gen-define-the-central-limit-theorem-and-give-one-short-worked-e-f54e

Define the central limit theorem and give one short worked example.

# Central Limit Theorem ## Definition The **Central Limit Theorem (CLT)** states that when independent random variables are summed, their normalized sum tends toward a normal distribution, even if the original variables themselves are not normally distributed. More precisely: If X₁, X₂, ..., Xₙ are independent random variables with mean μ and standard deviation σ, then as n becomes large, the sampling distribution of the sample mean approaches a normal distribution with: - Mean = μ - Standard deviation = σ/√n ## Worked Example **Problem:** A factory produces bolts with mean length 5 cm and standard deviation 0.4 cm. If we randomly sample 64 bolts and calculate their average length, what's the probability this average exceeds 5.1 cm? **Solution:** Given information: - Population mean: μ = 5 cm - Population std dev: σ = 0.4 cm - Sample size: n = 64 By CLT, the sample mean follows approximately N(5, 0.4/√64) = N(5, 0.05) Calculate the z-score: $$z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} = \frac{5.1 - 5}{0.05} = 2$$ From the standard normal table: P(Z > 2) = 0.0228 **Answer:** There's approximately a 2.28% probability that the average length exceeds 5.1 cm.