Do you ever test the pre-asymptotic behaviour of a variable before performing a comparison of means t-test? I understand that it is not the normality of a random variable that matters in a t-test, but rather the fact that the distribution of the mean follows a normal distribution for large samples. However, is it sometimes useful to test how large a sample must be before the assumption of a normal distribution is warranted? For example, if the number of samples required were very large (much larger than n=30), would assuming a normal distribution be inappropriate? If so, how would you go about checking this?
The inspiration for this question comes from reading about Nassim Nicholas Taleb's Kappa metric.

Of course, the main issue is for the t statistic to have
a t distribution with the appropriate degrees of freedom.
For normal data $\bar X$ and $S$ are stochastically independent. (One might argue that they
are not functionally independent because $S^2$ can be defined in terms of $\bar X.)$
By contrast, for exponential data $\bar X$ and $S$ are not independent.
So, technically, it not enough for $n$ to be large
enough that $\bar X$ is nearly normal. In order for the distribution of a t statistic, e.g, $T = \frac{\bar X-\mu_0}{S/\sqrt{n}},$ to have
the appropriate t distribution, its numerator and denominator should be independent.
Using goodness-of-fit tests to check data for normality
before doing a t test has been deprecated. But it is
a good idea to look at plots of the data to see if they are nearly symmetrical and free of far outliers in either direction before using the data for a t test.
For example, consider a sample of size $n = 100$ from a Pareto distribution with minimum value $1$ and shape parameter $\alpha = 10,$ and mean $\mu = 10/9.$ [Observations can be sampled as
$e^Y,$ where $Y\sim\mathsf{Exp}(\mathrm{rate}=10).]$
Each row of the matrix MAT below contains such a sample.
set.seed(2022)
m = 10^4;  n = 100
MAT = matrix(exp(rexp(m*n, 10)), nrow=m)

A boxplot of the first sample shows marked right-skewness and outliers.
boxplot(MAT[1,], horizontal=T, col="skyblue2")


a = rowMeans(MAT)
s = apply(MAT,1,sd)
cor(a, s)
[1] 0.7201542

Sample means and SDs are clearly correlated.
But one might consider that the means $\bar X$ (a in the code) are "close enough" to normal.
hist(a, prob=T, br=30, col="skyblue2")
 curve(dnorm(x,mean(a), sd(a)), add=T, col="red", lwd=2)


However, $T$ statistics are not distributed as $\mathsf{T}(\nu=99)$---especially not in the tails,
where values are used to decide whether to reject the null hypothesis. So one should not trust results of a
t test on such Pareto data.
t = (a - 10/9)/(s/sqrt(n))
hist(t, prob=T, br=30, col="skyblue2")
 curve(dt(x,n-1), add=T, lwd=2, col="red")


By contrast, samples of size $n = 50$ from a
standard uniform distribution behave well.
set.seed(129)
m = 10^4;  n = 50
MAT = matrix(runif(m*n), nrow=m)
a = rowMeans(MAT);  s = apply(MAT,1,sd)
cor(a, s)
[1] -0.01452768   # nearly independent

$T$ statistics are nearly distributed as $\mathsf{T}(\nu=49).$ So one could trust results of a
t test on such uniform data.
t = (a - 1/2)/(s/sqrt(50))
hist(t, prob=T, br=20, col="skyblue2")
 curve(dt(x,49), add=T, col="red")


