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,