Universal Hypothesis Testing Framework

🎯 ONE FRAMEWORK • INFINITE APPLICATIONS
This process works for ANY hypothesis about ANY parameter in ANY statistical test

Common Statistical Tests (All Use This Same Framework)

One-Sample Tests

Testing population mean, proportion, or variance against a specific value

Two-Sample Tests

Comparing means, proportions, or variances between two groups

Regression Tests

Testing slopes, intercepts, correlation coefficients

ANOVA

Comparing means across multiple groups

Chi-Square Tests

Testing independence or goodness-of-fit

And Many More...

F-tests, Mann-Whitney, Wilcoxon, etc.

Step 1: Set Up Your Hypothesis (Universal)

Choose any hypothesis about any parameter that makes a specific numerical claim:

H: parameter = specific value
Examples Across Different Tests:
  • One-sample mean: H: μ = 50
  • Two-sample means: H: μ₁ = μ₂ (or μ₁ - μ₂ = 0)
  • Proportions: H: p = 0.3 or H: p₁ = p₂
  • Regression slope: H: β₁ = 0
  • Correlation: H: ρ = 0
  • Variance: H: σ² = 25

Step 2: Collect Your Data (Universal)

Gather sample data and calculate your sample statistic relevant to your hypothesis.

Step 3: Calculate Test Statistic (Universal Pattern)

THE UNIVERSAL PATTERN:
Test Statistic = (Observed - Expected under H) / Standard Error
Examples Across Different Tests:
  • One-sample t-test: t = (x̄ - claimed mean) / (s/√n)
  • Two-sample t-test: t = (x̄₁ - x̄₂) / SE(x̄₁ - x̄₂)
  • Proportion test: z = (p̂ - claimed proportion) / √(claimed proportion × (1 - claimed proportion)/n)
  • Regression slope: t = (b₁ - 0) / SE(b₁)
  • Chi-square: χ² = Σ(Observed - Expected)² / Expected

Step 4: Compare to Appropriate Distribution (Test-Specific)

Distribution depends on test statistic:
  • t-statistic → t-distribution
  • z-statistic → Standard normal distribution
  • χ²-statistic → Chi-square distribution
  • F-statistic → F-distribution

Step 5: Look Up P-Value (Universal Process)

Use your test statistic and appropriate parameters to find the p-value in statistical tables or software.

Always the same meaning:
P-value = P(getting result this extreme or more | H is true)

Step 6: Interpret P-Value (Universal)

Universal Interpretation:
  • Low p-value: Data inconsistent with H → Evidence against the hypothesis
  • High p-value: Data consistent with H → No evidence against the hypothesis

Step 7: Make Decision (Universal)

Compare p-value to significance level α (commonly 0.05):

If p ≤ α:
Reject H
(statistically significant)
If p > α:
Fail to reject H
(not statistically significant)

What Stays Constant vs. What Changes

🟢 Always the Same

  • 7-step logical framework
  • Hypothesis structure (H: parameter = value)
  • Test statistic pattern: (Observed - Expected) / SE
  • P-value interpretation
  • Decision rule (compare p to α)
  • Statistical reasoning

🔴 Changes Between Tests

  • Specific parameter being tested
  • Exact formula for test statistic
  • Reference distribution (t, z, χ², F, etc.)
  • Degrees of freedom calculation
  • Standard error formula
  • Assumptions required

🎯 Key Insights About This Universal Framework

  1. Master Once, Use Forever: Learn this framework deeply and you understand ALL hypothesis tests
  2. Same Logic, Different Math: The reasoning is identical; only the calculations change
  3. P-values Always Mean the Same Thing: Probability of your result (or more extreme) if H were true
  4. Distribution Follows Statistic: Your test statistic determines which distribution to use
  5. Context Changes, Process Doesn't: Whether testing drug effectiveness or manufacturing quality, the steps are identical
  6. Software Does the Same Thing: Statistical software follows this exact same process automatically
🌟 Remember: Every Statistical Test is Just This Framework Applied
Different situations, same logical structure, same interpretation principles