# Frequentist Inference III – Confidence Intervals

Confidence Intervals are often used and are a great way to not just give a point estimate but to tell where else the point might be. We will focus on confidence intervals based on normal data today.

## z-confidence intervals for the mean

Suppose data $x_{ 1 },...,x_{ n } \sim N(\mu,\sigma^{ 2 })$ with unknown mean but known variance. The $(1-\alpha)$ confidence interval is then:

$[\overline{ x }-\frac{ z_{ a/2 }\sigma }{ \sqrt{ n } },\overline{ x }+\frac{ z_{ a/2 }\sigma }{ \sqrt{ n } }]$ where $z_{ a/2 }=P(Z>z_{ a/2 })=\frac{ \alpha }{ 2 }$

## t-confidence intervals for the mean

Suppose $x_{ 1 },...,x_{ n }\sim N(\mu,\sigma^{ 2 })$ where the mean and the variance are unknown. The $(1-\alpha)$ confidence interval for the mean is:

$[\overline{ x }-\frac{ t_{ a/2 }s}{ \sqrt{ n } },\overline{ x }+\frac{ t_{ a/2 }s }{ \sqrt{ n } }]$. Where $t_{ a/2 }=P(T>t_{ a/2 })=\frac{ \alpha }{ 2 }$ for T~t(n-1) and $s^{ 2 }$ is the sample variance.

## Chi-Square confidence intervals for the variance

Suppose $x_{ 1 },...,x_{ n } /sim N(\mu, \sigma^{ 2 })$  and both, the mean and the variance are unknown. The $(1-alpha)$ confidence interval is:

$[\frac{ (n-1)s^{ 2 } }{ c_{ a/2 } },\frac{ (n-1)s^{ 2 } }{ c_{ 1-a/2 } }]$ where $c_{ a/2 }=P(X^{ 2 }>c_{ a/2 })=\frac{ \alpha }{ 2 }$ for $X^{ 2 }/sim X^{ 2 }(n-1)$.

## Conservative normal confidence interval

Suppose $x_{ 1 },...,x_{ n }$ are i.i.d. Bin(p) distributed. The conservative normal $(1-\alpha)$ confidence interval is then:

$\overline{ x } \pm z_{ a/2 }\frac{ 1 }{ 2\sqrt{ n } }$