After formally stating the central limit theorem my statistics textbook says this:
Interpretation: Probability statements about the sample mean
$\overline{X}_n$ can be approximated using a Normal distribution. It's
the probability statements that we are approximating, not the random
variable itself.
What distinction are they trying to make here? What would it mean to approximate a random variable other than to approximate probability statements about it?
Best Answer
Under certain conditions, the probability $P(\bar{X}_n \in [-1, 1])$ (for example), which is a non-random number, can be approximated by $P(Y \in [-1, 1])$ for some normal random variable $Y$ with appropriate mean and variance, using the central limit theorem. To emphasize, you are approximating the probability $P(\bar{X}_n \in [-1, 1])$, and not the random variable $\bar{X}_n$ itself: you are not saying something is close to $\bar{X}_n$.