Jeff, you said your engine is a 92502, which means it has a VacuJet carburetor, and does not use a diaphragm between the fuel tank and carburetor, just a gasket. I suggest you check things out before you buy a diaphragm that you may not be able to use.
Your "new" $5 mower's engine is probably either a 92508 or a 92908. The third number tells you the type of carburetor - if it is a 92908 it has a PulsaJet carburetor, and uses a diaphragm instead of a gasket, between the tank and carburetor. The PulsaJet has a fuel pump, the VacuJet doesn't. The final number is 2 for a radial pull starter and 8 for a vertical pull starter. You said it is vertical pull, so the final number will be 8.
