This is called ta marbuta ة / ـة It has t sound but only if it is followed by a vowel. Also, short vowels that come immediately before a pause (like in the end of a sentence) are always omitted which causes the last ـة to be omitted as well

The computer is pronouncing it wrong a lot.

In formal Arabic, if she is the subject, "imra'a", "'amrikiyya", and "dhakiyya" must ALL end in "tun". If she is the object they all end in "tan". If your talking about something that belongs to her they should all end in "tin", as in "kalb-(u/a/i) imra'atin 'amrikiyyatin dhakiyyatin.

If you're talking about "THE smart American woman" the "n" disappears (and for this word the beginning changes a little too) it's al-mar'atul-'amrikiyyatul-dhakiyyatu / al-mar'ata l-'amrikiyyata l-dhakiyyata / al-mar'ati l-'amrikiyyat il-dhakiyyati.

And in vernacular speech, the "n" at the end and the vowels before it for subject object and genetive case aren't pronounced anyway, so it would just be 'imra'a 'amrikiyya dhakiyya, or al-mar'a l-'amrikiyya l-dhakiyya, except that dialects use different words for "woman".