Overview
This lecture covers how to find the mean and standard error of the sampling distribution of a sample proportion, and explains their role in statistical inference.
Finding the Mean of the Sampling Distribution
- The mean of the sampling distribution of sample proportions equals the population proportion (P).
- In the example, P = 0.65, so the mean is 0.65.
- The mean represents the "typical" or "expected" value for the sample proportion.
Calculating the Standard Error
- The standard error formula for the sampling distribution of a proportion is SE = sqrt[P(1−P)/n], where n is the sample size.
- For P = 0.65 and n = 500, substitute these into the formula: SE = sqrt[0.65×(1−0.65)/500].
- Calculating this gives SE ≈ 0.021.
Interpreting Statistical Inference
- Statistical inference involves making an estimate and acknowledging its uncertainty.
- The estimate is the mean (e.g., 65% of women get annual Pap tests).
- The uncertainty is expressed as "give or take" the standard error (e.g., ±2.1%).
- This process will be used throughout future chapters for various types of inference.
Key Terms & Definitions
- Sample Proportion (p-hat) — The proportion of a specific outcome in a sample.
- Population Proportion (P) — The proportion of the outcome in the entire population.
- Mean of Sampling Distribution — The average value of sample proportions across all possible random samples; equals P.
- Standard Error (SE) — The typical error or variability in sample proportions; calculated with SE = sqrt[P(1−P)/n].
- Statistical Inference — Drawing conclusions about a population using sample data, accounting for uncertainty.
Action Items / Next Steps
- Practice calculating the mean and standard error for sample proportions.
- Review the concept of statistical inference and its two components.
- Prepare for more examples of statistical inference in Chapters 7, 8, 9, and 11.