Good question.
Ten doctors and will never get a unanimous opinion.
Colored sputum doesn't mean a bacterial infection......viral infections may also colour the sputum. Even if it is infected by bacteria, it is not an indication for antibiotic therapy, as healthy adults can fight off the infection.
If one was diabetic or suffering from a debilitating underlying illness, then antibiotics may be necessary.
There is no rule of the thumb or "sound bite " therapy, each patient assessed on his/her merit.
You seem to be getting better now, so i feel that antibiotics are not required, carry on with the rest..fluids and paracetamol.
Other Drs would support the post of Eddie.