Introduction to Probability and Statistics for Engineers and Scientists

242 Chapter 7: Parameter Estimation

Since

x=

81

9

= 9

It follows, under the assumption that the values received are independent, that a 95 percent
confidence interval forμis

(

9 −1.96

σ

3

,9+1.96

σ

3

)

=(7.69, 10.31)

Hence, we are “95 percent confident” that the true message value lies between 7.69 and
10.31. ■

The interval in Equation 7.3.1 is called atwo-sided confidence interval. Sometimes,
however, we are interested in determining a value so that we can assert with, say, 95
percent confidence, thatμis at least as large as that value.
To determine such a value, note that ifZis a standard normal random variable then

P{Z<1.645}=.95

As a result,

P

{

√

n

(X−μ)

σ

<1.645

}

=.95

or

P

{

X−1.645

σ

√

n

<μ

}

=.95

Thus, a 95 percentone-sided upper confidence intervalforμis

(

x−1.645

σ

√

n

,∞

)

wherexis the observed value of the sample mean.
Aone-sided lower confidence intervalis obtained similarly; when the observed value of
the sample mean isx, then the 95 percent one-sided lower confidence interval forμis

(

−∞,x+1.645

σ

√

n

)

EXAMPLE 7.3b Determine the upper and lower 95 percent confidence interval estimates
ofμin Example 7.3a.