Is a confidence interval symmetric when constructed on an inverse prediction of a logistic regression? I have a dataset with a continuous variable paired with a binomial response.
We want a one-sided confidence bound to determine at what value of the continuous variable would we get a desired probability.
Can we use software that only calculates 2 sided confidence intervals and assume they're symmetric? For example, for a 1 sided confidence bound at 95%, we'd ask the software to calculate a 2 sided interval at 90% and only use the lower bound.
My understanding of the Central Limit Theorem would imply that we could, but a coworker says it's not symmetric. Neither of us are statisticians (we're engineers). Thanks!
 A: I'm not sure how meaningful it is to put a 'confidence interval' on an $x$ value.  In regression models (including logistic regression, as here), we consider the $x$ values to be fixed and known a-priori.  
Instead, I think I would fit your model and then compute the $90\%$ confidence band (or $95%$ one-sided confidence band) around the predicted probabilities.  I would then find the point where the lower limit of the band equaled whatever probability you were interested in.  If the function were monotonically increasing, your 'pseudo-confidence interval' would be all $x$ values greater than or equal to that point.  (If the fitted function were monotonically decreasing—i.e., a single variable with no additional functions of it, such as a squared term, and a negative coefficient—then the 'pseudo-confidence interval' would be all $x$ values less than or equal to that point.)  
If you are married to the idea of a confidence interval on $x$ itself, I suppose I would bootstrap.  (I have an outline of a basic bootstrapping algorithm here: How do I compare bootstrapped regression slopes?)  For this approach, I would find the $x$ value that corresponded to the predicted probability you were interested in on each iteration.  Then you could determine the confidence interval from the bootstrapped sampling distribution.  
