This comes from operations analysis. Often you have measurement of times, such as time between customers arriving at a store (e.g. Starbucks) or time it takes to process an order (e.g. a cappuccino).
Say you have two such variables X and Y, with X being time between customer arrival and Y being time required to process an order. Usually X is exponentially distributed (customers-per-hour would be the related Poisson distribution). In some processes, Y (time-to-process) is sometimes normal and sometimes exponential.
My question is this: if I observe a variable Y that intuitively could be normal or exponential, and I find that the mean equals the standard deviation (a characteristic of exponential distributions), can/should I assume that the distribution is exponential and not normal? Would either assumption work?
TL;DR More generally, if I have a variable for which the mean and standard deviation are equal, what is the general likelihood that the distribution is normal vs exponential? (I don't have actual data. This is more conceptual.)
[Edited: Replaced "probability" with "general likelihood" to be more specific.]
[Edited again: This question was motivated by an over-simplified textbook question that sparked my own conceptual question (being more deeply ingrained and interested in statistics than the class required). It has clarified the absurdity of the over-simplification. More precisely: any thing in operations measured as process time is not normally distributed. Exponential and lognormal distributions are often good candidates. This is true because you cannot have negative process time. However a normal approximation is often used because it is better understood and easy to teach. If I wrote the question again, I'd ask for examples where a normal approximation could result in a substantively bad analysis or recommendation. Thanks to all who helped!]