Suppose that $\displaystyle \theta \sim \text{Exp}(\lambda) $. An estimator for $\displaystyle \theta $ is $\displaystyle \bar{X} $. Suppose that we know that $\displaystyle \frac{2n \bar{X}}{\theta} \sim \chi^{2}_{2n} $. Construct a $\displaystyle 100(1-\alpha) \% $ confidence interval for $\displaystyle \theta $.

Would it be $\displaystyle \frac{2n \bar{X}}{\chi^{2}_{\alpha/2, 2n}} < \theta < \frac{2n \bar{X}}{\chi^{2}_{1-\alpha/2, 2n}} $?