Statistical Methods

ContentsPreface 1 DATA AND STATISTICS 1.1 Introduction Data Sources Using the Computer 1.2 Observations and Variables 1.3 Types of Measurements for Variables 1.4 Distributions Graphical Representation of Distributions 1.5 Numerical Descriptive Statistics Location Dispersion Other Measures Computing the Mean and Standard Deviationfrom a Frequency Distribution Change of Scale 1.6 Exploratory Data Analysis The Stem and Leaf Plot The Box Plot Comments 1.7 Bivariate Data Categorical Variables Categorical and Interval Variables Interval Variables 1.8 Populations, Samples, and Statistical Inference — A Preview 1.9 Chapter Summary Summary 1.10 Chapter Exercises Concept Questions Practice Exercises Exercises 2 PROBABILITY AND SAMPLING DISTRIBUTIONS 2.1 Introduction Chapter Preview 2.2 Probability Definitions and Concepts System Reliability Random Variables 2.3 Discrete Probability Distributions Properties of Discrete Probability Distributions Descriptive Measures for Probability Distributions The Discrete Uniform Distribution The Binomial Distribution The Poisson Distribution 2.4 Continuous Probability Distributions Characteristics of a Continuous Probability Distribution The Continuous Uniform Distribution The Normal Distribution Calculating Probabilities Using the Tableof the Normal Distribution 2.5 Sampling Distributions Sampling Distribution of the Mean Usefulness of the Sampling Distribution Sampling Distribution of a Proportion 2.6 Other Sampling Distributions The ÷2 Distribution Distribution of the Sample Variance The t Distribution Using the t Distribution The F Distribution Using of the F Distribution Relationships among the Distributions 2.7 Chapter Summary 2.8 Chapter Exercises Concept Questions Practice Exercises Exercises 3 PRINCIPLES OF INFERENCE 3.1 Introduction 3.2 Hypothesis Testing General Considerations The Hypotheses Rules for Making Decisions Possible Errors in Hypothesis TestingProbabilities of Making Errors The Trade-off between á and â Five-Step Procedure for Hypothesis Testing Why Do We Focus on the Type I Error? The Trade-off between á and â The Five Steps for Example 3.3 P Values Type II Error and Power Power Uniformly Most Powerful Tests One-Tailed Hypothesis Tests 3.3 Estimation Interpreting the Confidence Coefficient Relationship between Hypothesis Testingand Confidence Intervals 3.4 Sample Size 3.5 Assumptions Statistical Significance versus Practical Significance 3.6 Chapter Summary 3.7 Chapter Exercises Concept Questions Practice Exercises Multiple Choice Questions Exercises 4 INFERENCES ON A SINGLE POPULATION 4.1 Introduction 4.2 Inferences on the Population Mean Hypothesis Test on µ Estimation of µ Sample Size Degrees of Freedom 4.3 Inferences on a Proportion Hypothesis Test on p Estimation of p Sample Size 4.4 Inferences on the Variance of One Population Hypothesis Test on ó2 Estimation of ó2 4.5 Assumptions Required Assumptions and Sources of Violations Prevention of Violations Detection of Violations Tests for Normality If Assumptions Fail Alternate Methodology 4.6 Chapter Summary 4.7 Chapter Exercises Concept Questions Practice Exercises Exercises 5 INFERENCES FOR TWO POPULATIONS 5.1 Introduction 5.2 Inferences on the Difference between MeansUsing Independent Samples Sampling Distribution of a Linear Functionof Random Variables The Sampling Distribution of the Differencebetween Two Means Variances Known Variances UnKnown but Assumed Equal The Pooled Variance Estimate The "Pooled" t Test Variances Unknown but Not Equal 5.3 Inferences on Variances 5.4 Inferences on Means for Dependent Samples 5.5 Inferences on Proportions Comparing Proportions Using Independent Samples Comparing Proportions Using Paired Samples 5.6 Assumptions and Remedial Methods 5.7 Chapter Summary 5.8 Chapter Exercises Concept Questions Practice Exercises Exercises 6 INFERENCES FOR TWO OR MORE MEANS 6.1 Introduction Using the Computer 6.2 The Analysis of Variance Notation and Definitions Heuristic Justification for the Analysis of Variance Computational Formulas and the Partitioningof Sums of Squares The Sum of Squares among Means The Sum of Squares within Groups The Ratio of Variances Partitioning of the Sums of Squares 6.3 The Linear Model The Linear Model for a Single Population The Linear Model for Several Populations The Analysis of Variance Model Fixed and Random Effects Model The Hypotheses Expected Mean Squares Notes on Exercises 6.4 Assumptions Assumptions Required Detection of Violated Assumptions The Hartley F-Max Test Violated Assumptions Variance Stabilizing Transformations Notes on Exercises 6.5 Specific Comparisons Contrasts Orthogonal Contrasts Fitting Trends Lack of Fit Test Notes on Exercises Computer Hint Post Hoc Comparisons Comments Confidence Intervals 6.6 Random Models 6.7 Unequal Sample Sizes 6.8 Analysis of Means ANOM for Proportions Analysis of Means for Count Data 6.9 Chapter Summary 6.10 Chapter Exercises Concept Questions Exercises 7 LINEAR REGRESSION 7.1 Introduction Notes on Exercises 7.2 The Regression Model 7.3 Estimation of Parameters â0 and â1 A Note on Least Squares 7.4 Estimation of ó2 and the Partitioning of Sums of Squares 7.5 Inferences for Regression The Analysis of Variance Test for â1 The (Equivalent) t Test for â1 Confidence Interval for â1 Inferences on the Response Variable 7.6 Using the Computer 7.7 Correlation 7.8 Regression Diagnostics 7.9 Chapter Summary 7.10 Chapter Exercises Concept Questions Exercises 8 MULTIPLE REGRESSION Notes on Exercises 8.1 The Multiple Regression Model The Partial Regression Coefficient 8.2 Estimation of Coefficients Simple Linear Regression with Matrices Estimating the Parameters of a Multiple Regression Model Correcting for the Mean, an Alternative Calculating Method 8.3 Inferential Procedures Estimation of ó2 and the Partitioning of the Sums of Squares The Coefficient of Variation Inferences for Coefficients Tests Normally Provided by Computer Outputs The Equivalent t Statistic for Individual Coefficients Inferences on the Response Variable 8.4 Correlations Multiple Correlation How Useful is the R2 Statistic? Partial Correlation 8.5 Using the Computer 8.6 Special Models The Polynomial Model The Multiplicative Model Nonlinear Models 8.7 Multicollinearity Redefining Variables Other Methods 8.8 Variable Selection Other Selection Procedures 8.9 Detection of Outliers, Row Diagnostics 8.10 Chapter Summary 8.11 Chapter Exercises Concept Questions Exercises 9 FACTORIAL EXPERIMENTS 9.1 Introduction 9.2 Concepts and Definitions 9.3 The Two-Factor Factorial Experiment The Linear Model Notation Computations for the Analysis of Variance Between Cells Analysis The Factorial Analysis Expected Mean Squares Notes on Exercises 9.4 Specific Comparisons Preplanned Contrasts Computing Contrast Sums of Squares Polynomial Responses Lack of Fit Test Multiple Comparisons 9.5 No Replications 9.6 Three or More Factors Additional Considerations 9.7 Chapter Summary 9.8 Chapter Exercises Exercises 10 DESIGN OF EXPERIMENTS 10.1 Introduction Notes on Exercises 10.2 The Randomized Block Design The Linear Model Relative Efficiency Random Treatment Effects in the Randomized Block Design 10.3 Randomized Blocks with Sampling 10.4 Latin Square Design 10.5 Other Designs Factorial Experiments in a Randomized Block Design Nested Designs Split Plot Designs Additional Topics 10.6 Chapter Summary 10.7 Chapter Exercises Exercises 11 OTHER LINEAR MODELS 11.1 Introduction 11.2 The Dummy Variable Model 11.3 Unbalanced Data 11.4 Computer Implementation of the Dummy Variable Model 11.5 Models with Dummy and Interval Variables Analysis of Covariance Multiple Covariates Unequal Slopes 11.6 Extensions to Other Models 11.7 Binary Response Variables Linear Model with a Dichotomous Dependent Variable Weighted Least Squares Logistic Regression Other Methods 11.8 Chapter Summary An Example of Extremely Unbalanced Data Chapter Summary 11.9 Chapter Exercises Exercises 12 CATEGORICAL DATA 12.1 Introduction 12.2 Hypothesis Tests for a Multinomial Population 12.3 Goodness of Fit Using the ÷2 Test Test for a Discrete Distribution Test for a Continuous Distribution 12.4 Contingency Tables Computing the Test Statistic Test for Homogeneity Test for Independence Measures of Dependence Other Methods 12.5 Log linear Model 12.6 Chapter Summary 12.7 Chapter Exercises Exercises 13 NONPARAMETRIC METHODS 13.1 Introduction 13.2 One Sample 13.3 Two Independent Samples 13.4 More Than Two Samples 13.5 Randomized Block Design 13.6 Rank Correlation 13.7 Chapter Summary 13.8 Chapter Exercises Exercises 14 SAMPLING AND SAMPLE SURVEYS 14.1 Introduction 14.2 Some Practical Considerations 14.3 Simple Random Sampling Notation Sampling Procedure Estimation Systematic Sampling Sample Size 14.4 Stratified Sampling Estimation Sample Sizes Efficiency An Example Additional Topics in Stratified Sampling 14.5 Other Topics 14.6 Chapter Summary APPENDIX A A.1 The Normal Distribution—Probabilities Exceeding Z A.1A Selected Probability Values for the Normal Distribution—Values of Z Exceeded with Given Probability A.2 The t Distribution—Values of t Exceeded withGiven Probability A.3 ÷2 Distribution—÷2 Values Exceededwith Given Probability A.4 The F Distribution, p= 0.1 A.4A The F Distribution, p = 0.05 A.4B The F Distribution, p = 0.025 A.4C The F Distribution, p = 0.01 A.4D The F Distribution, p = 0.005 A.5 The Fmax Distribution—Percentage Pointsof Fmax = s2max/s2min A.6 Orthogonal Polynomials (Tables of Coefficientsfor Polynomial Trends) A.7 Percentage Points of the Studentized Range A.8 Percentage Points of the Duncan Multiple-Range Test A.9 Critical Values for the WilcoxonSigned Rank Test N= 5(1)50 A.10 The Mann-Whitney Two-Sample Test A.11 Exact Critical Values for use with the Analysis of Means APPENDIX B A BRIEF INTRODUCTION TO MATRICES Matrix Algebra Solving Linear Exercises REFERENCES SOLUTIONS TO SELECTED EXERCISES INDEX