Suppose a sample yields estimates (widehat{theta}_{1}=5, widehat{theta}_{2}=3), se (left[widehat{theta}_{1} ight]=2), and se (left[widehat{theta}_{2} ight]=1) and the correlation

Suppose a sample yields estimates \(\widehat{\theta}_{1}=5, \widehat{\theta}_{2}=3\), se \(\left[\widehat{\theta}_{1}\right]=2\), and se \(\left[\widehat{\theta}_{2}\right]=1\) and the correlation coefficient between \(\widehat{\theta}_{1}\) and \(\widehat{\theta}_{2}\) equals 0.5 . Perform the following tests at level 0.05 , assuming asymptotic normality of the parameter estimates.

(a) Test \(H_{0}: \theta_{1}=0\) against \(H_{a}: \theta_{1} eq 0\).

(b) Test \(H_{0}: \theta_{1}=2 \theta_{2}\) against \(H_{a}: \theta_{1} eq 2 \theta_{2}\).

(c) Test \(H_{0}: \theta_{1}=0, \theta_{2}=0\) against \(H_{a}\) : at least one of \(\theta_{1}, \theta_{2} eq 0\).