State the central limit theorem and use it to compute probabilities for the sample mean

What this dot point is asking

SCSA wants you to state the theorem precisely, recognise when it applies, and use it to find probabilities about the sample mean by standardising.

The statement

The remarkable part is the phrase regardless of shape. The population might be skewed, bimodal or discrete, yet the averaging in $\bar{X}$ smooths it toward a normal shape as $n$ increases. If the population is already normal, $\bar{X}$ is exactly normal for every $n$.

Why it matters

The theorem is what makes inference about a mean possible with normal-distribution tools. Without it we would need to know the exact population distribution; with it, the sample mean is approximately normal and we can use $z$-values and confidence intervals.

Standardising the sample mean

To find a probability about $\bar{X}$, convert to the standard normal:

The denominator is the standard error, not $\sigma$. Then probabilities such as $P(\bar{X} > c)$ become standard-normal probabilities.