Earlier quoted context omitted.
This is what P-values are for
Perhaps explain how P-values are to be used in this case, for the lay of us.
Given a statement and a sample with a mean and standard deviation, what is the probability that you manage to choose a sample with that mean and SD, assuming the statement is true?
For example, my statement is the average weight of chicken an american eats every month is 20 pounds and my sample gives me a mean of 16 and SD of 4. The p value is the probability of choosing a group of people who eat on average 16 pounds of chicken every month if the REAL mean is actually 20.
Edit to answer question: The p value doesn't "care" about the sample size - it is adjusted and remains accurate. And so if the p value if high for the sample in the study, that means it is a likely occurrence and the result is not very noteworthy.