I have come across two major problems recently, and couldn't solve them. Imagine we have measured an independent variable 2000 times and I'm interested to talk about the mean of the population: So as some textbooks suggested, I can perform some calculations like this:
Variance = Sum of squared/(2000-1)
then standard deviation =sqrt(variance)
Standard error(SE) = standard deviation / sqrt(2000)
mean value - SE and mean value + SE (for approximately 68.2% confidence interval)
But the first problem here is that why we don't put these 2000 measurements into several samples, then calculate the sampling distribution and instead of using sqrt(2000) in the SE denominator, using sqrt(Number of samples) then calculate the mean plus-minus the SE.
1- Which method is better? Is one of these methods wrong?
Apparently, when the sample size gets bigger, the interpretation of some tests like Shapiro-Wilk, Levene's test should be made cautiously (Also the significant results regarding the p-values). On the other hand, if the sample size is too small, the normality of data(or sampling distribution) won't be valid and some other issues. I found out that for example in psychology departments the sample size less than 30 considered small and greater than 200 will be considered very big, but in other fields it's not the case.
2- How can I be sure about the sample size which is not too big (or small)? Passing the normality test is enough to conclude that the sample size is not small?
Also, I can't partition my population into subpopulations at all. So having a large sample should be the same as having multiple samples with less sample size (approximately)?
Thank you so much in advance.