Critical values, p-values and the decision rule

In short

• Critical values (rejection points) come from $\alpha$ and the statistic's distribution. They mark the edge of the rejection region.
• Two-tailed test: $\alpha/2$ in each tail. One-tailed test: all of $\alpha$ in one tail, so its critical value is smaller in absolute value.
• A statistic beyond the critical value is statistically significant; otherwise we fail to reject.
• The p-value is the smallest significance level at which the null can be rejected. Reject when the p-value is below $\alpha$.
• In a two-sided test, rejecting at $\alpha$ is the same as the hypothesised value lying outside the $1 - \alpha$ confidence interval.