Is it proper for a woman to make ברכות out loud so that her family will hear her and answer אמן ?
Absolutely. How else are you going to make the ברכה?! The נביא says, והבושת אכלה את יגיע אבותינו (Yirmiyahu 3-24). The embarrassment, the bashfulness ate up the יגיע אבותינו. To be embarrassed with yiddishkeit?! והבושת אכלה את יגיע אבותינו. Being bashful can sometimes eat away at your עבודת ה׳. Don’t allow your bashfulness to take away from your service of Hashem. To be embarrassed when you thank Hashem?! Don’t be embarrassed! Certainly make a ברכה out loud.
Of course, if there are men present, let’s say, תלמידי חכמים, then you could make it quietly in another room. Nobody should make themselves conspicuous in front of תלמידי חכמים. But among the family, why not?
Everyone should make all the ברכות out loud. הודו לה׳ קראו בשמו. Thank Hashem and make His name great by proclaiming His name. What?! Are you going to proclaim His name by whispering, by mumbling?! You have to proclaim the name of Hashem aloud. Absolutely, the mother must always make ברכות out loud in the house.