Overview
This lecture introduces sampling distributions, focusing on sample proportions and how they are used to estimate population parameters in probability experiments involving categorical variables and numerical analysis.
Sampling Distributions
- A sampling distribution is formed by taking multiple samples and recording the outcome (sample statistic) from each.
- It is a type of probability distribution where outcomes are sample statistics, not individual data points.
- Outcomes are listed in a table, showing possible sample statistic values and their corresponding probabilities.
Sample Proportions (p-hat)
- Sample proportions ("p-hat") summarize categorical data, such as results from flipping coins or rolling dice.
- Each trial (sample) produces a sample proportion, representing the observed proportion in that sample.
- Population proportion ("p") is the theoretical or true proportion in the population, calculated without sampling.
Probability Distribution Table
- The probability distribution table's outcomes are values of the sample proportions from repeated samples.
- With more trials, the table would include repeated values and allow calculation of probabilities for each outcome.
- Sample proportions from repeated samples can estimate the population proportion.
Theoretical vs. Empirical Probability
- Theoretical probability uses known properties (e.g., a fair coin: probability of tails = 1/2).
- Empirical probability is calculated from actual experiments or trials.
Numerical Analysis of Sampling Distributions
- Sample proportions are numerical data, allowing analysis using descriptive statistics from Chapter 3: shape, center, and spread.
Key Terms & Definitions
- Sampling Distribution — Distribution of a statistic (e.g., sample proportion) calculated from many samples of the same size.
- Sample Proportion (p-hat) — Proportion of successes in a specific sample.
- Population Proportion (p) — True proportion of successes in the population.
- Theoretical Probability — Probability based on known characteristics, not experiments.
- Empirical Probability — Probability based on observed data from repeated trials.
- Categorical Variable — Variable with values that are categories (e.g., head/tail).
Action Items / Next Steps
- Review Chapter 6 for probability and sampling distribution basics.
- Recall Chapter 3 concepts: shape, center, and spread for analyzing numerical data.
- Practice constructing probability distribution tables for sample proportions.