That may still be copyright infringement, but it's a very nontrivial question. To be safe, you should either obtain a license, or restrict yourself to public domain material, or create similar-sounding snippets your self.
Within the limits of copyright law, only the copyright holder has the right to decide over public performances or reproductions of their work. Copyright does have limits, so there are some exceptions. For example, limited reproductions for educational purposes are typically fine. However, this does not imply a general permission to use a work just because your use is non-commercial or just because you gave attribution.
Specifically in the US, the fair use doctrine provides a broad copyright exception, though fair use should be understood more as a possible defense when accused of infringement than as a clear right. A fair use analysis must consider four factors:
- purpose and character of the use, for example whether the use is commercial or nonprofit educational
- nature of the work
- amount of the work used
- effect of the use on the potential market of the copyrighted work
It will probably not be fair use just because the use is non-commercial. However, other factors might help with the fair use defense.
A highly transformative use (remixing the audio snippets)
for noncommercial purposes,
using only short audio snippets that are unlikely to have any effect on the value of the original work (your performance doesn't compete with the original TED talk)
might indeed make it possible to rely on a fair use defense.
Indeed, music sampling is a common phenomenon, and there is a long history of debate on the question whether sampling is fair use. However, most (but not all) instances of sampling will require a license.
Note that other countries have completely different copyright exceptions. Fair use is an US-specific phenomenon. For example, Germany's copyright law has explicit protections for limited educational use or for highly transformative use, but all without the flexibility that fair use allows – and in particular without recognition of non-commercial use as a factor. Within German copyright law, even smallest recognizable audio snippets still enjoy copyright protection, and using such snippets will require a license.