In Canada, the provinces handle BMD records. If this daughter was born in Alberta, it will be difficult to find out about it since Alberta doesn't have any online BMD indexes. We often rely on newspaper birth announcements but in this case, she may not have wanted to publicize it.
Most other provinces, however, do list births up to about 100 years ago - the exact timeframe varies by province. Do you know what surname this child might have been registered under?
Here is the general link for Medicine Hat directories.
http://peel.library.ualberta.ca/bibliography/3895.htmlThe name Robert is confusing. I checked a little further into that 1934 directory and Anne is living with her daughter Mrs. Nellie Bennie at that address.
http://www.rootschat.com/links/0luo/In the 1932 directory, Anne is still listed as the widow of Robert and Nellie Benie is listed as the widow of George.
http://www.rootschat.com/links/0lup/In the 1923 directory, Mrs. E. Bennie and Mrs. R. McCabe are living at 1116 Seventh Ave.
Nellie was a common nickname for Eleanor and I suspect that was her actual name.
In the 1919 directory 1116 7th Ave. is vacant and there doesn't seem to be any sign of either Annie or Nellie.
You might wish to contact the Medicine Hat public library to see if they can provide Annie's obituary. That may have a few answers for you.
Linda