## Quick Reference

The simplest and most used of all statistical regression models. The model states that the random variable *Y* is related to the variable *x* by

*Y*=*α*+*βx*+*ε*,

where the parameters *α* and *β* correspond to the intercept and the slope of the line, respectively, and *ε* denotes a random error. With observations (*x*_{1}, *y*_{1}), (*x*_{2}, *y*_{2}),…, (*x** _{n}*,

*y*

*) the usual assumption is that the random errors are independent observations from a normal distribution with mean 0 and variance*

_{n}*σ*

^{2}. In this case the parameters are usually estimated using ordinary least squares. The estimates, denoted by

*α̂*and

*β̂*, are given by , where

*x̄*and

*ȳ*are the means of

*x*

_{1},

*x*

_{2},…,

*x*

*and*

_{n}*y*

_{1},

*y*

_{2},…,

*y*

*, respectively, and where . The variance*

_{n}*σ*

^{2}is estimated by , where .

A 100 (1-*θ*)% confidence interval for *β* is provided by , where *t** _{ν}*(

*θ*) is the upper 100

*θ*% point (see percentage point) of a

*t*-distribution with

*ν*degrees of freedom. A 100(1−

*θ*)% confidence interval for the expected value of

*Y*when

*x*=

*x*

_{0}is . A 100(1-

*θ*)% prediction interval for the value

*y*

_{0}of

*Y*when

*x*=

*x*

_{0}is See also multiple regression model; regression diagnostics; regression through the origin.

*Subjects:*
Probability and Statistics — Social Sciences.

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