Lec 14  Sampling Distribution/Central Limit Theorem
IOE 265 F11
1
1
Sampling Distribution
Central Limit Theorem
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Topics
I.
Concepts of a “Statistic”
II.
Sampling Distribution of Statistics
By Probability Rules
Distribution of sample mean and sample total
III.
Central Limit Theorem
IV.
Distribution of a Linear Combination
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Lec 14  Sampling Distribution/Central Limit Theorem
IOE 265 F11
2
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I. Concepts of a “Statistic”
Consider taking two samples of size
n
from
the
same
population distribution.
A: 30.7, 29.4, 31.1
Mean 30.4
B: 28.8, 30.0, 31.1
Mean 29.97
Propositions
Any sample statistic (e.g. sample mean, sample
variance) varies from sample to sample.
Therefore a sample statistic behaves like a
random variable!
4
Example: Minitab
Suppose X ~ Weibull (shape= 2, scale = 5)
E(X) = 4.4311;
V(X) = 5.365
Using Minitab obtain 6 samples of 10
observations each and calculate sample mean
and variance for each.
Sample
1
2
3
4
5
6
Mean
4.401
5.928
4.229
4.132
3.620
5.761
Median
4.360
6.144
4.608
3.857
3.221
6.342
Std Dev
2.642
2.062
1.611
2.124
1.678
2.496
Lec 14  Sampling Distribution/Central Limit Theorem
IOE 265 F11
3
5
Point Estimates / Sampling
Distributions
Point Estimate – value for a sample statistic from a
particular sample.
Statistic – rv whose value may be calculated from a
sample of data
 lowercase letter indicates the
calculated or observed value of the statistic.
;
S
s
Probability Distribution of a Statistic is known as its
Sampling Distribution.
x
X
value
computed
the
is
where
)
(
x
x
X
P
6
iid Random Samples
Sampling Distribution depends on several items:
Population Distribution (parameters)
Sample size, n
Method of Sampling (with or without replacement)
rv’s X
1
, X
2
, .. X
n
form a random sample of size n if:
1.
Xi’s are independent rv’s (
independent
)
2.
Every Xi has the same probability distribution (
identically distributed
)
If satisfy above two conditions
we say Xi’s are iid
sampling with replacement or from infinite population
iid
sampling w/o replacement requires sample sizes
n
much smaller
than population
N
to assume iid (rule: n/N <= 0.05).
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Lec 14  Sampling Distribution/Central Limit Theorem
IOE 265 F11
4
7
II. Deriving Sampling Distribution
of a Statistic
By Probability Rules
used for simple cases with a few Xi’s
cases where derivation is already done.
By Simulation (more common!)
typically used when derivation via probability rules
is complicated, or if:
Underlying distribution of interest in unknown
(assumed).
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Deriving Via Simple Probability
Example: Suppose you sell two brands of DVD players
for A: $150, and B: $200.
Sales records indicate the following:
A – 60% of Sales; B: 40% of Sales
Let X
1
– revenue from first DVD; X
2
revenue from second DVD
Suppose you take samples of size n=2.
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 Fall '07
 Jin
 Normal Distribution, Variance, Probability theory, Distribution/Central Limit Theorem

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