Day 17

Math 216: Statistical Thinking

Bastola

Central Limit Theorem: Foundation of Statistical Inference

Core Statistical Principle

Central Limit Theorem: For any population with finite mean μ and variance σ², the sampling distribution of the sample mean approaches a normal distribution as sample size increases.

Mathematical Formulation: \[\bar{x} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)\]

Practical Significance: This universal principle enables statistical inference across diverse population distributions.

Case Study 1: Steel Manufacturing

Industrial Context: Steel sheets produced with uniform thickness distribution (150-200 mm). Quality assurance requires monitoring average thickness from small samples.

Case Study 1: Steel Manufacturing

Numerical Analysis Summary:

  • Population Mean: 175.0 mm (theoretical)
  • Population Standard Deviation: 14.43 mm (theoretical)
  • Sample Size: 36 sheets per quality check
  • Standard Error: 2.41 mm (σ/√n = 14.43/√36)

Statistical Insight: Despite uniform population distribution, sample means follow normal distribution with predictable precision!

Case Study 2: Exponential Distribution Analysis

Context: Customer service call durations follow exponential distribution with mean = 5 minutes.

Statistical Insight: Even with highly skewed exponential population, sample means become normally distributed with n=40!

Case Study 3: Bimodal Distribution Analysis

Context: Product weights from two different production lines create bimodal distribution.

Statistical Insight: Complex bimodal populations still yield normally distributed sample means with adequate sample size!

Case Study 4: Left-Skewed Distribution Analysis

Context: Employee performance ratings with left-skewed distribution (most employees rated highly).

Statistical Insight: Left-skewed populations also produce normally distributed sample means with sufficient sample size!

Standard Error: Quantifying Estimation Precision

Standard Error Formula: \(SE = \frac{\sigma}{\sqrt{n}}\)

Applications of Central Limit Theorem: Political Polling

Context: Predicting election outcomes from voter samples

Statistical Findings:

  • Poll results follow normal distribution despite binary data
  • 95% of polls fall between 44.5% and 51.5%
  • CLT enables accurate predictions from small samples

Applications of Central Limit Theorem: Quality Control

Context: Monitoring production quality from small samples

Quality Control Analysis:

  • Uniform population → Normal sampling distribution
  • 95% of samples fall within 174.3-175.7mm
  • CLT works effectively with non-normal populations

Applications of Central Limit Theorem: Medical Research

Context: Testing drug effectiveness from clinical trials

Medical Research Analysis:

  • Skewed population → Normal sampling distribution
  • CLT functions with any population shape
  • Sample size is more critical than population distribution

Key Statistical Principles

Essential Statistical Concepts:

  1. Universal Normality: Sample means follow normal distribution regardless of population shape
  2. Precision Quantification: Standard error = σ/√n measures estimation accuracy
  3. Sample Size Effect: n ≥ 30 typically provides adequate CLT approximation
  4. Distribution Independence: Works with uniform, skewed, bimodal, and any finite-variance population

Statistical Guidelines:

  • Sample size ≥ 30 → Sampling distribution is approximately normal
  • Population shape → CLT works regardless of distribution
  • Standard error → Quantifies precision of estimates
  • Sample size increase → Improves estimation accuracy

Next Topic: Applying these principles to confidence intervals and hypothesis testing