Unit 5.5: Distributions of Differences Between Sample Means - Study Guide

KEY CONCEPTS

1. Distribution of Difference Between Sample Means

When comparing two independent groups, we analyze the difference between their sample means (x̄₁ - x̄₂).

Formula: μ(x̄₁-x̄₂) = μ₁ - μ₂

2. Standard Error of the Difference

Measures the variability of the difference between sample means.

Formula: SE(x̄₁-x̄₂) = √(s₁²/n₁ + s₂²/n₂)

Where:

- s₁, s₂ = sample standard deviations

- n₁, n₂ = sample sizes

3. Degrees of Freedom

For t-procedures with two samples, use the conservative approach:

df = smaller of (n₁ - 1) and (n₂ - 1)

4. Margin of Error

The margin of error accounts for sampling variability in estimating the difference.

Formula: E = t* × SE(x̄₁-x̄₂)

Where t* = critical value from t-distribution table based on desired confidence level and degrees of freedom

5. Confidence Interval

A confidence interval estimates the range for the true difference between population means.

Formula: (x̄₁ - x̄₂) ± E

Or: (x̄₁ - x̄₂) ± t* × SE(x̄₁-x̄₂)

6. Hypothesis Testing for Difference of Means

Test whether there is a significant difference between two population means.

Null Hypothesis (H₀): μ₁ - μ₂ = 0 (no difference)

Alternative Hypothesis (Hₐ):

- Two-sided: μ₁ - μ₂ ≠ 0

- One-sided: μ₁ - μ₂ > 0 or μ₁ - μ₂ < 0

INTERPRETING RESULTS

When interpreting confidence intervals:

- If the interval contains 0: No evidence of a significant difference

- If the interval is entirely positive: Evidence that μ₁ > μ₂

- If the interval is entirely negative: Evidence that μ₁ < μ₂

Remember: The confidence level represents the long-run proportion of intervals that would capture the true parameter if repeated many times.