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Module 9: Study Guide

Comprehensive reference for two-sample hypothesis testing

Module 9 Study Guide

Hypothesis Testing for Two Populations

Overview of Two-Sample Tests

This module covers hypothesis tests for comparing TWO populations. The key is identifying:

  1. Are you testing means or proportions?
  2. Are the samples independent or paired?
Test When to Use Test Statistic Distribution
Independent Two-Sample t-Test Compare means from two separate groups t t-distribution
Paired t-Test Compare means for same subjects (or matched pairs) t t-distribution
Two-Proportion z-Test Compare proportions from two groups z Normal (z)

1 Independent Two-Sample t-Test

Purpose

Compare means from two independent populations (μ₁ vs. μ₂)

Hypotheses

  • Two-tailed: H₀: μ₁ = μ₂, Hₐ: μ₁ ≠ μ₂
  • Right-tailed: H₀: μ₁ = μ₂, Hₐ: μ₁ > μ₂
  • Left-tailed: H₀: μ₁ = μ₂, Hₐ: μ₁ < μ₂

Conditions

  • Independent random samples
  • Both populations normally distributed OR both n ≥ 30
  • Independence within each sample

Test Statistic (Unpooled - Welch's t-test)

t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)

Degrees of freedom: Use Welch's approximation (use technology)

Test Statistic (Pooled Variance)

Pooled variance:

sp² = [(n₁ - 1)s₁² + (n₂ - 1)s₂²] / (n₁ + n₂ - 2)

Test statistic:

t = (x̄₁ - x̄₂) / (sp√(1/n₁ + 1/n₂))

Degrees of freedom: df = n₁ + n₂ - 2

When to Use Which?

  • Unpooled (Welch's): DEFAULT. Use when variances may be unequal or uncertain.
  • Pooled: Only when you have strong evidence that σ₁² = σ₂²

Confidence Interval for μ₁ - μ₂ (Unpooled)

(x̄₁ - x̄₂) ± t* × √(s₁²/n₁ + s₂²/n₂)

2 Paired t-Test

Purpose

Compare means when same subjects are measured twice OR matched pairs are used

Key Concept

A paired t-test is a one-sample t-test on the DIFFERENCES!
Calculate d = x₁ - x₂ for each pair, then test if mean difference = 0

When to Use

  • Before/after measurements on same subjects
  • Matched pairs (e.g., twins, siblings)
  • Same subject, two conditions (left vs. right hand)

Hypotheses

  • Two-tailed: H₀: μd = 0, Hₐ: μd ≠ 0
  • Right-tailed: H₀: μd = 0, Hₐ: μd > 0
  • Left-tailed: H₀: μd = 0, Hₐ: μd < 0

Process

  1. Calculate differences: d = x₁ - x₂ (or x₂ - x₁, be consistent)
  2. Find mean of differences: d̄ = Σd / n
  3. Find standard deviation of differences: sd
  4. Calculate test statistic

Test Statistic

t = (d̄ - μd₀) / (sd / √n)

where μd₀ = 0 (usually)
Degrees of freedom: df = n - 1 (n = number of PAIRS)

Conditions

  • Random sampling of pairs
  • Pairs are independent of each other
  • Differences are approximately normal OR n ≥ 30

Confidence Interval for μd

d̄ ± t* × (sd / √n)

Advantages of Paired Design

  • Controls for individual variability
  • More powerful (better at detecting real effects)
  • Requires fewer total subjects for same power

3 Two-Proportion z-Test

Purpose

Compare proportions from two independent populations (p₁ vs. p₂)

Hypotheses

  • Two-tailed: H₀: p₁ = p₂, Hₐ: p₁ ≠ p₂
  • Right-tailed: H₀: p₁ = p₂, Hₐ: p₁ > p₂
  • Left-tailed: H₀: p₁ = p₂, Hₐ: p₁ < p₂

Conditions

  • Independent random samples
  • Success-failure condition for BOTH samples:
    • n₁p̂₁ ≥ 10 and n₁(1 - p̂₁) ≥ 10
    • n₂p̂₂ ≥ 10 and n₂(1 - p̂₂) ≥ 10

Pooled Proportion (for Hypothesis Testing)

p̄ = (x₁ + x₂) / (n₁ + n₂)

or equivalently: p̄ = (n₁p̂₁ + n₂p̂₂) / (n₁ + n₂)

Test Statistic

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

Uses standard normal (z) distribution

Confidence Interval for p₁ - p₂ (NO pooling!)

(p̂₁ - p̂₂) ± z* × √[(p̂₁(1-p̂₁)/n₁) + (p̂₂(1-p̂₂)/n₂)]

Important Note

For hypothesis testing, use pooled proportion p̄
For confidence intervals, DO NOT pool - use individual p̂₁ and p̂₂

Decision Flowchart: Choosing the Right Test

Start Here ↓

QUESTION 1: Are you testing MEANS or PROPORTIONS?

If Testing MEANS
One sample? One-sample t-test (σ unknown) or z-test (σ known)
Two independent samples? Independent two-sample t-test
Same subjects twice or matched pairs? Paired t-test
If Testing PROPORTIONS
One sample? One-sample z-test for proportion
Two samples? Two-proportion z-test

Common Mistakes to Avoid

Mistake Correct Approach
Using independent test for paired data If same subjects measured twice → MUST use paired test
Confusing n for paired tests n = number of PAIRS (not total observations)
df = n - 1
Using pooled p̄ in confidence intervals Pooling ONLY for hypothesis tests, NOT for CIs
Forgetting to check conditions Always verify sample size, normality, independence
Not specifying direction of difference In paired tests, clearly state if d = Before - After or After - Before

Quick Reference Table

Test Hypotheses Test Statistic df
Independent t-test
(unpooled)		 H₀: μ₁ = μ₂
Hₐ: μ₁ ≠ μ₂ (or <, >)		 t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂) Welch's formula
(use technology)
Independent t-test
(pooled)		 H₀: μ₁ = μ₂
Hₐ: μ₁ ≠ μ₂ (or <, >)		 t = (x̄₁ - x̄₂) / (sp√(1/n₁ + 1/n₂)) n₁ + n₂ - 2
Paired t-test H₀: μd = 0
Hₐ: μd ≠ 0 (or <, >)		 t = (d̄ - 0) / (sd / √n) n - 1
Two-proportion z-test H₀: p₁ = p₂
Hₐ: p₁ ≠ p₂ (or <, >)		 z = (p̂₁ - p̂₂) / √[p̄(1-p̄)(1/n₁ + 1/n₂)] N/A (uses z)

Tips for Success

  • Identify the design first: Independent or paired? This is the most critical decision.
  • Check ALL conditions: Don't skip this step! Violating conditions invalidates results.
  • Use technology for df: Welch's approximation is complex - use calculators or software.
  • Interpret in context: Always relate statistical conclusions to the research question.
  • Remember the difference: Hypothesis tests pool proportions; confidence intervals don't.
  • Practice decision-making: Work through many scenarios until test selection is automatic.
  • Draw diagrams: Visualize the two distributions or before/after measurements.

Key Formulas Summary

Independent Two-Sample t-Test (Unpooled)

t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)

Pooled Variance

sp² = [(n₁-1)s₁² + (n₂-1)s₂²] / (n₁+n₂-2)

Independent Two-Sample t-Test (Pooled)

t = (x̄₁ - x̄₂) / (sp√(1/n₁ + 1/n₂))

Paired t-Test

t = (d̄ - 0) / (sd / √n)

Pooled Proportion

p̄ = (x₁ + x₂) / (n₁ + n₂)

Two-Proportion z-Test

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

You've got this! Review this guide, practice problems, and you'll master two-sample hypothesis testing!