As I have seen, there are many questions on this site concerning the rate of convergence of the central limit theorem: for example this, this, this, and this. In particular, the third question of this list concerns quantitative bounds on the convergence in distribution
$$ \frac{\overline{X}_n - \mu}{\sigma/\sqrt{n}} \to_D Z \sim \mathcal{N}(0,1). $$
One commenter points out the Berry-Essen theorem which gives a bound on the CDF of $(\overline{X}_n - \mu)/(\sigma/\sqrt{n})$ and the standard normal random variable based on the third absolute moment $\rho := \Bbb E[|X-\mu|^3]$. Specifically, if $F_n$ is the CDF of $(\overline{X}_n - \mu)/(\sigma/\sqrt{n})$ and $\Phi$ of the standard normal random variable, then the theorem states that the difference between these is bounded as
$$ |F_n(x) - \Phi(x)| \le \frac{C\rho}{\sigma^3\sqrt{n}}, \text{ for all } x \in \Bbb R, \text{ $C$ constant}. $$
However, despite the existence of a concrete quantitative error bound on the approximation of $(\overline{X}_n - \mu)/(\sigma/\sqrt{n})$, introductory statistics classes often treat $(\overline{X}_n - \mu)/(\sigma/\sqrt{n})$ as either unapproximable at all by a normal random variable or perfectly normal, with $n = 30$ as a seemingly arbitrary cutoff between the two. Presumably, whether a given sample should be approximated as normal depends highly on $\rho$ and $\sigma$, not just $n$.
My questions:
- Why isn't the error in the normal approximation taken into account when performing statistical inference or making confidence intervals using the CLT?
- Does computing
$$ \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i, \: s := \sqrt{ \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}, $$
and some unbiased consistent estimate $\hat{\rho}$ of $\rho$ and computing $C\hat{\rho}/s^3\sqrt{n}$ give a "better" heuristic for whether one should use a large sample CLT approximation than simply looking at $n\ge 30$? If so, why is $n\ge 30$ approximation so ubiquitous in introductory statistics classes?