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I understand that under central limit theorem, when certain conditions are met, the distribution of sample mean follows:

  • Normal distribution
  • Mean: population mean
  • SE: population std / square root of sample size

So 95% of the observed sample means will be within 1.96 SE of the actual population mean (When z-statistics is used).

But how does it leads to:

There is 95% chance that the actual population mean is within 1.96 SE (The range of confidence interval) of the observed sample mean?

How does this statement holds true when the position of population mean and sample mean is exchanged?

Soo Yeon Lee
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  • Hint: it doesn't. That would imply a probability distribution for the value of the actual population mean. But there isn't one. Only the data is considered to be random. – conjugateprior Mar 16 '21 at 09:49

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