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Module 9 Quick Reference Card

One-page cheat sheet for two-sample hypothesis testing

Module 9: Two-Sample Hypothesis Testing

Quick Reference Card

Which Test Should I Use?

Step 1: Means or Proportions?
  → MEANS → Go to Step 2
  → PROPORTIONS → Use Two-Proportion z-Test

Step 2 (for Means): Independent or Paired?
  → INDEPENDENT (two separate groups) → Independent Two-Sample t-Test
  → PAIRED (same subjects twice or matched pairs) → Paired t-Test

Independent Two-Sample t-Test

Use when: Comparing means from two separate groups

Hypotheses:

  • H₀: μ₁ = μ₂
  • Hₐ: μ₁ ≠ μ₂ (or <, >)

Test Statistic (Unpooled):

t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)

df: Welch's approximation (use technology)

Test Statistic (Pooled):

sp² = [(n₁-1)s₁² + (n₂-1)s₂²]/(n₁+n₂-2)
t = (x̄₁ - x̄₂) / (sp√(1/n₁ + 1/n₂))

df = n₁ + n₂ - 2

Conditions:

  • Independent random samples
  • Both normal OR both n ≥ 30

Paired t-Test

Use when: Same subjects measured twice or matched pairs

Hypotheses:

  • H₀: μd = 0
  • Hₐ: μd ≠ 0 (or <, >)

Process:

  1. Calculate d = x₁ - x₂ for each pair
  2. Find d̄ and sd
  3. Calculate test statistic

Test Statistic:

t = (d̄ - 0) / (sd / √n)

df = n - 1 (n = # of pairs)

Conditions:

  • Random pairs
  • Pairs independent
  • Differences normal OR n ≥ 30

Why use paired?

Controls individual variability → more power!

Two-Proportion z-Test

Use when: Comparing proportions from two independent groups

Hypotheses:

  • H₀: p₁ = p₂
  • Hₐ: p₁ ≠ p₂ (or <, >)

Pooled Proportion:

p̄ = (x₁+x₂)/(n₁+n₂)

(Use for hypothesis test only!)

Test Statistic:

z = (p̂₁-p̂₂) / √[p̄(1-p̄)(1/n₁+1/n₂)]

Use z-distribution

Conditions:

  • Independent samples
  • n₁p̂₁ ≥ 10, n₁(1-p̂₁) ≥ 10, n₂p̂₂ ≥ 10, n₂(1-p̂₂) ≥ 10

Important:

For confidence intervals: DON'T pool! Use (p̂₁-p̂₂) ± z*√[p̂₁(1-p̂₁)/n₁ + p̂₂(1-p̂₂)/n₂]

Comparison Table

Test Data Type Samples Distribution df
Indep. t-Test Means 2 independent t Welch's or n₁+n₂-2
Paired t-Test Means Paired t n - 1
Two-Prop z-Test Proportions 2 independent z N/A

Common Mistakes

WRONG

  • Independent test for before/after data
  • df = 2n for paired (should be n-1)
  • Pooling for confidence intervals
  • Skipping condition checks

CORRECT

  • Paired test for same subjects
  • df = n-1 where n = # of pairs
  • Pool only for hypothesis tests
  • Always verify conditions first

Common Critical Values

t-distribution (large df ≥ 30):

  • α = 0.10 (two-tail): ±1.697
  • α = 0.05 (two-tail): ±2.042
  • α = 0.01 (two-tail): ±2.750

z-distribution:

  • α = 0.10 (two-tail): ±1.645
  • α = 0.05 (two-tail): ±1.96
  • α = 0.01 (two-tail): ±2.576

For one-tailed tests, use positive value only

Quick Tips for Success

  • Identify data structure FIRST: Independent or paired?
  • Check conditions: Don't skip! Invalid conditions = invalid results
  • Use technology for complex calculations: Welch's df, p-values
  • State direction of difference: Is d = Before - After or After - Before?
  • Interpret in context: Always relate to the research question
  • Remember: Statistical significance ≠ practical significance!

Keep this card handy during exams and practice!