Confidence Intervals

Definition

Given a statistical model $\{p(x|\theta ):\theta \in \mathrm{\Theta }\}$$\large \{p(x | \theta): \theta \in \Theta\}$ and $0\ge \alpha \ge 1$$\large 0 \ge \alpha \ge 1$, an interval statistic $T(x)=[l(x),u(x)]$$\large T(x) = [l(x), u(x)]$, where $l(x)$$\large l(x)$ and $u(x)$$\large u(x)$ are lower and upper in is a $(1-\alpha )$$\large (1 - \alpha)$ confidence interval for ${\theta }_i$$\large \theta_i$, where ${\theta }_i$$\large \theta_i$ is a component of $\theta$$\large \theta$,

$z$$z$-Confidence Interval

Setup & Assumptions

• Samples: $x_1,x_2,...,x_n$$\Large x_1, x_2, ..., x_n$
• $x_i\sim N(\mu ,{\sigma }^2)$$\Large x_i \sim N(\mu, \sigma^2)$
• $\mu$$\Large \mu$ is unknown, ${\sigma }^2$$\Large \sigma^2$ is known

Objective

Interval

where $z_{\alpha /2}$$\large z_{\alpha / 2}$ is the right critical value such that,

$z$$z$-Confidence Interval with Two Independent Samples

Setup & Assumptions

• Samples: $x_1,x_2,...,x_n$$\Large x_1, x_2, ..., x_n$ and $y_1,y_2,...,y_m$$\Large y_1, y_2, ..., y_m$
• $x_i\sim N({\mu }_x,{\sigma }_x^2)$$\Large x_i \sim N(\mu_x, \sigma_x^2)$ amd $y_i\sim N({\mu }_y,{\sigma }_y^2)$$\Large y_i \sim N(\mu_y, \sigma_y^2)$
• ${\mu }_x,{\mu }_y$$\Large \mu_x, \mu_y$ are unknown, ${\sigma }_x^2,{\sigma }_y^2$$\Large \sigma_x^2, \sigma_y^2$ are known

Objective

Interval

where $z_{\alpha /2}$$\large z_{\alpha / 2}$ is the right critical value such that,

$t$$t$-Confidence Interval

Setup & Assumptions

• Samples: $x_1,x_2,...,x_n$$\Large x_1, x_2, ..., x_n$
• $x_i\sim N(\mu ,{\sigma }^2)$$\Large x_i \sim N(\mu, \sigma^2)$
• $\mu ,{\sigma }^2$$\Large \mu, \sigma^2$ are unknown

Objective

Interval

where $t_{\alpha /2}$$\large t_{\alpha / 2}$ is the right critical value such that,

$t$$t$-Confidence Interval with Two Independent Samples

Setup & Assumptions

• Samples: $x_1,x_2,...,x_n$$\Large x_1, x_2, ..., x_n$ and $y_1,y_2,...,y_m$$\Large y_1, y_2, ..., y_m$
• $x_i\sim N({\mu }_x,{\sigma }_x^2)$$\Large x_i \sim N(\mu_x, \sigma_x^2)$ amd $y_i\sim N({\mu }_y,{\sigma }_y^2)$$\Large y_i \sim N(\mu_y, \sigma_y^2)$
• ${\mu }_x,{\mu }_y,{\sigma }_x^2,{\sigma }_y^2$$\Large \mu_x, \mu_y, \sigma_x^2, \sigma_y^2$ are unknown

Objective

Interval

where $t_{\alpha /2}$$\large t_{\alpha / 2}$ is the right critical value such that,

${\chi }^2$$\chi^2$-Confidence Interval

Setup & Assumptions

• Samples: $x_1,x_2,...,x_n$$\Large x_1, x_2, ..., x_n$
• $x_i\sim N(\mu ,{\sigma }^2)$$\Large x_i \sim N(\mu, \sigma^2)$
• $\mu ,{\sigma }^2$$\Large \mu, \sigma^2$ are unknown

Objective

Interval

where $c_{\alpha /2}$$\large c_{\alpha / 2}$ and $c_{1-\alpha /2}$$\large c_{1 - \alpha / 2}$ are the right critical values such that,

respectively.