How Statistics Works: Introductory Notes

Preface

Chapter 1 Introduction
1.1 The Nature of Statistics
1.2 How Statistics Works
1.3 Mathematics and Statistics
1.3.1 American Community Service (ACS)
1.3.2 Current Population Survey
1.3.3 On the Benefit Effects of Taking Aspirin

Chapter 2 The Data
2.1 Variables
2.2 Variables and Measurement Levels
2.2.1 Examples of Data Sets of Several Variables
2.2.2 Another Example of a Multivariable Data Set
2.3 Selecting a Sample for an Observational Study
2.3.1 Simple Random Sampling
2.3.2 Selecting a Simple Random Variable
2.4 Other Sampling Methods
2.5 Observational Studies and Designed Experiments
2.5.1 Examples of Information Obtained by Observational Studies and from Experiments
2.6 The Importance of the Definition of the Variables
2.6.1 More Examples and Problems

Chapter 3 Description of Data
3.1 Frequency: Frequency Tables and Graphics
3.2 Graphic Representation of Quantitative Data: The Histogram
3.2.1 Example of the Construction of a Histogram
3.3 Descriptive Measures of Quantitative Data
3.4 Notation
3.5 Definitions and Examples
3.6 Descriptive Measures of Variability
3.7 Specific Formulas in the Case of Data Given in a Frequency Table
3.8 Descriptive Measures of Relative Standing
3.9 Summary of Formulas for This Chapter: Examples

Chapter 4 The Role of the Standard Deviation in the Description of Quantitative Data
4.1 Chebyshev’s Inequality
4.2 The Particular Quantitative Data We Call “Normal”
4.3 Skewness of Data
4.4 Parameters and Statistics
4.5 Examples

Chapter 5 Elements of Probability
5.1 Randomness and Probability
5.2 Probability Space
5.3 Operation between Events to Obtain New Events
5.4 Probability Formulas
5.4.1 Additive Rule
5.4.2 Multiplicative Rule and Conditional Probability
5.4.3 Examples
5.5 Odds and Probability
5.6 More Examples
5.7 Probability Tree
5.8 Summary of Probability Formulas
5.9 Counting Ways by Counting Formulas
5.9.1 Multiplicative Rule for Counting Formulas
5.9.2 More Examples of Counting Ways

Chapter 6 Discrete Random Variable
6.1 Formal Definition of a Discrete Random Variable
6.2 Expected Value of a Discrete Random Variable
6.3 Variance and Standard Deviation of a Discrete Random Variable
6.4 Variance and the Notion of Uncertainty
6.5 Binomial Experiment and Binomial Variable
6.5.1 Binomial Probability Formula
6.5.2 Mean and Variance of a Binomial Random Variable
6.5.3 Skewness of a Binomial Probability Distribution
6.5.4 Examples of Applications of Binomial Distribution

Chapter 7 Continuous Random Variables
7.1 Continuous Uniform Density
7.2 Standard Normal Probability Density and z Standard Normal
7.3 Standard Normal Table
7.4 General Normal Probability Density
7.5 Conversion Formula for a General Normal Variable x
7.6 Critical Values and Significance Level
7.7 Approximating a Binomial Probabilities by a Normal Density
7.7.1 Other Examples of the Approximation of a Binomial Probability by a Normal Distribution

Chapter 8 The Central Limit Theorem (CLT) and Applications
8.1 Role of the CLT in the Way Statistics Works
8.2 Sample Size and Sampling Error
8.3 Two Probability Distributions Derived from Normal Distribution
8.3.1 The t-random Variable
8.3.2 Examples of Reading the t-table
8.4 The Chi-Square Random Variable
8.4.1 Properties of Chi-Square Distribution

Chapter 9 Estimation
9.1 Introduction
9.2 Estimation of a Population Mean
9.3 The t-interval Procedure for the Estimation of a Population Mean When the Sigma is not Known
9.4 The z-interval Procedure for a Confidence Interval Estimation of a Population Proportion p
9.5 Minimum Sample Size for a Given Margin of Error E and a Given Level of Confidence
9.6 A Confidence Interval Estimate of a Population Variance: The Chi-Square Interval Procedure

Chapter 10 Test of a Hypothesis
10.1 Decision Making: Rejecting the Null Hypothesis or Failing to Reject It
10.2 Hypothesis Testing: Steps and Reasoning
10.3 Risk of Error and Important Symbols
10.3.1 Direction of Tests
10.4 Test Statistic: Formulas and Assumptions
10.4.1 Test Statistic for Hypothesis on the Population Mean
10.4.2 Test Statistic for Hypothesis on the Population Proportion
10.4.3 Test Statistic for Hypothesis on the Population Variance
10.5 Test of Hypothesis – Main Elements and Computations
10.5.1 Critical Values for the z-Distribution and Their Significance Levels
10.6 Steps in a Hypothesis Testing: The Use of P-value
10.7 More Examples
10.8 Concepts and More Examples about P-value

Chapter 11 Comparing Two Populations: Independent and Dependent Samples
11.1 Inferences for the Difference between Two Means: Independent Large Samples
11.2 Pooled Variance
11.3 Small Samples from Two Normal Populations: Assumptions and Formulas
11.4 Inferences for the Difference between Two Population Proportions
11.5 Inferences on the Differences of Means: Dependent Samples or Matched Pairs Design

Chapter 12 A Model in Science: The Simple Linear Model
12.1 Simple Linear Model
12.2 Estimation of the Parameters in the Simple Linear Model
12.3 Formulas to Estimate the Population Parameters
12.4 Scatter Plot: A Descriptive Statistical Tool
12.4.1 Inferential Interpretation of r and r²
12.4.2 More Inferences in the Simple Linear Model
12.4.3 Making Inference about the Slope bı
12.4.4 Applications of the Simple Linear Model: Estimation of the Mean of y for a Given x
12.5 An Excel Feature to Fit the Simple Linear Model
12.6 A Last Word about Modeling and How Statistics Works

Appendix
A.1 Recommended Readings
A.2 Current Developments
A.2.1 Nonparametric Statistics
A.2.2 Bayesian Statistics
A.2.3 New Developments in Meta-Analysis
A.3 Formulas
A.3.1 Probability Formulas
A.3.2 Interval Estimation Formulas—One Sample
A.3.3 Interval Estimation Formulas—Two Samples
A.3.4 Test of Hypothesis Formulas—One Sample
A.3.5 Test of Hypothesis Formulas—Two Samples
A.3.6 Simple Linear Model Formulas
A.4 Sample Tests