Transcript Chapter Nine - Proportions

CONFIDENCE
INTERVALS FOR
PROPORTIONS
Sec 9.3
THE PLAN;
FOR BUILDING A 95% CONFIDENCE INTERVAL

p̂
THE PLAN; FOR BUILDING A 95% CONFIDENCE INTERVAL

1.96(se)
p̂
1.96(se)
What do we use for (se)
WHAT TO USE FOR STANDARD ERROR,

The standard error for a sampling distribution of
proportions is normally;
p (1  p )
n

Sense we don’t know p we will use p̂ instead, where p̂ is
our sample proportion.
se 

p
ˆ (1  p
ˆ)
n
This will give us a strong approximation for the true
standard error.
THE PLAN; FOR BUILDING A 95% CONFIDENCE INTERVAL

1.96(se)
1.96(se)
p̂
pˆ  1.96(se)
pˆ  1.96(se)
THE PLAN;
FOR BUILDING A 95% CONFIDENCE
INTERVAL

In general: For proportional data

Our confidence interval is found by finding;
Sample
proportion
Z-score for
confidence level
pˆ  z(se)
or
p̂(1 - p̂)
p̂  z
n
Standard
Error
REMINDER ABOUT SAMPLE SIZE;


Since we are working with proportional
distributions we need to be sure that our sample
size is large enough for the central limit theorem
to give us a normal distribution to work with.
The sample size must be large enough so that;
npq  10
and
0.05 N  n
LETS TRY;
You are tasked with finding out
the proportion of American
households that will have
cranberries with their
Thanksgiving meal.
73.5% of Americans have
cranberries at
thanksgiving.
.735
Step one: Find a point estimate
1.96(se)
1.96(se)
Step two: Build a 95% confidence
interval
p̂
LETS TRY;
73.5%
Step One: Using a sample you
randomly survey 100 American
and find that 70 of plan on
having cranberries.
This gives you a point
estimate of .7000
1.96(se)
1.96(se)
p̂
0.7000
LETS TRY;
73.5%
Step Two: We need to find the
standard error in order to build
our intervals.
se 
se 
p
ˆ (1  p
ˆ)
n
(0.70)(0.30)
100
se 
se 
0.2100
100
0.0021
se  0.0458
1.96(se)
1.96(se)
0.7000
LETS TRY;
73.5%
Step Two: We need to find the
standard error in order to build
our intervals.
se 
se 
p
ˆ (1  p
ˆ)
n
(0.70)(0.30)
100
se 
se 
0.2100
100
0.0021
se  0.0458
1.96(.0458)
1.96(.0458)
0.7000
LETS TRY;
73.5%
Step Two: We need to find the
standard error in order to build
our intervals.
So our interval is;
1.96(.0458
0.0898 )
1.96(.0458
0.0898 )
0.7000  0.0898
(0.6102,0.7898)
The interval from 61.02% to
78.98%
Margin of
error
0.7000
LETS TRY;
73.5%
Therefore we are 95% confident that
between 61% and 79% of Americans
will be having cranberries with
their Thanksgiving meal.
0.6102
0.7898
YOU TRY;

A sample of 1500 dogs finds that only 1037 of
them are “altered”. Find a 95% confidence
interval for the population proportion of altered
dogs.
OTHER CONFIDENCE LEVELS:
We can find confidence intervals with other confidence levels. The
most common are 90%, 95%, and 99%.
The only difference is the z-score we will use needs to be changed to
model the new confidence level.
Here are the z-scores for these common confidence levels
YOU TRY;

A sample of 1500 dogs finds that only 1037 of
them are “altered”. Find a 99% confidence
interval for the population proportion of altered
dogs.
POSSIBILITIES;
Is it possible that our
confidence interval does not
contain the population
proportion?

1.96(se)
1.96(se)
Oh yes, and this will
happen 5% of the time with
a 95% confidence interval.
p̂