If you have a multimeter or can borrow one you can separate the motherboard from the PSU to test the PSU. I would do this first.
The following procedure involves working with components inside the case and live voltages ranging from +3.3V to +12V. You need to follow the instructions carefully to avoid damage to the components inside the case and keep yourself safe.
The motherboard usually initiates the start or the PSU when the power button is pressed. This shorts the two wires from the power switch which send a signal to the PSU to start it. This signal travel back to the PSU via the green wire in the 24 wire connector.
Make sure the computer is turned off before going further. With the computer turned off disconnect the power cord from the PSU from the wall receptacle.
Open the side of the case. Important: Before touching any of the components inside the case you need to discharge any static electricity in your body by touch the metal of the inside of the case. It only takes as little as 10Volts to kill integrated circuits like the ones on your RAM modules, the human body can discharge amount of voltage in access of 2K Volts.
The PSU has a 24 pin connector which is attached to the motherboard, this needs to be removed from the motherboard. This connector has a clip on the front side of the connector which needs to be depressed to in order to disconnect the connector from the motherboard.
With the connector free you will need to place a jumper between the socket of the green wire (there's only the one in this bundle) and any black wire socket. A small piece of wire or a paper clip can be used for this, just make sure the jumper isn't touching anything else.
With the jumper in place it's time to turn the computer on. All you need to do is plug the power cord back into the receptacle, this will start the PSU.
With the PSU running you should have at least the case fan and the PSU fan running. When the power is on use a multimeter to read the +3.3V, +5V, and +12V rail voltages.
Readings should not have variances larger than +/- five percent.
If you are able to do this and have voltages within their normal ranges the PSU should be good.
To restore the computer to its operational condition unplug the power cord, remove the jumper from the 24 pin connector and reattach it to the motherboard. Plug the PSU back in and the computer can be started with the power button.