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Maintenance of precision in NIntegrate

Consider integrating a function arg[] with a stated PrecisionGoal, say 10:

NIntegrate[arg[x], PrecisionGoal->10, ...]

The integrand arg[x] is a numerical function of the form:

arg[x] := (bunch of elementary functions) + NIntegrate[arg2[y],...]

arg2[y] is, at best, computable to MachinePrecision.

However, I don't know the best choices for
PrecisionGoal or related options for the second integral (the inside 
in order to minimize my computation runtime. So, let's assume I do not use
these options at all in the inside integral.

My question: will Mathematica then automatically request the inside
integral to perform at a high enough PrecisionGoal in order to
meet my PrecisionGoal target of 10 in the first (outside) integral?



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