So, at its most basic, one sample t-test uses some given, presumably true number/mu, and tests your sample against that number. This conceptual example illustrates this via the naturally occurring sex ratio in humans (your mu) versus 2006-8 sex ratio data from different states in India (your sample data). Why look at this data? Social pressure, like dowries, high rates of sexual violence against women in India, etc. make male offspring more attractive than female offspring to some families. And the data provides evidence that this is leading to disturbing demographic shifts. See the table below from The Economist:
If you would like, you could even eyeball and enter the data for the different states and calculate your t-test.
I like this example because it illustrates how a very simple statistical test could be used to provide evidence that something fishy is going on in India. A tiny example of forensic statistics in the spirit of Freakonomics.
The data is from the UN and demonstrates how data collection is used by enormous organizations to uncover concerns in given countries/regions.
This data also demonstrates a global issue. The article itself goes into greater detail about this complicated issue: Why it is occurring and possible outcomes (increased sex trafficking, violence against women) that may result from this sex imbalance.