Praise be to Allah the Lord of the Worlds.
It is recommended for the one praying alone to recite aloud during the prayer in which the Quran is recited aloud and silently during silent prayer, for men and women. This is as reported by the Khalaf (Those who came after the Salaf) from the Salaf (Righteous predecessors or ancestors). However, if there are non-Mahram men present, she shouldn`t recite out loud. If she did, her prayer is valid, but it is disliked for her to do so. If she was praying alone, or in the presence of her Mahrams, or with a group of women, then it is recommended that she recites out loud, but lower than men. An-Nawawi said, "According to our fellow scholars, it is recommended that she recites out loud if she was praying alone, or with a group of women, or in the presence of her Mahrams, whether she was leading women in prayer or was praying alone. But, if there are non-Mahrams, then she should recite silently."{Al-Majmou`, 3/390}. As for her saying "Ameen" out loud, it takes the same ruling of reciting out loud mentioned above. If non-Mahrams are present, then she should say it silently, but if she was praying alone, or with a group of women, or in the presence of her Mahrams, then she is permitted to say it out loud. It is stated in "Hashiyat Al-Bajirmi Ala Al-Khateeb, 2/29": "In conclusion, it is recommended for the person praying behind an Imam or alone to say "Ameen" out loud during loud prayer and silently during silent prayer." And Allah the Almighty knows best.