Decision criteria in hypothesis test with diagram

In hypothesis testing, the decision is made by comparing the calculated value of the test statistic with the critical value(s), which separate the acceptance region from the rejection (critical) region. The size of the rejection region is the level of significance α.

Decision rule

• If the calculated statistic falls in the rejection region, reject H₀ (the result is significant).
• If it falls in the acceptance region, do not reject H₀ (accept H₀; the result is not significant).
• Equivalent p-value rule: reject H₀ if p-value ≤ α.

Two-tailed test (H₁: μ ≠ μ₀)

                 Acceptance region (area 1 − α)
                   .-"""""""""""""""-.
                 .'                   '.
                /                       \
              .'                         '.
          _.-'                             '-._
  ___.--''                                     ''--.___
 |////////|                                 |////////|
 |////////|                                 |////////|
 +--------+---------------------------------+--------+--
        −Zα/2              0                Zα/2
 Reject H0 (area α/2)                 Reject H0 (area α/2)

Reject H₀ if Z < −Z_α/2 or Z > Z_α/2 (±1.96 at α = 0.05).

One-tailed tests

 Right-tailed (H1: μ > μ0)            Left-tailed (H1: μ < μ0)

          .-"""-.                             .-"""-.
        .'       '.                         .'       '.
      .'           '.                     .'           '.
  _.-'                '-._            _.-'                '-._
 ___.--'              |////|          |////|              '--.___
 ---------------------+----+--        --+----+---------------------
                       Zα  (area α)      −Zα  (area α)

• Right-tailed: reject H₀ if Z > Z_α (1.645 at 5%).
• Left-tailed: reject H₀ if Z < −Z_α (−1.645 at 5%).

Steps in applying the criterion

1. State H₀ and H₁. 2. Fix α. 3. Compute the test statistic. 4. Find the critical value from the table. 5. Compare and decide. 6. State the conclusion.