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Revision as of 20:50, October 13, 2007
, 20:50, October 13, 2007in case the computer dies
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 ''x''ϵ[0,1].
<math>\Pr(|\tilde \mu - \mu| < y)=x</math>
==Example==
Define the following:
<math>\tilde \mu=\frac{\sum_{i=1}^n X_i}{n}.</math>
Where ''mu-tilde'' has the following [[Gaussian distribution|Gaussian sampling distribution]]
<math>\tilde \mu \sim G(\mu, \frac{\sigma}{\sqrt n})</math>
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:
<math>\Pr(|\tilde \mu - \mu| < 100)=2\Pr(\tilde \mu - \mu > 100)</math>
<math>1 - \Pr(\tilde \mu - \mu < 100)</math>
<math>1 - \Pr(\frac{\tilde \mu - \mu}{\sigma / \sqrt n} > \frac {100}{\sigma / \sqrt n})</math>
<math>
==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 ''x''ϵ[0,1].
<math>\Pr(|\tilde \mu - \mu| < y)=x</math>
==Example==
Define the following:
<math>\tilde \mu=\frac{\sum_{i=1}^n X_i}{n}.</math>
Where ''mu-tilde'' has the following [[Gaussian distribution|Gaussian sampling distribution]]
<math>\tilde \mu \sim G(\mu, \frac{\sigma}{\sqrt n})</math>
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:
<math>\Pr(|\tilde \mu - \mu| < 100)=2\Pr(\tilde \mu - \mu > 100)</math>
<math>1 - \Pr(\tilde \mu - \mu < 100)</math>
<math>1 - \Pr(\frac{\tilde \mu - \mu}{\sigma / \sqrt n} > \frac {100}{\sigma / \sqrt n})</math>
<math>