1. Definition

  • A critical value is the cutoff point on a statistical distribution that defines the boundary for rejecting the null hypothesis (H₀).
  • It separates the “acceptance region” (where we fail to reject H₀) from the “rejection region” (where we reject H₀).
  • It depends on:
    • The chosen significance level (α),
    • Whether the test is one-tailed or two-tailed,
    • The distribution used (Z, t, χ², F).

2. General Rule

  • If the test statistic falls beyond the critical value → reject H₀.
  • If the test statistic is inside the critical value → fail to reject H₀.

3. Examples by Distribution

(a) Z-distribution (large samples, known σ)

  • Two-tailed test, α = 0.05 → critical z = ±1.96.
  • One-tailed test, α = 0.05 → critical z = 1.645.

(b) t-distribution (small samples, unknown σ)

  • Two-tailed, α = 0.05, df = 10 → critical t ≈ ±2.228.
  • As sample size grows, t critical values approach z-values.

(c) Chi-square (χ² test)

  • Goodness-of-fit or independence tests.
  • Example: α = 0.05, df = 4 → χ² critical ≈ 9.49.

(d) F-distribution

  • ANOVA tests.
  • Example: α = 0.05, df1 = 3, df2 = 20 → F critical ≈ 3.10.

4. Critical Value in Confidence Intervals

  • Confidence Interval formula:
    • $\text{Estimate} \pm \text{Critical Value} \times \text{Standard Error}$
  • Example: 95% CI for mean with z:
    • Critical z = 1.96
    • If mean = 100, SE = 2 → CI = [100 ± (1.96 × 2)] = [96.08, 103.92].

5. Example – Hypothesis Test

Suppose we test H₀: μ = 50 vs H₁: μ ≠ 50.

  • Sample mean = 53, σ = 10, n = 100.
  • Test statistic: $z = \frac{53 – 50}{10/\sqrt{100}} = \frac{3}{1} = 3.0$
  • At α = 0.05 (two-tailed), critical value = ±1.96.
  • Since z = 3.0 > 1.96 → reject H₀.

6. Key Takeaways

  • A critical value is the threshold beyond which results are considered “extreme” under H₀.
  • It is determined by α, tails of the test, and distribution type.
  • Used in both hypothesis testing (reject/fail to reject) and confidence intervals (width of interval).

In short:
The critical value is the cutoff point from the chosen distribution that marks the boundary of the rejection region. If your test statistic exceeds it (in magnitude), you reject the null hypothesis.