Suppose (left(Y_{i}, X_{i} ight)) satisfy the least squares assumptions in Key Concept 4.3 and, in addition, (u_{i})

Suppose \(\left(Y_{i}, X_{i}\right)\) satisfy the least squares assumptions in Key Concept 4.3 and, in addition, \(u_{i}\) is \(N\left(0, \sigma_{u}^{2}\right)\) and is independent of \(X_{i}\). A sample of size \(n=30\) yields

\[ \begin{equation*} \hat{Y}=43.2+61.5 X, R^{2}=0.54, S E R=1.52 \tag{10.2} \end{equation*} \]

where the numbers in parentheses are the homoskedastic-only standard errors for the regression coefficients.

a. Construct a \(95 \%\) confidence interval for \(\beta_{0}\).

b. Test \(H_{0}: \beta_{1}=55\) vs. \(H_{1}: \beta_{1} eq 55\) at the \(5 \%\) level.

c. Test \(H_{0}: \beta_{1}=55\) vs. \(H_{1}: \beta_{1}>55\) at the \(5 \%\) level.