Confidence Intervals for Linear Regression Slope Parameters

Constructing t-based confidence intervals for the true slope parameter β₁ of a linear regression model, using either the given standard error of the slope or the residual standard error computed from the sum of squared residuals.

Step 1 of 119%

Tutorial

Confidence Interval for the Slope

For a linear regression of $Y$ on $X,$ the population model is

Y = \beta_0 + \beta_1 X + \varepsilon.

We estimate the true slope $\beta_1$ with the sample slope $b_1$ obtained by least squares. To quantify the uncertainty in this estimate, we use a confidence interval for $\beta_1$ :

b_1 \pm t^* \cdot SE(b_1),

where $SE(b_1)$ is the standard error of the slope estimate and $t^*$ is the critical value from a $t$ -distribution with $n-2$ degrees of freedom. Two degrees of freedom are lost because both $\beta_0$ and $\beta_1$ are estimated from the data.

For example, suppose a regression with $n = 15$ produces $b_1 = 1.4$ and $SE(b_1) = 0.5.$ Using $t^* = 2.160$ (the $95\%$ critical value for $13$ degrees of freedom),

\begin{align*} \text{ME} &= t^* \cdot SE(b_1) \\[3pt] &= 2.160 \cdot 0.5 \\[3pt] &= 1.080, \end{align*}

and the $95\%$ confidence interval is

1.4 \pm 1.080 = (0.320,\, 2.480).