How does Hogwarts know when a witch or a wizard is born in a muggle family? In the case of Hermoine. What clues are there?

Mayb the answer is that they can sense magic, as we learned in Hbp, although they cannont sense who is doin it. So if they sense magic comking from a muggle house, they must look into it and discover one of the muggle is a witch/wizard

Well, muggle-borns must have some sort of witch and/or wizard in their blood line somewhere- even if it's REALLY far back- to go to Hogwarts/be a witch or wizard. Maybe they do a sort of 'track' on that family to wait for the next witch or wizard?

the underage witch or wizard even from the muggle borns have traces on them before they turn seventeen so i think that's how they detect if muggle borns have magic.

There's a book that magically records when a witch or wizard is born, and sends out letters when it's time for them to come to school. (JK Rowling has said so.) It's hard to say "how" they know, exactly--it's magic.

However, like has been said, they'd have a Trace on them, so the Wizarding World would know that there was a wizard child living there.

Wouldn't it be a bit like the trace? When magic is sensed, the ministry knows. but becuase you're underage, they can't arrest you?? That's what I thought anyway!

Well I guess that if the Ministry of Magic knows when magic is performed somewhere in the muggle world, therefore, it could identify muggle witches and wizards. The Ministry and Hogwarts have been known to work together at sometimes if not always for many centuries...
I think that despite what happened OotP and HBP, that part of the relation between the school and the Ministry remained unchanged,

Well, I think that maybe the Ministry or The Head Master could since magic. Even tho a Wizard is born into the muggle family the Ministry or some one of high authority could sense that there was magic some where inside this Muggle-Born. Its just like asking, how does Hogwarts Send its letters? And How do they know this Boy/Girl is a Witch/Wizard? I mean, It all falls right in the same catagory. So i Guess they can sense the magic within the air, and tell if the child has some magic blood in them or not.

Well, maybe when the muggle born grows, he or she, produces some kind of unintentional magic(like Harry and the glass in the Zoo), and the Ministry trace it and so, that way they know.

This could be a random guess, and probibly doesn't make any sense but is it possible that the sorting hat knows when wizards or witches are born.

Just like MissRandom said, Muggle born witches and wizards have a wizard somewhere in their genealogy. I read an interview with JKR saying so. So, at least we know how they get their powers.

Ron said that it was very hard to find a whole pureblood family because if wizards and witches kept marrying purebloods they would be extinct.

I believe that, maybe the different Ministries around the world have spells, like the 'Trace' placed around the grounds they cover. That way they can manage to locate them when they turn 11 and invite them to join each school or academy of wizardry. That's all I can imagine.