Condition "Sample size > 30" for infering population proportion or mean One of the conditions to use statistical inference, when estimating the proportion of a population based on the sample proportion, is that:

The data's individual observations have to display normality. This can be verified mathematically with the following definition:
Let $\displaystyle n$ be the sample size of a given random sample and let $\displaystyle {\hat {p}}$ be its sample proportion. If $\displaystyle n{\hat {p}}\geq 10$ and $\displaystyle n(1-{\hat {p}})\geq 10$, then the data's individual observations display normality.

In other source, it says that the sample size $n \ge 30$, which

this rule-of-thumb was developed by having a computer do what are called “Monte Carlo simulations”

So far, I haven't found a source that formalize any of these assumptions.
Could someone provide some references (articles, books) about this?
 A: This rule-of-thumb is meaningless without specification of further details
I remember this same assertion being bandied around when I was first learning statistics, and really, it is meaningless without some specification of the conditions of assessing the approximation.  The classical CLT applies to any underlying sequence of random variables that are IID from some distribution with a finite variance.  This wide scope allows consideration of a huge number of possible underlying distributions, which vary substantially in how close they already are to the normal distribution (i.e., how good the accuracy is when $n=1$).
In order to specify a minimum required number of data points for "good approximation" by the normal distribution (even undertaking a simulation study or other analysis) you would need to specify two things:


*

*How different to the normal distribution is the underlying distribution of the data?

*How close to the normal distribution is "good enough" for approximation purposes?
Any attempt to formalise a rule-of-thumb for this approximation would need to specify these two things, and then show that the specified number of data points achieves the specified minimum level of accuracy for underlying data coming from the specified distribution.
Depending on how you specify the above two things, the minimum number of data points in the resulting "rule of thumb" will be different.  If the underlying data is already close in shape to a normal distribution then the number of data points required for "good" approximation" will be lower; if the underlying data is substantially different in shape to a normal distribution then the number of data points required for "good approximation" will be higher.  Similarly, if "good approximation" requires a very small "distance" from the normal distribution then the number of data points required for "good" approximation" will be higher; if "good approximation" is taken a bit more liberally, as allowing a higher "distance" from the normal distribution, then the number of data points required for "good" approximation" will be lower.
A: One quote I like to bring up about the greater than 30 rule for the Central Limit Theorem (implying normality) is from Rand Wilcox, 2017, Modern Statistics for the Social and Behavioral Sciences.  Section 7.3.4.

Three Modern Insights Regarding Methods for Comparing Means
There have been three modern insights regarding methods for comparing
  means, each of which has already been described. But these insights
  are of such fundamental importance that it is worth summarizing them
  here.
• Resorting to the central limit theorem in order to justify the
  normality assumption can be highly unsatisfactory when working with
  means. Under general conditions, hundreds of observations might be
  needed to get reasonably accurate confidence intervals and good
  control over the probability of a Type I error. Or in the context of
  Tukey's three-decision rule, hundreds of observations might be needed
  to be reasonably certain which group has the largest mean. When using
  Student's T, rather than Welch's test, concerns arise regardless of
  how large the sample sizes might be.
• Practical concerns about heteroscedasticity (unequal variances) have
  been found to much more serious than once thought. All indications are
  that it is generally better to use a method that allows unequal
  variances.
• When comparing means, power can be very low relative to other
  methods that might be used. Both differences in skewness and outliers
  can result in relatively low power. Even if no outliers are found,
  differences in skewness might create practical problems. Certainly
  there are exceptions. But all indications are that it is prudent not
  to assume that these concerns can be ignored.
Despite the negative features just listed, there is one positive
  feature of Student's T is worth stressing. If the groups being
  compared do not differ in any manner, meaning that they have identical
  distributions, so in particular the groups have equal means, equal
  variances, and the same amount of skewness, Student's T appears to
  control the probability of a Type I error reasonably well under
  nonnormality. That is, when Student's T rejects, it is reasonable to
  conclude that the groups differ in some manner, but the nature of the
  difference, or the main reason Student's T rejected, is unclear. Also
  note that from the point of view of Tukey's three-decision rule,
  testing and rejecting the hypothesis of identical distributions is not
  very interesting.

