Typically, a purpose statement comes out of a problem statement. The problem statement lays out the problem and the purpose statement outlines what you as a researcher intend to do to study the problem.
Let's say that you are a social scientist and you want to study the negative campaign ads in elections. Your problem statement might talk about how many campaign ads are negative. It might talk about how anecdotal evidence suggests that such campaign ads make people disillusioned about politics. You would then set out the purpose of your research. You might say that you propose to study the actual effects of negative campaign ads on people's attitudes towards the political process. You would then need to lay out the steps that you will take to measure those attitudes and the impact of the negative campaign ads.
In all types of research the investigator has to be very clear as far as what is at stake, what is being investigated, and what is the reason for the investigation. Moreover, you also need to specify to what point your investigation will have any significance once it is completed. All of these factors are important to prove for a research or investigation to get funding, resources and time granted.
When you state the PROBLEM you are basically saying WHAT is the person, place, thing, or idea (theory) that you will be studying, or testing. You have to define the problem stating a rationale for choosing it. The relevance of this problem is also important to know. For instance, if you are a chemistry major looking to become approved for a research, the problem that you choose should be relevant and related to your field of study, not to another field, such as cooking, or arts.
When you state the PURPOSE of an investigation you are going to answer three specific questions: WHY? HOW? WHEN?
You will use action words and active tense to voice your purpose. For example, you are going to "determine", or "disprove", "describe", "contrast" the problem that you described before. This is the place where you say what exactly you plan to do about the problem.
It is important to do all of this prior to naming your investigation. A common mistake that research students do is title their work and then end up doing something other than what they said that they will do. This is because the purpose is not yet well-identified.
In the purpose part you should also add the methodology that you will employ to conduct the research: surveys, phenomenology, and pre/post testing are examples of these methods.
When you do all of this you will be prompted to talk about how your project is significant. This section is separate from the problem and purpose, but its goal is to determine how your project actually gears toward making a difference in the problem that you defined. Moreover, it also will help you as a researcher put the problem in a context and decide whether it is worth your time to research that problem, or if another problem is more important for you and your field.