Definition:Confidence Interval/Definition 1

Definition

Let $X$ be a random sample from this population.

Let $I = \openint {\map f X} {\map g X}$ for some real-valued functions $f$, $g$.

$I$ is said to be a $100 \gamma \%$ confidence interval for $\theta$ if:

$\map \Pr {\theta \in I} = \gamma$

where $0 < \gamma < 1$.

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

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:

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

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 $H_1: \theta \ne \theta_0$.