CLT: sample mean between 67 and 83

Central Limit Theorem (CLT): if X₁, X₂, …, Xₙ is a random sample of size n from any population with mean μ and finite variance σ², then for large n (n ≥ 30) the sampling distribution of the sample mean x̄ is approximately normal with mean μ and standard deviation σ/√n, whatever the shape of the population:

x̄ ~ N(μ, σ²/n), and z = (x̄ − μ)/(σ/√n) ~ N(0, 1).

If the population itself is normal, x̄ is exactly normal for every n.

Numerical

Given μ = 75, σ² = 256, so σ = 16, and n = 100 (large, so CLT applies).

Standard error = σ/√n = 16/√100 = 1.6

x̄ ~ N(75, 1.6²)

z₁ = (67 − 75)/1.6 = -5.00 and z₂ = (83 − 75)/1.6 = 5.00

P(67 < x̄ < 83) = P(−5 < Z < 5) = 2 × P(0 < Z < 5) ≈ 2 × 0.4999997 = 0.9999994

Answer: the probability that the sample mean lies between 67 and 83 is about 0.9999994, i.e. practically 1 (certain).