It really depends because so many factors come into play.
But I will say this:
If you have a well designed website that is more tightly focused on a niche and then you have a site that is trying to be all things to all people then your 'niche' site is definitely going to convert better, all other things being equal.
Web surfers are really no different than shoppers on the street. They are going to go to whatever vendor is PERCEIVED to the 'best' at whatever they're selling.