Question

"if we have a distribution of x values that is more or less mound-shaped and somewhat symmetrical, what is the sample size n needed to claim that the distribution of sample means x from random samples of that size is approximately normal"

Answer 1

Answer: 30 or greater.

Explanation:
The Central Limit Theorem (CLT) states that the distribution of sample means is normally distributed.

If the distribution is already known to be Gaussian, then smaller sample sizes than 30 will be fine to use.

If the distribution is not known, then a minimum sample size of 30 is needed to guarantee that the CLT holds.

Typically,
$\bar{x} = \mu \\ s = \frac{\sigma}{ \sqrt{n} }$
where
$\bar{x}$ =  sample average
μ = population mean
s = sample standard deviation
σ = population standard deviation
n = sample size