What is the intuition behind the generalized confidence interval? What is the intuition behind the generalized confidence interval? My best description on GCI that it is the way to derive a formula to calcuate the area of the center region in a asymetry distribution where it has the two-sided-equal-tailed regions in which the area of the two-sided-equal-tailed regions are the same in the n dimensional case.
A paper on the GCI: http://www.stat.colostate.edu/statresearch/stattechreports/Technical%20Reports/2002/02_10.pdf
Also: http://www3.stat.sinica.edu.tw/statistica/oldpdf/a10n420.pdf
 A: 
"We can never know the true proportion in a population, and we can
  only know the proportion within our sample. The Central Limit Theorem
  suggests that we are much more likely to get a value close to the true
  population value from our sample value because the distribution of
  sample proportions will follow a normal distribution. Weird samples
  are less probable than representative samples. The degree to which
  your sample proportion will differ from the population proportion will
  depend on the variation in the population and the size of the sample. 
The Central Limit Theorem allows us to say that the average
  (proportion) of the sample follows a normal distribution centered on
  the true population average with standard deviation  where sigma is
  the standard deviation of the population and $n$ is the size of the
  sample.
To increase accuracy, you can take a larger sample, but there are
  diminishing returns to  increasing sample size because we are taking
  its square root. The more diversity there is in the population, the
  less accurate your sample proportion will be. You determine the size
  of your sample based on your resources and desired accuracy. Most
  polls are in the ballpark of $n=1,000$."

Quoted from: Confidence Intervals and Hypothesis Testing
Also,

"A confidence interval is a range of values that we hope includes the
  true population value, for now that is the population mean.  We might
  think of it as an interval of guesses of the values of the population
  mean.  The interval is based on the information we randomly select
  from the population and include in the sample.  The size of this
  interval depends on the confidence that we want to have that the
  interval includes the population mean, the size of the sample we
  select, and the standard deviation of dispersion of the population
  values (usually we estimate this standard deviation with the standard
  deviation of the sample of values we have collected)."

Quoted from: Further Expositions on Intuition and Confidence Intervals 
