There is a video for problem 30, which goes over one approach to determine the claim. Here is the link to that video: http://www.statsprofessor.com/video.php?chapterId=16&id=324#ptop . The part that discusses the claim is from: 2:48 - 4:10. I explain a different approach below.
There is a video fro problem 33, which shows the approach I have explained below. The part that discusses the claim in detail is: 2:16 - 3:33. Here is the link: http://www.statsprofessor.com/video.php?chapterId=16&id=325#ptop
As for example 31, the problem says the claim is, "the program increases core strength." This means that post the program, we would be have higher numbers for the fitness test, which would mean a larger average. Isn't that correct? This is the first step of the process. You must think about what the claim is saying. It will always say that one mean is bigger than the other, or it will say they are equal (or not equal).
Next, you have to decide how you are going to do the subtraction. Let's assume that you are going to always choose to do the subtraction as you see the data laid out it in the problem: Pre - Post. Then, you must set up a statement that has the two means in the same earlier separated by an inequality sign: µpre < µpost . We have the pre mean less than the post mean because the claim says, "the program increases core strength." Now from here, we simply move the mean on the right of our statement to the left (remember when a term crosses an equal sign in Algebra, we change its sign): µpre - µpost < 0. Lastly, this difference can be written as µd < 0.
You'll notice that you will get the exact opposite claim, µd > 0, if you perform the subtraction Post - Pre (like I did in the answer to this problem).
I hope that helps,