A confidence interval is a mathematical interval between two values, based on a parameter. The purpose of the interval is to find the probability that the actual value of a parameter falls within the interval. They are often used to help prove the likelihood of causation. A "strong" confidence interval will have a well-defined, reasonable range, and the values of the parameter will fall in the interval frequently. While "frequently" is differently defined, it is often accepted as a 95% likelihood (sometimes phrased as "19 out of 20 times").
Confidence intervals can be calculated for any parameter within a statistical population. For this example of notation, assume that mu is the mean, and mu-tilde is the estimator of mu. The probability that the mean and its estimator are less than some value y is equal to x, where y is some non-negative real number, and xϵ[0,1].
Define the following mean estimator:
Where mu-tilde has the following Gaussian sampling distribution
Assume the standard notation (sigma is the standard deviation, n is the sample size). You are given that sigma/sqrt(n) is 67.5. Assume we want to find the probability that the difference between the mean and its estimator is less than 100. Then:
, where Phi is the cumulative distribution function for the G(0,1) Gaussian distribution.
Then we conclude that there is a 13.8% chance that the average is within the interval
In another practice, the confidence interval is used to find the interval as opposed to the probability. In the above process, you are given the probability (say, 95%) that a parameter falls within an interval, and from there, the interval itself must be found. Then, one can conclude that 95% of times, the parameter will fall within a given interval.