7.3Interval Estimates 243
SOLUTION Since
1.645
σ
√
n
=
3.29
3
=1.097
the 95 percent upper confidence interval is
(9−1.097,∞)=(7.903,∞)
and the 95 percent lower confidence interval is
(−∞,9+1.097)=(−∞, 10.097) ■
We can also obtain confidence intervals of any specified level of confidence. To do so,
recall thatzαis such that
P{Z>zα}=α
whenZis a standard normal random variable. But this implies (see Figure 7.1) that for
anyα
P{−zα/2<Z<zα/2}= 1 −α
As a result, we see that
P
{
−zα/2<
√
n
(X−μ)
σ
<zα/2
}
= 1 −α
or
P
{
−zα/2
σ
√
n
<X−μ<zα/2
σ
√
n
}
= 1 −α
or
P
{
−zα/2
σ
√
n
<μ−X<zα/2
σ
√
n
}
= 1 −α
That is,
P
{
X−zα/2
σ
√
n
<μ<X+zα/2
σ
√
n
}
= 1 −α
Area = –
−za/2 0 za/2
a
Area = – 2
a
2
FIGURE 7.1 P{−zα/2<Z<zα/2}= 1 −α.