Hi Ali,
No, a necessary assumption, aka the correct answer to an Assumption question, doesn't by itself prove the conclusion true. That's the job of a sufficient assumption, aka a Justify answer. The relationship the two types of assumptions have to the conclusion is one of sufficient or necessary conditions for it:
Justify AC True (SA)
Conclusion True
If the justify answer choice is true, the conclusion must be true. Pretty straightforward. Where it gets a little trickier is with necessary assumptions:
Conclusion True
Assumption True
This requires us to take the contrapositive to test necessary assumption answer choices, in a process we call the Assumption Negation test:
Assumption True Conclusion True
So we need to first negate our contending answer choices, then negate the conclusion and see if, when paired with the premises given in the stimulus, the negated assumption would logically lead to the negated conclusion.
In this case, the conclusion is that talking on a cell phone while driving is more dangerous than talking to a passenger while driving, with the premise being that being physically present leads passengers to be either quiet or even helpful when driving becomes dangerous, while someone on a cell phone won't know to react to dangerous driving conditions and continue talking. The logical gap that the stimulus needs to work is that talking during dangerous driving conditions is itself dangerous. This is almost certainly factually true, so it's an easy assumption for a test taker to make unthinkingly, but unless it's stated as a premise in the stimulus, it's an assumption being made.
(A) gives us this assumption, which we can test by negating to:
Speaking to a driver during dangerous situations
does not increase the risk of an accident
which would then lead to the negated conclusion:
Conversing on a cell phone while driving isn't necessarily more dangerous than talking to a passenger while driving
Works perfectly, making it the correct answer.
Hope this clears things up!