Schaum's Outline of Theory and Problems of Elements of Statistics II: Inferential Statistics | Schaum's Outline

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Schaum's Outline of Theory and Problems of Elements of Statistics II: Inferential Statistics

Schaum's Outline of Theory and Problems of Elements of Statistics II: Inferential Statistics

Authors: Stephen Bernstein and Ruth Bernstein

Published: 1999

ISBN: 9780071346375 0071346376

CONTENTS

CHAPTER 11 DISCRETE PROBABILITY DISTRIBUTIONS

11.1 Discrete Probability Distributions and Probability Mass Functions

11.2 Bernoulli Experiments and trials

11.3 Binomial Random Variables, Experiments, and Probability Functions

11.4 The Binomial Coefficient

11.5 The Binomial Probability Function

11.6 Mean, Variance, and Standard Deviation of the Binomial Probability Distribution

11.7 The Binomial Expansion and the Binomial Theorem

11.8 Pascal's Triangle and the Binomial Coefficient

11.9 The Family of Binomial Distributions

11.10 The Cumulative Binomial Probability Table

11.11 Lot-Acceptance Sampling

11.12 Consumer's Risk and Producer's Risk

11.13 Multivariate Probability Distributions and Joint Probability Distributions

11.14 The Multinomial Experiment

11.15 The Multinomial Coefficient

11.16 The Multinomial Probability Function

11.17 The Family of Multinomial Probability Distributions

11.18 The Means of the Multinomial Probability Distribution

11.19 The Multinomial Expansion and the Multinomial Theorem

11.20 The Hypergeometric Experiment

11.21 The Hypergeometric Probability Function

11.22 The Family of Hypergeometric Probability Distributions

11.23 The Mean, Variance, and Standard Deviation of the Hypergeometric Probability Distribution

11.24 The Generalization of the Hypergeometric Probability Distribution

11.25 The Binomial and Multinomial Approximations to the Hypergeometric Distribution

11.26 Poisson Processes, Random Variables, and Experiments

11.27 The Poisson Probability Function

11.28 The Family of Poisson Probability Distributions

11.29 The Mean, Variance, and Standard Deviation of the Poisson Probability Distribution

11.30 The Cumulative Poisson Probability Table

11.31 The Poisson Distribution as an Approximation to the Binomial Distribution

CHAPTER 12 The Normal Distribution and Other Continuous Probability Distributions

12.1 Continuous Probability Distributions

12.2 The Normal Probability Distributions and the Normal Probability Density Function

12.3 The Family of Normal Probability Distributions

12.4 The Normal Distribution: Relationship between the Mean (μ), the Median ( ), and the Mode

12.5 Kurtosis

12.6 The Standard Normal Distribution

12.7 Relationship Between the Standard Normal Distribution and the Standard Normal Variable

12.8 Table of Areas in the Standard Normal Distribution

12.9 Finding Probabilities Within any Normal Distribution by Applying the Z Transformation

12.10 One-tailed Probabilities

12.11 Two-tailed Probabilities

12.12 The Normal Approximation to the Binomial Distribution

12.13 The Normal Approximation to the Poisson Distribution

12.14 The Discrete Uniform Probability Distribution

12.15 The Continuous Uniform Probability Distribution

12.16 The Exponential Probability Distribution

12.17 Relationship between the Exponential Distribution and the Poisson Distribution

CHAPTER 13 SAMPLING DISTRIBUTIONS

13.1 Simple Random Sampling Revisited

13.2 Independent Random Variables

13.3 Mathematical and Nonmathematical Definitions of Simple Random Sampling

13.4 Assumptions of the Sampling Technique

13.5 The Random Variable X

13.6 Theoretical and Empirical Sampling Distributions of the Mean

13.7 The Mean of the Sampling Distribution of the Mean

13.8 The Accuracy of an Estimator

13.9 The Variance of the Sampling Distribution of the Mean: Infinite Population or Sampling with Replacement

13.10 The Variance of the Sampling Distribution of the Mean: Finite Population Sampled without Replacement

13.11 The Standard Error of the Mean

13.12 The Precision of An Estimator

13.13 Determining Probabilities with a Discrete Sampling Distribution of the Mean

13.14 Determining Probabilities with a Normally Distributed Sampling Distribution of the Mean

13.15 The Central Limit Theorem: Sampling from a Finite Population with Replacement

13.16 The Central Limit Theorem: Sampling from an Infinite Population

13.17 The Central Limit Theorem: Sampling from a Finite Population without Replacement

13.18 How Large is " Sufficiently Large?''

13.19 The Sampling Distribution of the Sample Sum

13.20 Applying the Central Limit Theorem to the Sampling Distribution of the Sample Sum

13.21 Sampling from a Binomial Population

13.22 Sampling Distribution of the Number of Successes

13.23 Sampling Distribution of the Proportion

13.24 Applying the Central Limit Theorem to the Sampling Distribution of the Number of Successes

13.25 Applying the Central Limit Theorem to the Sampling Distribution of the Proportion

13.26 Determining Probabilities with a Normal Approximation to the Sampling Distribution of the Proportion

CHAPTER 14 ONE-SAMPLE ESTIMATION OF THE POPULATION MEAN

14.1 Estimation

14.2 Criteria for Selecting the Optimal Estimator

14.3 The Estimated Standard Error of the Mean Sx

14.4 Point Estimates

14.5 Reporting and Evaluating the Point Estimate

14.6 Relationship between Point Estimates and Interval Estimates

14.7 Deriving

14.8 Deriving

14.9 Confidence Interval for the Population Mean μ: Known Standard Deviation σ , Normally Distributed Population

14.10 Presenting Confidence Limits

14.11 Precision of the Confidence Interval

14.12 Determining Sample Size when the Standard Deviation is Known

14.13 Confidence Interval for the Population Mean μ: Known Standard Deviation σ , Large Sample (n ≥ 30) from any Population Distribution

14.14 Determining Confidence Intervals for the Population Mean μ when the Population Standard Deviation σ is Unknown

14.15 The t Distribution

14.16 Relationship between the t Distribution and the Standard Normal Distribution

14.17 Degrees of Freedom

14.18 The Term " Student's t Distribution''

14.19 Critical Values of the t Distribution

14.20 Table A.6: Critical Values of the t Distribution

14.21 Confidence Interval for the Population Mean μ: Standard Deviation σ not known, Small Sample (n < 30) from a Normally Distributed Population

14.22 Determining Sample Size: Unknown Standard Deviation, Small Sample from a Normally Distributed Population

14.23 Confidence Interval for the Population Mean μ: Standard Deviation σ not known, large sample (n ≥ 30) from a Normally Distributed Population

14.24 Confidence Interval for the Population Mean μ: Standard Deviation σ not known, Large Sample (n ≥ 30) from a Population that is not Normally Distributed

14.25 Confidence Interval for the Population Mean μ: Small Sample (n < 30) from a Population that is not Normally Distributed

CHAPTER 15 ONE-SAMPLE ESTIMATION OF THE POPULATION VARIANCE, STANDARD DEVIATION, AND PROPORTION

15.1 Optimal Estimators of Variance, Standard Deviation, and Proportion

15.2 The Chi-Square Statistic and the Chi-Square Distribution

15.3 Critical Values of the Chi-Square Distribution

15.4 Table A.7: Critical Values of the Chi-Square Distribution

15.5 Deriving the Confidence Interval for the Variance σ2 of a Normally Distributed Population

15.6 Presenting Confidence Limits

15.7 Precision of the Confidence Interval for the Variance

15.8 Determining Sample Size Necessary to Achieve a Desired Quality-of-Estimate for the Variance

15.9 Using Normal-Approximation Techniques To Determine Confidence Intervals for the Variance

15.10 Using the Sampling Distribution of the Sample Variance to Approximate a Confidence Interval for the Population Variance

15.11 Confidence Interval for the Standard Deviation σ of a Normally Distributed Population

15.12 Using the Sampling Distribution of the Sample Standard Deviation to Approximate a Confidence Interval for the Population Standard Deviation

15.13 The Optimal Estimator for the Proportion p of a Binomial Population

15.14 Deriving the Approximate Confidence Interval for the Proportion p of a Binomial Population

15.15 Estimating the Parameter p

15.16 Deciding when n is "Sufficiently Large'', p not known

15.17 Approximate Confidence Intervals for the Binomial Parameter p When Sampling From a Finite Population without Replacement

15.18 The Exact Confidence Interval for the Binomial Parameter p

15.19 Precision of the Approximate Confidence-Interval Estimate of the Binomial Parameter p

15.20 Determining Sample Size for the Confidence Interval of the Binomial Parameter p

15.21 Approximate Confidence Interval for the Percentage of a Binomial Population

15.22 Approximate Confidence Interval for the Total Number in a Category of a Binomial Population

15.23 The Capture±Recapture Method for Estimating Population Size N

CHAPTER 16 ONE-SAMPLE HYPOTHESIS TESTING

16.1 Statistical Hypothesis Testing

16.2 The Null Hypothesis and the Alternative Hypothesis

16.3 Testing the Null Hypothesis

16.4 Two-Sided Versus One-Sided Hypothesis Tests

16.5 Testing Hypotheses about the Population Mean μ: Known Standard Deviation σ, Normally Distributed Population

16.6 The P Value

16.7 Type I Error versus Type II Error

16.8 Critical Values and Critical Regions

16.9 The Level of Significance

16.10 Decision Rules for Statistical Hypothesis Tests

16.11 Selecting Statistical Hypotheses

16.12 The Probability of a Type II Error

16.13 Consumer's Risk and Producer's Risk

16.14 Why It is Not Possible to Prove the Null Hypothesis

16.15 Classical Inference Versus Bayesian Inference

16.16 Procedure for Testing the Null Hypothesis

16.17 Hypothesis Testing Using X as the Test Statistic

16.18 The Power of a Test, Operating Characteristic Curves, and Power Curves

16.19 Testing Hypothesis about the Population Mean μ: Standard Deviation σ Not Known, Small Sample (n < 30) from a Normally Distributed Population

16.20 The P Value for the t Statistic

16.21 Decision Rules for Hypothesis Tests with the t Statistic

16.22 β, 1 – β, Power Curves, and OC Curves

16.23 Testing Hypotheses about the Population Mean μ: Large Sample (n ≥ 30) from any Population Distribution

16.24 Assumptions of One-Sample Parametric Hypothesis Testing

16.25 When the Assumptions are Violated

16.26 Testing Hypothesis about the Variance σ2 of a Normally Distributed Population

16.27 Testing Hypotheses about the Standard Deviation σ of a Normally Distributed Population

16.28 Testing Hypotheses about the Proportion p of a Binomial Population: Large Samples

16.29 Testing Hypotheses about the Proportion p of a Binomial Population: Small Samples

CHAPTER 17 TWO-SAMPLE ESTIMATION AND HYPOTHESIS TESTING

17.1 Independent Samples Versus Paired Samples

17.2 The Optimal Estimator of the Difference Between Two Population Means (μ1 – μ2)

17.3 The Theoretical Sampling Distribution of the Difference Between Two Means

17.4 Confidence Interval for the Difference Between Means (μ1 – μ2): Standard Deviations (σ1 and σ2) Known, Independent Samples from Normally Distributed Populations

17.5 Testing Hypotheses about the Difference Between Means (μ1 – μ2): Standard Deviations (σ1 and σ2) known, Independent Samples from Normally Distributed Populations

17.6 The Estimated Standard Error of the Difference Between Two Means

17.7 Confidence Interval for the Difference Between Means (μ1 – μ2): Standard Deviations not known but Assumed Equal (σ1 = σ2), Small (n1 < 30 and n2 < 30) Independent Samples from Normally Distributed Populations

17.8 Testing Hypotheses about the Difference Between Means (μ1 – μ2): Standard Deviations not Known but Assumed Equal (σ1 = σ2), Small (n1 < 30 and n2 < 30) Independent Samples from Normally Distributed Populations

17.9 Confidence Interval for the Difference Between Means (μ1 – μ2): Standard Deviations (σ1 and σ2) not Known, Large (n1 ≥ 30 and n2 ≥ 30) Independent Samples from any Population Distributions

17.10 Testing Hypotheses about the Difference Between Means (μ1 – μ2): Standard Deviations (σ1 and σ2), not known, Large (n1 ≥ 30 and n2 ≥ 30) Independent Samples from any Populations Distributions

17.11 Confidence Interval for the Difference Between Means (μ1 – μ2): Paired Samples

17.12 Testing Hypotheses about the Difference Between Means (μ1 – μ2): Paired Samples

17.13 Assumptions of Two-Sample Parametric Estimation and Hypothesis Testing about Means

17.14 When the Assumptions are Violated

17.15 Comparing Independent-Sampling and Paired-Sampling Techniques on Precision and Power

17.16 The F Statistic

17.17 The F Distribution

17.18 Critical Values of the F Distribution

17.19 Table A.8: Critical Values of the F Distribution

17.20 Confidence Interval for the Ratio of Variances (σ21/σ22 ): Parameters (σ21, σ1, μ1 and σ22, σ2, μ2) Not Known, Independent Samples From Normally Distributed Populations

17.21 Testing Hypotheses about the Ratio of Variances (σ21/σ22): Parameters (σ21, σ1, μ1 and σ22, σ2, μ2) not known, Independent Samples from Normally Distributed Populations

17.22 When to Test for Homogeneity of Variance

17.23 The Optimal Estimator of the Difference Between Proportions (p1 – p2): Large Independent Samples

17.24 The Theoretical Sampling Distribution of the Difference Between Two Proportions

17.25 Approximate Confidence Interval for the Difference Between Proportions from Two Binomial Populations (p1 – p2): Large Independent Samples

17.26 Testing Hypotheses about the Difference Between Proportions from Two Binomial Populations (p1 – p2): Large Independent Samples

CHAPTER 18 MULTISAMPLE ESTIMATION AND HYPOTHESIS TESTING

18.1 Multisample Inferences

18.2 The Analysis of Variance

18.3 ANOVA: One-Way, Two-Way, or Multiway

18.4 One-Way ANOVA: Fixed-Effects or Random Effects

18.5 One-way, Fixed-Effects ANOVA: The Assumptions

18.6 Equal-Samples, One-Way, Fixed-Effects ANOVA: H0 and H1

18.7 Equal-Samples, One-Way, Fixed-Effects ANOVA: Organizing the Data

18.8 Equal-Samples, One-Way, Fixed-Effects ANOVA: the Basic Rationale

18.9 SST = SSA + SSW

18.10 Computational Formulas for SST and SSA

18.11 Degrees of Freedom and Mean Squares

18.12 The F Test

18.13 The ANOVA Table

18.14 Multiple Comparison Tests

18.15 Duncan's Multiple-Range Test

18.16 Confidence-Interval Calculations Following Multiple Comparisons

18.17 Testing for Homogeneity of Variance

18.18 One-Way, Fixed-Effects ANOVA: Equal or Unequal Sample Sizes

18.19 General-Procedure, One-Way, Fixed-effects ANOVA: Organizing the Data

18.20 General-Procedure, One-Way, Fixed-effects ANOVA: Sum of Squares

18.21 General-Procedure, One-Way, Fixed-Effects ANOVA Degrees of Freedom and Mean Squares

18.22 General-Procedure, One-Way, Fixed-Effects ANOVA: the F Test

18.23 General-Procedure, One-Way, Fixed-Effects ANOVA: Multiple Comparisons

18.24 General-Procedure, One-Way, Fixed-Effects ANOVA: Calculating Confidence Intervals and Testing for Homogeneity of Variance

18.25 Violations of ANOVA Assumptions

CHAPTER 19 REGRESSION AND CORRELATION

19.1 Analyzing the Relationship between Two Variables

19.2 The Simple Linear Regression Model

19.3 The Least-Squares Regression Line

19.4 The Estimator of the Variance σ2Y.X

19.5 Mean and Variance of the y Intercept â and the Slope b

19.6 Confidence Intervals for the y Intercept a and the Slope b

19.7 Confidence Interval for the Variance σ2Y.X

19.8 Prediction Intervals for Expected Values of Y

19.9 Testing Hypotheses about the Slope b

19.10 Comparing Simple Linear Regression Equations from Two or More Samples

19.11 Multiple Linear Regression

19.12 Simple Linear Correlation

19.13 Derivation of the Correlation Coefficient r

19.14 Confidence Intervals for the Population Correlation Coefficient ρ

19.15 Using the r Distribution to Test Hypotheses about the Population Correlation Coefficient ρ

19.16 Using the t Distribution to Test Hypotheses about ρ

19.17 Using the Z Distribution to Test the Hypothesis ρ = c

19.18 Interpreting the Sample Correlation Coefficient r

19.19 Multiple Correlation and Partial Correlation

CHAPTER 20 NONPARAMETRIC TECHNIQUES

20.1 Nonparametric vs. Parametric Techniques

20.2 Chi-Square Tests

20.3 Chi-Square Test for Goodness-of-fit

20.4 Chi-Square Test for Independence: Contingency Table Analysis

20.5 Chi-Square Test for Homogeneity Among k Binomial Proportions

20.6 Rank Order Tests

20.7 One-Sample Tests: The Wilcoxon Signed-Rank Test

20.8 Two-Sample Tests: the Wilcoxon Signed-Rank Test for Dependent Samples

20.9 Two-Sample Tests: the Mann-Whitney U Test for Independent Samples

20.10 Multisample Tests: the Kruskal-Wallis H Test for k Independent Samples

20.11 The Spearman Test of Rank Correlation

Appendix

Table A.3 Cumulative Binomial Probabilities

Table A.4 Cumulative Poisson Probabilities

Table A.5 Areas of the Standard Normal Distribution

Table A.6 Critical Values of the t Distribution

Table A.7 Critical Values of the Chi-Square Distribution

Table A.8 Critical Values of the F Distribution

Table A.9 Least Significant Studentized Ranges rp

Table A.10 Transformation of r to zr

Table A.11 Critical Values of the Pearson Product-Moment Correlation Coefficient r

Table A.12 Critical Values of the Wilcoxon W

Table A.13 Critical Values of the Mann-Whitney U

Table A.14 Critical Values of the Kruskal-Wallis H

Table A.15 Critical Values of the Spearman rS

Index