Introduction to Probability and Statistics for Engineers and Scientists

244 Chapter 7: Parameter Estimation

Hence, a 100(1−α) percent two-sided confidence interval forμis

(

x−zα/2

σ

√

n

, x+zα/2

σ

√

n

)

wherexis the observed sample mean.

Similarly, knowing thatZ =

√
n(X−σμ)is a standard normal random variable, along
with the identities

P{Z>zα}=α

and

P{Z<−zα}=α

results in one-sided confidence intervals of any desired level of confidence. Specifically, we
obtain that
(
x−zα

σ

√

n

,∞

)

and
(
−∞,x+zα

σ

√

n

)

are, respectively, 100(1−α) percent one-sided upper and 100(1−α) percent one-sided
lower confidence intervals forμ.

EXAMPLE 7.3c Use the data of Example 7.3a to obtain a 99 percent confidence interval
estimate ofμ, along with 99 percent one-sided upper and lower intervals.

SOLUTION Sincez.005=2.58, and

2.58

α

√

n

=

5.16

3

=1.72

it follows that a 99 percent confidence interval forμis

9 ±1.72

That is, the 99 percent confidence interval estimate is (7.28, 10.72).
Also, sincez.01=2.33, a 99 percent upper confidence interval is

(9−2.33(2/3),∞)=(7.447,∞)

Similarly, a 99 percent lower confidence interval is

(−∞,9+2.33(2/3))=(−∞, 10.553) ■