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For example if it's possible to dish out damage between 100 to 120 based on your attack stat, luck will help you roll the higher end of the spectrum more often, so you'd more often do 120 or close to it than you would 100 or close to that.
So in a way, yes, it effects damage. But no it's not going to make your max damage output higher than possible from your attack/strength stat.
Depending on your level and luck, your physical damage range can be either 0.8 ~ 0.9, 0.9 ~ 1.1, or 1.1 ~ 1.2, so you'll still need a high Strength stat, especially due to high enemy defense values.
https://www.reddit.com/r/dragonquest/comments/1gy2mym/dq3_hd2d_remake_base_luck_and_physical_damage/
You WILL notice mediocre Strength in the Hero.