Final answer:
According to mythology, Athena founded the Panathenaic Festival, a yearly celebration in Athens where the garment worn by the statue of Athena was ritually changed.
Step-by-step explanation:
The Athenian institution that Athena founded, according to mythology, is the Panathenaic Festival. This festival was held annually in Athens and included a procession leading to the Acropolis where the peplos (garment) worn by the statue of Athena was ritually changed. The Panathenaic Festival was of great mythological significance and celebrated Athena as the patron goddess of Athens.