Calculating mean confidence intervals from a sample with low SEM

Question

Given that:

• I. The distribution of means of large enough samples converge to a normal distribution (by CLT).
• II. The Standard Error of the Mean (SEM) of a sample approaches the Standard Deviation (SD) of the means as the sample size increases (also, SEM is low for larger samples).

Does that mean that if the SEM of a single sample is low I can assume there is 95% prob of the pop mean to be nearly between sample mean +- 1.96*SEM ?

Rational behind the question:

• A random sample ${z}$ of largely enough size ${n}$ selected out of a population ${p}$ has a SEM of $\dfrac{std({z})}{\sqrt[2]{n}}$
• Suppose SEM is very low, thus mean(${z}$) approaches mean(${p}$)
• Let ${c}$ be a set constituted by the means of enough samples of size ${n}$ taken from ${p}$; thus ${c}$ is normally distributed (I.)
• Remember that for a large enough sample SEM approaches ${std(c)}$ (II.)
• Since as stated by CLT mean(${c}$) also approaches mean(${p}$), and mean(${p}$) is close to mean(${z}$) (low SEM) can we assume that there is approximately 95% probability that ${mean(p)}$ is equal to ${mean(z) +/- SEM*1.96}$ ?

E.g: Python dice simulation

Relates to Calculating the confidence interval for the mean value from a sample

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