Confidence Intervals
with Proportions
Chapter 9
Suppose we wanted to estimate
the proportion of the earth that is
covered by water?
Point Estimate
• Use a single statistic based on
sample data to estimate a
population parameter
• Simplest approach
• But not always very precise due to
variation in the sampling
distribution
Confidence intervals
• Are used to estimate the
unknown population parameter
• Formula:
statistic + margin of error
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



Rate your confidence
0 - 100
• Guess 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?
Your confidence level decreases
with smaller intervals
%
%
%
%
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
Critical value (z*)
• Found from the confidence level
• The upper z-score with probability p lying to
its left 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
Confidence interval for a
But do we know the
population proportion:
population proportion?

pˆ  z *

pˆ1  ppˆ 
n
Statistic + Critical value × Standard deviation of the statistic
Margin of error
What are the steps for performing a
confidence interval?
1.) Identify the interval by name or
formula (CI for one-sample proportion)
2.) Assumptions
•
•
•
SRS of context
Approximate Normal distribution because
np > 10 & n(1-p) > 10
Population is at least 10n
3.) Calculations
4.) Conclusion (in context of problem)
Conclusion Statement:
(memorize!!)
We are ________% confident
that the true proportion context
is between ______ and ______.
Suppose we wanted to estimate the
proportion of the earth that is covered by
water? Let’s “throw” the earth around
and record the number of times that we
point to water or land. Repeat the
sampling for 50 trials. Calculate a 90%
confidence interval for the amount of
water on earth.
Calculate a 95% confidence interval
for the true proportion of water on the
earth.
What do
you
notice?
Calculate a 99% confidence interval
for the true proportion of water on the
earth.
• As the confidence level increases, do the intervals
generally get wider or more narrow? Explain.
wider
• As the sample size increases, do the intervals
generally get wider or more narrow? Explain.
More narrow
•When 100 confidence intervals are generated, why
are they all different?
Sampling variability
• If the confidence level selected is 90%, about how
many of 100 intervals will cover the true percentage of
orange balls? Will exactly this number of intervals
cover the true percentage each time 100 intervals are
created? Explain.
Assumptions:
Where are the last two
assumptions from?
• SRS of context
• Approximate Normal
distribution because
np > 10 & n(1-p) > 10
• Population is at least 10n
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 ghost.
Assumptions:
Step 1: check assumptions!
•Have an SRS of adults
•np =1012(.38) = 384.56 & n(1-p) = 1012(.62) = 627.44
Since both are greater than 10, the distribution can be
approximated by a normal curve
2:10,120.
make
•Population of adults isStep
at least
calculations
 .38(.62)
 p 1  p  
  .38  1.96
Pˆ  z * 



n
1012




  .35,.41 


Step 3: conclusion in context
We are 95% confident that the true proportion of
adults who believe in ghosts is between 35% and
41%.
The manager of the dairy section of a
large supermarket took a random
sample of 250 egg cartons and found
that 40 cartons had at least one broken
egg. Find a 90% confidence interval for
the true proportion of egg
cartons with at least one
broken egg.
Assumptions:
Step 1: check assumptions!
•Have an SRS of egg cartons
•np =250(.16) = 40 & n(1-p) = 250(.84) = 210 Since
both are greater than 10, the distribution can be
approximated by a normal curve
2: make
•Population of cartons Step
is at least
2500.
calculations
 .16(.84) 
  .122,.198
.16  1.645


250


Step 3: conclusion in context
We are 90% confident that the true proportion of egg
cartons with at least one broken egg is between
12.2% and 19.8%.
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
Remember
p-hat (p)that,
do you
in a binomial
use when
distribution,
withfor
thea
trying
to findthe
thehistogram
sample size
largest
deviation
givenstandard
margin of
error? was the one
for probability of success of 0.5.
.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




.5.5  
n 
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