Stat 312: Lecture 08
Large sample confidence intervals
Moo K. Chung
September 27, 2004
1. The sample size is inversely related to the width
of confidence interval. Example 7.4.
2. Central Limit Theorem. Let X1,··· , Xnbe a
random sample with mean µand variance σ2. For
sufficiently large n,
Z=¯
X−µ
σ/√n∼N(0,1).
3. Let X1,··· , Xnbe a random sample with mean
µ. For sufficiently large n,
Z=¯
X−µ
S/√n∼N(0,1)
where Sis the sample standard deviation. If n
is sufficiently large, approximate 100(1 −α)%
confidence interval for µis
¯x±zα/2
s
√n,
where sis the sample standard deviation.
4. General large sample confidence interval. Sup-
pose ˆ
θis an unbiased estimator of some parame-
ter θ, Then 100(1 −α)% confidence interval is
ˆ
θ+zα/2pVˆ
θ.
In many applications, Vˆ
θis a function of θwhich
makes computation of CI complicated. In this sit-
uation, we need to estimate Vˆ
θfurther.
Example. Toss n= 100 biased coins with
P(H) = p. Suppose you observe 38 heads. Con-
struct 95%CI of p.
> X<-rbinom(100,1,0.4)
> X
[1]1000000010110000
[17]0011010110101010
[33]1100001010000010
[49]0100001101011001
[65]0010010001001000
[81]1010011000100111
[97] 0 1 1 0
> sqrt(0.38*(1-0.38)/100)*1.96
[1] 0.09513574
> 0.38+0.095
[1] 0.475
> 0.38-0.095
[1] 0.285
5. One-sided confidence interval: An 100(1 −α)%
upper confidence bound for θis
θ < ¯x+zαpVˆ
θ
and a lower confidence bound for µis
θ > ¯x−zαpVˆ
θ.
Review Problems. Example 7.8, 7.10.