Summary points

The main points from this lecture are

1. A confidence interval estimate for \(\mathbf{b}^T\boldsymbol{\beta}\) with confidence \(c\) is

\[\mathbf{b}^T\boldsymbol{\hat{\beta}}\pm t\left(n-p; \frac{1+c}{2}\right)\sqrt{\frac{RSS}{n-p}(\mathbf{b}^T(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{b})}.\]

2. A prediction interval for \(y_f\) is

\[\mathbf{b}^T\boldsymbol{\hat{\beta}}\pm t\left(n-p; \frac{1+c}{2}\right)\sqrt{\frac{RSS}{n-p}(1+\mathbf{b}^T(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{b})}.\]