Stat 312: Lecture 7
Confidence Intervals III.
Moo K. Chung
February 11, 2003
Concepts
1. If nis sufficiently large, approximate
100(1 −α)% confidence interval for µis
¯x±zα/2s/√n, where sis the sample stan-
dard deviation.
2. Let pdenote the proportion of an individual
with a specified property. 100(1 −α)% CI
for a population proportion pis
ˆp±zα/2qˆpˆq/n.
3. One-sided confidence interval: An upper
confidence bound for µis
µ < ¯x+zαs/√n
and a lower confidence bound for µis
µ > ¯x−zαs/√n.
In-class problems
Ex. Toss n= 100 biased coins with P(H) =
p. Suppose you observe 38 heads. Construct
95% CI of p.rbinom(n,1,p) will generate a
Bernoulli random sample of size nwith P(Xi=
1) = 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
> 0.38+1.96*0.49/sqrt(100)
[1] 0.47604
> 0.38-1.96*0.49/sqrt(100)
[1] 0.28396
Exercise 7.23. Use Concept 2.
When 37 helmets were subjected to a certain im-
pact test, 24 showed damage. Let pdenote the
proportion of helmets that would show damage
under the test. Find a 99% CI for p.
Self-study problems
Example 7.8., 7.8. Use Concept 2.