Sampling Distribution of p̂₁ − p̂₂ When we compare two population proportions p 1 {p_1} p 1 p 2 {p_2} p 2 p ^ 1 − p ^ 2 {\hat{p}_1 - \hat{p}_2} p ^ 1 − p ^ 2
Setup:
Group 1: population proportion p 1 {p_1} p 1 n 1 {n_1} n 1 Group 2: population proportion p 2 {p_2} p 2 n 2 {n_2} n 2 Groups are independent Sampling distribution of p ^ 1 − p ^ 2 {\hat{p}_1 - \hat{p}_2} p ^ 1 − p ^ 2
Shape: Approximately normal when each group satisfies the large counts condition:
n 1 p 1 ≥ 10 , n 1 ( 1 − p 1 ) ≥ 10 , n 2 p 2 ≥ 10 , n 2 ( 1 − p 2 ) ≥ 10 n_1 p_1 \geq 10,\quad n_1(1-p_1) \geq 10,\quad n_2 p_2 \geq 10,\quad n_2(1-p_2) \geq 10 n 1 p 1 ≥ 10 , n 1 ( 1 − p 1 ) ≥ 10 , n 2 p 2 ≥ 10 , n 2 ( 1 − p 2 ) ≥ 10 Center (mean):
μ p ^ 1 − p ^ 2 = p 1 − p 2 \mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2 μ p ^ 1 − p ^ 2 = p 1 − p 2 Spread (standard deviation):
σ p ^ 1 − p ^ 2 = p 1 ( 1 − p 1 ) n 1 + p 2 ( 1 − p 2 ) n 2 \sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}} σ p ^ 1 − p ^ 2 = n 1 p 1 ( 1 − p 1 ) + n 2 p 2 ( 1 − p 2 ) 
Stellar