They can check the wreckage to see if the engine was turning over at impact. (Both external and internal cues will tell the investigators this). If it wasn't, they look at why it stopped.
Was fuel present? OK, was it the right kind of fuel? (The 300 HP engine of this Cherokee Six would probably run badly on auto gas. It would stop running if the fuel was contaminated with water [as mentioned by billorites -- it usually gets in to the fuel by condensing inside the tank] or jet fuel [usually gets in because someone messed up]).
Other contaminants that get into fuel are alcohol (found in some auto fuels, but unsafe in aircraft fuel systems), and debris from the rubber tank bladders if they get old and start decomposing (or if some chemical, like the above-mentioned ethanol, was in the fuel).
Like billorites said, during preflight inspection of this particular machine there are three places where the pilot drains fuel into a cup or a tube. If he sees water, he keeps draining till it's all out (water is heavier than gas and settles to the bottom, where the drains are). There is one drain below each wing tank and one underneath the engine.
Anyway, after an accident there are a lot of things the investigators do. Any remaining fuel is checked for the kind of contamination "bill" and I described. The fuel filter will also be examined closely.
The most usual reason for an aero motor to stop turning is operator error -- ran it out of gas, or switched to an empty fuel tank, or pulled the mixture knob when trying to chop the throttle. Second most common is carburetor ice which is unlikely to be a factor in this area (it's pretty arid) in this season (spring), but it will certainly be considered by the investigators.
The same procedures are used by ATSB, America's NTSB, Britain's AAIB and France's BEA, etc. Accident investigation is both an art and a science, but everybody who does it worldwide shares information and techniques. After all, if we can learn something from these accidents these unfortunate souls did not die in vain.
d.o.l.
Criminal Number 18F
I don't know what the situation was, or what was available to land on. Maybe the guy tried to make an emergency landing and didn't succeed. Maybe conditions made it impossible.
But, when all is said and done they will probably hang it on the pilot.
d.o.l.
Criminal Number 18F