z Confidence Interval for the Mean

x ˉ \pm z_{α /2} \frac{σ}{n}

Two-sided CI for μ when σ is known (or n is large). z_{α/2} is 1.645 (90%), 1.96 (95%), 2.58 (99%). The half-width z σ/√n shrinks as n grows. About 95% of such intervals cover μ in repeated sampling.

$x ˉ$: Sample mean
$z_{α /2}$: Standard-normal critical value
$σ$: Population standard deviation
$n$: Sample size

Step-by-step solved example

x̄ = 100, σ = 15, n = 25. Construct the 95% CI for μ.

x̄ ± z σ/√n as a band on the normal.

1. Standard error
σ/√n = 15/5 = 3.
$SE = 3$
2. Margin and interval
z_{0.025} = 1.96. ME = 5.88. CI: 100 ± 5.88 → (94.12, 105.88).
$100 \pm 5.88$

Answer: 94.1 to 105.9

10 practice questions

0/10 correct

1.σ = 20, n = 25. SE = σ/√n equals

2.For a 95% two-sided interval, z_{α/2} ≈

3.Increasing n makes the CI

4.x̄ = 50, z = 2, σ = 10, n = 16. Margin of error is

5.A 90% two-sided z is nearest

6.99% two-sided z_{α/2} is nearest

7.Quadrupling n multiplies SE by

8.When σ is known, a mean CI uses the

9.The half-width of the interval is

10.A 95% CI means that