gen-state-the-central-limit-theorem-precisely-and-explain-why-it-1f4b

State the central limit theorem precisely and explain why it matters in 2-3 sentences.

# Central Limit Theorem **Precise Statement:** Let X₁, X₂, ..., Xₙ be independent and identically distributed random variables with mean μ and finite variance σ². Then the distribution of the standardized sample mean $$\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} = \frac{\sqrt{n}(\bar{X}_n - \mu)}{\sigma}$$ converges in distribution to a standard normal distribution N(0,1) as n → ∞. **Why It Matters:** The CLT explains why normal distributions appear everywhere in nature and statistics—aggregated effects tend toward normality regardless of the underlying distribution. This justifies using normal-based inference (confidence intervals, hypothesis tests) for sample means even when we don't know the population distribution, making it the foundation of practical statistical analysis.