Central Limit Theorem interpretation confusion

Question

I was wonder in some problems, the central limit theorem is Z = root(n)((xbar - mu)/sigma), however in some cases, the root n is not there. In this case per se, the root n is not there as I checked the answer online. If possible can someone explain why there isn't a root n in this case.

https://www.slader.com/discussion/question/let-y-denote-the-sum-of-the-observations-of-a-random-sample-of-size-12-from-a-distribution-having-pm/

Answer 1

Because we are interested in $Y=\sum X_i$ having Variance $\operatorname{Var}(Y)=\sum_i \operatorname{Var} (X_i)=12(2.92)=n\sigma^2$.

If instead we were interested in the mean $\bar Y=\frac Y {12}$ it would have variance $\frac {\operatorname{Var}(Y)} {12^2}=\frac{2.92}{12}=\frac{\sigma^2}{n}$.

The root n comes from the second part where $\frac{\bar X-\mu}{\sigma/\sqrt n}=\sqrt n\frac{\bar X-\mu}{\sigma}$ if we are dealing with sample mean but here we are dealing with a sum furthermore we have already incorporated the $\sqrt n$ in the calculation of the variance of $Y$ so that it why does not appear again in part 2.

So depending on if you are interested in the sum of a sample or the mean of the sample, the $n$ will be in different places. The central limit theorem applies to both cases.