1. The problem states the convergence in distribution of the standardized sample mean to a standard normal random variable. 2. The formula given is $$\frac{\overline{X}_n - \mu}{\sigma/\sqrt{n}} \xrightarrow{n \to \infty} Z \sim N(0,1)$$ which expresses the Central Limit Theorem (CLT). 3. Important rules: - $\overline{X}_n$ is the sample mean of $n$ independent and identically distributed random variables with mean $\mu$ and variance $\sigma^2$. - As $n$ grows large, the distribution of the standardized sample mean approaches the standard normal distribution $N(0,1)$. 4. This means that for large $n$, the random variable $$Z_n = \frac{\overline{X}_n - \mu}{\sigma/\sqrt{n}}$$ behaves approximately like a standard normal variable. 5. This result is fundamental in statistics for inference about population means when the population distribution is unknown or not normal. Final answer: The standardized sample mean converges in distribution to a standard normal variable as $n \to \infty$ according to the Central Limit Theorem.