User:ConserveATory/test

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").

Notation

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].

$\Pr(|{\tilde {\mu }}-\mu |<y)=x$

Example

Define the following:

${\tilde {\mu }}={\frac {\sum _{i=1}^{n}X_{i}}{n}}.$

Where mu-tilde has the following Gaussian sampling distribution

${\tilde {\mu }}\sim G(\mu ,{\frac {\sigma }{\sqrt {n}}})$

Assume the standard notation (sigma is the standard deviation, n is the sample size). Assume we want to find the probability that the difference between the mean and its estimator is less than 100. Then:

$\Pr(|{\tilde {\mu }}-\mu |<100)=2\Pr({\tilde {\mu }}-\mu >100)$

$1-\Pr({\tilde {\mu }}-\mu <100)$ $1-\Pr({\frac {{\tilde {\mu }}-\mu }{\sigma /{\sqrt {n}}}}>{\frac {100}{\sigma /{\sqrt {n}}}})$ <math>