CLT: average resistance of 25 resistors

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 = 100 Ω, σ = 10 Ω, n = 25, population normal.

SE = σ/√n = 10/5 = 2 Ω

z = (95 − 100)/2 = -2.50

P(x̄ < 95) = P(Z < −2.50) = 0.5 − 0.4938 = 0.0062

Answer: the probability that the average resistance is below 95 Ω is 0.0062 (about 0.62%).