If $X_i \sim \text{EXP}(\lambda)$ are i.i.d for $1 \leq i \leq n$ and $Y_j \sim \text{EXP}(\mu)$ for $1 \leq j \leq m$ and the two datasets are independent, find an equal-tail confidence interval for $\tau = \lambda/\mu$. ($\lambda$ and $\mu$ are the means.)
This is part three of a problem. In the first two parts, I found that the likelihood ratio test for testing $H_0: \lambda = \mu$ against $H_A: H_0 \text{ is false}$ could be given by rejecting if $\frac{\hat{\lambda}^n \hat{\mu}^m}{\hat{\theta}^{n + m}} \leq k$ for appropriately chosen $k$, when $\hat \lambda = \bar x$, $\hat \mu = \bar y$, and $\hat \theta = \frac{1}{n + m} (n \bar x + m \bar y)$ (that is, $\hat \theta$ is the pooled estimate). It also holds that $-2 \log\left(\frac{\hat{\lambda}^n \hat{\mu}^m}{\hat{\theta}^{n + m}}\right) \approx \chi^2(1)$.
My guess is that the likelihood ratio test can be turned into a confidence interval, but I'm not sure how I would do that.