One reason I tell my students as to why they need to learn mathematics is that knowledge of mathematics gives a sense of power. It often gives a neat understanding of problems and aha insights into solutions. Here are a list of popular problems that students can handle with confidence provided they have a clear idea of how regression works.
(1) Was the coffee spiked with Barbiturates?
Forensic scientists regularly need to handle situation where they have to verify if some chemical is present in a given sample. The absorption spectrometry gives a set of linear equations connecting chemical concentration to absorbance. The best fit solution would yield the chemical concentrations in the sample.
(2) Does higher rain correlate with high temperatures?
Folklore says that harsher a summer is, better will be the monsoon. With data available online one may solve a least square fit for a model and figure out if the folklore makes sense.
(3) Best way to mix.
In a industrial set-up you want to mix several fluids and do it in energy efficient manner. A standard way is to compare the performance of different types of mixers and for each of they fit a curve to motor speed and energy used. The minimum of the curve gives an operating speed for the mixer for which energy usage is minimum.
(4) Do people living in big cities use less energy compared to people in towns?
This and similar other questions have been analyzed by Prof. Geoffrey West and his colleagues using best fit on data of (say) population density and electricity usage. You may check for yourself if city dwellers pay lesser electricity bills.
Often not only one is able to answer interesting questions using regression on data but also gain insights into efficient designs and behavior of complex systems. If you like to think about data and modeling do watch this TED talk. Mathematical skills are powerful tools, May the power be with you.