I don't know what the hospitals do nowdays, but when i had my baby early 19 years ago, they did some test and told me exactly how early they thought she was. Per my previous due date, she was a month early. Per their test 5 weeks, 3 days.
Yes this has been discussed. Everyone's babies weigh differently at different times (mine was just short of 7 lbs). The biggest question, as i have seen addressed here, would be why would you care WHEN an unmarried woman was having a child?? It would have to fit with WHO she wanted to say the father was. At least i can't think of another reason. jmo
ETA: just to address the 10 lb baby post above, my MIL weighs 110 pounds and is 5 foot 2 inches...and she had 3 babies, all around 10 pounders....i know....ugh.....but just to point out the variance in babies, parents sizes, term of gestation, etc.