One-sided t-test

Problem

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

\[y_i=β_1+β_2x_i+ε_i i=1, \ldots ,n\]

The hypothesis

\[H_0:β_2=0\]

is called a two-sided hypothesis (we use two tails). You can also test a one-sided hypothesis such as

\[H_0:β_2>0\]

The \(t\) -value for this hypothesis is the same but the critical value is \(t_{α,n-2}\) instead of \(t_{α/2,n-2}\) and you reject only in the left tail, that is, if \(t<-t_{α,n-2}\) .

Similarly, for the one-sided hypothesis

\[H_0:β_2<0\]

you reject only in the right tail, that is, if \(t>t_{α,n-2}\) .

You observe \(b_2=1.2\) , \(SE\left( b_2 \right)=0.5\) , \(n=6\) and \(α=0.05\) .

1. Test \(H_0:β_2=0\)
2. Test \(H_0:β_2>0\)
3. Test \(H_0:β_2<0\)