Of course the 5% elbirdo quotes actually means 50 batting points....
Take a .250 hitter after 248 AB (not PA). The 95% CI is given by [.196, .304], because
.250 + 1.96 SQRT(.25*.75/248) or [.250 - 0.054, .250 + 0.054].
So it's true that it is 5% in absolute terms, but it can represent 20% of the hitter's batting average.