Question

at what age does the sex of a baby is determined?

Answer 1

The gender is determined at conception. or at the time of intercourse

At conception, gender is determined by chromosome characteristics - and it will be the male 
(or rather the male's sperm) that dictates whether the baby will be a boy or a girl.

Prior to conception, the unfertilized egg carries an X chromosome while the sperm can carry either an X or a Y chromosome. The gender of the baby comes down to one simple event:

If the sperm carrying an X chromosome fertilizes the egg, a girl will be conceived.

If the sperm carrying a Y chromosome fertilizes the egg, a boy will be conceived.