Statistical Reporting and Interpretation
6. The p-value summarizes results of tests of the same null vs. the same alternative hypothesis at all possible values of a. Tests using values of a no smaller than the p-value lead to rejection of the null hypothesis, while tests at any other values of a lead to retention of the null hypothesis. The p-value is a useful way to report hypothesis tests because readers who require different standards of evidence in order to reject the null hypothesis may use the same p-value to see whether the evidence meets their particular standards. The p-value can also be used as a measure of the compatibility of the data with the null hypothesis, with a p-value close to 1 indicating strong compatibility with the null hypothesis, and a p-value close to 0 indicating strong incompatibility.
The confidence interval, on the other hand, summarizes the results of tests of all possible assumed "null" values of a given parameter, including an actual null value of 0 and all specific non-null values (e.g. 1, 2, -5, 10) representing effects of particular direction and strength, at the same value of a = 1-(confidence coefficient of the interval).