I am stuck at CI. Intuitively, a confidence interval (CI) is a type of interval estimate, computed from the statistics of the observed data ( random samples), that might contain the true value (mean) of an unknown population parameter. (Wikipedia)
CLT states that plotting means of sufficiently large no samples of sample size n will follow a normal distribution and the mean of sampling distribution will be approximately the same as the mean of a population. The standard deviation of sampling mean distribution (also called Standard Error) will be sigma/sqrt(n).
Assume Standard deviation of Population is known. Now, if we have sufficiently large no of samples, why do we need Confidence Intervals Is confidence interval used to tell the interval of the true mean from A SINGLE SAMPLE OF MEAN X_bar and Standard Error (sigma/sqrt(n)) or Is CI used to tell the Interval of the true population mean based on the X_bar of Sampling Distribution and the S.E (sigma/sqrt(n)).
Is the purpose of CI to tell us in a quantifiable manner how Sample Size (assuming large no of samples) impacts the finding of the true mean from sampling distribution?
Any help is highly appreciated.
TL:DR - While Calculating Confidence Interval (Population Std.Dev is known), whether SAMPLING DISTRIBUTION is used or a RANDOM SAMPLE is used?