The distribution of the sample mean, , will be normally distributed if the sample is obtained from a population that is normally distributed, regardless of the sample size.

Isn’t it fascinating how the Central Limit Theorem (CLT) plays a superhero role here? It tells us that even if your underlying population distribution isn’t normal, as long as you have a sufficiently large sample size—usually n ≥ 30—your sample mean will trend toward a normal distribution. This makes it easier to make inferences about the population mean, even when the data might be all over the place!

When it comes to real-world applications, think about it in terms of quality control! When a factory checks the average weight of its packaged cookies, they don’t weigh every single cookie—who has the time for that? Instead, they take a sample, compute the mean, and rely on that sample mean’s distribution to determine if they’re meeting their standards. Thanks to the properties of the sample mean, they can confidently ensure that consumers get the tasty snacks they expect, with minimal hassle!