Sampling Distribution of x̄₁ − x̄₂ When comparing two population means μ 1 {\mu_1} μ 1 μ 2 {\mu_2} μ 2 X ˉ 1 − X ˉ 2 {\bar{X}_1 - \bar{X}_2} X ˉ 1 − X ˉ 2
Setup:
Group 1: population mean μ 1 {\mu_1} μ 1 σ 1 {\sigma_1} σ 1 n 1 {n_1} n 1 Group 2: population mean μ 2 {\mu_2} μ 2 σ 2 {\sigma_2} σ 2 n 2 {n_2} n 2 Groups are independent Sampling distribution of X ˉ 1 − X ˉ 2 {\bar{X}_1 - \bar{X}_2} X ˉ 1 − X ˉ 2
Shape: Approximately normal when each group satisfies its normality condition (normal population or n ≥ 30 {n \geq 30} n ≥ 30
Center:
μ X ˉ 1 − X ˉ 2 = μ 1 − μ 2 \mu_{\bar{X}_1 - \bar{X}_2} = \mu_1 - \mu_2 μ X ˉ 1 − X ˉ 2 = μ 1 − μ 2 Spread (Standard Error):
σ X ˉ 1 − X ˉ 2 = σ 1 2 n 1 + σ 2 2 n 2 \sigma_{\bar{X}_1 - \bar{X}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} σ X ˉ 1 − X ˉ 2 = n 1 σ 1 2 + n 2 σ 2 2 Summary: X ˉ 1 − X ˉ 2 ≈ N ( μ 1 − μ 2 , σ 1 2 n 1 + σ 2 2 n 2 ) {\bar{X}_1 - \bar{X}_2 \approx N\!\left(\mu_1 - \mu_2,\ \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}\right)} X ˉ 1 − X ˉ 2 ≈ N ( μ 1 − μ 2 , n 1 σ 1 2 + n 2 σ 2 2 )
Stellar