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250 CHAPTER 8 STATISTICAL INTERVALS FOR A SINGLE SAMPLE

The end-points or bounds l and u are called the lower- and upper-confidence limits, respec-
tively, and 1 is called the confidence coefficient.
In our problem situation, because Z 1X 2 1 1n2 has a standard normal distribu-
tion, we may write

X
P e z 2 z 2 f 1
1n

Now manipulate the quantities inside the brackets by (1) multiplying through by 1n , (2)
subtracting X from each term, and (3) multiplying through by 1. This results in

P eX z 2 X z 2 f 1 (8-6)
1n 1n
From consideration of Equation 8-4, the lower and upper limits of the inequalities in Equation
8-6 are the lower- and upper-confidence limits L and U, respectively. This leads to the fol-
lowing definition.

Definition
x
If is the sample mean of a random sample of size n from a normal population with
2
known variance , a 100(1 )% CI on is given by
x z 2 1n x z 2 1n (8-7)

where z 2 is the upper 100 2 percentage point of the standard normal distribution.

EXAMPLE 8-1 ASTM Standard E23 defines 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 decreasing
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 1J. We want to find a 95% CI for , the mean
z 1.96, n 10, 1, and
impact energy. The required quantities are z 2 0.025
x 64.46. The resulting 95% CI is found from Equation 8-7 as follows:

x z 2 x z 2
1n 1n
1 1
64.46 1.96 64.46 1.96
110 110
63.84 65.08

That is, based on the sample data, a range of highly plausible vaules for mean impact energy
for A238 steel at 60°C is 63.84J 65.08J.

Interpreting a Confidence Interval
How does one interpret a confidence interval? In the impact energy estimation problem in
Example 8-1 the 95% CI is 63.84 65.08, so it is tempting to conclude that is within