Under the Central Limit Theorem, the distribution of the sample mean Xbar of i.i.d. with finite mean mu and variance sigma^2 is approximately normally distributed for large n with what mean and variance?

Under the Central Limit Theorem, the distribution of the sample mean Xbar of i.i.d. with finite mean mu and variance sigma^2 is approximately normally distributed for large n with what mean and variance?

• A. Mean mu; variance sigma^2
• B. Mean mu; variance sigma^2/n
• C. Mean mu; variance sigma^2/n^2
• D. Mean mu; variance sigma^2 * n

Answer: (B)

The sample mean behaves like a normal variable centered at the true mean, with its variability shrinking as you average more observations. Specifically, E[X̄] = μ and Var(X̄) = σ²/n, so X̄ is approximately N(μ, σ²/n) for large n. The standard deviation of X̄ is σ/√n, showing how the spread decreases with n. This is why increasing the sample size makes the average estimate more precise.

The sample mean behaves like a normal variable centered at the true mean, with its variability shrinking as you average more observations. Specifically, E[X̄] = μ and Var(X̄) = σ²/n, so X̄ is approximately N(μ, σ²/n) for large n. The standard deviation of X̄ is σ/√n, showing how the spread decreases with n. This is why increasing the sample size makes the average estimate more precise.