The dependence on sample size is not exponential, it's sublinear. The heuristic rate of convergence to keep in mind is the square root of the sample size, i.e.

The dependence on sample size is not exponential, it's sublinear. The heuristic rate of convergence to keep in mind is the square root of the sample size, i.e. getting 10x more samples shrinks the margin of error (in a multiplicative sense) by sqrt(10) ≈ 3ish. The exponential bit applies to the probability densities as a function of the bounds themselves, i.e. how likely you are to fall x units away from the mean typically decreases exponentially with (some polynomial in) x. Of course, this is all assuming a whole bunch of standard conditions on the data you're looking at (independence, identically distributed, bounded variance, etc.) and may not hold if these are violated.