$10 is the average in our area. It does vary, region to region.
Also, I looked on your profile, but didn't see how many of your own kids factor into this picture. If you have young children at home, the rate might change, depending on how many kids and their level of need. Add to this, you need to decide if you are comfortable with providing sick care for this family, knowing that your kids could become sick if exposed to this child.
Do you expect them to have back-up care? That's a good conversation to have.
Since you are new to the babysitting world, here are a few other tips:
Have a written agreement which details: days of the week you provide care; start/end hours; food arrangements (they should pay more if they don't provide food for their child at your house); parental preferences (nap? no nap? no tv? -- good to talk about beforehand); contingency plan for if your car breaks down or your kids are sick at home and you can't take her to preschool; name of doctor/dentist in case of emergency; two emergency contacts on their end in case of medical emergency (what if parents are in meetings or have their phones off? someone still needs to come); and very importantly, terms of payment (do they pay at the end of the day or every two weeks, every month... YOU need to know this). A written agreement or contract is a good friend to have. A lot of sitters get screwed over.
And remember, if you feel that they don't want to meet you on price (like, want dirt-cheap from you), don't do it. It's not your obligation to cut them a huge break. What they are basically asking for is private car with door-to-door transportations. Some providers would do this, some won't. So if the price they offer seems to low, be cautious. I personally think that $7 an hour is the lowest you should go, but that's me.
And remember, "babysitting" looks different to everyone. In home care providers usually have more than one family enrolled and can afford to charge less for their time per family because it's being doubled or tripled at the least.