Central Limit Theorem - a naive question

[#1]

This question does not relate to JMP itself, but rather represents a basic question of interpretation. I understand that, due to the the phenomenon we call the CLT, sampling of a population will always strive towards a normal distribution of the resulting means with the overall mean and SD approaching the population mean and SD, irrespective of how the population data is distributed (skewed or normal), but provided a sufficient number of repeat samples are taken (≥30). Furthermore, as the size of each sample is increased, the precision of the estimated mean and SD increases. I also understand that, one does not need to perform such multiple sampling as the CLT is accomodated in parametric testing. What I'm trying to understand is whether there is a cut off point of sample size, above which normality can categorically be assumed without testing. Secondly, and perhaps unrelated, with the call by the The American Statistician journal to drop significance level thresholds (p<0.05) in favour of reporting of actual p-values and to refrain from the use of the term "statistically significant", how does one then determine the outcome in such matters as confirming normal distribution, for example through Shapiro-Wilk testing? Is there a grey line there too?

[#2]

@34South: Regarding my earlier post where I mentioned 'mononumerosis', when I taught statistical methods to scientists and engineers in an industrial problem solving or product/process development framework I surrounded my mention of the disease with something I called 'The Gap'. "The Gap" recognizes that in hypothesis there are two types of risk involved in ANY decision making process. I think the list of ASA '...Not To...' are strongly aligned with "The Gap". The two types of risk are:

1. Statistical risk; Which is the risk we can quantify and structurally address through techniques such as sample size, population variance assumptions, beta risk, delta to detect etc. Hypothesis tests culminate in p-values to guide decision making and the statement '...statistical signficance.'

2. Representation risk; Which are ALL the other cumulative effects of system characteristics that impart 'risk' associated with making a decision. This family of risk, in my experience often SWAMPS statistical risk...and is often impossible to control or quantify using statistical methods. Representation risk can only be addressed by rational, thoughtful, knowledgeable domain expertise. For example, in my industry days, we often ran experiments on pilot equipment with a goal of determining product design specifications. But there was almost ALWAYS a huge issue...what we learned on pilot equipment was quite often, just not scalable to production scale equipment. Hence we had a "Gap" in understanding that was impossible to overcome with methods that ONLY involve statistical risk.

So my point on 'mononumerosis' was always, if all you report is a p-value in isolation, and don't incorporate representation risk in your decision making...well you've in all likelihood grossly underrepresented TOTAL risk of making an decision making error.

I hope this helps?