If you had a data set which exhibited both non-normally distributed and normally distributed data, which statistical test would you use and why?
Which car would you use? That question in it self, harbours too little information to come with a qualified answer. Choice of car depends on context - racing, rock climbing, family vacation, etc.
The same goes for choice of statistical test. You will have to supply us some context 
You can start with the following information:
- What is the data description?
- How do you know some part of the data is normal, where the rest is not? How have you defined these "parts"?
- What is the problem of interest?
- What do you want to test?
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