Ch. 18  Sampling Distributions
Expanded def’n: A parameter
is:  a numerical value describing some aspect of a pop’n
 usually regarded as constant
 usually unknown
A statistic
is:  a numerical value describing some aspect of a sample
 regarded as random before sample is selected
 observed after sample is selected
The observed value depends on the particular sample selected from the population;
typically, it varies from sample to sample.
This variability is called sampling variability
.
The distribution of all the values of a statistic is called its sampling distribution
.
Def’n:
p
ˆ = proportion of ppl with a specific characteristic in a random sample of size
n
p
= population proportion of ppl with a specific characteristic
The standard deviation of a sampling distribution is called a standard error
.
General Properties of the Sampling Distribution of
p
ˆ :
Let
p
ˆ and
p
be as above.
Also,
p
ˆ
µ
and
p
ˆ
σ
are the mean and standard deviation for the
distribution of
p
ˆ .
Then the following rules hold:
Rule 1
:
p
ˆ
µ
=
p
.
(Textbook uses
)
ˆ
(
p
µ
)
Rule 2
:
n
pq
n
p
p
p
=
−
=
)
1
(
ˆ
σ
.
(standard error
Æ
p
ˆ
ˆ
σ
)
e.g. Suppose the population proportion is 0.5.
a) What is the standard deviation of
p
ˆ for a sample size of 4?
25
.
0
4
)
5
.
0
1
(
5
.
0
)
1
(
ˆ
=
−
=
−
=
n
p
p
p
σ
b) How large must
n
(sample size) be so that the sample proportion has a standard
deviation of at most 0.125?
p
p
p
n
ˆ
)
1
(
σ
−
=
Æ
16
125
.
0
)
5
.
0
1
(
5
.
0
)
1
(
2
2
ˆ
=
−
=
−
=
p
p
p
n
σ
Rule 3
: When
n
is large and
p
is not too near 0 or 1, the sampling distribution of
p
ˆ is
approximately normal.
The farther from
p
= 0.5, the larger
n
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 Winter '10
 PaulCartledge
 Normal Distribution, Standard Deviation, Standard Error, Variance, σp

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