Large-Sample Distribution of the Sample Mean

The central limit theorem describes the approximate shape of the sampling distribution for large $n$.

The mean and variance of the sample mean carry over exactly; only the shape becomes approximately normal.

1. 1. Apply the central limit theorem

   For large $n$, the standardized sample mean converges in distribution to a standard normal.

2. 2. Restate in the original scale

   Undoing the standardization gives the approximate distribution of the sample mean itself.

   $$\bar{X}\stackrel{\text{approx}}{\sim }N\left(\mu ,\frac{{\sigma }^2}{n}\right)$$

Answer: Approximately N(mu, sigma^2/n)