Commutative Algebra and Algebraic Geometry are of relevance to Statistics and in recent years there was quite a lot of activity on this.

See e.g. http://en.wikipedia.org/wiki/Algebraic_statistics (and scroll down, the beginning is perhaps also interesting for your purpose, but what I mean is rather at the end of the page).

For example there is this book L. Pachter and B. Sturmfels. Algebraic Statistics for Computational Biology from 2005.

And there is a fairly recent (I believe) Activity Group of SIAM (Society for Applied and Industrial Mathematics) for Algebraic Geometry (which perhaps is close enough CA), about to hold its first conference http://www.siam.org/meetings/ag11/ (looking up the planery speakers should yield further details; there is a considerable intersection with names I. Rivin gives).

Another topic at the borderline of commutative algebra and number theory is Elliptic Curve Cryptography see http://en.wikipedia.org/wiki/Elliptic_curve_cryptography and also other cryptographic problems, but in part they feeel perhaps too number theoretic for you.

Finally, not really your question, but apparently the motivation: to convince your friends, depending on the background of your friends, I suggest to explain them the (simple) congruence arithmetic behind the final digit of the ISBN numbers. This was the only thing that I found that I felt had some real impact on the opinion of some of my friends on the usefulnes of pure mathematics.