Deriving a $(1-\alpha)$100% confidence interval for $\theta$ pivotal quantity

Question

In the question, $u = \frac{\sqrt{n}(\bar{y}-\theta)}{\sqrt{\theta}}$ is given, where $\theta$ is the variable and unlike the other questions, I can't find a way to solve for $\theta$.

Answer 1

1. If you're finding an interval for $\theta$ you should strictly be dealing with an inequality in $\theta$ (the interval for the pivotal quantity would be all the values for that where a particular inequality is true; you're trying to get the corresponding inequality for $\theta$).

   However, you can deal with the equation part of the inequality if you're careful to check that what you think is the interior of the resulting interval actually is (it's no good just assuming that a solution to the equation of $\theta=\theta_L,\theta_U$ means that $\theta_L<\theta<\theta_U$ is true; in some situations it might just turn out that the resulting interval is actually in two parts $\theta<\theta_L$ and $\theta>\theta_U$). In some situations there can be other things you might need to be concerned about -- jump discontinuities, for example, might in some situaitons cause problems if you're trying to find a root.

   (I wasn't watching equality signs in my inequalities there; substitute them in yourself as appropriate -- the basic point should still be clear)

2. If you multiply through by $\sqrt\theta$ (as Tavrock indicated), it's quite straightforward to solve, since it's quadratic in $\sqrt{\theta}$.