Normal and Sampling Distributions in Your Back Pocket 37

Recognizing the VIP Distribution — the Normal 38

Characterizing the normal 38

Standardizing to the standard normal (Z-) distribution 38

Using the normal table 40

Finding probabilities for the normal distribution 41

Finally Getting Comfortable with Sampling Distributions 42

The mean and standard error of a sampling distribution 42

Sampling distribution of X 43

Sampling distribution of ˆp 44

Heads Up! Building Confidence Intervals and Hypothesis Tests 45

Confidence interval for the population mean 45

Confidence interval for the population proportion 46

Hypothesis test for population mean 46

Hypothesis test for the population proportion 47

Chapter 4: Reviewing Confidence Intervals and Hypothesis Tests 49

Estimating Parameters by Using Confidence Intervals 50

Getting the basics: The general form of a confidence interval 50

Finding the confidence interval for a population mean 51

What changes the margin of error? 52

Interpreting a confidence interval 55

What’s the Hype about Hypothesis Tests? 56

What Ho and Ha really represent 56

Gathering your evidence into a test statistic 57

Determining strength of evidence with a p-value 57

False alarms and missed opportunities: Type I and II errors 58

The power of a hypothesis test 60

Part 2: Using Different Types of Regression to Make Predictions 65

Chapter 5: Getting in Line with Simple Linear Regression 67

Exploring Relationships with Scatterplots and Correlations 68

Using scatterplots to explore relationships 69

Collating the information by using the correlation coefficient 70

Building a Simple Linear Regression Model 71

Finding the best-fitting line to model your data 72

The y-intercept of the regression line 73

The slope of the regression line 74

Making point estimates by using the regression line 75

No Conclusion Left Behind: Tests and Confidence Intervals for Regression 75

Scrutinizing the slope 76

Inspecting the y-intercept 78

Building confidence intervals for the average response 80

Making the band with prediction intervals 81

Checking the Model’s Fit (The Data, Not the Clothes!) 83

Defining the conditions 84

Finding and exploring the residuals 85

Using r2 to measure model fit 89

Scoping for outliers 90

Knowing the Limitations of Your Regression Analysis 92

Avoiding slipping into cause-and-effect mode 92

Extrapolation: The ultimate no-no 93

Sometimes you need more than one variable 94

Chapter 6: Multiple Regression with Two X Variables 95

Getting to Know the Multiple Regression Model 96

Discovering the uses of multiple regression 96

Looking at the general form of the multiple regression model 96

Stepping through the analysis 97

Looking at x’s and y’s 97

Collecting the Data 98

Pinpointing Possible Relationships 100

Making scatterplots 100

Correlations: Examining the bond 101

Checking for Multicolinearity 104

Finding the Best-Fitting Model for Two x Variables 105

Getting the multiple regression coefficients 106

Interpreting the coefficients 107

Testing the coefficients 108

Predicting y by Using the x Variables 110

Checking the Fit of the Multiple Regression Model 111

Noting the conditions 112

Plotting a plan to check the conditions 112

Checking the three conditions 114

Chapter 7: How Can I Miss You If You Won’t Leave? Regression Model Selection 117

Getting a Kick out of Estimating Punt Distance 118

Brainstorming variables and collecting data 118

Examining scatterplots and correlations 120

Just Like Buying Shoes: The Model Looks Nice, But Does It Fit? 123

Assessing the fit of multiple regression models 124

Model selection procedures 125

Chapter 8: Getting Ahead of the Learning Curve with Nonlinear Regression 129

Anticipating Nonlinear Regression 130

Starting Out with Scatterplots 131

Handling Curves in the Road with Polynomials 133

Bringing back polynomials 134

Searching for the best polynomial model 136

Using a second-degree polynomial to pass the quiz 138

Assessing the fit of a polynomial model 141

Making predictions 143

Going Up? Going Down? Go Exponential! 145

Recollecting exponential models 145

Searching for the best exponential model 146

Spreading secrets at an exponential rate 148

Chapter 9: Yes, No, Maybe So: Making Predictions by Using Logistic Regression 153

Understanding a Logistic Regression Model 154

How is logistic regression different from other regressions? 154

Using an S-curve to estimate probabilities 155

Interpreting the coefficients of the logistic regression model 156

The logistic regression model in action 157

Carrying Out a Logistic Regression Analysis 158

Running the analysis in Minitab 158

Finding the coefficients and making the model 160

Estimating p 161

Checking the fit of the model 162

Fitting the movie model 162

Part 3: Analyzing Variance with Anova 167

Chapter 10: Testing Lots of Means? Come On Over to ANOVA! 169

Comparing Two Means with a t-Test 170

Evaluating More Means with ANOVA 171

Spitting seeds: A situation just waiting for ANOVA 172

Walking through the steps of ANOVA 173

Checking the Conditions 174

Verifying independence 174

Looking for what’s normal 174

Taking note of spread 176

Setting Up the Hypotheses 178

Doing the F-Test 179

Running ANOVA in Minitab 180

Breaking down the variance into sums of squares 180

Locating those mean sums of squares 182

Figuring the F-statistic 183

Making conclusions from ANOVA 184

What’s next? 186

Checking the Fit of the ANOVA Model 186

Chapter 11: Sorting Out the Means with Multiple Comparisons 189

Following Up after ANOVA 190

Comparing cellphone minutes: An example 190

Setting the stage for multiple comparison procedures 192

Pinpointing Differing Means with Fisher and Tukey       .193

Fishing for differences with Fisher’s LSD 194

Separating the turkeys with Tukey’s test 197

Examining the Output to Determine the Analysis 198

So Many Other Procedures, So Little Time! 199

Controlling for baloney with the Bonferroni adjustment 200

Comparing combinations by using Scheffé’s method 201

Finding out whodunit with Dunnett’s test 202

Staying cool with Student Newman-Keuls 202

Duncan’s multiple range test 202

Chapter 12: Finding Your Way through Two-Way ANOVA 205

Setting Up the Two-Way ANOVA Model 206

Determining the treatments 206

Stepping through the sums of squares 207

Understanding Interaction Effects 209

What is interaction, anyway? 209

Interacting with interaction plots 210

Testing the Terms in Two-Way ANOVA             .213

Running the Two-Way ANOVA Table 214

Interpreting the results: Numbers and graphs 214

Are Whites Whiter in Hot Water? Two-Way ANOVA Investigates 217

Chapter 13: Regression and ANOVA: Surprise Relatives! 221

Seeing Regression through the Eyes of Variation 222

Spotting variability and finding an “x-planation” 222

Getting results with regression 223

Assessing the fit of the regression model 225

Regression and ANOVA: A Meeting of the Models 226

Comparing sums of squares 226

Dividing up the degrees of freedom 228

Bringing regression to the ANOVA table 229

Relating the F- and t-statistics: The final frontier 230

Part 4: Building Strong Connections with Chi-Square Tests and Nonparametrics 233

Chapter 14: Forming Associations with Two-Way Tables 235

Breaking Down a Two-Way Table 236

Organizing data into a two-way table 236

Filling in the cell counts 237

Making marginal totals 238

Breaking Down the Probabilities 239

Marginal probabilities 239

Joint probabilities 241

Conditional probabilities 242

Trying To Be Independent 247

Checking for independence between two categories 247

Checking for independence between two variables 249

Demystifying Simpson’s Paradox 250

Experiencing Simpson’s Paradox 250

Figuring out why Simpson’s Paradox occurs 253

Keeping one eye open for Simpson’s Paradox 254

Chapter 15: Being Independent Enough for the Chi-Square Test 257

The Chi-Square Test for Independence 258

Collecting and organizing the data 259

Determining the hypotheses 261

Figuring expected cell counts 261

Checking the conditions for the test 262

Calculating the Chi-square test statistic 263

Finding your results on the Chi-square table 266

Drawing your conclusions 269

Putting the Chi-square to the test 271

Comparing Two Tests for Comparing Two Proportions 272

Getting reacquainted with the Z-test for two population proportions 273

Equating Chi-square tests and Z-tests for a two-by-two table 274

Chapter 16: Using Chi-Square Tests for Goodness-of-Fit (Your Data, Not Your Jeans) 279

Finding the Goodness-of-Fit Statistic 280

What’s observed versus what’s expected 280

Calculating the goodness-of-fit statistic 282

Interpreting the Goodness-of-Fit Statistic Using a Chi-Square 284

Checking the conditions before you start 285

The steps of the Chi-square goodness-of-fit test 286

Chapter 17: Rebels Without a Distribution — Nonparametric Procedures 291

Arguing for Nonparametric Statistics 292

No need to fret if conditions aren’t met 292

The median’s in the spotlight for a change 293

So, what’s the catch? 295

Mastering the Basics of Nonparametric Statistics 296

Sign 296

Chapter 18: All Signs Point to the Sign Test 299

Reading the Signs: The Sign Test 300

Testing the median in real estate 302

Estimating the median 304

Testing matched pairs 306

Part 5: Putting it all Together: Multi-Stage Analysis of A Large Data Set 309

Chapter 19: Conducting a Multi-Stage Analysis of a Large Data Set 311

Steps Involved in Working with a Large Data Set 311

Wrangling Data 313

Discovery 313

Structuring 314

Cleaning 315

Enriching 315

Validating 316

Publishing 317

Visualizing Data 317

Exploring the Data 319

Looking for Relationships 319

Building Models and Making Inferences 320

Sharing the Story 321

Who is the audience? 322

Make an outline 322

Include an executive summary 323

Check your writing 323

Chapter 20: A Statistician Watches the Movies 325

Examining the Movie Variables and Asking Questions 326

Visualizing the Movie Data 327

Categorical movie variables 328

Quantitative movie variables 329

Doing Descriptive Dirty Work 332

Looking for Relationships 333

Relationships between quantitative movie variables 333

Relationships between two categorical variables 337

Relationships between quantitative and categorical variables 338

Building a Model for Predicting U.S Revenue 340

Writing It Up 343

Chapter 21: Looking Inside the Refrigerator 347

Refrigerator Data — The Variables 348

Exploring the Data 348

Analyzing the Data 350

Writing It Up 358

Part 6: The Part of Tens 361

Chapter 22: Ten Common Errors in Statistical Conclusions 363

Claiming These Statistics Prove 363

It’s Not Technically Statistically Significant, But 364

Concluding That x Causes y 365

Assuming the Data Was Normal 366

Only Reporting “Important” Results 366

Assuming a Bigger Sample Is Always Better 367

It’s Not Technically Random, But 369

Assuming That 1,000 Responses Is 1,000 Responses 369

Of Course the Results Apply to the General Population 371

Deciding Just to Leave It Out 372

Chapter 23: Ten Ways to Get Ahead by Knowing Statistics 375

Asking the Right Questions 375

Being Skeptical 376

Collecting and Analyzing Data Correctly 377

Calling for Help 378

Retracing Someone Else’s Steps 379

Putting the Pieces Together 379

Checking Your Answers 380

Explaining the Output 381

Making Convincing Recommendations 382

Establishing Yourself as the Statistics Go-To Person 383

Chapter 24: Ten Cool Jobs That Use Statistics 385

Pollster 386

Data Scientist 387

Ornithologist (Bird Watcher) 387

Sportscaster or Sportswriter 388

Journalist 390

Crime Fighter 390

Medical Professional 391

Marketing Executive 392

Lawyer 393

Appendix A: Reference Tables 395

Index 409