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:

1. Unbiasedness: \(\mu_{\bar{X}} = \mu\)
   
2. 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