Stat 312: Lecture 08
Large sample confidence intervals
Moo K. Chung
[email protected]
September 27, 2004
1.
The sample size is inversely related to the width
of confidence interval. Example 7.4.
2.
Central Limit Theorem.
Let
X
1
,
· · ·
, X
n
be a
random sample with mean
μ
and variance
σ
2
. For
sufficiently large
n
,
Z
=
¯
X

μ
σ/
√
n
∼
N
(0
,
1)
.
3.
Let
X
1
,
· · ·
, X
n
be a random sample with mean
μ
. For sufficiently large
n
,
Z
=
¯
X

μ
S/
√
n
∼
N
(0
,
1)
where
S
is the sample standard deviation.
If
n
is sufficiently large, approximate
100(1

α
)%
confidence interval for
μ
is
¯
x
±
z
α/
2
s
√
n
,
where
s
is the sample standard deviation.
4.
General large sample confidence interval.
Sup
pose
ˆ
θ
is an unbiased estimator of some parame
ter
θ
, Then
100(1

α
)%
confidence interval is
ˆ
θ
+
z
α/
2
p
V
ˆ
θ.
In many applications,
V
ˆ
θ
is a function of
θ
which
makes computation of CI complicated. In this sit
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