Why cant we decide based on the Standard Error differences of the two models?
Ans given by question :
"To actually determine the explanatory power for sales itself, fitted values for the log-linear trend would have to be determined and then compared to the original data. The given information does not allow for such a comparison. "
Simply put, the dependent variables are different-- one model is predicting Sales and the other model is predicting the natural log of Sales. You cannot directly compare model fit for these models because the left-hand sides (dependent variables) are not the same.
To make the comparison, you would need to get the predicted Y-values from the Ln(Sales) model, giving you Ln(y-hat) values. Then, you would take the inverse of the natural log for each of these values. From here, you could compute an R-squared value for the natural log model, a pseudo R-squared. You could compare this pseudo R-squared to the R-squared of the model predicting Sales.
Hope this helps!
Edit: Also, to answer your original question-- you can’t compare the standard errors in the same way that you can’t compare r-squared or the f-tests for these models (or any models with different Dependent Variables)— each of these model-based statistics is a function of Y-values and predicted Y-values. The models don’t use the same Y-values and predicted Y-values (as I mentioned previously, it’d be like comparing oranges to apples).