Can you name this film?

I can't remember the name of this film and it's driving me mad! Can any body help because I'd love to see it again.
I'ts a real b/ w Classic in the early Hitchcock vein and is about an ordinary American middle class family whose daughter is murdered. Her sister assists the detective in finding the perpetrator. The sister wears glasses and some of the late scenes are reflected in these.
All I can remember apart from this is that the final scenes take place in a fairground on some sort of island. The daughter, I think goes to meet the perpetrator in a sort of arranged 'sting'.
I'd say it's date is early 1950's from what I can recall about the decor and fashions ( the men for instance wear trilby hats and roomy loose suits. Is it Hichcock?