The characters in DBZ use their ki to amplify ALL of their abilities, including perception, speed, strength, destructive capacity, durability, etc.
This is why most of the time, they will fight with ki bulking their strength and speed equally, but they are able to bulk up just their speed, sacrificing strength, or vice-versa. They have total control over it.
That's also why they can exert so much raw destructive force in the form of ki blasts, after they charge them up by standing still for a bit. Just like how Vegeta was keeping up with Cell, as Super Vegeta, but when he sacrificed his speed to try and do some damage, he managed to blow off half of Cell's body.
So yes, "power", as in ki capacity, DOES determine a persons speed. But strength does not. Only ki does.
Also, I'm assuming we're only talking DBZ here, and not any other anime's. That's why I'm relying on ki to explain here.