Confidence
Intervals for
Proportions
Chapter 19
Rate your confidence
0 - 100
• Name my age within 10 years?
•
within 5 years?
•
within 1 year?
• Shooting a basketball at a wading pool, will
make basket?
• Shooting the ball at a large trash can, will
make basket?
• Shooting the ball at a carnival, will make
basket?
What happens to your
confidence as the interval
gets smaller?
The larger your confidence,
the wider the interval.
Confidence intervals
• Are used to estimate the
unknown population proportion
• Formula:
estimate + margin of error
Confidence Interval Conditions:
Plausible Independence Condition: Is there any
reason to believe that the data values somehow
affect each other?
Randomization Condition: Was the data sampled
at random?
10% Condition: Was the sample less than 10%
of the population?
Success/Failure Condition: Do both np and nq ≥ 10?
If these conditions are satisfied, then I can use a
Normal model to find a one-proportion z-interval.
Margin of error
• Shows how accurate we believe our
estimate is
• The smaller the margin of error, the
more precise our estimate of the true
parameter
• Formula:
 critical
m  
 value
  standard deviation
  
  of the statistic



Confidence level
• Is the success rate of the method
used to construct the interval
• Using this method, ____% of the
time the intervals constructed will
contain the true population
parameter
What does it mean to be 95%
confident?
• 95% chance that p is contained in
the confidence interval
• The probability that the interval
contains p is 95%
• The method used to construct the
interval will produce intervals that
contain p 95% of the time.
Critical value (z*)
• Found from the confidence level
• The upper z-score with probability p lying to
its right under the standard normal curve
Confidence level
90%
95%
99%
z*=1.645
tail area z*=1.96
z*=2.576z*
.05
.025
.005
1.645
.05
.025 1.96
.005
2.576
Formula for Confidence interval:
CI  statistic  critical value SD of statistic
p̂  z *
 p 1  p  




n


Note: For confidence intervals, we DO NOT know p –
so we MUST substitute p-hat for p in both the SD &
when checking assumptions.

Steps for doing a confidence interval:
1) State the parameter
2) Assumptions –
1)SRS from population
2) Success/Failure Condition (Large enough sample)
np  10 and n(1-p)  10
3 10% rule – the sample is less then 10% of the
population
3) Calculate the interval
4) Write a statement about the interval in the
context of the problem.
Statement: (memorize!!)
We are ________% confident
that the true proportion context
lies within the interval ______
and ______.
A May 2000 Gallup Poll found that
38% of a random sample of 1012
adults said that they believe in
ghosts. Find a 95% confidence
interval for the true proportion of
adults who believe in ghosts.
State the parameter
p = the true proportion of adults who believe in ghosts
Justify the confidence interval needed (state
assumptions)
1) The sample must be random which is stated in the problem.
2) The sample should be large. Since np =1012(.38) =
384.56 > 10 and n(1-p) = 1012(.62) = 627.44 > 10, the
sample is large enough.
3) The sample should be less than 10% of the population.
The population should be at least 10,120 adults which we will
assume.
Since the conditions are satisfied a CI for proportions is
appropriate.
Calculate the confidence interval.
pˆ  .38
n  1012
95% CI

pˆ  z * 


 .38(.62) 
p 1  p  
  .38  1.96 

.35,.41






n
1012 


Explain the interval in the context of the problem.
We are 95% confident that the true proportion of
adults who believe in ghosts is between 35% and
41%.
If we wanted a 99% confidence interval for the
previous problem, what would change?
pˆ  .38
n  1012
99% CI

pˆ  z * 


 .38(.62) 
p 1  p  
  .38  2.576 

.34,.42






n
1012 


The interval is larger. Only the margin of error changed.
Another Gallop Poll is taken
To findtosample
size: the
in order
measure
proportion of adults
 pwho



1

p

m z *


approve of attempts
to
clone
n


humans.
What
size
is a
However,
since sample
we have not
yet taken
sample, we
know a +
p-hat
(or p)
necessary
todobenotwithin
0.04
oftothe
use!
true proportion of adults who
approve of attempts to clone
humans with a 95% Confidence
Interval?
What p-hat (p) do you use when
trying to find the sample size for a
given margin of error?
.1(.9) = .09
.2(.8) = .16
.3(.7) = .21
.4(.6) = .24
.5(.5) = .25
By using .5 for p-hat,
we are using the worstcase scenario and
using the largest SD in
our calculations.
Another Gallop Poll is taken in order to measure the
proportion of adults who approve of attempts to clone
humans. What sample size is necessary to be within +
0.04 of the true proportion of adults who approve of
attempts to clone humans with a 95% Confidence
 p 1  p  
Interval?
m z *



.04  1.96




n

.5.5  
n 
.5.5 
n
.04

1.96
2
.25
 .04 



n
 1.96 
n  600 .25  601
Use p-hat = .5
Divide by 1.96
Square both sides
Round up on sample size