For population proportion p and sample proportion p̂, the 95% confidence interval for p is:
p̂ − 1.96·√(p̂(1−p̂)/n) ≦ p ≦ p̂ + 1.96·√(p̂(1−p̂)/n)
Example ① Of n=400 people, 240 support a proposal. Find the 95% confidence interval for the population approval rate p.
Solution:p̂ = 240/400 = 0.6.
√(0.6·0.4/400) = √0.0006 ≈ 0.0245.
0.6 ± 1.96·0.0245 = 0.6 ± 0.048
0.552 ≦ p ≦ 0.648
Solution:p̂ = 240/400 = 0.6.
√(0.6·0.4/400) = √0.0006 ≈ 0.0245.
0.6 ± 1.96·0.0245 = 0.6 ± 0.048
0.552 ≦ p ≦ 0.648
Key Point
- Opinion polls’ “approval rating ± an error of ○○%” uses this idea
Practice Problems
- When n=100 and p̂=0.5, find the half-width of the 95% confidence interval.
Solutions and Explanations
- Solution:approximately 0.098
1.96·√(0.5·0.5/100)=1.96·0.05=0.098.