Central Limit Theorem and the Sample Sum This 27.1 The Theorem lists two equations for Z:
$$
Z=\frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt n}}\\
Z = \frac{\sum_{i=1}^{n} X_i-n\mu}{\sqrt{n}\sigma}
$$
Is it correct to say that the second equation above is equivalent to the first but is simply expressed in terms of the sample sum rather than the sample mean, so every term is multiplied by $n$?
In other words, does what the Central Limit Theorem says about the sample mean also apply to the sample sum?

Yes, you are correct. Consider:
$$
\begin{aligned}
Z &= \frac{\sum_{i=1}^{n} X_i-n\mu}{\sqrt{n}\sigma}\\
&=\frac{n\bar{X}-n\mu}{n\frac{1}{\sqrt n}\sigma}\\
&=\frac{n}{n}\frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt n}}\\
&=\frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt n}}
\end{aligned}
$$
Although, as Dave points out, the CLT technically relates to a standardized or normalized sum of random variables and not the sample mean.
