Statistical Methods by Wilson

 Author: Rudolf J. Freund & William J. Wilson Published in: Academic Press Release Year: 2003 ISBN: 0-12-267651-3 Pages: 694 Edition: Second Edition File Size: 3 MB File Type: pdf Language: English

Description of Statistical Methods

The objective of Statistical Methods, Second Edition, is to provide students with a working introduction to statistical methods. Courses using statistical Methods books are normally taken by advanced undergraduate statistics students and graduate students from various disciplines. Statistical Method is an upper-level requirement for undergraduate degrees in disciplines emphasizing quantitative skills, or a requirement for graduate degrees in disciplines where statistics is an important research tool.
Statistical Methods book is intended to be used for this type of course. The material statistical Methods book provides an overview of a wide range of applications and normally requires two semesters, although a limited knowledge of statistical methods is provided in the first semester. Many students will continue with several additional courses in specialized statistics applications.
Traditionally, textbooks used for statistical methods courses have emphasized plugging numbers into formulas, with computer usage as an afterthought. This approach has led to much mind-numbing drill, which obscures the real issues. The increased usage of computers and the availability of comprehensive statistical software packages would seem to imply that statistical methods should now be taught in terms of implementing such software.
This approach is likely to make the computer appear as a black box into which one pours data files and automatically receives the correct answers. However, a computer does not know whether it is doing the correct analysis and is capable of a beautifully annotated execution of incorrect analysis. Also, a computer cannot interpret results and write a report.

Content of Statistical Methods

1 DATA AND STATISTICS 1
1.1 Introduction 1
Data Sources 4
Using Computer 5
1.2 Observations and Variables 6
1.3 Types of Measurements for Variables 10
1.4 Distributions 12
Graphical Representation of Distributions 14
1.5 Numerical Descriptive Statistics 19
Location 20
Dispersion 23
Other Measures 28
Computing the Mean and Standard Deviation
from a Frequency Distribution 30
Change of Scale 30
1.6 Exploratory Data Analysis 32
The Stem and Leaf Plot 32
The Box Plot 35
1.7 Bivariate Data 38
Categorical Variables 39
Categorical and Interval Variables 40
Interval Variables 42
1.8 Populations, Samples, and Statistical Inference
A Preview 42
1.9 Chapter Summary 44
Summary 48
1.10 Chapter Exercises 49
Concept Questions 49
Practice Exercises 51
Exercises 52

2 PROBABILITY AND SAMPLING DISTRIBUTIONS 62

2.1 Introduction 63
Chapter Preview 65
2.2 Probability 66
Definitions and Concepts 66
System Reliability 70
Random Variables 71
2.3 Discrete Probability Distributions 73
Properties of Discrete Probability Distributions 74
Descriptive Measures for Probability Distributions 74
The Discrete Uniform Distribution 76
The Binomial Distribution 77
The Poisson Distribution 79
2.4 Continuous Probability Distributions 81
Characteristics of a Continuous Probability Distribution 81
The Continuous Uniform Distribution 82
The Normal Distribution 83
Calculating Probabilities Using the Table
of the Normal Distribution 86
2.5 Sampling Distributions 91
Sampling Distribution of the Mean 92
The usefulness of the Sampling Distribution 96
Sampling Distribution of a Proportion of 99
2.6 Other Sampling Distributions 101
The χ2 Distribution 102
Distribution of the Sample Variance 103
The t Distribution 104
Using the t Distribution 105
The F Distribution 106
Using of the F Distribution 106
Relationships among the Distributions 108
2.7 Chapter Summary 108
2.8 Chapter Exercises 109
Concept Questions 109
Practice Exercises 109
Exercises 110

3 PRINCIPLES OF INFERENCE 117

3.1 Introduction 117
3.2 Hypothesis Testing 118
General Considerations 119
The Hypotheses 120
Rules for Making Decisions 121
Possible Errors in Hypothesis Testing 122
Probabilities of Making Errors 123
Choosing between α and β 125
Five-Step Procedure for Hypothesis Testing 125
Why Do We Focus on the Type I Error? 126
Choosing α 127
The Five Steps for Example 3.3 131
P Values 132
Type II Error and Power 134
Power 136
Uniformly Most Powerful Tests 137
One-Tailed Hypothesis Tests 138
3.3 Estimation of 139
Interpreting the Confidence Coefficient of 141
Relationship between Hypothesis Testing
and Confidence Intervals 143

3.4 Sample Size 144
3.5 Assumptions 147
Statistical Significance versus Practical Significance 148
3.6 Chapter Summary 150
3.7 Chapter Exercises 152
Concept Questions 152
Practice Exercises 153
Multiple Choice Questions 154
Exercises 155

4 INFERENCES ON A SINGLE POPULATION 159
4.1 Introduction 159
4.2 Inferences on the Population Mean 161
Hypothesis Test on μ 161
Estimation of μ 164
Sample Size 165
Degrees of Freedom 166
4.3 Inferences on a Proportion 166
Hypothesis Test on p 167
Estimation of p 168
Sample Size 169
4.4 Inferences on the Variance of One Population 169
Hypothesis Test on σ2 170
Estimation of σ2 171
4.5 Assumptions 172
Required Assumptions and Sources of Violations 173
Prevention of Violations 173
Detection of Violations 173
Tests for Normality 175
If Assumptions Fail 176
Alternate Methodology 177
4.6 Chapter Summary 179
4.7 Chapter Exercises 180
Concept Questions 180
Practice Exercises 180
Exercises 181

5 INFERENCES FOR TWO POPULATIONS 185
5.1 Introduction 185
5.2 Inferences on the Difference between Means
Using Independent Samples 188
Sampling Distribution of a Linear Function
of Random Variables 188
The Sampling Distribution of the Difference
between Two Means 188
Variances Known 189
Variances Unknown but Assumed Equal 191
The Pooled Variance Estimate 191
The “Pooled” t-Test 192
Variances Unknown but Not Equal 194
5.3 Inferences on Variances 197
5.4 Inferences on Means for Dependent Samples 200
5.5 Inferences on Proportions 205
Comparing Proportions Using Independent Samples 205
Comparing Proportions Using Paired Samples 207
5.6 Assumptions and Remedial Methods 208
5.7 Chapter Summary 211
5.8 Chapter Exercises 213
Concept Questions 213
Practice Exercises 214
Exercises 215

6 INFERENCES FOR TWO OR MORE MEANS 219
6.1 Introduction 219
Using Computer 220
6.2 Analysis of Variance 221
Notation and Definitions 222
Heuristic Justification for the Analysis of Variance 225
Computational Formulas and the Partitioning
of Sums of Squares 228
The Sum of Squares among Means 228
The Sum of Squares within Groups 229
The Ratio of Variances 229
Partitioning of the Sums of Squares 229

6.3 The Linear Model 232
The Linear Model for a Single Population 232
The Linear Model for Several Populations 233
The Analysis of Variance Model 233
Fixed and Random Effects Model 234
The Hypotheses 234
Expected Mean Squares 235
Notes on Exercises 236
6.4 Assumptions 236
Assumptions Required 236
Detection of Violated Assumptions 237
The Hartley F -Max Test 238
Violated Assumptions 239
Variance Stabilizing Transformations 239
Notes on Exercises 242
6.5 Specific Comparisons 242
Contrasts 243
Orthogonal Contrasts 246
Fitting Trends 249
Lack of Fit Test 252
Notes on Exercises 253
Computer Hint 253
Post Hoc Comparisons 253
Confidence Intervals 263
6.6 Random Models 267
6.7 Unequal Sample Sizes 270
6.8 Analysis of Means 270
ANOM for Proportions 273
Analysis of Means for Count Data 275
6.9 Chapter Summary 277
6.10 Chapter Exercises 279
Concept Questions 279
Exercises 280

7 LINEAR REGRESSION 287
7.1 Introduction 287
Notes on Exercises 290
7.2 The Regression Model 290
7.3 Estimation of Parameters β0 and β1 294
A Note on Least Squares 297
7.4 Estimation of σ2 and the Partitioning of Sums of Squares 297
7.5 Inferences for Regression 301
The Analysis of Variance Test for β1 301
The (Equivalent) t-Test for β1 302
Confidence Interval for β1 303
Inferences on the Response Variable 304
7.6 Using the Computer 312
7.7 Correlation 316
7.8 Regression Diagnostics 319
7.9 Chapter Summary 324
7.10 Chapter Exercises 326
Concept Questions 326
Exercises 327

8 MULTIPLE REGRESSION 333
Notes on Exercises 335
8.1 The Multiple Regression Model 336
The Partial Regression Coefficient of 337
8.2 Estimation of Coefficients 338
Simple Linear Regression with Matrices 339
Estimating the Parameters of a Multiple Regression Model 343
Correcting for the Mean, an Alternative Calculating Method 344
8.3 Inferential Procedures 351
Estimation of σ2 and the Partitioning of the Sums of Squares 351
The Coefficient of Variation 352
Inferences for Coefficients 353
Tests Normally Provided by Computer Outputs 355
The Equivalent t Statistic for Individual Coefficients 358
Inferences on the Response Variable 359
8.4 Correlations 362
Multiple Correlation 363
How Useful Is the R2 Statistic? 363
Partial Correlation of 364
8.5 Using the Computer 366
8.6 Special Models 370
The Polynomial Model 370
The Multiplicative Model 374
Nonlinear Models 378
8.7 Multicollinearity 379
Redefining Variables 382
Other Methods 383
8.8 Variable Selection 384
Other Selection Procedures 387
8.9 Detection of Outliers, Row Diagnostics 388
8.10 Chapter Summary 395
8.11 Chapter Exercises 399
Concept Questions 399
Exercises 400

9 FACTORIAL EXPERIMENTS 417
9.1 Introduction 417
9.2 Concepts and Definitions 419
9.3 The Two-Factor Factorial Experiment 422
The Linear Model 422
Notation 423
Computations for the Analysis of Variance 424
Between Cells Analysis 424
The Factorial Analysis 425
Expected Mean Squares 426
Notes on Exercises 431
9.4 Specific Comparisons 431
Preplanned Contrasts 432
Computing Contrast Sums of Squares 432
Polynomial Responses 435
Lack of Fit Test 442
Multiple Comparisons 443
9.5 No Replications 448
9.6 Three or More Factors 448
9.7 Chapter Summary 451
9.8 Chapter Exercises 454
Exercises 454

10 DESIGN OF EXPERIMENTS 461
10.1 Introduction 462
Notes on Exercises 463
10.2 The Randomized Block Design 464
The Linear Model 466
Relative Efficiency 469
Random Treatment Effects in the Randomized Block Design 470
10.3 Randomized Blocks with Sampling 471
10.4 Latin Square Design 476
10.5 Other Designs 480
Factorial Experiments in a Randomized Block Design 481
Nested Designs 484
Split Plot Designs 488
10.6 Chapter Summary 492
10.7 Chapter Exercises 498
Exercises 498

11 OTHER LINEAR MODELS 508
11.1 Introduction 508
11.2 The Dummy Variable Model 510
11.3 Unbalanced Data 514
11.4 Computer Implementation of the Dummy Variable Model 516
11.5 Models with Dummy and Interval Variables 517
Analysis of Covariance 518
Multiple Covariates 522
Unequal Slopes 524
11.6 Extensions to Other Models 526
11.7 Binary Response Variables 527
Linear Model with a Dichotomous Dependent Variable 528
Weighted Least Squares 530
Logistic Regression 536
Other Methods 540
11.8 Chapter Summary 542
An Example of Extremely Unbalanced Data 543
11.9 Chapter Exercises 547
Exercises 547

12 CATEGORICAL DATA 557
12.1 Introduction 557
12.2 Hypothesis Tests for a Multinomial Population 558
12.3 Goodness of Fit Using the χ2 Test 561
Test for a Discrete Distribution 561
Test for a Continuous Distribution 562
12.4 Contingency Tables 564
Computing the Test Statistic 565
Test for Homogeneity 566
Test for Independence 568
Measures of Dependence 570
Other Methods 570
12.5 Log-linear Model 571
12.6 Chapter Summary 575
12.7 Chapter Exercises 576
Exercises 576

13 NONPARAMETRIC METHODS 581
13.1 Introduction 581
13.2 One Sample 586
13.3 Two Independent Samples 588
13.4 More Than Two Samples 590
13.5 Randomized Block Design 593
13.6 Rank Correlation 595
13.7 Chapter Summary 597
13.8 Chapter Exercises 599
Exercises 599

14 SAMPLING AND SAMPLE SURVEYS 602
14.1 Introduction 602
14.2 Some Practical Considerations 604
14.3 Simple Random Sampling 606
Notation 606
Sampling Procedure 607
Estimation of 607
Systematic Sampling of 608
Sample Size 608
14.4 Stratified Sampling 609
Estimation of 609
Sample Sizes 610
Efficiency of 612
An Example 612
Additional Topics in Stratified Sampling 615
14.5 Other Topics 616
14.6 Chapter Summary 617

APPENDIX A 618
A.1 The Normal Distribution—Probabilities Exceeding Z 618
A.1A Selected Probability Values for the Normal Distribution—
Values of Z Exceeded with Given Probability 622
A.2 The t Distribution—Values of t Exceeded with
Given Probability 623
A.3 χ2 Distribution—χ2 Values Exceeded
with Given Probability 624
A.4 The F Distribution, p = 0.1 625
A.4A The F Distribution, p = 0.05 627
A.4B The F Distribution, p = 0.025 629
A.4C The F Distribution, p = 0.01 631
A.4D The F Distribution, p = 0.005 633
A.5 The Fmax Distribution—Percentage Points
of Fmax = s2
max/s2
min 635
A.6 Orthogonal Polynomials (Tables of Coefficients
for Polynomial Trends) 636
A.7 Percentage Points of the Studentized Range 637
A.8 Percentage Points of the Duncan Multiple Range Test 639
A.9 Critical Values for the Wilcoxon
Signed Rank Test N = 5(1)50 641
A.10 The Mann–Whitney Two-Sample Test 642
A.11 Exact Critical Values for Use with the Analysis of Means 643

APPENDIX B A BRIEF INTRODUCTION TO MATRICES 645
Matrix Algebra 646
Solving Linear Exercises 649
REFERENCES 651

SOLUTIONS TO SELECTED EXERCISES 656

INDEX 668