(Solved): 2) Suppose we have the simple linear regression model: \[ Y_{1}=\beta_{1}+\beta_{2} X_{1}+u_{i} \] ...

2) Suppose we have the simple linear regression model: \[ Y_{1}=\beta_{1}+\beta_{2} X_{1}+u_{i} \] Using a sample of size \( n=30 \) observations, we obtain the OLS estimate \( \hat{\beta}_{2}=-1.5 \) and its associated standard error, s.e. \( \left(\hat{\beta}_{2}\right)=0.7 \). We want to test the null hypothesis, \( H_{0}: \beta_{2}=0 \), against the alternative hypothesis: \( H_{1}: \beta_{2} \neq 0 \). a) What is the value of the \( t \) statistic for this test? b) What are the degrees of freedom for this test? c) What is the \( 5 \% \) critical value for this test? d) Can we reject the null hypothesis at the \( 5 \% \) level? c) What was the probability of a Type I error for the test you performed in part (d)?