Simple t-test: extensions

Summary

• Setup:
• a linear regression model \(y=Xβ+ε\) with a random sample
• \(ε_i|x_i \sim N(0,σ^2)\)
• Null hypothesis: \(H_0:β_j=β_j^0\) where \(β_j^0\) is a given constant.
• Under \(H_0\) :

\[ \frac{b_j-β_j^0}{SE\left( b_j \right)} \sim t_{n-k}\]

• Reject \(H_0\) if

\[\left| \frac{b_j-β_j^0}{SE\left( b_j \right)} \right|>t_{n-k,α/2}\]

• Null hypothesis: \(H_0:β_j<0\) (one-sided test)
• Reject \(H_0\) if

\[t_j= \frac{b_j}{SE\left( b_j \right)}>t_{n-k,α}\]

• Null hypothesis: \(H_0:β_j>0\) (one-sided test)
• Reject \(H_0\) if

\[t_j= \frac{b_j}{SE\left( b_j \right)}<-t_{n-k,α}\]