Matakuliah
Tahun
: I0134-Metode Statistika
: 2007
Pertemuan 14
Peubah Acak Normal
1
Outline Materi:
• Sebaran rata-rata sampling
• Sebaran proporsi sampling
2
Sampling Distribution
• Theoretical Probability Distribution of a Sample Statistic
• Sample Statistic is a Random Variable
– Sample mean, sample proportion
• Results from Taking All Possible Samples of the Same
Size
3
Developing Sampling Distributions
• Suppose There is a Population …
• Population Size N=4
B
• Random Variable, X,
is Age of Individuals
C
• Values of X: 18, 20,
22, 24 Measured in
Years
D
A
4
(continued)
Developing Sampling Distributions
Summary Measures for the Population Distribution
N

X
i 1
P(X)
i
.3
N
18  20  22  24

 21
4
N
 
 X
i 1
i
N

.2
.1
0
2
 2.236
A
B
C
D
(18)
(20)
(22)
(24)
X
Uniform Distribution
5
Developing Sampling Distributions
(continued)
All Possible Samples of Size n=2
1st
Obs
2nd Observation
18
20
22
24
18 18,18 18,20 18,22 18,24
20 20,18 20,20 20,22 20,24
16 Sample Means
22 22,18 22,20 22,22 22,24
1st 2nd Observation
Obs 18 20 22 24
24 24,18 24,20 24,22 24,24
18 18 19 20 21
16 Samples Taken
with Replacement
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
6
Developing Sampling Distributions
(continued)
Sampling Distribution of All Sample Means
Sample Means
Distribution
16 Sample Means
1st 2nd Observation
Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
.3
P X 
.2
.1
0
_
18 19
20 21 22 23
X
24
7
Developing Sampling Distributions
(continued)
Summary Measures of Sampling Distribution
N
X 
X
i 1
N
i
18  19  19 

16
N
X 
 X
i 1
i
 X 
 21
2
N
18  21  19  21
2

 24
16
2

  24  21
2
 1.58
8
Comparing the Population with Its
Sampling Distribution
Population
N=4
  21
P X 
  2.236
Sample Means Distribution
n=2
 X  21
.3
.3
.2
.2
.1
.1
0
0
A
B
C
(18)
(20)
(22)
D X
P X 
 X  1.58
_
18 19
20 21 22 23
24
X
(24)
9
Properties of Summary Measures
•
X  
– I.e., X
is unbiased
• Standard Error (Standard Deviation) of the Sampling  X
Distribution
is Less Than the Standard Error of Other
Unbiased Estimators
• For Sampling with Replacement or without Replacement
 or Infinite Populations:
from Large
X 
n
X
– As n increases,
decreases
10
Unbiasedness (
f X 
Unbiased

X  
)
Biased
X
X
11
Less Variability
Standard Error (Standard Deviation) of the
Sampling Distribution  X is Less Than the
Standard Error of Other Unbiased Estimators
f  X  Sampling
Distribution
of Median
Sampling
Distribution of
Mean

X
12
Effect of Large Sample
For sampling with replacement:
As n increases,  X decreases
f X 
Larger
sample size
Smaller
sample size

X
13
When the Population is Normal
Population Distribution
Central Tendency
X  
Variation
X 

n
  10
  50
Sampling Distributions
n4
n  16
X 5
 X  2.5
 X  50
X
14
When the Population is
Not Normal
Population Distribution
Central Tendency
X  
Variation
X 

n
  10
  50
Sampling Distributions
n4
n  30
X 5
 X  1.8
 X  50
X
15
Central Limit Theorem
As Sample
Size Gets
Large
Enough
Sampling
Distribution
Becomes
Almost
Normal
Regardless
of Shape of
Population
X
16
How Large is Large Enough?
• For Most Distributions, n>30
• For Fairly Symmetric Distributions, n>15
• For Normal Distribution, the Sampling Distribution of the
Mean is Always Normally Distributed Regardless of the
Sample Size
– This is a property of sampling from a normal population
distribution and is NOT a result of the central limit theorem
17
Example:
 7.8  8
X  X
8.2  8 
P  7.8  X  8.2   P 



X
2 / 25 
 2 / 25
 P  .5  Z  .5   .3830
Standardized
Normal Distribution
Sampling Distribution
2
X 
 .4
25
Z 1
.1915
7.8
8.2
X  8
X
0.5
Z  0
0.5
Z
18
 p
Population Proportions
• Categorical Variable
– E.g., Gender, Voted for Bush, College Degree
• Proportion of Population Having a Characteristic
• Sample Proportion Provides an Estimate
–
 p
• If Two Outcomes, X Has a Binomial Distribution
– Possess or do not possess characteristic
X number of successes
pS  
n
sample size
19
Sampling Distribution of
Sample Proportion
• Approximated by
Normal Distribution
–
np  5
f(ps)
– Mean:
•
n 1  p   5
– Standard error:
•
Sampling Distribution
.3
.2
.1
0
0
p  p
.2
.4
.6
8
ps
1
S
p 
S
p 1  p 
n
p = population proportion
20
Standardizing Sampling Distribution of
Proportion
Z
pS   pS
p
S
p 1  p 
n
Standardized
Normal Distribution
Sampling Distribution
p

pS  p
Z 1
S
p
S
pS
Z  0
Z
21
Example:
n  200
p  .4
P  pS  .43  ?

 p 
.43  .4
S
pS

P  pS  .43  P

  pS
.4 1  .4 

200

Standardized
Normal Distribution
Sampling Distribution
p


  P  Z  .87   .8078



Z 1
S
 p .43
S
pS
0 .87
Z
22