terminology – What is a Problem Statement?

Typically Problem Statements are used to define and clarify the problem to be solved in the context of a UX design activity. There are a number of ways to structure or write this, but as long as the problem is written in a simple and clear statement then it is sufficient.

In contrast, a Hypothesis Statement is used to propose an explanation or a suggested solution to a particular question. It is used to help frame the way that the research is carried out, and typically in scientific research one would try to prove the hypothesis to be incorrect so as to avoid certain biases in the methodology and analysis. For example, if your hypothesis is that changing the button colour will improve usability, then you would typically set up some experiments to show that changing the button colour has no effect on usability. And if your data shows that it does have an effect then your hypothesis will have been proven to be valid.

You can define problem statements and hypothesis statements at different levels of detail, depending on the size of the project and the nature of the problems that you are trying to solve. So this can mean one general problem statement for the whole project, and different problems statements for major milestones or challenges to be solved.

Writing user stories is one of the techniques used in agile software development to help with requirements definition. From the Atlassian Playbook:

A user story is an informal, general explanation of a software feature
written from the perspective of the end user. Its purpose is to
articulate how a software feature will provide value to the customer.

Basically it is a good way to help frame the particular design problem from the perspective of the end-user, in terms of the goals that they want to achieve and without prompting what the solution should look like.