Statistical Inference

Statistical inference is the process of using data from a sample to draw conclusions or make predictions about the larger population from which the sample was drawn. Since studying an entire population is often impractical or impossible, inference provides the formal methods for generalizing from a part to the whole. This branch of statistics primarily involves two approaches: estimation, which uses sample data to determine a likely range of values for a population characteristic (e.g., a confidence interval for the average income), and hypothesis testing, which assesses evidence to make a decision about a specific claim regarding the population (e.g., whether a new drug is effective). Crucially, all statistical inferences are grounded in probability theory, allowing us to quantify the uncertainty inherent in making conclusions based on incomplete data.

1. Foundations of Statistical Inference
   1. The Goal of Inference
      1. Generalizing from Sample to Population
      2. Quantifying Uncertainty
      3. Making Data-Driven Decisions
   2. The Role of Probability Theory
      1. Probability as a Measure of Uncertainty
      2. Random Variables and Probability Distributions
      3. Law of Large Numbers
      4. Central Limit Theorem Preview
      5. Probability in Statistical Modeling
   3. Key Terminology and Concepts
      1. Population and Sample
         1. Definition of Population
         2. Definition of Sample
         3. Census vs Sample Survey
         4. Target Population vs Sampled Population
      2. Parameter and Statistic
         1. Definition of Parameter
         2. Definition of Statistic
         3. Notation Conventions
         4. Examples of Parameters and Statistics
      3. Statistical Models
         1. Model Specification
         2. Assumptions in Statistical Models
         3. Parametric vs Non-parametric Models
         4. Model Selection Criteria
   4. Types of Statistical Inference
      1. Estimation
         1. Point Estimation
         2. Interval Estimation
      2. Hypothesis Testing
         1. Null and Alternative Hypotheses
         2. Test Statistics
         3. Decision Making
      3. Prediction
         1. Predictive Models
         2. Assessing Predictive Accuracy
         3. Prediction vs Inference