My solution:

I created a new Scala project with IntelliJ IDEA and set the Scala home to the manually downloaded and unpacked Scala folder for Scala 2.10.2.

Now you can set the Scala compiler in the real project: Switch to the real project, open the project structure settings and then Modules | your module name | Scala | compiler library. Switch to your new version. Set the language level to 2.10.

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