Our in-tank fuel pump was replaced last summer. I know that when you turn the key on you should hear the pump hum and do its thing. How long should you hear this humming sound?

 

I hear mine for 3-5 seconds when I turn the key on.

 

The timer is for one second but one second is longer than most think.

If longer than one second it may mean the caps are starting to go. In other words it is taking the clock a second or so to get going.
If the clock does not start then the fuel pumps will continue to run and the fuel injectors will not be grounded (or fired) by the computer.