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I am attempting to use simple linear regression to construct a 95% prediction interval for a continuous response variable (Y) using a continuous input variable (X). When examining my data, I realized that if I log-transform both X and Y, the assumptions of equal variance and normality are essentially met, but if I do not log transform both variables, they are not. I'm able to run all of the R code needed to get the prediction interval for the transformed data, and I've read online and elsewhere how to interpret this interval. However, is there a way to get a 95% prediction interval on the untransformed scale using the 95% prediction interval on the log-transformed scale? In other words, I'm interested in talking about the conditional mean response instead of the median response (which, according to what I've read, is how I must interpret my prediction interval). I could exponentiate the interval endpoints, but I'm fairly sure that's incorrect because E(e^{Y}) is usually not equal to e^{E(Y)}.

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Back transform the endpoints. Quantiles are preserved if the transformation is monotonically increasing, so the probability coverage of $[e^L,e^U]$ on the untransformed scale is equal to the coverage of $[L,U]$ on the log scale.

The interval will not be the shortest possible interval, but it will have the required coverage.

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