Mathematics

11/3848–12FE Reference Handbook 10.6 · Engineering Probability and Statistics · p. 76 z and Zα/2 table

Handbook formula

z Confidence Interval for the Mean

x ˉ \pm z_{α /2} \frac{σ}{n}

When σ is known, p. 76 prints z = (X̄ − μ)/(σ/√n). Rearrangement is the one-sample interval x̄ ± z_{α/2} σ/√n. The boxed interval on that page is for the difference of two means, not one mean. Printed Zα/2 values: 90% 1.6449, 95% 1.9600, 99% 2.5758 (also 80% 1.2816, 96% 2.0537, 98% 2.3263).

$x ˉ$: 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