CLT: bulb mean life between 1100 and 1300

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

μ = 1200 h, σ = 150 h, n = 36 (large enough for CLT).

SE = σ/√n = 150/6 = 25 hours, so x̄ ~ N(1200, 25²)

• z₁ = (1100 − 1200)/25 = -4.00
• z₂ = (1300 − 1200)/25 = 4.00

P(1100 < x̄ < 1300) = P(−4 < Z < 4) = 2 × P(0 < Z < 4) ≈ 2 × 0.49997 = 0.99994

Answer: the probability is about 0.99994, almost certain.