Does the amount of levels spent on enchanting affect the magnitude of potential enchants?

As far as I could tell from experimenting, the enchantments provide random effects, but the level of the effects is based on experience spent. So, an exp-level-5 enchantment is much more likely to be a level II or III version rather than a level I version. A sword with "Sharpness V" does a lot more damage than one with "Sharpness I", for example. It'll be a much higher-level enchantment spell, too.

Some enchantment spells create multiple effects, and I think the total level of all effects is equal to the spell level. So, a Level-I sharpness plus a Level-II Fire Aspect would be something like 7 exp levels (2 for sharpness, plus 5 for fire aspect). I need to figure out the actual exp level to enchantment level curve, but I think each level (I, II, II, IV, V) requires substantially more experience.