Sampling Distribution
Sampling Distributions and the Central Limit Theorem
When estimating population parameters like \(\mu\), we use sample statistics \(\bar{X}\). Different samples of size \(n\) yield different means - these sample means are random variables with their own distribution. The sampling distribution describes all possible values of \(\bar{X}\) from repeated sampling. Key properties:
- Unbiasedness: \(\mu_{\bar{X}} = \mu\)
- Precision: \(\sigma_{\bar{X}} = \sigma/\sqrt{n}\)
The Central Limit Theorem (CLT) reveals the shape:
- Exact normality: If population is normal (any \(n\))
- Approximate normality: For any population with \(n \geq 30\)
Population Distribution
Exponential distribution with population mean
Sampling Distribution
CLT convergence despite non-normal parent distribution
Key Observations
- Sample means cluster around population mean (unbiasedness)
- Spread decreases with larger \(n\) (precision improvement)
- Normal shape emerges despite skewed population (CLT in action)
Critical Implications
Enables inference even with unknown population distributions
Justifies common statistical procedures (confidence intervals, hypothesis tests)
Fails if:
‣ Small samples (\(n < 30\)) from non-normal populations
‣ Population variance is infinite ## Lab Prep Connection
Before starting Lab 5, experiment with the sampling distribution app linked on this page. Focus on how the center, spread, and shape of the sampling distribution change with sample size and population shape.