Comment by Sniffnoy

No. If you have a good hash function, that means it's computationally infeasible to determine anything about x based only on y. It's not statistically impossible at all; "statistically" doesn't concern itself with computational difficulties.

This is similar to how, e.g., we generally assume that AES is unbreakable from a computational point of view, but if you want a statistically unbreakable cipher, your only (IINM) option is a one-time pad.