A problem statement is a statement you make in order to prove that your problem can be solved through research. Your problem statement must contain the context of what you are researching, what you are focusing on in your research, how your research will be significant, and how you will gather information and form a conclusion.
When your research is quantitative you are looking to describe the relationships or differences between variables. Your hypothesis must identify the variables you are researching and what you expect the relationship between the variables to be.
An example of a quantitative problem statement would be "There is a positive correlation between the weatherman predicting rain in the forecast and people carrying umbrellas with them when they walk to work." Here you have identified two variables, the weather prediction and umbrellas. You have also identified that there will be a positive correlation between these two variables showing that when one occurs in higher numbers the other will also occur in higher numbers.