Overview
This lecture focused on summarizing numerical data using the concepts of shape, center, and spread, specifically within the context of sampling distributions of sample proportions.
Summarizing Numerical Data
- Numerical data can be summarized using shape, center, and spread.
- Sampling distributions allow us to analyze the distribution of many sample proportions.
Shape of Sampling Distributions
- The shape of a sampling distribution of proportions is normal (bell-shaped).
- A normal (symmetric) distribution means the center is the mean and the spread is the standard deviation.
Center of Sampling Distributions
- The mean of all sample proportions (mean of sampling distribution) equals the population proportion, P.
- This is true regardless of the specific sample values, as long as sampling is random and from a large population.
Spread (Standard Error) of Sampling Distributions
- The standard deviation of the sampling distribution is called the standard error.
- The standard error formula: √[P × (1 − P) / n], where P is population proportion and n is sample size.
- A larger sample size (n) leads to a smaller standard error (less spread).
Population Size Consideration
- The size of the population does not affect the shape, mean, or standard error of the sampling distribution, as long as the population is much larger than the sample.
- Rule of thumb: Population size should be at least 10 times the sample size for these formulas to apply.
Key Terms & Definitions
- Shape — The visual form of a data distribution, here specifically normal (bell-shaped) for sampling distributions.
- Mean (Center) — The average value; for sampling distributions, it equals the population proportion, P.
- Standard Deviation (Spread) — Measure of variability; for sampling distributions, called standard error.
- Standard Error — The standard deviation of the sampling distribution, calculated as √[P × (1 − P) / n].
- Population Proportion (P) — The true proportion of a characteristic in the whole population.
- Sampling Distribution — The distribution of sample statistics (e.g., sample proportions) from many samples.
Action Items / Next Steps
- Add the formula for standard error and mean of sampling distributions to your note sheet for the exam.
- Ensure you understand when to use these formulas—only when the population is at least 10 times the sample size.