For a random sample of size n from a population with mean μ and variance σ^2, what is the approximate distribution of the sample mean as n grows large?

• A. Uniform distribution
• B. Approximately Normal with mean μ and variance σ^2/n
• C. Exponential distribution with rate 1/μ
• D. Binomial distribution

Answer: (B)

The central limit theorem tells us that when we average many independent observations with finite mean and variance, the distribution of that average becomes approximately normal. For a random sample of size n from a population with mean μ and variance σ^2, the sample mean X̄ has E[X̄] = μ and Var(X̄) = σ^2/n. Hence, for large n, X̄ is approximately Normal with mean μ and variance σ^2/n (standard deviation σ/√n). This explains why the distribution tightens around μ as n grows. Other distributions like Uniform, Exponential, or Binomial describe different underlying processes and do not generally describe the limiting distribution of the sample mean for a broad population.