Hypothesis Testing: Small Samples

When working with small samples (\(n < 30\)), the Central Limit Theorem no longer guarantees normality of sample means. This requires careful assumption checking and often necessitates nonparametric methods. Below, we outline a systematic framework and demonstrate its application with real-world examples.

Framework for Small-Sample Inference

  1. State hypotheses: Define \(H_0\) (null) and \(H_1\) (alternative)
  2. Check normality: Use graphical (Q-Q plots) and statistical tests (Shapiro-Wilk/Anderson-Darling)
  3. Choose test:
    • If normality holds → Parametric \(t\)-test
    • If normality violated or \(n\) very small → Nonparametric test (e.g., Wilcoxon, sign test)
  4. Calculate test statistic and p-value
  5. Compare p-value to \(\alpha\) (typically 0.05)
  6. Make reject/fail to reject decision
  7. Interpret results in context

Example 1: Optical Density Safety Threshold (Sign Test)

Scenario: A chemical batch with \(n = 6\) measurements must verify if its median optical density exceeds 0.01.

Hypotheses:
\(H_0\): M \(= 0.01\)
\(H_1\): M \(> 0.01\)

Analysis (Sign Test):

Results:

  • All 6 observations exceed 0.01 → Test statistic \(s = 6\)
  • Exact binomial p-value = \(0.0156\)
  • 95% CI for median: \((0.1531, \infty)\)

Conclusion: Reject \(H_0\) (\(p < 0.05\)). Strong evidence that the true median exceeds 0.01.

Example 2: Insect Spray Efficacy (\(t\)-Test with Normality Check)

Scenario: Test if Spray C (\(n = 12\)) reduces insect counts below 15.

Hypotheses:
\(H_0\): \(\mu = 15\)
\(H_1\): \(\mu < 15\)

Analysis (\(t\)-test):

Results:
- Sample mean = \(2.08\)
- \(t = -22.65\), df = 11
- 95% upper bound = \(3.11\)
- p-value ≈ \(7.0 \times 10^{-11}\)

Conclusion: Extreme significance (\(p \approx 0\)) despite mild non-normality. Spray C is highly effective.

Example 3: CO₂ Uptake in Quebec Plants (Wilcoxon Test)

Scenario: Test if median CO₂ uptake differs from 25 μmol/m² (\(n = 21\)).

Hypotheses:
\(H_0\): M = 25
\(H_1\): M ≠ 25

Analysis (Wilcoxon Signed-Rank):

Results:

  • Test statistic \(V = 794.5\)
  • p-value ≈ \(1.85 \times 10^{-5}\) (with ties adjustment)

Conclusion: Reject \(H_0\). Nonparametric methods are essential here due to non-normality.