Many include the capability to set the time for the system to come on. Simply disable that function.
Or go into the Power Management section and set to user define and then disable all possible options that can turn the computer on or bring it out of sleep mode, in the event that they actually are putting it into sleep mode when they think that they are shutting down.
Be aware that with ATX sytems the power supply DOES NOT TURN OFF when you shut down with a normal Windows shutdown. The Main Rails [voltages, 5 volts, 12 volts, 3.3 volts and the minus 5 and minus 12] go off however a 5 volt standby voltage remains on "forever". This voltage is what allows such automatic turn on features such as ON by LAN, On by MODEM [when the phone rings], ON by time and date etc etc.
If you have the computer plugged in to a surge protector with a switch, you can turn that standby voltage off by turning the surge protector off. You can also, in most BIOSes then have the system turn back on as soon as you turn the surge protector switch on if you prefer.
Many BIOSes are set to put the system into sleep mode [standby] if one momentarily depresses the power switch, while to turn it off one holds the switch depressed for 4 to 5 seconds. But either way the 5 volt standby voltage is always there.
You don't have a power supply problem. Simply not a full understanding of the features of ATX type systems and the settings to make in the BIOS..
My friend?s computer power supply will just suddenly come on or shut off for no reason. In the middle of the night they can hear it come on. When this happens the drives seem to try to come on. I checked the supply with a power supply tester with everything unplugged from it. I could turn the switch on the tester to on and it did not come on. When it did come on it was fine by the tester and I did a voltmeter check on it also and that was fine. When they were using it, it started making a funny noise which was the processor fan. It does not seem to be running good. Then it says that there is no mouse attached. Sometimes it boots up, sometimes it does not. I unplugged the case switch from the mother board and tested the supply again and it still does what it wants to. Any advice??? Thanks in advance for any help!