Samples of size n=36 are randomly selected from a population with mean μ = 125 and variance 12. Find the variance of the distribution of sample means.

For any population with finite variance σ², the Central Limit Theorem states that the sampling distribution of the sample mean Ẋ (for simple random samples of size n) is approximately normal with E(Ẋ) = μ and Var(Ẋ) = σ² / n. Given σ² = 12 and n = 36, Var(Ẋ) = 12 / 36 = 1/3 ≈ 0.333. Thus the variance of the distribution of sample means is 0.333, option A.

B. 0.577 → This is √0.333, the standard deviation, not the variance.
C. 2   → Uses σ / √n (≈2) but mixes variance with standard deviation.
D. 3.464 → This is √12, the population standard deviation, not the sampling‐mean variance.
E. 12  → Gives the population variance, ignoring the 1/n reduction for sample means.

1. Montgomery

D. C. & Runger

G. C. “Applied Statistics and Probability for Engineers”

7th ed.

Wiley

2019

Section 5.5

Eq. (5-8)

p. 219.

2. Walpole

R. E.

Myers

R. H.

Myers

S. L.

& Ye

K. “Probability & Statistics for Engineers and Scientists”

9th ed.

Pearson

2012

Sec. 6.2

p. 235.

3. MIT OpenCourseWare 18.05 (“Introduction to Probability and Statistics”)

Lecture Notes: “Sampling Distributions”

p. 3 (Var(Ẋ)=σ²/n).