STAT 350: Introduction to Statistics

Orientation

  • Course Introduction & Overview
    • Why Statistics? Why Now?
    • Office Hours
    • Course Roadmap
    • A Clear Note on Using AI
    • Getting Started: A Checklist
      • Homework Assignments on Edfinity
      • Miscellaneous Tips

Exam Information

  • Course Examinations
    • General Exam Policies
      • Exam 1
        • Exam Locations by Section
        • Exam Coverage
        • Preparation Materials
      • Exam 2
        • Exam Locations by Section
        • Exam Coverage
        • Preparation Materials
      • Final Exam
        • About the Final Exam
        • Required Review Materials
        • Study Guide Resource

Worksheets

  • Course Worksheets
    • Pedagogical Philosophy
    • Implementation Guidelines
    • Why These Worksheets Matter
    • The Critical Role of Simulation
    • Worksheets
      • Worksheet 1: Exploring Data with R
        • Introduction
        • Part 1: Loading and Understanding the Dataset
        • Part 2: Initial Data Exploration
        • Part 3: Frequency Tables
        • Part 4: Univariate Analysis of Uptake
        • Part 5: Grouped Statistics with tapply
        • Part 6: Comparative Visualization by Type
        • Part 7: Exploring the Concentration Effect
        • Part 8: Advanced Visualization with Multiple Categories
        • Reference: Key Functions
        • Troubleshooting Guide
        • Additional Resources
      • Worksheet 2: Set Theory and Probability Fundamentals
        • Introduction
        • Part 1: Set Theory Foundations
        • Part 2: Probability Axioms
        • Part 3: Applying Probability Rules
        • Part 4: The Inclusion-Exclusion Principle
        • Key Takeaways
        • Submission Guidelines
      • Worksheet 3: Conditional Probability and Bayes’ Theorem
        • Introduction
        • Part 1: Understanding Conditional Probability
        • Part 2: Tree Diagrams and Sequential Sampling
        • Part 3: Bayes’ Theorem and Sequential Updating
        • Key Takeaways
      • Worksheet 4: Independence and Random Variables
        • Part 1: Independence Property
        • Part 2: Independent vs. Mutually Exclusive Events
        • Part 3: Introduction to Random Variables
        • Part 4: Probability Mass Functions
        • Part 5: Joint Probability Mass Functions
        • Key Takeaways
      • Worksheet 5: Expected Value and Variance
        • Introduction
        • Part 1: Expected Value and LOTUS
        • Part 2: Variance and Its Properties
        • Part 3: Sums of Random Variables
        • Part 4: Joint Probability Mass Functions
        • Key Takeaways
      • Worksheet 6: Named Discrete Distributions
        • Introduction
        • Part 1: The Bernoulli and Binomial Distributions
        • The Binomial Distribution
        • Part 2: The Poisson Distribution
        • Part 3: Other Named Discrete Distributions
        • Key Takeaways
      • Worksheet 7: Continuous Random Variables
        • Introduction
        • Part 1: Probability Density Functions
        • Part 2: Finding Constants for Valid PDFs
        • Part 3: Expected Value and Variance
        • Part 4: Cumulative Distribution Functions
        • Key Takeaways
      • Worksheet 8: Uniform and Exponential Distributions
        • Introduction
        • Part 1: The Uniform Distribution
        • Part 2: The Exponential Distribution
        • Key Takeaways
      • Worksheet 9: The Normal Distribution
        • Introduction
        • Part 1: The Normal Distribution
        • Part 2: The Standard Normal Table
        • Part 3: Z-Score Transformation
        • Key Takeaways
      • Worksheet 10: Checking Normality and Introduction to Sampling Distributions
        • Part 1: Checking Normality
        • Part 2: Introduction to Sampling Distributions
        • Key Takeaways
      • Worksheet 11: The Central Limit Theorem
        • Introduction
        • Tutorial: Generating the Sampling Distribution of \(\bar{X}\)
        • Part 1: Exploring CLT with Skewed Distributions
        • Part 2: Application of the CLT
        • Part 3: Beyond Mean and Sum
        • Part 4: Exploring CLT Generalizability with AI Assistance (Exploration)
        • Key Takeaways
      • Worksheet 12: Point Estimators and Unbiased Estimation
        • Introduction
        • Part 1: Estimating Parameters of the Exponential Distribution
        • Part 2: Estimating the Maximum of a Uniform Distribution
        • Part 3: Minimum Variance Unbiased Estimators (MVUE)
        • Key Takeaways
      • Worksheet 13: Introduction to Confidence Intervals
        • Introduction
        • Part 1: Pivotal Quantities and Deriving Confidence Intervals
        • Part 2: Sample Size Determination
        • Part 3: One-Sided Confidence Bounds
        • Part 4: Confidence Intervals When σ is Unknown
        • Key Takeaways
      • Worksheet 14: Student’s t-Distribution and Statistical Power
        • Introduction
        • Assumptions for t-Distribution Inference
        • Part 1: Analyzing Chick Growth with Confidence Intervals
        • Part 2: Introduction to Hypothesis Testing
        • Part 3: Sample Size Determination and Power Analysis
        • Key Takeaways
      • Worksheet 15: Hypothesis Testing for a Single Mean
        • The Hypothesis Testing Framework
        • The Test Statistic (Known σ)
        • Distribution of the Test Statistic
        • The p-value
        • Question 1a: Simulation When Null Hypothesis is True
        • Question 1b: Simulation When Null Hypothesis is False
        • Question 1c: Comparing the Two Scenarios
        • The t-Test Statistic
        • Properties of the t-Distribution
        • Question 2a: Assumptions and Exploratory Analysis
        • Question 2b: Full Hypothesis Testing Procedure
        • Question 2c: Manual Verification
        • Question 2d: Confidence Bound
        • Question 2e: Power Calculation
      • Worksheet 16: Two-Sample Inference
        • Introduction
        • Part 1: Independent vs. Paired Designs
        • Part 2: Independent Samples with Known Variances
        • Part 3: Pooled Two-Sample t-Test (Equal Variances)
        • Part 4: Welch’s Two-Sample t-Test (Unequal Variances)
        • Key Takeaways
      • Worksheet 17: Paired Sample Inference
        • Introduction
        • Part 1: Theory and Procedure for Paired Samples
        • Part 2: Sleep Study Analysis
        • Part 3: Sample Size Calculation for Paired Designs
        • Key Takeaways
      • Worksheet 18: One-Way ANOVA (Analysis of Variance)
        • Introduction
        • Why Use One-Way ANOVA?
        • One-Way ANOVA Assumptions
        • One-Way ANOVA F-test Statistic and Sources of Variation
        • Notation and Setup
        • Part 1: Computing the Overall Sample Mean
        • Part 2: Checking the Equal Variance Assumption
        • Part 3: Pooled Variance Estimator
        • Part 4: Within-Group Variability (Sum of Squares Within, SSE)
        • Part 5: Between-Group Variability (Sum of Squares Among, SSA)
        • Part 6: Total Variability and Partitioning
        • Part 7: Degrees of Freedom
        • Part 8: Mean Squares
        • Part 9: The F-Test Statistic
        • Part 10: Computing the P-Value and Making a Decision
        • Part 11: Completing the ANOVA Table
        • Part 12: Limitations and Follow-Up Questions
        • Key Takeaways
      • Worksheet 19: Multiple Comparisons and Post-Hoc Testing
        • Introduction
        • The Multiple Comparisons Problem
        • Methods to Control the Family-Wise Error Rate
        • Part 1: Simulation Study - Type I Error with Multiple Comparison Methods
        • Part 2: Applying Tukey’s HSD to Worksheet 18 Data
        • Part 3: Comprehensive ANOVA Analysis - ToothGrowth Dataset
        • Key Takeaways
      • Worksheet 20: Introduction to Simple Linear Regression
        • Introduction
        • From ANOVA to Regression: Same Core Ideas, New Application
        • The Simple Linear Regression Model
        • Least Squares
        • Part 1: Deriving Least Squares Estimators
        • Part 2: The General Simple Linear Regression Estimators
        • Part 3: Sampling Distributions of Regression Estimators
        • Part 4: Practice Problems
        • Key Takeaways
      • Worksheet 21: Inference for Simple Linear Regression
        • Introduction
        • Part 1: Sampling Distributions of Regression Estimators
        • Part 2: Checking Regression Assumptions
        • Part 3: Bird Data Analysis
        • Part 4: Regression ANOVA
        • Key Takeaways
      • Worksheet 22: Model Assessment and Prediction in Regression
        • Introduction
        • Part 1: The Coefficient of Determination
        • Part 2: Confidence Intervals vs. Prediction Intervals
        • Part 3: Simulation Study — CI vs. PI Coverage
        • Part 4: Prediction with the Bird Data
        • Part 5: Diamonds Data Analysis
        • Key Takeaways

Computer Assignments

  • R / RStudio Guide and Function Reference
    • Getting Started with R and RStudio
      • Quick Start: R / RStudio Setup
      • RStudio Orientation
      • Packages / Libraries (Course Set)
      • Getting Started with swirl
        • Setup and Use swirl
    • Course Datasets
      • Primary Course Dataset - AppRating
      • Tutorial Support Datasets
      • Loading Datasets in R
      • Built-in R Datasets Used in Course
      • Data Download and Organization
    • Computer Assignments and Tutorials
      • Overview
      • Assignment Structure
      • Assignment Tutorials (Links)
      • Course Pipeline (At a Glance)
      • More Help and Reference Pages
    • Best Practices & Common Pitfalls
      • Data Import & Validation
      • Statistical Assumptions
      • Common Errors to Avoid
      • Workflow Template
    • Graphics (ggplot2)
      • Core Components
      • Geoms (Geometric Objects)
      • Categorical Data Visualization
      • Plot Customization
      • Tables & Reporting
    • Function Reference Part 1
      • Data I/O & Housekeeping
      • Data Structures & Creation
      • Data Wrangling & Utilities
      • Descriptive Statistics & Correlation
      • Probability & Distributions
      • Simulation Functions
    • Function Reference Part 2: Inference Functions
      • Diagnostic Plots for Assumptions
    • Alternative Resources (Tutorials)
      • General R & Data Science
      • ggplot2 Visualization
      • Tidyverse
      • Video Content
      • Quick Reference
      • Base R

Chapters

  • 1. Introduction to Statistics
    • 1.1. What Is Statistics?
    • 1.2. Probability & Statistical Inference: How Are They Associated?
  • 2. Graphical Summaries
    • 2.1. Data Set Structure and Variable Types
    • 2.2. Tools for Categorical (Qualitative) Data
    • 2.3. Tools for Numerical (Quantitative) Data
    • 2.4. Exploring Quantitative Distributions: Modality, Skewness & Outliers
  • 3. Numerical Summaries
    • 3.1. Introduction to Numerical Summaries: Notation and Terminology
    • 3.2. Measures of Central Tendency
    • 3.3. Measures of Variability - Range, Variance, and Standard Deviation
    • 3.4. Measures of Variability - Interquartile Range and Five-Number Summary
    • 3.5. Choosing the Right Measure & Comparing Measures Across Data Sets
  • 4. Probability
    • 4.1. Basic Set Theory
    • 4.2. Probability
    • 4.3. Conditional Probability
    • 4.4. Law of Total Probability and Bayes’ Rule
    • 4.5. Sequential Bayesian Updating
    • 4.6. Independence of Events
  • 5. Discrete Distributions
    • 5.1. Discrete Random Variables and Probability Mass Distributions
    • 5.2. Joint Probability Mass Functions
    • 5.3. Expected Value of a Discrete Random Variable
    • 5.4. Variance of a Discrete Random Variable
    • 5.5. Covariance of Dependent Random Variables
    • 5.6. The Binomial Distribution
    • 5.7. The Poisson Distribution
  • 6. Continuous Distributions
    • 6.1. Continuous Random Variables and Probability Density Functions
    • 6.2. Expected Value and Variance of Continuous Random Variables
    • 6.3. Cumulative Distribution Functions
    • 6.4. Normal Distribution
    • 6.5. Uniform Distribution
    • 6.6. Exponential Distribution
  • 7. Sampling Distributions
    • 7.1. Statistics and Sampling Distributions
    • 7.2. Sampling Distribution for the Sample Mean
    • 7.3. The Central Limit Theorem (CLT)
    • 7.4. Understanding Binomial and Poisson Distributions through CLT
  • 8. Experimental Design
    • 8.1. Experimental and Sampling Designs
    • 8.2. Experimental Design Principles
    • 8.3. Basic Types of Experimental Design
    • 8.4. Addressing Potential Flaws in Experimental Design
    • 8.5. Examples of Experimental Design
    • 8.6. Sampling Design
    • 8.7. Sampling Bias
  • 9. Confidence Intervals and Bounds
    • 9.1. Introduction to Statistical Inference
    • 9.2. Exercises: Introduction to Statistical Inference
    • 9.3. Confidence Intervals for the Population Mean, When σ is Known
    • 9.4. Exercises: Confidence Intervals for μ (σ Known)
    • 9.5. Precision of a Confidence Interval
    • 9.6. Exercises: Precision of CI and Sample Size Calculation
    • 9.7. Confidence Bounds for the Population Mean When σ is Known
    • 9.8. Confidence Intervals and Bounds When σ is Unknown
  • 10. Hypothesis Testing
    • 10.1. The Foundation of Hypothesis Testing
    • 10.2. Hypothesis Test for the Population Mean When σ is Known
    • 10.3. Connecting CI and HT; t-Test for μ When σ Is Unknown
    • 10.4. The Four Steps to Hypothesis Testing and Understanding the Result
  • 11. Two Sample Procedures
    • 11.1. Statistical Inference for Two Samples
    • 11.2. Independent Two-Sample Analysis When Population Variances Are Known
    • 11.3. Independent Two-Sample Analysis - Pooled Variance Estimator
    • 11.4. Independent Two-Sample Analysis - No Equal Variance Assumption
    • 11.5. Paired Two-Sample Analysis
  • 12. ANOVA
    • 12.1. Introduction to One-Way ANOVA
    • 12.2. Different Sources of Variability in ANOVA
    • 12.3. ANOVA F-Test and Its Relationship to Two-Sample t-Tests
    • 12.4. Multiple Comparison Procedures
  • 13. Simple Linear Regression
    • 13.1. Introduction to Linear Regression
    • 13.2. Simple Linear Regression
    • 13.3. Model Diagnostics and Statistical Inference
    • 13.4. Prediction and Robustness
STAT 350: Introduction to Statistics
  • 9. Confidence Intervals and Bounds
  • View page source

9. Confidence Intervals and Bounds

  • 9.1. Introduction to Statistical Inference
  • 9.2. Exercises: Introduction to Statistical Inference
  • 9.3. Confidence Intervals for the Population Mean, When σ is Known
  • 9.4. Exercises: Confidence Intervals for μ (σ Known)
  • 9.5. Precision of a Confidence Interval
  • 9.6. Exercises: Precision of CI and Sample Size Calculation
  • 9.7. Confidence Bounds for the Population Mean When σ is Known
  • 9.8. Confidence Intervals and Bounds When σ is Unknown
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