Confidence interval for

Summary

• Setup:
• a linear regression model \(y=Xβ+ε\) with a random sample
• \(ε_i|x_i \sim N(0,σ^2)\)
• Definition: A confidence interval for \(β_j\) with a level of significance \(α\) is the collection of all numbers \(β_j^0\) for which the hypothesis \(H_0:β_j=β_j^0\) cannot be rejected .
• Result: The \(\left( 1-α \right)100%\) confidence interval for \(β_j\) is given by

\[\left[ b_j-t_{n-k,α/2}SE\left( b_j \right),b_j+t_{n-k,α/2}SE\left( b_j \right) \right]\]