Definition:Confidence Interval/Definition 2

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Definition

Let $X$ be a random variable.

Let $\theta$ be a population parameter of $X$ whose distribution is unknown.

A $100 \paren {1 - \alpha}$ percent confidence interval for $\theta$ is an interval formed by a rule which ensures that, in the long run, $100 \paren {1 - \alpha}$ percent of such intervals will include $\theta$.

This confidence interval is derived from the information obtained from a random sample of observations of $X$.


Confidence Limit

The endpoints of a confidence interval are known as confidence limits.


Examples

$95 \%$ Confidence Interval

A $95 \%$ confidence interval is a confidence interval whose $\alpha$ parameter is:

$\alpha = 0 \cdotp 05$

Let $\bar x$ be the mean of a sample of $n$ observations from a normal distribution with unknown mean $\mu$ and known standard deviation $\sigma$.

Then a $95 \%$ confidence interval for $\mu$ is:

$\closedint {\bar x - \dfrac {1 \cdotp 96 \sigma} {\sqrt n} } {\bar x + \dfrac {1 \cdotp 96 \sigma} {\sqrt n} }$


Motivation

A $100 \paren {1 - \alpha}$ percent confidence interval for a population parameter $\theta$ derived from a given sample covers all values of $\theta_0$ of that parameter that would be accepted at significance level $\alpha$ in a hypothesis test of:

the null hypothesis $H_0: \theta = \theta_0$

against:

the alternative hypothesis $H_1: \theta \ne \theta_0$.


Also see

  • Results about confidence intervals can be found here.


Sources

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