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.