Page 296 - Applied statistics and probability for engineers

274 Chapter 8/Statistical intervals for a single sample

In our problem situation, because Z = ( X − μ) σ ( / n) has a standard normal distribution,

we may write
⎧ X − μ ⎫
P − z α/2 Ð ≤ z α/2 ⎬ = − α1
⎨
⎩ σ / n ⎭
Now manipulate the quantities inside the brackets by (1) multiplying through by σ n, (2)
subtracting X from each term, and (3) multiplying through by −1. This results in
⎧ σ σ ⎫
P X − z α/2 Ð μ ≤ X+ z α/2 ⎬ = − ≠1 (8-4)
⎨
⎩ n n ⎭
This is a random interval because the end-points X ± Z α σ n involve the random vari-
/2
able X. From consideration of Equation 8-4, the lower and upper end-points or limits of the
inequalities in Equation 8-4 are the lower- and upper-coni dence limits L and U, respectively.
This leads to the following dei nition.

Coni dence Interval
on the Mean, Variance If x is the sample mean of a random sample of size n from a normal population with
2
Known known variance σ , a 100 1( − ≠)% CI on μ is given by
x − z / È 2 n ≤ μ ≤ x + z / σ n (8-5)
α
α 2
where z /α 2 is the upper 100α / 2 percentage point of the standard normal distribution.

The development of this CI assumed that we are sampling from a normal population. The CI
is quite robust to this assumption. That is, moderate departures from normality are of no seri-
ous concern. From a practical viewpoint, this implies that an advertised 95% CI might have

actual conidence of 93% or 94%.
Example 8-1 Metallic Material Transition ASTM Standard E23 dei nes standard test methods for notched
bar impact testing of metallic materials. The Charpy V-notch (CVN) technique measures impact
energy and is often used to determine whether or not a material experiences a ductile-to-brittle transition with decreas-
ing temperature. Ten measurements of impact energy (J ) on specimens of A238 steel cut at 60ºC are as follows: 64.1,
64.7, 64.5, 64.6, 64.5, 64.3, 64.6, 64.8, 64.2, and 64.3. Assume that impact energy is normally distributed with σ = 1J.
We want to ind a 95% CI for μ, the mean impact energy. The required quantities are z α = z 0.025 = 1.96, n = 10, σ = 1,

/ 2
and x = 64 46. . The resulting 95% CI is found from Equation 8-5 as follows:
σ σ
x − z α/2 ≤ μ Ð x + z α/2
n n
+
.
.
.
− .
64 46 1 96 1 ≤ μ ≤ 64 46 1 96 1
10 10
63 84 ≤ μ ≤ 65 08
.
.
Practical Interpretation: Based on the sample data, a range of highly plausible values for mean impact energy for
A238 steel at 60°C is 63 84 ≤ μ ≤ 65 08J.
.
.
J
Interpreting a Confi dence Interval
How does one interpret a coni dence interval? In the impact energy estimation problem in
.
Example 8-1, the 95% CI is 63 84 ≤ μ ≤ 65.08, so it is tempting to conclude that μ is within