Introduction to Probability and Statistics for Engineers and Scientists

7.5Approximate Confidence Interval for the Mean of a Bernoulli Random Variable 261

TABLE 7.2 100(1−σ)Percent Confidence Intervals forμ 1 −μ 2

X 1 ,...,Xn∼N(μ 1 ,σ 12 )

Y 1 ,...,Ym∼N(μ 2 ,σ 22 )

X=

∑n

i= 1

Xi/n, S^21 =

∑n

i= 1

(Xi−X)^2 /(n−1)

Y=

∑m

i= 1

Yi/n, S 22 =

∑m

i= 1

(Yi−Y)^2 /(m−1)

Assumption Confidence Interval

σ 1 ,σ 2 known X−Y±zα/2

√

σ 12 /n+σ 22 /m

σ 1 ,σ 2 unknown but equal X−Y±tα/2,n+m− 2

√(

1

n

+^1

m

)(n−1)S 2

1 +(m−1)S

2

2

n+m− 2

Assumption Lower Confidence Interval

σ 1 ,σ 2 known (−∞,X−Y+zα

√

σ 12 /n+σ 22 /m)

σ 1 ,σ 2 unknown but equal



−∞,X−Y+tα,n+m− 2

√(

1

n+

1

m

)(n−1)S 2

1 +(m−1)S 22

n+m− 2





Note: Upper confidence intervals forμ 1 −μ 2 are obtained from lower confidence intervals forμ 2 −μ 1.

the normal approximation to the binomial thatXis approximately normally distributed
with meannpand variancenp(1−p). Hence,

X−np

√

np(1−p)

·

∼N(0, 1) (7.5.1)

where∼· means “is approximately distributed as.” Therefore, for anyα∈(0, 1),

P

{

−zα/2<

X−np

√

np(1−p)

<zα/2

}

≈ 1 −α

and so ifXis observed to equalx, then an approximate 100(1−α) percent confidence
regionforpis
{
p:−zα/2<

x−np

√

np(1−p)

<zα/2

}

The foregoing region, however, is not an interval. To obtain a confidenceintervalfor
p, letpˆ=X/nbe the fraction of the items that meet the standards. From Example 7.2a,