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In their bestselling MATHEMATICAL STATISTICS WITH APPLICATIONS, premiere authors Dennis Wackerly, William Mendenhall, and Richard L. Scheaffer present a solid foundation in statistical theory while conveying the relevance and importance of the theory in solving practical problems in the real world. The authors' use of practical applications and excellent exercises helps you discover the nature of statistics and understand its essential role in scientific research.
Table of Contents
Preface xiii
Note to the Student xxi
What Is Statistics? 1 (19)
Introduction 1 (2)
Characterizing a Set of Measurements: 3 (5)
Graphical Methods
Characterizing a Set of Measurements: 8 (5)
Numerical Methods
How Inferences Are Made 13 (1)
Theory and Reality 14 (1)
Summary 15 (5)
Probability 20 (66)
Introduction 20 (1)
Probability and Inference 21 (2)
A Review of Set Notation 23 (3)
A Probabilistic Model for an Experiment: 26 (9)
The Discrete Case
Calculating the Probability of an Event: 35 (5)
The Sample-Point Method
Tools for Counting Sample Points 40 (11)
Conditional Probability and the 51 (6)
Independence of Events
Two Laws of Probability 57 (5)
Calculating the Probability of an Event: 62 (8)
The Event-Composition Method
The Law of Total Probability and Bayes' 70 (5)
Rule
Numerical Events and Random Variables 75 (2)
Random Sampling 77 (2)
Summary 79 (7)
Discrete Random Variables and Their 86 (71)
Probability Distributions
Basic Definition 86 (1)
The Probability Distribution for a 87 (4)
Discrete Random Variable
The Expected Value of a Random Variable 91 (9)
or a Function of a Random Variable
The Binomial Probability Distribution 100 (14)
The Geometric Probability Distribution 114 (7)
The Negative Binomial Probability 121 (4)
Distribution (Optional)
The Hypergeometric Probability 125 (6)
Distribution
The Poisson Probability Distribution 131 (7)
Moments and Moment-Generating Functions 138 (5)
Probability-Generating Functions 143 (3)
(Optional)
Tchebysheff's Theorem 146 (3)
Summary 149 (8)
Continuous Variables and Their Probability 157 (66)
Distributions
Introduction 157 (1)
The Probability Distribution for a 158 (12)
Continuous Random Variable
Expected Values for Continuous Random 170 (4)
Variables
The Uniform Probability Distribution 174 (4)
The Normal Probability Distribution 178 (7)
The Gamma Probability Distribution 185 (9)
The Beta Probability Distribution 194 (7)
Some General Comments 201 (1)
Other Expected Values 202 (5)
Tchebysheff's Theorem 207 (3)
Expectations of Discontinuous Functions 210 (4)
and Mixed Probability Distributions
(Optional)
Summary 214 (9)
Multivariate Probability Distributions 223 (73)
Introduction 223 (1)
Bivariate and Multivariate Probability 224 (11)
Distributions
Marginal and Conditional Probability 235 (12)
Distributions
Independent Random Variables 247 (8)
The Expected Value of a Function of 255 (3)
Random Variables
Special Theorems 258 (6)
The Covariance of Two Random Variables 264 (6)
The Expected Value and Variance of Linear 270 (9)
Functions of Random Variables
The Multinomial Probability Distribution 279 (4)
The Bivariate Normal Distribution 283 (2)
(Optional)
Conditional Expectations 285 (5)
Summary 290 (6)
Functions of Random Variables 296 (50)
Introduction 296 (1)
Finding the Probability Distribution of a 297 (1)
Function of Random Variables
The Method of Distribution Functions 298 (12)
The Method of Transformations 310 (8)
The Method of Moment-Generating Functions 318 (7)
Multivariable Transformations Using 325 (8)
Jacobians (Optional)
Order Statistics 333 (8)
Summary 341 (5)
Sampling Distributions and the Central 346 (44)
Limit Theorem
Introduction 346 (7)
Sampling Distributions Related to the 353 (17)
Normal Distribution
The Central Limit Theorem 370 (7)
A Proof of the Central Limit Theorem 377 (1)
(Optional)
The Normal Approximation to the Binomial 378 (7)
Distribution
Summary 385 (5)
Estimation 390 (54)
Introduction 390 (2)
The Bias and Mean Square Error of Point 392 (4)
Estimators
Some Common Unbiased Point Estimators 396 (3)
Evaluating the Goodness of a Point 399 (7)
Estimator
Confidence Intervals 406 (5)
Large-Sample Confidence Intervals 411 (10)
Selecting the Sample Size 421 (4)
Small-Sample Confidence Intervals for 425 (9)
μ and μ1 -- μ2
Confidence Intervals for σ2 434 (3)
Summary 437 (7)
Properties of Point Estimators and Methods 444 (44)
of Estimation
Introduction 444 (1)
Relative Efficiency 445 (3)
Consistency 448 (11)
Sufficiency 459 (5)
The Rao-Blackwell Theorem and 464 (8)
Minimum-Variance Unbiased Estimation
The Method of Moments 472 (4)
The Method of Maximum Likelihood 476 (7)
Some Large-Sample Properties of 483 (2)
Maximum-Likelihood Estimators (Optional)
Summary 485 (3)
Hypothesis Testing 488 (75)
Introduction 488 (1)
Elements of a Statistical Test 489 (7)
Common Large-Sample Tests 496 (11)
Calculating Type II Error Probabilities 507 (4)
and Finding the Sample Size for Z Tests
Relationships Between Hypothesis-Testing 511 (2)
Procedures and Confidence Intervals
Another Way to Report the Results of a 513 (5)
Statistical Test: Attained Significance
Levels, or p-Values
Some Comments on the Theory of Hypothesis 518 (2)
Testing
Small-Sample Hypothesis Testing for μ 520 (10)
and μ1 -- μ2
Testing Hypotheses Concerning Variances 530 (10)
Power of Tests and the Neyman--Pearson 540 (9)
Lemma
Likelihood Ratio Tests 549 (7)
Summary 556 (7)
Linear Models and Estimation by Least 563 (77)
Squares
Introduction 564 (2)
Linear Statistical Models 566 (3)
The Method of Least Squares 569 (8)
Properties of the Least-Squares 577 (7)
Estimators: Simple Linear Regression
Inferences Concerning the Parameters 584 (5)
βi
Inferences Concerning Linear Functions of 589 (4)
the Model Parameters: Simple Linear
Regression
Predicting a Particular Value of Y by 593 (5)
Using Simple Linear Regression
Correlation 598 (6)
Some Practical Examples 604 (5)
Fitting the Linear Model by Using Matrices 609 (6)
Linear Functions of the Model Parameters: 615 (1)
Multiple Linear Regression
Inferences Concerning Linear Functions of 616 (6)
the Model Parameters: Multiple Linear
Regression
Predicting a Particular Value of Y by 622 (2)
Using Multiple Regression
A Test for H0 : βg+1 = βg+2 = . 624 (9)
. . = βk = 0
Summary and Concluding Remarks 633 (7)
Considerations in Designing Experiments 640 (21)
The Elements Affecting the Information in 640 (1)
a Sample
Designing Experiments to Increase Accuracy 641 (3)
The Matched-Pairs Experiment 644 (7)
Some Elementary Experimental Designs 651 (6)
Summary 657 (4)
The Analysis of Variance 661 (52)
Introduction 661 (1)
The Analysis of Variance Procedure 662 (5)
Comparison of More Than Two Means: 667 (4)
Analysis of Variance for a One-Way Layout
An Analysis of Variance Table for a 671 (6)
One-Way Layout
A Statistical Model for the One-Way Layout 677 (2)
Proof of Additivity of the Sums of 679 (2)
Squares and E(MST) for a One-Way Layout
(Optional)
Estimation in the One-Way Layout 681 (5)
A Statistical Model for the Randomized 686 (2)
Block Design
The Analysis of Variance for a Randomized 688 (7)
Block Design
Estimation in the Randomized Block Design 695 (1)
Selecting the Sample Size 696 (2)
Simultaneous Confidence Intervals for 698 (3)
More Than One Parameter
Analysis of Variance Using Linear Models 701 (4)
Summary 705 (8)
Analysis of Categorical Data 713 (28)
A Description of the Experiment 713 (1)
The Chi-Square Test 714 (2)
A Test of a Hypothesis Concerning 716 (5)
Specified Cell Probabilities: A
Goodness-of-Fit Test
Contingency Tables 721 (8)
r x c Tables with Fixed Row or Column 729 (5)
Totals
Other Applications 734 (2)
Summary and Concluding Remarks 736 (5)
Nonparametric Statistics 741 (55)
Introduction 741 (1)
A General Two-Sample Shift Model 742 (2)
The Sign Test for a Matched-Pairs 744 (6)
Experiment
The Wilcoxon Signed-Rank Test for a 750 (5)
Matched-Pairs Experiment
Using Ranks for Comparing Two Population 755 (3)
Distributions: Independent Random Samples
The Mann--Whitney U Test: Independent 758 (7)
Random Samples
The Kruskal--Wallis Test for the One-Way 765 (6)
Layout
The Friedman Test for Randomized Block 771 (6)
Designs
The Runs Test: A Test for Randomness 777 (6)
Rank Correlation Coefficient 783 (6)
Some General Comments on Nonparametric 789 (7)
Statistical Tests
Introduction to Bayesian Methods for 796 (25)
Inference
Introduction 796 (1)
Bayesian Priors, Posteriors, and 797 (11)
Estimators
Bayesian Credible Intervals 808 (5)
Bayesian Tests of Hypotheses 813 (3)
Summary and Additional Comments 816 (5)
Appendix 1 Matrices and Other Useful 821 (16)
Mathematical Results
Matrices and Matrix Algebra 821 (1)
Addition of Matrices 822 (1)
Multiplication of a Matrix by a Real 823 (1)
Number
Matrix Multiplication 823 (2)
Identity Elements 825 (2)
The Inverse of a Matrix 827 (1)
The Transpose of a Matrix 828 (1)
A Matrix Expression for a System of 828 (2)
Simultaneous Linear Equations
Inverting a Matrix 830 (4)
Solving a System of Simultaneous Linear 834 (1)
Equations
Other Useful Mathematical Results 835 (2)
Appendix 2 Common Probability Distributions, 837 (2)
Means, Variances, and Moment-Generating
Functions
Table 1 Discrete Distributions 837 (1)
Table 2 Continuous Distributions 838 (1)
Appendix 3 Tables 839 (38)
Table 1 Binomial Probabilities 839 (3)
Table 2 Table of e-x 842 (1)
Table 3 Poisson Probabilities 843 (5)
Table 4 Normal Curve Areas 848 (1)
Table 5 Percentage Points of the t 849 (1)
Distributions
Table 6 Percentage Points of the X2 850 (2)
Distributions
Table 7 Percentage Points of the F 852 (10)
Distributions
Table 8 Distribution Function of U 862 (6)
Table 9 Critical Values of T in the 868 (2)
Wilcoxon Matched-Pairs, Signed-Ranks
Test; n = 5(1)50
Table 10 Distribution of the Total Number 870 (2)
of Runs R in Samples of Size (n1, n2);
P(R ≤ a)
Table 11 Critical Values of Spearman's 872 (1)
Rank Correlation Coefficient
Table 12 Random Numbers 873 (4)
Answers to Exercises 877 (19)
Index 896
