 Prediction interval for a ratio of log-normal responses

Suppose independent future log responses satisfy $Z_j=x_j^T\beta +{\epsilon }_j$, where the errors have variance ${\sigma }^2$. Conditional on a fitted normal linear model, the predicted log ratio $Z_1-Z_2$ has estimated mean $(x_1-x_2{)}^T\hat{\beta }$ and estimated varianceA Student t interval on this logarithmic scale can be exponentiated to obtain a prediction interval for the positive ratio.