This is a hard one to track, I've found a few alternate spellings that might be related, but the one I'm leaning toward is a Simitz or Semitz from Bohemia that I've found in Johnson County.Just can't find them yet in 1885 or 1895 when Anna should have been with her parents if they were still living.
The Anna and Emory moved to South Dakota by 1920, but I haven't found them yet in 1930.
Finally found the family in 1895, still unable to find them in 1880 or 1885.
1895 Iowa State Census Barbra Shimitz 49 Bohemia White Johnson Third Ward Iowa City Iowa Annia Shimitz 16 Iowa City IA White Johnson Third Ward Iowa City Iowa Rosa Shimitz 14 Iowa City IA White Johnson Third Ward Iowa City Iowa Frank Shimitz 7 Iowa City IA White Johnson Third Ward Iowa City Iowa I think Annia is the one you want.
1925 Iowa State Census, Barbara is living with her son Frank Her parents are listed as Frank Vachotuky, b. Bohemia and Katherine Shonka, also b. Bohemia and married there Frank Shimitz' father is listed as Frank Shimitz, b. Bohemia, married to Barbara in Iowa
In looking for a birth record, I would suggest going to the clerk's office for Johnson county yourself, so you can scan all of the "s" births for several years around 1881.Look for any name that sounds good.