PREFACE 

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CHAPTER 1: QUICK SUMMARY 

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1.1 Overview of this book 


1  (1) 


2  (2) 


4  (1) 

1.4 Quick summary: choosing and performing an ANOVA 


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CHAPTER 2: UNDERSTANDING THE BASICS 

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2.1 The basic logic and assumptions of ANOVA 


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2.1.1 A 'model' that describes and predicts some data 


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2.1.2 An example: data and a structural model 


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2.1.3 The null hypothesis 


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2.1.4 The assumptions of ANOVA 


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10  (2) 

2.1.6 Expected mean squares (EMS) 


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2.2 The calculations behind a oneway ANOVA (one BS factor) 


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2.2.1 Calculations using means (preferred) or totals 


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2.2.2 Sums of squares: calculating SStotal, SStreatment, and SSerror 


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16  (1) 


17  (1) 


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2.2.6 ANOVA summary table 


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2.2.7 SStreatment for unequal sample sizes 


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2.2.8 Pictorial representation 


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2.3 Regression ANOVA: the other way to understand the logic 


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2.3.1 Linear regression in terms of sums of squares 


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2.3.2 Pictorial representation 


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2.3.3 Linear regression as an ANOVA 


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2.4 Factors versus covariates 


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2.5 Assumptions of ANOVA involving covariates 


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2.6 ANOVA with two betweensubjects factors 


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2.6.1 Main effects, interactions, simple effects, and a structural model 


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2.6.2 Expected mean squares 


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30  (1) 


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2.6.5 Relating SS calculations to the structural model 


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2.6.7 Pictorial representation 


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2.7 Withinsubjects (repeated measures) ANOVA 


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35  (1) 


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2.7.4 EMS and ANOVA summary table 


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2.8 Assumptions of withinsubjects ANOVA: 'sphericity' 


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39  (1) 


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2.9 Missing data in designs involving withinsubjects factors 


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2.10 Mixed ANOVA (with both BS and WS factors) 


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2.10.2 Degrees of freedom 


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2.11 Fixed and random factors 


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2.12 Additional material (ADVANCED) 


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2.12.1 Notation for variances and mean squares in EMS expressions 


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2.12.2 Expected value of F 


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2.12.3 A χ² distribution is the sum of squared z scores 


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2.12.4 Relationship between the sample variance and the χ² distribution 


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2.12.5 The F distribution 


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2.12.6 Comparing two variances with an F test 


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2.12.7 ANOVA: comparing two meansquare values with an F test 


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2.12.8 Relating SS calculations to the model for oneway ANOVA 


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CHAPTER 3: PRACTICAL ANALYSIS 

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3.1 Reminder: assumptions of ANOVA 


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3.2 Reminder: assumption of ANOVA with WS factors 


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3.3 Consequences of violating the assumptions of ANOVA 


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3.4 Exploratory data analysis and transformations 


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66  (1) 


67  (4) 


71  (1) 


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3.7 Further analysis: after the ANOVA has been run 


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3.7.1 Main effects, interactions, and simple effects revisited 


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3.7.2 Conducting simpleeffects analysis 


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3.7.3 A fallacy to avoid: when A differs from C but B doesn't 


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3.7.4 A fallacy to avoid: simple effects without interactions 


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3.7.5 Determining the effects of a factor with >2 levels 


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3.7.6 Multiple comparisons: a problem 


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3.7.7 Post hoc tests: a problem 


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3.7.8 The special case of three groups: multiple t tests are OK 


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3.7.9 Otherwise: a variety of post hoc tests 


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3.7.10 Post hoc tests for withinsubject factors 


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3.7.11 A priori tests: planned contrasts 


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3.7.12 Apparent inconsistency between the F test and post hoc tests 


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3.7.13 SPSS's default pairwise comparison post hoc tests 


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3.8 Drawing pictures: error bars for different comparisons 


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3.8.1 Error bars for t tests: betweensubjects comparisons 


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3.8.2 Error bars for t tests: withinsubjects comparisons 


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3.8.3 Error bars for an ANOVA: betweensubjects designs 


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3.8.4 Error bars for an ANOVA: effects in mixed designs 


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3.9 Summarizing your methods: a writing guide 


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3.10 Additional material (ADVANCED) 


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3.10.1 Error bars for t tests: betweensubjects comparisons: SEMs 


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3.10.2 Error bars for t tests: betweensubjects comparisons: CIs 


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3.10.3 Error bars for t tests: betweensubjects comparisons: SDs 


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3.10.4 Obtaining SEDs from an ANOVA table 


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CHAPTER 4: PITFALLS AND COMMON ISSUES 

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4.1 Time in withinsubjects (repeated measures) designs 


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4.2 Analysis of pretest versus posttest data 


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4.3 Observing subjects repeatedly to increase power 


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4.4 'It's significant in this subject...' 


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4.5 Should I add/remove a factor? Full and reduced models 


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4.6 Should I add/remove/collapse over levels of a factor? 


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4.6.1 Adding and removing levels by adding new observations 


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4.6.2 Collapsing over or subdividing levels 


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CHAPTER 5: USING SPSS FOR ANOVA 

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5.1 Running ANOVAs using SPSS 


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5.1.1 Analysis of variance 


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5.1.2 Organizing and reorganizing your data 


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120  (1) 


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5.1.5 Options, including homogeneityofvariance tests 


121  (2) 


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5.2 Interpreting the output 


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Tip: pairwise comparisons for interactions 


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5.3. Further analysis: selecting cases 


139  (3) 

5.4 The 'intercept', 'total', and 'corrected total' terms 


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CHAPTER 6: CONTRASTS AND TRENDS 

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6.1.1 About linear contrasts 


147  (1) 

6.1.2 Type I error rates with planned contrasts 


148  (2) 

6.1.3 Orthogonal contrasts 


150  (1) 

6.1.4 Linear contrasts in SPSS 


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6.1.5 Contrasts in multifactor designs—an overview 


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6.2 Trend analysis: the effects of quantitative factors 


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6.2.2 Trend analysis in SPSS 


157  (1) 

6.2.3 Trend analysis, multiple regression, and polynomial ANCOVA 


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CHAPTER 7: ADVANCED TOPICS 

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7.1 Rules for calculating sums of squares 


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7.1.1 Partitioning sums of squares 


161  (1) 

7.1.2 General rule for calculating sums of squares 


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7.2 Rules for calculating degrees of freedom 


163  (1) 

7.3 Expected mean squares (EMS) and error terms 


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7.3.1 Rules for obtaining expected mean squares (EMS) 


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7.3.2 Choosing an error term 


168  (2) 

7.3.3 Error terms in models including random factors (complicated) 


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7.3.4 Pooling error terms 


173  (1) 

7.4 Unequal group sizes and nonorthogonal sums of squares 


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7.4.1 Proportional cell frequencies 


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7.4.2 Disproportionate cell frequencies—a problem 


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7.4.3 Correlated predictors in general—a problem 


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7.5 How computers perform ANOVA: general linear models 


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7.5.1 The basic idea of a GLM, illustrated with multiple regression 


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7.5.2 Using a GLM for simple ANOVA: the design matrix 


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7.5.3 Example of a GLM for a oneway ANOVA 


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7.5.4 GLM for twoway ANOVA and beyond 


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7.5.5 F statistics for GLMs: comparing full and reduced models 


187  (1) 

7.5.6 An overview of GLM designs 


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7.5.7 GLM designs involving random effects 


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7.5.8 A hint at multivariate analysis: MANOVA 


195  (1) 

7.5.9 Linear contrasts with a GLM 


196  (1) 

7.5.10 GLMs and custom contrasts in SPSS 


197  (5) 


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7.6.1 Effect size in the language of multiple regression 


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7.6.2 Effect size in the language of ANOVA 


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CHAPTER 8: SPECIFIC DESIGNS 

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8.1 One betweensubjects (BS) factor 


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222  (3) 


225  (3) 

8.4 One withinsubjects (WS) factor 


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238  (6) 

8.7 One BS and one WS factor 


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8.8 Two BS factors and one WS factor 


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8.9 One BS factor and two WS factors 


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8.10 Other ANOVA designs with BS and/or WS factors 


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8.11 One BS covariate (linear regression) 


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8.12 One BS covariate and one BS factor 


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8.12.1 The covariate and factor do not interact 


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8.12.2 The covariate and factor interact 


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8.13 One BS covariate and two BS factors 


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8.14 Two or more BS covariates (multiple regression) 


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8.15 Two or more BS covariates and one or more BS factors 


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8.17 One WS covariate and one BS factor 


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8.17.1 The covariate and factor do not interact 


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8.17.2 The covariate and factor interact 


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8.18 Hierarchical designs 


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8.18.1 Subjects within groups within treatments (S/G/A) 


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8.18.2 Groups versus individuals 


311  (1) 

8.18.3 Adding a further withingroup, BS variable (S/GB/A) 


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8.18.4 Adding a withinsubjects variable (US/GB/A) 


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8.18.5 Nesting withinsubjects variables, such as V/US/A 


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8.18.6 The splitsplit plot design 


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8.18.7 Three levels of relatedness 


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8.19 Latin square designs 


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8.19.1 Latin squares in experimental design 


328  (1) 

8.19.2 The analysis of a basic Latin square 


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8.19.3 A x B interactions in a single Latin square 


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8.19.4 More subjects than rows: (a) using several squares 


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8.19.5 More subjects than rows: (b) using one square several times 


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8.19.6 BS designs using Latin squares (fractional factorial designs) 


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8.19.7 Severalsquares design with a BS factor 


344  (3) 

8.19.8 Replicatedsquares design with a BS factor 


347  (4) 

8.20 Agricultural terminology and designs 


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CHAPTER 9: MATHEMATICS 

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367  (7) 


367  (2) 


369  (3) 

9.1.3 The inverse of a matrix 


372  (1) 

9.1.4 Matrix transposition 


373  (1) 


374  (5) 


374  (1) 

9.2.2 Simple, nontrigonometric derivatives 


375  (1) 

9.2.3 Rules for differentiation 


375  (1) 

9.2.4 Derivatives of a vector function 


376  (1) 

9.2.5 Partial derivatives 


376  (1) 

9.2.6 The chain rule for partial derivatives 


377  (1) 

9.2.7 Illustrations of partial derivatives 


377  (2) 

9.3 Solving a GLM (an overdetermined system of equations) 


379  (3) 

9.4 Singular value decomposition to solve GLMs 


382  (5) 

9.4.1 Eigenvectors and eigenvalues 


384  (1) 

9.4.2 Singular value decomposition 


385  (1) 

9.4.3 An underdetermined set of equations: the role of expectations 


386  (1) 

9.5 Random variables, means, and variances 


387  (10) 


387  (1) 

9.5.2 Random variables; definition of mean and variance 


388  (1) 

9.5.3 Continuous random variables 


389  (1) 


390  (1) 


391  (1) 

9.5.6 Distribution of a set of means: the standard error of the mean 


392  (3) 

9.5.7 The sample mean and SD are unbiased estimators of μ and σ² 


395  (2) 


397  (1) 

9.7 Rules for powers and logarithms 


397  (1) 


398  (5) 

Basic notation in probability 


398  (1) 

Basic laws of probability 


398  (1) 


399  (1) 

Bayes' theorem and Bayesian inference 


400  (3) 
CHAPTER 10: STATISTICAL TABLES 

403  (6) 

10.1 Critical values of t 


403  (1) 

10.2 Critical values of F 


404  (4) 

10.3 Polynomial trend coefficients 


408  (1) 
GLOSSARY 

409  (24) 


409  (2) 


411  (1) 


412  (21) 
FURTHER READING 

433  (2) 
REFERENCES 

435  (4) 
INDEX 

439  