Sampling Distribution Models
Population
Parameter: p
Population – all items
of interest.
Random
selection
Sample – a
few items from
the population.
Inference
Sample
Statistic: p̂
1
Sampling Distribution of p̂

Shape: Approximately Normal
Center: The mean is p.
 Spread: The standard deviation
is
p1  p 

n
2
Sampling Distribution of p̂

Conditions:
10% Condition: The size of the
sample should be less than 10%
of the size of the population.
 Success/Failure Condition: np
and n(1 – p) should both be
greater than 10.

3
68 – 95 – 99.7 Rule
p3
pq
n
p2
pq
n
p
pq
n
p
p 1
pq
n
p2
pq
n
p3
pq
n
4
Probability

If the population proportion, p,
is known, we can find the
probability or chance that p̂
takes on certain values using a
normal model.
5
Inference
In practice the population
parameter, p, is not known and
we would like to use a sample
to tell us something about p.
 Use the sample proportion, p̂ ,
to make inferences about the
population proportion p.

6
Example
Population: All adults in the U.S.
 Parameter: Proportion of all
adults in the U.S. who think
that abortion should be legal.
Unknown!

7
Example


Sample: 1,772 randomly selected
registered voters nationwide.
Quinnipiac University Poll, Jan. 30 –
Feb. 4, 2013.
Statistic: 992 of the 1,772
registered voters in the sample
(56%) answered that abortion
should be legal.
8
68-95-99.7 Rule

95% of the time the sample
proportion, p̂ , will be between
p(1  p)
p(1  p)
p2
and p  2
n
n
9
68-95-99.7 Rule

95% of the time the sample
proportion, p̂ , will be within
p(1  p)
2
n
two standard deviations of p.
10
Standard Deviation

Because p, the population
proportion is not known, the
standard deviation
SD( pˆ ) 
is also unknown.
p(1  p)
n
11
Standard Error
Substitute p̂ as our estimate
(best guess) of p.
 The standard error of p̂ is:

SE( pˆ ) 
pˆ (1  pˆ )
n
12

About 95% of the time the
sample proportion, p̂ , will be
within
pˆ (1  pˆ )
2SE( pˆ )  2
n
two standard errors of p.
13

About 95% of the time the
population proportion, p, will be
within
pˆ (1  pˆ )
2SE( pˆ )  2
n
two standard errors of p̂ .
14
Confidence Interval for p

We are 95% confident that p
will fall between
pˆ (1  pˆ )
pˆ (1  pˆ )
pˆ  2
and pˆ  2
n
n
15
Example
pˆ  0.56
pˆ (1  pˆ )
0.560.44

 0.012
n
1772
0.56  20.012 to 0.56  20.012
0.536 to 0.584
16
Interpretation

We are 95% confident that the
population proportion of all
adults in the U.S. who would
answer that abortion should be
legal is between 53.6% and
58.4%.
17
Interpretation
Plausible values for the
population parameter p.
 95% confidence in the process
that produced this interval.

18
95% Confidence

If one were to repeatedly
sample at random 1,772 adults
and compute a 95% confidence
interval for each sample, 95%
of the intervals produced would
contain, or capture, the
population proportion p.
19
Simulation
http://statweb.calpoly.edu/chanc
e/applets/Confsim/Confsim.html
20
21
Margin of Error
pˆ (1  pˆ )
2SE( pˆ )  2
n
Is called the Margin of Error
(ME).
This is the furthest p̂ can be
from p, with 95% confidence.
22
Margin of Error

What if we want to be 99.7%
confident?
pˆ (1  pˆ )
ME  3SE( pˆ )  3
n
23
Margin of Error
pˆ (1  pˆ )
ME  z * SE( pˆ )  z *
n
Confidence
z*
80%
90%
95%
98%
99%
1.282 1.645 2 or 1.96 2.326 2.576
24
Another Example


Pew Research Center/USA Today
Poll, Feb. 13 – 18, 2013. Asked of
1,504 randomly selected adults
nationwide.
“Do you favor or oppose setting
stricter emission limits on power
plants in order to address climate
change?”
25
Another Example
n=1,504 randomly selected adults.
Favor
Oppose
52%
28%
Unsure/
Refused
10%
26
Another Example

90% confidence interval for p, the
proportion of the population of all
adults in the U.S. who favor emission
limits on power plants in order to
address climate change.
27
Calculation
pˆ 1  pˆ 
pˆ  0.52 SE( pˆ ) 
 0.013 z*  1.645
n
0.52  1.6450.013 to 0.52  1.6450.013
0.52  0.021 to 0.52  0.021
0.499 to 0.541
28
What Sample Size?

Conservative Formula

The sample size to be 95%
confident that p̂ , the sample
proportion, will be within ME of
the population proportion, p.
1
n
2
ME
29
Example

Suppose we want to be 95%
confident that our sample
proportion will be within 0.02 of
the population proportion.
1
1
n

n


2
,
500
2
2
ME
0.02 
30