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STAT 400
(Chapter 5.6)
Spring 2012
Population:
mean
,
standard deviation
.
Random
Sample:
X
1
, X
2
, … , X
n
. (or we say X
i
’s are
i.i.d.)
E
(
X
1
+ X
2
+ … + X
n
)
=
n
,
SD
(
X
1
+ X
2
+ … + X
n
)
=
σ
n
.
If the sampling is done
without
replacement
from a
finite
population of size
N
,
then
SD
(
X
)
=
1
N
N
n
n
σ
.
The
sample
mean
n
n
X
...
X
X
2
1
X
.
E
(
X
)
=
,
SD
(
X
)
=
n
σ
.
If the sampling is done
without
replacement
from a
finite
population of size
N
,
then
SD
(
X
)
=
1
N
N
n
n
σ
.
X
=
+ chance error.
L
AW
OF
L
ARGE
N
UMBERS
(
L
AW
OF
A
VERAGES
)
:
As the sample size,
n
,
increases,
the sample mean,
X
,
“tends to gets closer and
closer” to the population mean
.
For Bernoulli trials,
as # of trials
n
increases,
the sample proportion of “successes”,
n
X
,
“tends to gets closer and closer” to the probability of “success”

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