Stat 312: Lecture 6
Confidence Intervals II.
Moo K. Chung
February 6, 2003
Concepts
1. Let Xi∼N(µ, σ2)with known σ2and unknown
µ.100(1 −α)% confidence interval for µis.
ˆµL= ¯x−zα/2·σ/√n, ˆµU= ¯x+zα/2·σ/√n.
2. The sample size is inversely related to the width of
confidence interval.
3. Central Limit Theorem. Let X1,··· ,Xnbe a ran-
dom sample with mean µand variance σ2. For
large n,
Z=¯
X−µ
σ/√n∼N(0,1).
4. If nis sufficiently large, approximate 100(1 −α)%
confidence interval for µis ¯x±zα/2s/√n, where
sis the sample standard deviation.
In-class problems
Continuing Exercise 6.25, construct 98% confidence in-
terval.
Example 7.4. Response time ∼N(µ, σ 2), σ = 25. Find
the sample size nthat ensures 95%CI with a width of
10.
Example 7.6. Alternating current (AC) voltage data
> data(xmp07.06)
> attach(xmp07.06)
> str(xmp07.06)
‘data.frame’:48 obs. of 1 variable:
$ C1: int 62 50 53 57 ...
> boxplot(C1)
> mean(C1)
[1] 54.70833
> sd(C1)
40 45 50 55 60 65
[1] 5.230672
>mean(C1)-qnorm(0.975)*sd(C1)/sqrt(length(C1))
[1] 53.2286
>mean(C1)+qnorm(0.975)*sd(C1)/sqrt(length(C1))
[1] 56.18807
Ex. Toss n= 100 biased coins with P(H) = p. Sup-
pose you observe 38 heads. Construct 95%CI of p.
> X<-rbinom(100,1,0.4)
> X
[1] 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0
[17] 0 0 1 1 0 1 0 1 1 0 1 0 1 0 1 0
[33] 1 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0
[49] 0 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1
[65] 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0
[81] 1 0 1 0 0 1 1 0 0 0 1 0 0 1 1 1
[97] 0 1 1 0
> sd(X)
[1] 0.4878317
> sqrt(0.38*(1-0.38)/(100-1))
[1] 0.04878317
> 0.38+1.96*0.049/sqrt(100)
[1] 0.389604
> 0.38-1.96*0.049/sqrt(100)
[1] 0.370396
Self-study problems
Example 7.8., Exercise 7.13., 7.19., 7.25.
In the above coin tossing example, check if ¯
Xis MLE.