I have this problem and solution, and Ive been doing a lot of these problems and I still can't seem to understand the reasoning and wording.
For the true statement in problem 8, is it saying that there is a small probability that h0 (u=0) is very small? and thus its highly probable that h1 is true and it is out of calibration?
The $p-$value for a test of $H_{0} : \mu = \mu_{0}$ against the alternative $H1 : \mu \neq \mu_0$ is the probability, under $H_0$, that the variable you are looking for takes a value at least as extreme as your observed data, hence a small $p-$value means you have a small probability to get more extreme values, hence your observed data is big(or small, depending on the null hypothesis).
But assuming the null hypothesis, you usually want to impose some behaviour on the random variable - for example make it follow the normal distribution or the $t-$distribution, hence smaller $p-$values are problematic since your observed data is big(small).
Not sure that helped since is a bit general, but think of $p-$values as indicators whether your observed data is big(or small) which means it's not as you would expect them to be