Lectur
e
Reading
s
Objectives
Assignment
25
8.18.5
Sampling distributions of
sample proportion and sample mean
Calculations with sampling distributions
Stats:
P526 115 odd
P551 1725 odd
DEFINITION:
The
sampling distribution of the sample proportion
is
the distribution of values of the sample proportion in all
possible random samples of the same size
n
taken from
the same population.
Example
Proportion of Women
We assume that 50% of the population are women,
p =
0.50.
Take a simple random sample of size
n
=4 people
from this population and observe the proportion of women
in the sample.
Then this sampling process will be
repeated many times to examine the possible values for
the sample proportion and see how these possible values
vary. ( seed 91, 1= woman, and 0= man)
NUMBER
OF WOMEN
SAMPLE
PROPORTION TALLY FREQUENCY
PROPORTION OF
ALL TRIALS
0
1
2
3
4
0.00
50
0.25
0.75
0.50
1.00
4
16
10
16
4
4/50=.08
4/50=.08
16/50=.32
16/50=.32
10/50=.20
50/50=1.00
Chapter 8  1
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View Full DocumentProperties of the
Distribution of the Sample Proportion
Chapter 8  2
If simple random samples of size
n
are taken from a population where the
proportion of "successes" is
p
, then the sampling distribution of
p
ˆ
has the
following properties:
1.
( )
E p
p
=
)
The average of all possible
p
ˆ
values is equal to the parameter
p
.
In other words,
p
ˆ
is an
unbiased
estimator of
p
.
2.
(1
)
( )
p
p
p
n
σ

=
)
The
standard deviation
for
p
ˆ
decreases as the sample size
n
increases.
3.
If
n
is “sufficiently” large
, the distribution of
p
ˆ
eventually looks like a
normal distribution
with mean and standard deviation as given in 1 and 2
above.
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