When the population variance σ² is known, the 95% confidence interval for the population mean μ is:
X̄ − 1.96·σ/√n ≦ μ ≦ X̄ + 1.96·σ/√n
(For a 99% confidence interval, replace 1.96 with 2.58.)
Example ① If σ=10, n=25, and X̄=50, find the 95% confidence interval for μ.
Solution:Standard error σ/√n = 10/5 = 2.
50 ± 1.96·2 = 50 ± 3.92
46.08 ≦ μ ≦ 53.92
Solution:Standard error σ/√n = 10/5 = 2.
50 ± 1.96·2 = 50 ± 3.92
46.08 ≦ μ ≦ 53.92
Key Point
- “95% confidence” means that if the same procedure is repeated many times, the interval will contain μ in 95% of the cases
Practice Problems
- Find the 95% confidence interval when σ=4, n=64, and X̄=20.
Solutions and Explanations
- Solution:19.02 ≦ μ ≦ 20.98
σ/√n=0.5、20±0.98。