Introduction to Probability and Statistics for Engineers and Scientists

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.