A p-value is what the level of significance (a.k.a. a) would have been if your test statistic were the critical value. Usually it is compared against the level of significance in your problem to decide whether or not to accept or reject the null hypothesis. If your p-value is less than a, that means your test statistic is in the rejection region and so you reject H0. In strict numerical terms it is the tail area of your test statistic in the direction of the alternative hypothesis for a 1-tail test, and 2 times the tail area for a 2-tail test.
Example 1: Suppose your hypotheses were:
H0: m = 10
H1: m < 10
and a = 0.05 and the sample size is large
enough to use Z. Then you would reject H0 if
Example 2: Suppose your hypotheses were:
H0: m = 10
H1: m > 10
and a = 0.05 and the sample size is large
enough to use Z. Then you would reject H0 if
Example 3: Suppose your hypotheses were:
H0: m = 10
H1: m not = 10
and a = 0.05 and the sample size is large
enough to use Z. Then you would reject H0 if
The other thing to keep in mind with the first 2 examples is that if we had chosen any level of significance less than 0.0136 (or 0.0272 in the third example), such as 0.01, we would have accepted the null hypothesis. What this means in terms of testing hypotheses is this:
high p-values support
H0
low p-values support H1
So, if you get a p-value like 0.543, you can pretty well bet the farm that the null hypothesis is right. On the other hand, if you get a p-value like 0.00002 (it happens), the evidence is overwhelming in favor of the alternative hypothesis.