t-test by hand

Problem

Setup: a linear regression model with a random sample, the Gauss-Markov assumptions hold and the errors are normally distributed,

\[y_i=β_1+β_2x_i+ε_i i=1, \ldots ,n\]

You want to test \(H_0:β_2=0\) . We find that \(b_2=1.8\) , \(SE\left( b_2 \right)=2.0\) , \(n=6\) and \(α=0.05\) .

  1. Find the \(t\) -value
  2. Find the critical value
  3. Should you reject the null-hypothesis?

Solution

  1. \(t=b_2/SE\left( b_2 \right)=1.8/2.0=0.9\)
  2. \(t_{0.025,4}=2.776\) (for example, “=T.INV.2T(0.05,4)” in Excel)
  3. You should not.