Check that the CPU and heatsink/fan are well seated. The CPU is always in a socket and can be jostled loose during shipping.
Also, there is a handy device made called a Power Supply Tester. It's about 3.5 x 2.5 x 0.5 with connectors on each end and lights for each supply voltage. They cost from $5 to $40 when you can find them. They will find major problems like one output dead, but will miss subtle things such as high ripple or noise.
Your school probably has folks that maintain their PCs. Try taking your PC to them and asking for a quick power supply test. This is a case where you supply the manual labor of transport, removing covers, etc. for a free test. You may be able to pick their brains for other ideas while you are there too.
You could also try at a local computer shop, pleading student poverty of course.