PC keeps shutting down randomly?

Boy do I have a wierd one here.

I have a PC that a local shop build a couple of years ago for my grandmother, and it keeps shutting down at random times, just like I held in the power button. Sometimes it stops seconds after start up, and other times hours after it has been active. The only other thing that I can add is that she used to smoke.

--It had a 350 watt power supply, so I replaced it with a 500 watt and it still kept rebooting.
-- I removed the side pannel, left it unplugged for a day,and then restarted it...yet it still kept rebooting.
--There did not appear to be anytyhing wrong with the CPU fan, but I replaced it anyway with another fan with a coper core, and I added silver thermal paste....no help.
--The motherboard has 2 DDR 400 dimms, so I put the RAM stick into dimm2 and still nothing.
--I replaced the ram with a new DDR 400 stick... still shuts down.
--I completely removed the hard drive and tried to boot up off of the CD... nothing.
Any ideas??

ASROCK P4i4SE Motherboard
Intel 1.8 Ghz processor
512mb Elixir 3200 DDR
500 Watt Power Tek power supply
1 DVD/cdr combo drive
1 floppy drive
1 rear case fan
1 top case fan
1 side case fan