Statistical tests are used to assess how well a mathematical model matches a real world situation.
- a real world problem is observed
- a statistical model is devised
- the model is used to make predictions
- experimental or observational data are collected
- comparisons made against the model
- statistical techniques are used to test how well the model describes a real-world problem
- the model is refined, then go back to step 3 and repeat
To be honest, mathematical modelling is just useful.