Editing Prediction interval (section)

== In regression analysis ==
{{further|Regression analysis#Prediction (interpolation and extrapolation)|Mean and predicted outcome}}

A common application of prediction intervals is to [[regression analysis]].
Suppose the data is being modeled by a straight line ([[simple linear regression]]):
:<math>y_i=\alpha+\beta x_i +\varepsilon_i\,</math>
where <math>y_i</math> is the [[response variable]], <math>x_i</math> is the [[explanatory variable]], ''ε<sub>i</sub>'' is a random error term, and <math>\alpha</math> and <math>\beta</math> are parameters.

Given estimates <math>\hat \alpha</math> and <math>\hat \beta</math> for the parameters, such as from a [[ordinary least squares]], the predicted response value ''y''<sub>''d''</sub> for a given explanatory value ''x''<sub>''d''</sub> is

:<math>\hat{y}_d=\hat\alpha+\hat\beta x_d ,</math>

(the point on the regression line), while the actual response would be

:<math>y_d=\alpha+\beta x_d +\varepsilon_d.  \,</math>

The [[point estimate]] <math>\hat{y}_d</math> is called the ''[[mean response]]'', and is an estimate of the [[expected value]] of ''y''<sub>''d''</sub>, <math>E(y\mid x_d).</math>

A prediction interval instead gives an interval in which one expects ''y''<sub>''d''</sub> to fall; this is not necessary if the actual parameters ''α'' and ''β'' are known (together with the error term ''ε<sub>i</sub>''), but if one is estimating from a [[Sampling (statistics)|sample]], then one may use the [[standard error]] of the estimates for the intercept and slope (<math>\hat\alpha</math> and <math>\hat\beta</math>), as well as their correlation, to compute a prediction interval.

In regression, {{Harvtxt|Faraway|2002|p=39}} makes a distinction between intervals for predictions of the mean response vs. for predictions of observed response—affecting essentially the inclusion or not of the unity term within the square root in the expansion factors [[#Unknown mean, unknown variance|above]]; for details, see {{Harvtxt|Faraway|2002}}.