What is the intuition behind the generalized confidence interval?

Question

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

Answer 1

"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