Confidence Intervals for Linear Regression Intercept Parameters

In simple linear regression, we model the relationship between an explanatory variable $X$ and a response variable $Y$ as

$$Y = \beta_0 + \beta_1 X + \varepsilon,$$

where $\varepsilon \sim N(0, \sigma^2)$. The intercept $\beta_0$ is an unknown population parameter. Just as we built a confidence interval around the sample mean $\bar{x}$ to estimate the population mean $\mu$, we can build one around the sample estimate $\hat{\beta}_0$ to estimate $\beta_0$.

A $(1 - \alpha) \cdot 100\%$ confidence interval for the intercept $\beta_0$ is given by

$$\hat{\beta}_0 \;\pm\; t_{\alpha/2,\, n-2} \cdot SE(\hat{\beta}_0),$$

where $SE(\hat{\beta}_0)$ is the standard error of $\hat{\beta}_0$ and $t_{\alpha/2,\, n-2}$ is the critical value from the Student $t$-distribution with $n - 2$ degrees of freedom.

We use $n - 2$ degrees of freedom because two parameters, $\beta_0$ and $\beta_1$, must be estimated from the data.

For instance, suppose a regression on $n = 14$ observations gives $\hat{\beta}_0 = 10$ and $SE(\hat{\beta}_0) = 2$. For a $95\%$ CI, $\alpha = 0.05$, so we use $t_{0.025,\, 12} = 2.179$. The margin of error is $2.179 \cdot 2 = 4.358$, giving the CI

$$10 \pm 4.358 \;=\; (5.64,\; 14.36).$$