Confidence Interval for the Slope, Step by Step

The four steps

State: "$\beta$ is the true slope of the population regression line relating [$y$] to [$x$]; estimate it with a $95\%$ $t$-interval." Plan: check LINER — linear scatter, independent observations (random sample, $10\%$), normal residuals (no strong skew or outliers in the residual plot or histogram), equal spread in the residual plot, random data. Do:

with $b$ and $SE_b$ read straight from the slope row of the output. Conclude in context: "we are $95\%$ confident that the true slope — the change in mean [$y$] for each additional unit of [$x$] — is between $L$ and $U$."

A worked reading of output

Output for $n=22$ cars: slope $0{,}0523$, SE Coef $0{,}0112$. Then $df=20$, $t^*=2{,}086$, and the interval is $0{,}0523\pm2{,}086(0{,}0112)=0{,}0523\pm0{,}0234$, i.e. $(0{,}029;\,0{,}076)$. Because $0$ is not inside, the data give convincing evidence of a positive linear relationship — the same verdict a two-sided test at $\alpha=0{,}05$ would reach. Never use $df=n-1$ here; the model estimates two parameters, so $df=n-2$.

When the model was transformed

If the regression was fitted to $\ln y$, the interval is for the slope of $\ln y$ on $x$. To speak in original units, exponentiate both endpoints: an interval $(0{,}029;\,0{,}076)$ for the log-slope means the mean $y$ is multiplied by between $e^{0{,}029}\approx1{,}03$ and $e^{0{,}076}\approx1{,}08$ per unit of $x$. The interpretation is multiplicative, and the width in original units is no longer symmetric.