I don't know your machine, but is it possible to remove the existing module or is it fixed?
If you can remove it, try installing the new module in the old socket, if you see what I mean. Run the tests and that should help to pinpoint if it's a socket or module problem.
You can also try putting the old module in the other slot (if the BIOS supports leaving the original slot empty), and testing that.
The memory module is more likely, on balance, to be the problem (especially if bought 2nd hand), than the laptop. That isn't saying that it cannot be the laptop! If so, I hope it is still under warranty!
SD