Engineers solve problems. A Problem Statement is a description of the problem. A project proposal is more general. A Problem Statement is a contract negotiated between the engineering and the client or instructor.
Most students make the mistake of talking about solutions instead of making a problem statement.
Avoid talking about a solution.
Another name for problem statement is "project proposal" but this context is vague and doesn't not sharpen the focus on context,specifications, and outcomes.
Context
Context refers to the unspoken assumptions, unknown facts, the beginning situation, the definition of completion, the people involved,the customers, and the institutional processes affected. Establishing the context is one of the hardest tasks associated with the project. Ask a thousand questions from the most creative part of your mind. Keep the words flowing and people talking. Spend time with the people the project is being financed by and the people that are going to be affected by the changes it makes. The terms "Scope, Scale and Specifications" are more formal ways of describing project context.
Specifications
Specifications are a short, 10 second sound bytes that describe what the project is about. Everyone should agree. The goal is to be as precise as possible. Good specifications outline the engineering requirements.
The goal is to gather what specifications exist. Don't stall starting the project until everything is clear. Nothing is ever clear enough. For example a PI (Client, principal investigator) did not know what physiology of the body to measure in order to figure out why bones shrink and fluid builds up in astronauts. The engineer started the project by purchasing generic A/D converters and instrument amplifiers rather than forcing the specification of the transducers.
Outcomes
Describe, visualize the change. How much more efficient? What will the customer have to do? Will training be required? Will nobody notice but profit increase?
Convince the customer to change.
Describe a test procedure that can measure value to the customer.
If outcomes can not done, a justification may be needed.
Successful Project
Projects are judged successful in relationship to their problem statement. Did the team solve the problem they attempted? Success can be making significant progress on a difficult problem. Or success can be completely solving the problem.
Problem statements can evolve or change during the project. However, negotiate changes with your client or instructor. They may not agree to the changes.