A Y-DNA test would be valid from one male in the male line to the previous males.
mtDNA is transmitted from the mother to both sons and daughters.
So, in the first case, you and your father back to gggg grandfather and beyond in the Hay line should have the same Y-DNA.
In the second, your sister and your mother would have the same mtDNA back to their gggg grandmother and beyond.
You might want to take one of Spencer Wells books, "The Journey of Man" or "Deep Ancestry Inside the Genographic Project" out of the library. The 2nd book is the more recent. There are also videos of the material in each book, if you are interested.