Mathematics

3/3848–12FE Reference Handbook 10.6 · Engineering Probability and Statistics · p. 68 unit normal z; p. 77 table

Handbook formula

Standard Normal z-Score

z = \frac{x - μ}{σ}

z = (x − μ)/σ converts any normal to the unit normal (Handbook 10.6 p. 68). The p. 77 table columns are x, f(x), F(x) (area from −∞ to x), R(x) (area from x to ∞), 2R(x) (both tails), and W(x) (area between −x and x). F(−x) = 1 − F(x). A right-tail probability is R(z), not F(z) or 2R(z).

$x$: Observed value
$μ$: Population mean
$σ$: Population standard deviation

Step-by-step solved example

Concrete strength is N(4000 psi, 300 psi). What is P(X > 4450)?

Standard normal: z measures how many σ the sample sits from μ.

1. z-score
z = (4450 − 4000)/300 = 1.50.
$z = 1.50$
2. Tail area
On p. 77 at x = 1.50: F = 0.9332, R = 0.0668, 2R = 0.1336, W = 0.8664. The right tail is R(1.50) = 0.0668 ≈ 6.7%. Do not read F or 2R.

Answer: 6.7%

10 practice questions

0/10 correct

1.z for x = 18, μ = 12, σ = 3?

2.If z = −1 and σ = 5, μ = 20, then x =

3.Empirical rule: P(μ − σ < X < μ + σ) ≈

4.P(Z < 0) for standard normal is

5.A value 2σ below the mean has z =

6.Variance if σ = 4 is

7.Sample mean of 4, 6, 8, 10 is

8.Coefficient of variation is

9.For n = 25, σ = 10, SE of the mean is

10.95% of a normal population lies within about