Overview
This lecture introduces the two main parts of the course: probability (starting from a known population) and statistical inference (starting from a sample to estimate population characteristics).
Probability: Population to Sample
- Probability starts from a known population with a model describing its characteristics.
- A sample is a subset of the population selected for study.
- Probabilistic models (like parameters) describe uncertainty and predict chances of events in samples.
- Example: If 36% of all Americans have passports, probability can compute the chance that 10 out of 20 randomly sampled people have passports.
- The number of observations in a sample is denoted as n.
Statistical Inference: Sample to Population
- Statistical inference uses only a sample to infer unknown population characteristics.
- Assumes sample is representative and follows certain properties.
- Estimates population parameters and quantifies uncertainty (provides error bounds or regions of plausibility).
- Uses concepts and models from probability to make logical inferences about the broader population.
- Example: If 8 of 20 sampled Americans have passports, infer (with uncertainty) the likely percentage for all Americans.
Comparing Probability and Inference
- Probability: Population known → Calculate sample likelihoods.
- Inference: Population unknown → Use sample to estimate population parameters.
- Probability provides tools needed for statistical inference.
Key Terms & Definitions
- Population — Entire collection of individuals or objects under study.
- Sample — Subset of the population selected for analysis.
- Parameter — Numeric characteristic describing a population.
- Probabilistic Model — Mathematical description of random processes within the population.
- Statistical Inference — Process of using a sample to estimate or draw conclusions about a population.
- Uncertainty Quantification — Process of measuring how certain or uncertain an estimate is.
Action Items / Next Steps
- Review the basics of probability and sampling models.
- Understand the differences between probability calculations and statistical inference.
- Prepare to apply probability tools before moving to inference problems.