To Reject or not to reject, that is the question...
The question of rejecting the null (and risk committing a Type I error) or failing to reject the null (and risk committing a Type II error) is a ponderous thing, but as we have learned there are 3 methods by which we can come to the same conclusion.
Compare the test-statistic against some z (or t) value
Calculate a confidence interval and compare if the sample mean is contained within it, or
Compare the p-value against α.
All three are dependent upon α, although if our test statistic (z-cal) is greater than the z-value, which is based on α or α/2, then we conclude we must reject the null, while the p-value < α also indicates we reject. This change in inequalities is odd, so hopefully the attached notes will make all three methods relationship (and the direction of the inequalities) more clear.