The Reasoning of Statistical Estimation
Where does the formula for computing a confidence interval come from?
The true value of the population mean is never known – it can only be
approximated or estimated.
The best way to do this is to select a large number of random samples of the
same size from the population.
The mean from each random sample will be slightly different.
The average of these sample means is the population mean.
For instance, the mean for the sample in the example was 80, but if another
sample was selected the mean might be 78 or 83.
If a large number of sample means were represented graphically, they would
have a Normal distribution.
The mean of this distribution is the same as the sample mean, but the
standard deviation of this distribution is equal to the standard deviation of
the variable in the population divided by the square root of the sample size.
This is the reason that the standard deviation is divided by the square root of
n in the formula, instead of the simple standard deviation, because this
formula represents the standard deviation of the distribution of many
sample means.
When working with real data it may not be feasible to select a very large
number of random samples, but if researchers were able to do so, the
samples would form a Normal distribution.