Sampling Distribution of p̂ For a sample of size
n {n} n from a population with proportion
p {p} p , the sample proportion is:
p ^ = number of successes n \hat{p} = \frac{\text{number of successes}}{n} p ^ = n number of successes Sampling distribution of p ^ {\hat{p}} p ^
Shape: Approximately normal when:
n p ≥ 10 {np \geq 10} n p ≥ 10 n ( 1 − p ) ≥ 10 {n(1-p) \geq 10} n ( 1 − p ) ≥ 10 Center (mean):
μ p ^ = p \mu_{\hat{p}} = p μ p ^ = p
p ^ {\hat{p}} p ^ is an unbiased estimator of
p {p} p Spread (standard deviation):
σ p ^ = p ( 1 − p ) n \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} σ p ^ = n p ( 1 − p ) Summary: When
n p ≥ 10 {np \geq 10} n p ≥ 10 and
n ( 1 − p ) ≥ 10 {n(1-p) \geq 10} n ( 1 − p ) ≥ 10 :
p ^ ≈ N ( p , p ( 1 − p ) n ) \hat{p} \approx N\!\left(p,\ \sqrt{\frac{p(1-p)}{n}}\right) p ^ ≈ N ( p , n p ( 1 − p ) ) 
Stellar