= Simple linear regression
{wiki}
Simple linear regression uses one predictor and an <intercept>, $Y_i=\alpha+\beta x_i+\varepsilon_i$, with $\mathbb E(\varepsilon_i\mid x_i)=0$. The <regression coefficients> describe a straight conditional-mean line: $\alpha$ is its value at zero and $\beta$ its rate of change per unit of the predictor. When zero is outside the observed range, <predictor centering> can make the <intercept> refer to a more representative predictor value.
For fixed predictors and uncorrelated mean-zero errors of common <variance> $\sigma^2$, with $S_{xx}>0$,
$$
\operatorname{Var}(\widehat\beta)=\frac{\sigma^2}{S_{xx}},\qquad
\operatorname{Var}(\widehat\alpha)=\sigma^2\left(\frac1n+\frac{\bar x^2}{S_{xx}}\right).
$$
A larger predictor spread improves slope precision under this model. In the worked example $S_{xx}=10$ and $n=5$; the estimated error <variance> is $s^2=\operatorname{RSS}/(n-2)=0.8/3$. A <normal linear model> gives the slope pivot $(\widehat\beta-\beta)/(s/\sqrt{S_{xx}})$ a <Student t-distribution> with $n-2$ <degrees of freedom>. The distinction between a fitted line and a reliable uncertainty statement is precisely the role of the error assumptions.
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