A more common source of error than disagreement in gender is disagreement in number. They, their, theirs, and them are plural, but are often improperly used when only singular pronouns should be used. The cause of the error is failure to realize the true antecedent.
If ANYBODY makes that statement, THEY are misinformed. This sentence is wrong. Anybody refers to only one person; both any and body, the parts of the word, denote the singular. The sentence should read, If ANYBODY makes that statement, HE is misinformed. Similarly, Let EVERYBODY keep THEIR peace, should read, Let EVERYBODY keep HIS peace.