Final answer:
In Arthur Miller's play 'The Crucible', the 'sickly wife' given a doll with a needle in it by Mary Warren is Elizabeth Proctor, which causes her arrest for witchcraft.
Step-by-step explanation:
The "sickly wife" who was given a doll with a needle in it by Mary Warren is Elizabeth Proctor from the play The Crucible by Arthur Miller. In the story, this incident is crucial as it leads to Elizabeth's arrest for witchcraft. Mary Warren, a servant in the Proctor household and a member of the Salem community, gives Elizabeth the doll, which later becomes evidence against her when a needle is found embedded in the doll's fabric, mirroring a wound on another character, Abigail Williams, who accuses Elizabeth of using the doll for witchcraft.