Are you sure that the sentence was correctly written? The sentence, although grammatical, doesn't make any sense. It seems that you meant to say "although" instead of "because." Am I correct?

She didn't take an umbrella because it was raining. Not very sensible but quite correct. I sometimes enjoy the feeling of the rain on a warm day and will not wear a raincoat, perhaps she feels the same. This sentence is about choice and though they may not be logical, we have the freedom to choose. A sensible choice would be: She used an umbrella because it was raining or she took an umbrella because it was raining.

The book is correct. She took an umbrella not because it was raining...but for another reason. Eg She took the unbrella not because it was raining but because she feared it would by the time she had finished working.