•
Any unit
1,2, … , ?
to be selected with probability
1/?
•
The sampled unit is then replaced in the population, and a second
unit is randomly selected with again probability
1/?
•
In sampling no additional information is obtained from selecting a
unit several times.
•
Thus we prefer to sample without replacement.
•
A
simple random sample without replacement
(SRS):
n
unites are selected so that every possible subset of n distinct
units in has the same probability of being selected as the
sample.
•
Each sample
?
of size n is selected with probability
Pr
?
= 1/
?
𝑛
•
Each unit
𝑖
is in the sample with probability
π
𝑖
= 𝑛/?
•
A parameter
ϑ
:
Any real valued function of the population values
•
Examples:
•
?
?
=
?
1
+ ?
2
+⋯+?
𝑁
?
the
population mean
.
•
?
2
=
1
?−1
𝑖=1
?
?
𝑖
−
?
?
2
the
population variance
.
•
S
=
1
?−1
𝑖=1
?
?
𝑖
−
?
?
2
the
population standard deviation
.
•
A
statistic T
: any real valued function of the values
?
1
, … , ?
?
taken
from the sample S.
•
For sample S we write
?
𝑆
to denote the values of the statistic
calculated from the sample.
•
A
estimator
ϑ
:
just a statistic used to estimate the parameter
ϑ
.
•
The
sampling distribution of a statistic T
:
A probability distribution
that determines
Pr
? = ?
𝑖
.
Illustration map
?
1
?
𝑖
1
?
𝑖
2
?
4
?
?
?
?
.. 2

1
0
1
2 … ϑ
?
?−1
?
2
?
3
Population U with their measured
values
ϑ
Sample
S
Parameter
ϑ
: a real number
ϑ
𝑆
Example
•
Suppose the value of
?
𝑖
for each of the N = 8 units in the whole
population is given by
•
Then
Pr
2, 4, 6, 7
= 1/70
.
•
Suppose the total
t = ?
1
+ ?
2
+ ⋯ ?
?
, is the parameter of interest.
•
We may choose
?
=
? ?
𝑆
= 2(?
1
+ ?
2
+ ?
3
+?
4
)
as an estimator of
?
.
𝒊
1
2
3
4
5
6
7
8
?
𝑖
1
2
4
4
7
7
7
8
k
?
Sampling distribution of the statistic
?
More definitions related to an estimator
•
The expected value of an estimator is
•
𝐸
ϑ
=
𝑘=1
?
kPr(
ϑ = k) =
𝑆
ϑ
𝑆
𝑃?(?)
•
The
bias
of
ϑ
is B
𝑖??
ϑ
= 𝐸
ϑ
− ϑ
•
The estimator is said to be unbiased if B
𝑖??
ϑ
= 0
•
AN IMPORTANT NOTE:
the bias of an estimator is not the same as the
selection bias or the measurement bias.
How to measure the quality of an estimator?
•
The
mean squared error
of
ϑ
is
??𝐸
ϑ
= 𝐸((
𝜗 − ϑ)
2
)
•
An estimator
ϑ
is
accurate
if
??𝐸
ϑ
is small.
•
An estimator
ϑ
is
precise
if
𝑉(
ϑ)
is small.
•
Note that
??𝐸
ϑ
= 𝑉
ϑ
+ [Bias(
ϑ)]
2
•
Also note that
𝑉
ϑ
=
𝑆
Pr(?)[
ϑ
𝑠
− 𝐸
ϑ ]
2
•
For the previous example
𝑉
?
=
1
70
22 − 40
2
+ ⋯ +
1
70
58 − 40
2
= 54.86
Illustration
•
A: Unbiased but not precise
•
B:
Biased but precise
•
C: Accurate and
precise
The Proportion
•
Suppose we are interested in the proportion of the population that
the
has characteristic of the interest.
•
If we assign the value
?
𝑖
= 1
when
𝑖
has the characteristic and 0
otherwise then the proportion of the population that has the
characteristic is
𝑝 =
?
1
+ ?
2
+ ⋯ + ?
?
?
=
?
?
•
We can take
𝑝
=
?
Example
•
The U.S. government conducts a Census of Agriculture every five years.
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 Summer '19
 representative, Lohr, Sharon L