CLT: sample mean between 75 and 78

Central limit theorem

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 (usually n ≥ 30) the sampling distribution of the sample mean x̄ is approximately normal with mean μ and variance σ²/n, whatever the shape of the parent population. That is,

Z = (x̄ − μ)/(σ/√n) → N(0, 1) as n → ∞.

Numerical

Given: n = 100, μ = 76, σ² = 256, so σ = 16.

Since n is large, x̄ ~ N(μ, σ²/n):

• Mean of x̄ = 76
• Standard error σ/√n = 16/√100 = 16/10 = 1.6

Standardise:

• x̄ = 75: Z₁ = (75 − 76)/1.6 = -0.625
• x̄ = 78: Z₂ = (78 − 76)/1.6 = 1.250

P(75 < x̄ < 78) = P(-0.625 < Z < 1.250) = Φ(1.250) − Φ(-0.625) = 0.8944 − 0.2660 = 0.6284

Answer: P(75 < x̄ < 78) ≈ 0.6284 (about 62.8%).