It is hard to look at the universe and not wonder about the meaning, of, well, everything. A natural question is whether what we see is a sign of intelligent design. The antithesis of design would be a random universe or, assuming laws of physics, one whose fundamental physical parameters were randomly selected, but conditioned on life (ourselves) being here to observe it. In unpublished work, the British physicist Dennis Sciama argued that such a randomly selected universe would display a statistical signature. He concluded that a random universe would almost certainly have parameters only just allowing for the possibility of life. Here we consider whether this signature is definitive. We find that with plausible additional assumptions Sciama's signature would appear to reverse: Were our universe random, it could give the false impression of being intelligently designed, with the fundamental constants appearing to be fine-tuned to a strong probability for life to emerge and be maintained.