Pareto

When you trying to do establish a correlation, you can run a correlation or a regression test and use the p-value or the r-square values (in correlation, you can use the Pearson’s correlation) to see if indeed there is a strong relation or not.
Cheers,
A.

If you have identified the root cause from your Pareto, you should implement some type of corrective action to prevent or reduce this cause from occurring. After this action has been implemented, start collecting data on the occurence of the defect. Then you could run a hypothesis test on the before and after data to see if there is a significant difference between the two data sets. This would validate your corrective action as the reason for the improvement of your defect rate.

I think it depends on the nature of the problem and the kind of data involved. You have several tools to do that, such as DOE, correlation analysis, means tests, variance tests, etc. but that will strongly depend on the type of data (discrete, continuous, etc.), the distribution (z test, t test, nonparametric tests) the physical relation between variables (linnear, cuadratic, etc)
All you agree? More comments please!!!