Math 6010 • Linear Models

Let \(X_1,\ldots,X_n\) be a random sample from \(N(\theta,\theta)\) (i.e., normal with mean and variance both equal to \(\theta\)), where \(\theta>0\).
\begin{parts}
\item Give three pivotal quantities (or pivots) involving the entire sample, which respectively have a standard normal distribution, a \(t\)-distribution, and a chi-squared distribution, and indicate the numbers of degrees of freedom. (5 points)
\item Use the normal pivotal quantity in part (a) to obtain a \(100(1-\alpha)\) percent confidence interval for \(\theta\). (5 points)
\end{parts}