Decision procedure by critical value approach

In the critical value (classical) approach, the sample test statistic is compared with a critical value fixed from the level of significance α. The critical value separates the acceptance region from the rejection (critical) region of the sampling distribution under H0.

Steps

1. State H0 and H1. The form of H1 (≠, >, <) fixes whether the test is two-tailed, right-tailed or left-tailed.
2. Fix α, the probability of a Type I error (commonly 0.05 or 0.01).
3. Choose the test statistic and its distribution under H0: z, t, χ² or F, depending on the problem.
4. Find the critical value(s) from the table for α and the degrees of freedom.

Test          Rejection region
Two-tailed    |stat| > critical value (α/2 in each tail)
Right-tailed  stat > critical value
Left-tailed   stat < −critical value

5. Compute the test statistic from the sample data.
6. Decision rule.
   • If the calculated value lies in the rejection region, reject H0.
   • If it lies in the acceptance region, do not reject H0 (the evidence is not enough to reject it).
7. Conclusion. State the decision in the words of the problem.

Example

For a two-tailed z-test at α = 0.05 the critical values are ±1.96. If the calculated z = 2.4, then |z| > 1.96, so H0 is rejected. If z = 1.2, H0 is not rejected.

      reject    |   accept H0    |    reject
  ------|-------|-------|-------|-------|-----
      −1.96             0            +1.96

The p-value approach gives the same decision: reject H0 when p-value < α.