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Introductory Statistics
David Diez, Mine Çetinkaya-Rundel, Christopher D Barr
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Front Matter
Colophon
Note on this PreTeXt Edition
Preface
Acknowledgements
1
Introduction to data
1.1
Case study: using stents to prevent strokes
1.1
Exercises
1.2
Data basics
1.2.1
Observations, variables, and data matrices
1.2.2
Types of variables
1.2.3
Relationships between variables
1.2.4
Explanatory and response variables
1.2.5
Introducing observational studies and experiments
1.2.6
Exercises
1.3
Sampling principles and strategies
1.3.1
Populations and samples
1.3.2
Anecdotal evidence
1.3.3
Sampling from a population
1.3.4
Observational studies
1.3.5
Four sampling methods
1.3.6
Exercises
1.4
Experiments
1.4.1
Principles of experimental design
1.4.2
Reducing bias in human experiments
1.4.3
Exercises
1.5
Review exercises
1.5
Review exercises
2
Summarizing data
2.1
Examining numerical data
2.1.1
Scatterplots for paired data
2.1.2
Dot plots and the mean
2.1.3
Histograms and shape
2.1.4
Variance and standard deviation
2.1.5
Box plots, quartiles, and the median
2.1.6
Robust statistics
2.1.7
Transforming data (special topic)
2.1.8
Mapping data (special topic)
2.1.9
Exercises
2.2
Considering categorical data
2.2.1
Contingency tables and bar plots
2.2.2
Row and column proportions
2.2.3
Using a bar plot with two variables
2.2.4
Mosaic plots
2.2.5
The only pie chart you will see in this book
2.2.6
Comparing numerical data across groups
2.2.7
Exercises
2.3
Case study: malaria vaccine
2.3.1
Variability within data
2.3.2
Simulating the study
2.3.3
Checking for independence
2.3.4
Exercises
2.4
Chapter 2 Review Exercises
2.4
Chapter Review
3
Probability
3.1
Defining probability
3.1.1
Introductory examples
3.1.2
Probability
3.1.3
Disjoint or mutually exclusive outcomes
3.1.4
Probabilities when events are not disjoint
3.1.5
Probability distributions
3.1.6
Complement of an event
3.1.7
Independence
3.1.8
Exercises
3.2
Conditional probability
3.2.1
Exploring probabilities with a contingency table
3.2.2
Marginal and joint probabilities
3.2.3
Defining conditional probability
3.2.4
Smallpox in Boston, 1721
3.2.5
General multiplication rule
3.2.6
Independence considerations in conditional probability
3.2.7
Tree diagrams
3.2.8
Bayes’ Theorem
3.2.9
Exercises
3.3
Sampling from a small population
3.3
Exercises
3.4
Random variables
3.4.1
Expectation
3.4.2
Variability in random variables
3.4.3
Linear combinations of random variables
3.4.4
Exercises
3.5
Continuous distributions
3.5.1
From histograms to continuous distributions
3.5.2
Probabilities from continuous distributions
3.5.3
Exercises
3.6
Chapter 3 Review Exercises
3.6
Chapter Review
4
Distributions of random variables
4.1
Normal distribution
4.1.1
Normal distribution model
4.1.2
Standardizing with Z-scores
4.1.3
Finding tail areas
4.1.4
Normal probability examples
4.1.5
68-95-99.7 rule
4.1.6
Exercises
4.2
Geometric distribution
4.2.1
Bernoulli distribution
4.2.2
Geometric distribution
4.2.3
Exercises
4.3
Binomial distribution
4.3.1
The binomial distribution
4.3.2
Normal approximation to the binomial distribution
4.3.3
The normal approximation breaks down on small intervals
4.3.4
Exercises
4.4
Negative binomial distribution
4.4
Section 4.4 Exercises
4.5
Poisson distribution
4.5
Section 4.5 Exercises
4.6
Chapter review
4.6
Chapter review
5
Foundations for Inference
5.1
Point Estimates and Sampling Variability
5.1.1
Point Estimates and Error
5.1.2
Understanding the Variability of a Point Estimate
5.1.3
Central Limit Theorem
5.1.4
Applying the Central Limit Theorem to a real-world setting
5.1.5
More details regarding the Central Limit Theorem
5.1.6
Extending the framework for other statistics
5.1.7
Section 5.1 Exercises
5.2
Confidence Intervals for a Proportion
5.2.1
Capturing the Population Parameter
5.2.2
Constructing a 95% Confidence Interval
5.2.3
Changing the Confidence Level
5.2.4
More Case Studies
5.2.5
Interpreting Confidence Intervals
5.2.6
Section 5.2 Exercises
5.3
Hypothesis Testing for a Proportion
5.3.1
Hypothesis testing framework
5.3.2
Testing hypotheses using confidence intervals
5.3.3
Decision errors
5.3.4
Formal Testing Using P-values
5.3.5
Choosing a Significance Level
5.3.6
Statistical Significance versus Practical Significance
5.3.7
One-Sided Hypothesis Tests (Special Topic)
5.3.8
Section 5.3 Exercises
5.4
Review Exercises
5.4
Exercises
6
Inference for Categorical Data
6.1
Inference for a single proportion
6.1.1
Identifying when the sample proportion is nearly normal
6.1.2
Confidence intervals for a proportion
6.1.3
Hypothesis testing for a proportion
6.1.4
When one or more conditions aren’t met
6.1.5
Choosing a sample size when estimating a proportion
6.1.6
Section 6.1 Exercises
6.2
Difference of two proportions
6.2.1
Sampling distribution of the difference of two proportions
6.2.2
Confidence intervals for
\(p_1 - p_2\)
6.2.3
Hypothesis tests for the difference of two proportions
6.2.4
More on 2-proportion hypothesis tests (special topic)
6.2.5
Examining the standard error formula (special topic)
6.2.6
Section 6.2 Exercises
6.3
Testing for goodness of fit using chi-square
6.3.1
Creating a test statistic for one-way tables
6.3.2
The chi-square test statistic
6.3.3
The chi-square distribution and finding areas
6.3.4
Finding a p-value for a chi-square distribution
6.3.5
Evaluating goodness of fit for a distribution
6.3.6
Section 6.3 Exercises
6.4
Testing for independence in two-way tables
6.4.1
Expected counts in two-way tables
6.4.2
The chi-square test for two-way tables
6.4.3
Section Exercises
6.5
Chapter 6 Review Exercises
6.5
Exercises
7
Inference for Numerical Data
7.1
One-sample means with the
\(t\)
-distribution
7.1.1
The sampling distribution of
\(\bar{x}\)
7.1.2
Evaluating the two conditions required for modeling
\(\bar{x}\)
7.1.3
Introducing the
\(t\)
-distribution
7.1.4
One sample
\(t\)
-confidence intervals
7.1.5
One sample
\(t\)
-tests
7.1.6
Section Exercises
7.2
Paired data
7.2.1
Paired observations
7.2.2
Inference for paired data
7.2.3
Section Exercises
7.3
Difference of two means
7.3.1
Confidence interval for a difference of means
7.3.2
Hypothesis tests for the difference of two means
7.3.3
Case study: two versions of a course exam
7.3.4
Pooled standard deviation estimate (special topic)
7.3.5
Exercises
7.4
Power calculations for a difference of means
7.4.1
Going through the motions of a test
7.4.2
Computing the power for a 2-sample test
7.4.3
Determining a proper sample size
7.4.4
Exercises
7.5
Comparing many means with ANOVA
7.5.1
Core ideas of ANOVA
7.5.2
Is batting performance related to player position in MLB?
7.5.3
Analysis of variance (ANOVA) and the
\(F\)
-test
7.5.4
Reading an ANOVA table from software
7.5.5
Graphical diagnostics for an ANOVA analysis
7.5.6
Multiple comparisons and controlling Type 1 Error rate
7.5.7
Exercises
7.6
Chapter 7 Review Exercises
7.6
Chapter Review
8
Introduction to Linear Regression
8.1
Fitting a line, residuals, and correlation
8.1.1
Fitting a line to data
8.1.2
Using linear regression to predict possum head lengths
8.1.3
Residuals
8.1.4
Describing linear relationships with correlation
8.1.5
Section 8.1 Exercises
8.2
Least squares regression
8.2.1
Gift aid for freshman at Elmhurst College
8.2.2
An objective measure for finding the best line
8.2.3
Conditions for the least squares line
8.2.4
Finding the least squares line
8.2.5
Interpreting regression model parameter estimates
8.2.6
Extrapolation is treacherous
8.2.7
Using
\(R^2\)
to describe the strength of a fit
8.2.8
Categorical predictors with two levels
8.2.9
Exercises
8.3
Types of outliers in linear regression
8.3
Section Exercises
8.4
Inference for linear regression
8.4.1
Midterm elections and unemployment
8.4.2
Understanding regression output from software
8.4.3
Confidence interval for a coefficient
8.4.4
Section Exercises
8.5
Chapter 8 Review
8.5
Chapter Review
9
Multiple and Logistic Regression
9.1
Introduction to Multiple Regression
9.1.1
Indicator and categorical variables as predictors
9.1.2
Including and assessing many variables in a model
9.1.3
Adjusted
\(R^2\)
as a better tool for multiple regression
9.1.4
Exercises
9.2
Model Selection
9.2.1
Identifying variables in the model that may not be helpful
9.2.2
Two model selection strategies
9.2.3
The p-value approach, an alternative to adjusted
\(R^2\)
9.2.4
Exercises
9.3
Checking model conditions using graphs
9.3.1
Diagnostic plots
9.3.2
Options for improving the model fit
9.3.3
Exercises
9.4
Multiple regression case study: Mario Kart
9.4.1
Data set and the full model
9.4.2
Model selection
9.4.3
Checking model conditions using graphs
9.5
Introduction to Logistic Regression
9.5.1
Resume data
9.5.2
Modeling the probability of an event
9.5.3
Building the logistic model with many variables
9.5.4
Diagnostics for the callback rate model
9.5.5
Exploring discriminations between groups of different sizes
9.5.6
Exercises
9.5.6
Exercises
9.6
Chapter Review Exercises
9.6
Exercises
Back Matter
A
Exercise Solutions
A.1
Introduction To Data
A.2
Summarizing Data
A.3
Probability
A.4
Distributions Of Random Variables
A.5
Foundations For Inference
A.6
Inference For Categorical Data
A.7
Inference For Numerical Data
A.8
Introduction To Linear Regression
A.9
Multiple And Logistic Regression
B
Data Sets within the Text
B.1
Introduction to Data
B.2
Summarizing Data
B.3
Probability
B.4
Distributions
B.5
Foundations for Inference
B.6
Inference for Proportions
B.7
Inference for Means
B.8
Simple Linear Regression
B.9
Multiple and Logistic Regression
C
Distribution Tables
C.1
Normal Probability Table
C.2
\(t\)
-Probability Table
C.3
Chi-Square Probability Table
Index
Colophon
Appendix
A
Exercise Solutions
This appendix contains solutions to selected odd-numbered exercises from each chapter. These solutions are provided to help you check your work and understand the problem-solving process.
🔗
A.1
Introduction To Data
A.2
Summarizing Data
A.3
Probability
A.4
Distributions Of Random Variables
A.5
Foundations For Inference
A.6
Inference For Categorical Data
A.7
Inference For Numerical Data
A.8
Introduction To Linear Regression
A.9
Multiple And Logistic Regression
🔗
