Hypothesis testing: simple t-test

Summary

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

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

• \( \frac{b_j}{SE\left( b_j \right)}\) is called the t-value for variable \(j\) , denoted by \(t_j\) . Under \(H_0:t_j \sim t_{n-k}\)
• Reject \(H_0\) if

\[\left| t_j \right|>t_{n-k,α/2}\]

• where \(t_{n-k,α/2}\) is the critical value for the level of significance \(0<α<1/2\) .