Applications of the CLT

Probability · Axiom Academy

LESSON Applications of the Central Limit Theorem From theory to practice: using the CLT for inference and approximation 1. Approximating Distributions The CLT allows us to approximate the distribution of sums and averages using the normal distribution. For i.i.d. random variables X₁, ..., Xₙ with mean μ and variance σ²: This means we can compute probabilities using the normal CDF: 2. Constructing Confidence Intervals The CLT is the foundation for confidence intervals. Since X̄ₙ is approximately normal for large n: Rearranging, we get a 95% confidence interval for μ: This interval captures the true population mean μ in approximately 95% of samples. The CLT enables z-tests for population means. To test H₀: μ = μ₀ vs. Hₐ: μ ≠ μ₀, we compute: Under H₀ and by the CLT, Z ~ N(0,1) for large n. We reject H₀ if |Z| > 1.96 (for α = 0.05).

This is the written version of the interactive lesson above. See the full Probability course.