LECTURE UNIT 5
Confidence Intervals
(application of the Central Limit
Theorem)
Sections 5.1, 5.2
Introduction and Confidence Intervals
for Proportions
Lecture Unit 5 Objectives
1. Construct confidence intervals for
population proportions and population
means based on the information contained
in a single sample.
2. Perform hypothesis tests for population
proportions and populations means based
on the information contained in a single
sample.
Concepts of Estimation
• The objective of estimation is to estimate
the unknown value of a population
parameter, like a population proportion p,
on the basis of a sample statistic calculated
from sample data.
 e.g., NCSU student affairs office may want
to estimate the proportion of students that
want more campus weekend activities
• There are two types of estimates
– Point Estimate
– Interval estimate
Point Estimate of p
x
• p^ = n
, the sample proportion of x
successes in a sample of size n, is the best
point estimate of the unknown value of the
population proportion p
Example: Estimating an
unknown population proportion p
• Is Sidney Lowe’s departure good or bad
for State's men's basketball team?
(Technician poll; not scientifically valid!!)
• In a sample of 1000 students, 590 say that
Lowe’s departure is good for the bb team.
• ^p = 590/1000 = .59 is the point estimate of
the unknown population proportion p that
think Lowe’s departure is good.
Shortcoming of Point Estimates
• ^p = 590/1000 = .59, best estimate of
population proportion p
BUT
How good is this best estimate?
No measure of reliability
Another type of estimate
Interval Estimator
A confidence interval is a range (or an
interval) of values used to estimate the
unknown value of a population parameter .
http://abcnews.go.com/US/PollVault/
Tool for Constructing Confidence
Intervals: The Central Limit
Theorem
• If a random sample of n observations is
selected from a population (any population),
and x “successes” are observed, then when n
is sufficiently large, the sampling distribution
of the sample proportion p will be
approximately a normal distribution.
• (n is large when np ≥ 10 and nq ≥ 10).
Standard Normal
P(-1.96  z  1.96) =. 95
95% Confidence Interval for p
x
ˆ
Use p  to constructa 95% confidenceinterval
n
for p :
ˆ (1  pˆ )
ˆ (1  pˆ )
p
p
)
, pˆ  1.96
( pˆ  1.96
n
n
written
ˆ (1  pˆ )
p
pˆ  1.96
n
Sampling distribution
model for p̂
Confidence level
.95
pq
p  1.96
n
p
pq
p  1.96
n
ˆ will be in this interval
95% of the time p
Therefore, the interval

pq
pq 
, pˆ  1.96
 pˆ 1.96

n
n 

will "capture" p 95% of the time
Example (Gallup Polls)
Voter preference polls typically sample
approximat ely 1600 voters; suppose pˆ  .52.
Then if we desire a 95% confidence interval
for p we calculate
pˆ qˆ
(.52)(. 48)
pˆ  1.96
 .52  1.96
n
1600
 .52  .024  (.496, .544)
http://abcnews.go.com/US/PollVault/story?id=1
45373&page=1
• Confidence intervals other than
95% confidence intervals are also
used
Standard Normal
98% Confidence Intervals
For p :

pˆ qˆ
pˆ qˆ 
 pˆ  2.33

ˆ  2.33
,
p


n
n


written
pˆ qˆ
pˆ  2.33
n
Four Commonly Used
Confidence Levels
Confidence Level
.90
.95
.98
.99
Multiplier
1.645
1.96
2.33
2.58
Medication side effects (confidence
interval for p)
Arthritis is a painful, chronic inflammation of the joints.
An experiment on the side effects of pain relievers
examined arthritis patients to find the proportion of
patients who suffer side effects.
What are some side effects of ibuprofen?
Serious side effects (seek medical attention immediately):
Allergic reaction (difficulty breathing, swelling, or hives),
Muscle cramps, numbness, or tingling,
Ulcers (open sores) in the mouth,
Rapid weight gain (fluid retention),
Seizures,
Black, bloody, or tarry stools,
Blood in your urine or vomit,
Decreased hearing or ringing in the ears,
Jaundice (yellowing of the skin or eyes), or
Abdominal cramping, indigestion, or heartburn,
Less serious side effects (discuss with your doctor):
Dizziness or headache,
Nausea, gaseousness, diarrhea, or constipation,
Depression,
Fatigue or weakness,
Dry mouth, or
Irregular menstrual periods
440 subjects with chronic arthritis were given ibuprofen for pain relief;
23 subjects suffered from adverse side effects.
Calculate a 90% confidence interval for the population proportion p of
arthritis patients who suffer some “adverse symptoms.”
ˆ  z*
p
ˆˆ
pq
n
What is the sample proportion p̂ ?
pˆ 
23
 0.052
440
For a 90% confidence level, z* = 1.645.
pˆ  z *
ˆˆ
pq
n
.052(1  .052)
440
.052  1.645(0.011)
.052  1.645
90% CI for p :
0.052  0.018  (.034,.070)
.052  .018
 We are 90% confident that the interval (.034, .070) contains the true
proportion of arthritis patients that experience some adverse symptoms when
taking ibuprofen.
Example: impact of sample size
90 % CI : .052  1.645
.052 (1  .052 )
 .052  .018  (.034 ,.070 )
440
90 % CI : .052  1.645
.052 (1  .052 )
 .052  .007  (.045 ,.059 )
1000
n=440: width of 90% CI: 2*.018 = .036
n=1000: width of 90% CI: 2*.007=.014
When the sample size is increased, the 90% CI is narrower
IMPORTANT
• The higher the confidence level, the wider
the interval
• Increasing the sample size n will make a
confidence interval with the same
confidence level narrower (i.e., more
precise)
Example
• Find a 95% confidence interval for p, the
proportion of NCSU students that strongly
favor the current lottery system for
awarding tickets to football and men’s
basketball games, if a random sample of
1000 students found that 50 strongly favor
the current system.
Example (solution)
Interpreting Confidence Intervals
• Previous example: .05±.014(.036, .064)
• Correct: We are 95% confident that the interval from
.036 to .064 actually does contain the true value of p.
This means that if we were to select many different
samples of size 1000 and construct a 95% CI from each
sample, 95% of the resulting intervals would contain the
value of the population proportion p. (.036, .064) is one
such interval. (Note that 95% refers to the procedure we
used to construct the interval; it does not refer to the
population proportion p)
• Wrong: There is a 95% chance that the population
proportion p falls between .036 and .064. (Note that p is
not random, it is a fixed but unknown number)
Confidence Interval Interpretation
7 x 5 = 42
She achieved 95% accuracy, so she would
answer 95 out of 100 correctly, say.
95% confidence intervals
behave the same way. An
individual confidence
interval either captures p or it
doesn’t…
Is there a 95% chance that
7x5=42?
…but in a group of many 95%
confidence intervals, about
95% of them will capture p.