Unlikely. You would actually have to chance the laws of thermodynamics to
make a difference. One area you might not be considering is that it's not
only against code, it's also more expensive to heat the home. Bath (and
kitchen) fans carry a tremendous amount of moisture. Dump this into the
attic in cool/cold weather will cause this moisture to settle into the
insulation; degrading the performance and moving the dew point closer to the
heated side. It's not out-of-the question to consider a R-value loss of 50%
or more (one test showed much more) on cold days.
Your right, I didn't get that from what you originally wrote. But that
defeats the whole purpose of ventilation, doesn't it? The purpose it to
ensure use year round. But I have to admit, you are correct. If you don't
use the fan, not much moisture will get trapped in the attic.
Less to be certain. But just the air by itself carries a lot more water
vapor than can dissipate in cold air, and a fan will draw air from outside
the half-bath. It still will be moisture-laden air and that moisture will
collect on the insulation. Not as much or as fast, but it will affect the R
value to some extent. Best it goes outside in the winter (whenever
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