Day 16

Math 216: Statistical Thinking

Bastola

The Concept of a Sampling Distribution

• Background: Traditionally, the exact probability distribution of a population is unknown, and we rely on sample data to make inferences.
• Definitions:
  • Parameter: A fixed, often unknown characteristic of a population.
  • Statistic: A characteristic of a sample, used as an estimate of the population parameter.

Calculating Sample Statistics

• Point Estimators:
  • Sample mean (\(\bar{x}\)): \(\bar{x}=\frac{\sum x_i}{n}\)
  • Sample variance (\(s^2\)): \(s^2=\frac{\sum(x_i-\bar{x})^2}{n-1}\)
• Usage: These estimators help approximate the population mean (\(\mu\)) and variance (\(\sigma^2\)).

Unbiasedness and Minimum Variance

• Unbiased Estimators: An estimator is unbiased if it consistently represents the population parameter across different samples.
• Minimum Variance: Among unbiased estimators, preference is given to those with smaller variances, as they tend to produce more reliable estimates.

Comparing Sampling Distributions

List of relevant parameters

Standard Error and Estimation Error

• Standard Error: The standard deviation of the sampling distribution, providing a measure of the spread of the estimates.
• Estimation Error: Reflects how much an estimate derived from a sample statistic is expected to differ from the actual population parameter.

Comparing Sampling Distributions

• Visual Example: Two sampling distributions for statistics A and B might show different spreads (standard deviations), indicating the reliability of the estimates.
• Selection Criterion: Statistic with a smaller standard error is preferred as it is likely to be closer to the population parameter.

Practical Implications

• Inference: Sampling distributions allow us to make educated guesses about population parameters.
• Application: Essential in fields like epidemiology, economics, and quality control where population parameters are not directly observable.