The problem here is the negative word "without". Take that word away and the
sentence is easier to respond to
Will you attend an educational institution without being a full-time
student?
If you are going to be a full-time student, then the answer would be "No, I
will be a full-time student." Answering "yes" or "no" without qualifying it
would be confusing.
In practice, most English speakers would not phrase the sentence like this
because it sounds awkward. Better to ask "When you attend an educational
institution will you be part time or full time?"
Colin Evans
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