# Book:Stephen Bernstein/Elements of Statistics II: Inferential Statistics

## Stephen Bernstein and Ruth Bernstein: Elements of Statistics II: Inferential Statistics

Published $1999$, Schaum's Outlines

ISBN 0-07-134637-6.

### 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 $(\mu)$, Median $(\overline \mu)$, 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 $\overline X$
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 $S_{\overline x}$
14.4 Point Estimates
14.5 Reporting and Evaluating the Point Estimate
14.6 Relationship between Point Estimates and Interval Estimates
14.7 Deriving $P \left({\overline x_{1 - \alpha/2} \le \overline X \le \overline x_{\alpha/2}}\right) = P \left({-z_{\alpha/2} \le Z \le z_{\alpha/2}}\right) = 1 - \alpha$
14.8 Deriving $P \left({X - z_{\alpha/2} \sigma_{\overline x} \le \mu \le \overline X + z_{\alpha/2} \sigma_{\overline x}}\right) = 1 - \alpha$
14.9 Confidence Interval for the Population Mean $\mu$: Known Standard Deviation $\sigma$, 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 $\mu$: Known Standard Deviation $\sigma$, Large Sample $(n \ge 30)$ from any Population Distribution
14.14 Determining Confidence Intervals for the Population Mean $\mu$ when the Population Standard Deviation $\sigma$ 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 $\mu$: Standard Deviation $\sigma$ 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 $\mu$: Standard Deviation $\sigma$ not known, large sample $(n \ge 30)$ from a Normally Distributed Population
14.24 Confidence Interval for the Population Mean $\mu$: Standard Deviation $\sigma$ not known, large sample $(n \ge 30)$ from a Population that is not Normally Distributed
14.25 Confidence Interval for the Population Mean $\mu$: Standard Deviation $\sigma$ not known, 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 qf the Chi-square Distribution
15.5 Deriving the Confidence Interval for the Variance $\sigma^2$ of a Normally Distributed Population
15.6 Presenting Confidence Limits
15.7 Precision of the Confidence Interval for the Variance
15.8 Determining Samble 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 $\sigma$ 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 fof 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 Internal 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 $\mu$: Known Standard Deviation $\sigma$, 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 Testily the Null Hypothesis
16.17 Hypothesis Testing Using $\overline X$ 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 $\mu$: Standard Deviation $\sigma$ 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 $\beta$, $1 - \beta$, Power Curves, and $OC$ Curves
16.23 Testing Hypotheses about the Population Mean $\mu$: Large Sample $(n \ge 30)$ 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 $\sigma^2$ of a Normally Distributed Population
16.27 Testing Hypotheses about the Standard Deviation $\sigma$ 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 $(\mu_1 - \mu_2)$
17.3 The Theoretical Sampling Distribution of the Difference Between Two Means
17.4 Confidence Interval for the Difference Between Means $(\mu_1 - \mu_2)$: Standard Deviations ($\sigma_1$ and $\sigma_2$) Known, Independent Samples from Normally Distributed Populations
17.5 Testing Hypotheses about the Difference Between Means $(\mu_1 - \mu_2)$: Standard Deviations ($\sigma_1$ and $\sigma_2$) 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 $(\mu_1 - \mu_2)$: Standard Deviations not known but Assumed Equal ($\sigma_1 = \sigma_2$), Small ($n_1 < 30$ and $n_2 < 30$) Independent Samples from Normally Distributed Populations
17.8 Testing Hypotheses about the Difference Between Means $(\mu_1 - \mu_2)$: Standard Deviations not Known but Assumed Equal ($\sigma_1 = \sigma_2$), Small ($n_1 < 30$ and $n_2 < 30$) Independent Samples from Normally Distributed Populations
17.9 Confidence Interval for the Difference Between Means $(\mu_1 - \mu_2)$: Standard Deviations ($\sigma_1$ and $\sigma_2$) not Known, Large ($n_1 \ge 30$ and $n_2 \ge 30$) Independent Samples from any Population Distributions
17.10 Testing Hypotheses about the Difference Between Means $(\mu_1 - \mu_2)$: Standard Deviations ($\sigma_1$ and $\sigma_2$), not known, Large ($n_1 \ge 30$ and $n_2 \ge 30$) Independent Samples from any Populations Distributions
17.11 Confidence Interval for the Difference Between Means $(\mu_1 - \mu_2)$: Paired Samples
17.12 Testing Hypotheses about the Difference Between Means $(\mu_1 - \mu_2)$: 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 $\left({\sigma_1^2 / \sigma_2^2}\right)$: Parameters ($\sigma_1^2, \sigma_1, \mu_1$ and $\sigma_2^2, \sigma_2, \mu_2$) Not Known, Independent Samples From Normally Distributed Populations
17.21 Testing Hypotheses about the Ratio of Variances $\left({\sigma_1^2 / \sigma_2^2}\right)$: Parameters ($\sigma_1^2, \sigma_1, \mu_1$ and $\sigma_2^2, \sigma_2, \mu_2$) 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 $(p_1 - p_2)$: 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 $(p_1 - p_2)$: Large Independent Samples
17.26 Testing Hypotheses about the Difference Between Proportions from Two Binomial Populations $(p_1 - p_2)$: 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: $H_0$ and $H_1$
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: Organising the Data
18.20 General-Procedure, One-Way, Fixed-effects ANOVA: Sum of Squares
18.21 General-Procedure, One-Way, Fixed-Erects 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-Erects 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 $\sigma^2_{Y \cdot X}$
19.5 Mean and Variance of the $y$ Intercept $\hat a$ and the Slope $\hat b$
19.6 Confidence Intervals for the $y$ Intercept $a$ and the Slope $b$
19.7 Confidence Interval for the Variance $\sigma^2_{Y \cdot X}$
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 $\rho$
19.15 Using the $r$ Distribution to Test Hypotheses about the Population Correlation coefficient $\rho$
19.16 Using the $f$ Distribution to Test Hypotheses about $\rho$
19.17 Using the $Z$ Distribution to Test the Hypothesis $\rho = c$
19.18 Interpreting the Sample Correlation Coefficient $r$
19.19 Multiple Correlation and Partial Correlation
CHAPTER 20 NONPARAMETRIC TECHNIQUES
20.1 Nonpnmmetric 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 $r_p$
Table A.10 Transformation of $r$ to $z_r$
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 $r_s$
Index