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List of Examples | |

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Preface | |

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Acknowledgments | |

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Authors | |

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Introduction: Probability, Statistics, and Science | |

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Reality, Nature, Science, and Models | |

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Statistical Processes: Nature, Design and Measurement, and Data | |

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Models | |

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Deterministic Models | |

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Variability | |

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Parameters | |

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Purely Probabilistic Statistical Models | |

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Statistical Models with Both Deterministic and Probabilistic Components | |

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Statistical Inference | |

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Good and Bad Models | |

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Uses of Probability Models | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Random Variables and Their Probability Distributions | |

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Introduction | |

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Types of Random Variables: Nominal, Ordinal, and Continuous | |

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Discrete Probability Distribution Functions | |

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Continuous Probability Distribution Functions | |

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Some Calculus-Derivatives and Least Squares | |

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More Calculus-Integrals and Cumulative Distribution Functions | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Probability Calculation and Simulation | |

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Introduction | |

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Analytic Calculations, Discrete and Continuous Cases | |

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Simulation-Based Approximation | |

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Generating Random Numbers | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Identifying Distributions | |

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Introduction | |

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Identifying Distributions from Theory Alone | |

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Using Data: Estimating Distributions via the Histogram | |

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Quantiles: Theoretical and Data-Based Estimates | |

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Using Data: Comparing Distributions via the Quantile-Quantile Plot | |

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Effect of Randomness on Histograms and q-q Plots | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Conditional Distributions and Independence | |

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Introduction | |

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Conditional Discrete Distributions | |

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Estimating Conditional Discrete Distributions | |

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Conditional Continuous Distributions | |

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Estimating Conditional Continuous Distributions | |

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Independence | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Marginal Distributions, Joint Distributions, Independence, and Bayes' Theorem | |

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Introduction | |

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Joint and Marginal Distributions | |

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Estimating and Visualizing Joint Distributions | |

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Conditional Distributions from Joint Distributions | |

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Joint Distributions When Variables Are Independent | |

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Bayes' Theorem | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Sampling from Populations and Processes | |

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Introduction | |

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Sampling from Populations | |

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Critique of the Population Interpretation of Probability Models | |

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Even When Data Are Sampled from a Population | |

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Point 1: Nature Defines the Population, Not Vice Versa | |

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Point 2: The Population Is Not Well Defined | |

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Point 3: Population Conditional Distributions Are Discontinuous | |

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Point 4: The Conditional Population Distribution p(yx) Does Not Exist for Many x | |

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Point 5: The Population Model Ignores Design and Measurement Effects | |

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The Process Model versus the Population Model | |

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Independent and Identically Distributed Random Variables and Other Models | |

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Checking the iid Assumption | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Expected Value and the Law of Large Numbers | |

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Introduction | |

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Discrete Case | |

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Continuous Case | |

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Law of Large Numbers | |

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Law of Large Numbers for the Bernoulli Distribution | |

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Keeping the Terminology Straight: Mean, Average, Sample Mean, Sample Average, and Expected Value | |

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Bootstrap Distribution and the Plug-In Principle | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Functions of Random Variables: Their Distributions and Expected Values | |

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Introduction | |

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Distributions of Functions: The Discrete Case | |

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Distributions of Functions: The Continuous Case | |

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Expected Values of Functions and the Law of the Unconscious Statistician | |

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Linearity and Additivity Properties | |

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Nonlinear Functions and Jensen's Inequality | |

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Variance | |

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Standard Deviation, Mean Absolute Deviation, and Chebyshev's Inequality | |

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Linearity Property of Variance | |

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Skewness and Kurtosis | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Distributions of Totals | |

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Introduction | |

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Additivity Property of Variance | |

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Covariance and Correlation | |

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Central Limit Theorem | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Estimation: Unbiasedness, Consistency, and Efficiency | |

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Introduction | |

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Biased and Unbiased Estimators | |

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Bias of the Plug-In Estimator of Variance | |

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Removing the Bias of the Plug-In Estimator of Variance | |

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The Joke Is on Us: The Standard Deviation Estimator Is Biased after All | |

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Consistency of Estimators | |

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Efficiency of Estimators | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Likelihood Function and Maximum Likelihood Estimates | |

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Introduction | |

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Likelihood Function | |

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Maximum Likelihood Estimates | |

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Wald Standard Error | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Bayesian Statistics | |

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Introduction: Play a Game with Hans! | |

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Prior Information and Posterior Knowledge | |

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Case of the Unknown Survey | |

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Bayesian Statistics: The Overview | |

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Bayesian Analysis of the Bernoulli Parameter | |

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Bayesian Analysis Using Simulation | |

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What Good Is Bayes? | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Frequentist Statistical Methods | |

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Introduction | |

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Large-Sample Approximate Frequentist Confidence Interval for the Process Mean | |

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What Does Approximate Really Mean for an Interval Range? | |

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Comparing the Bayesian and Frequentist Paradigms | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Are Your Results Explainable by Chance Alone? | |

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Introduction | |

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What Does by Chance Alone Mean? | |

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The p-Value | |

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The Extremely Ugly "pv ≤ 0.05" Rule of Thumb | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Chi-Squared, Student's t, and F-Distributions, with Applications | |

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Introduction | |

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Linearity and Additivity Properties of the Normal Distribution | |

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Effect of Using an Estimate of ï¿½ | |

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Chi-Squared Distribution | |

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Frequentist Confidence Interval for ï¿½ | |

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Student's t-Distribution | |

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Comparing Two Independent Samples Using a Confidence Interval | |

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Comparing Two Independent Homoscedastic Normal Samples via Hypothesis Testing | |

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F-Distribution and ANOVA Test | |

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F-Distribution and Comparing Variances of Two Independent Groups | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Likelihood Ratio Tests | |

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Introduction | |

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Likelihood Ratio Method for Constructing Test Statistics | |

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Evaluating the Statistical Significance of Likelihood Ratio Test Statistics | |

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Likelihood Ratio Goodness-of-Fit Tests | |

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Cross-Classification Frequency Tables and Tests of Independence | |

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Comparing Non-Nested Models via the AIC Statistic | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Sample Size and Power | |

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Introduction | |

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Choosing a Sample Size for a Prespecified Accuracy Margin | |

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Power | |

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Noncentral Distributions | |

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Choosing a Sample Size for Prespecified Power | |

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Post Hoc Power: A Useless Statistic | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Robustness and Nonparametric Methods | |

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Introduction | |

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Nonparametric Tests Based on the Rank Transformation | |

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Randomization Tests | |

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Level and Power Robustness | |

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Bootstrap Percentile-t Confidence Interval | |

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Vocabulary and Formula Summaries | |

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Exercises | |

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Final Words | |

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Index | |