Suppose a sample yields estimates \(\widehat{\theta}_{1}=5, \widehat{\theta}_{2}=3\) with asymptotic variance estimates 4 and 2 and the correlation coefficient between \(\widehat{\theta}_{1}\) and \(\widehat{\theta}_{2}\) equals 0.5 . Assume asymptotic normality of the parameter estimates.

(a) Test \(H_{0}: \theta_{1} e^{\theta_{2}}=100\) against \(H_{a}: \theta_{1} eq 100\) at level 0.05 .

(b) Obtain a \(95 \%\) confidence interval for \(\gamma=\theta_{1} e^{\theta_{2}}\).