Embed Size (px)
DESCRIPTION
3 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
Citation preview
1
Pertemuan 14Peubah Acak Normal
Matakuliah : I0134-Metode StatistikaTahun : 2007
2
Outline Materi:• Sebaran rata-rata sampling• Sebaran proporsi sampling
3
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
4
Developing Sampling Distributions• Suppose There is a Population …• Population Size N=4• Random Variable, X,
is Age of Individuals• Values of X: 18, 20,
22, 24 Measured inYears
A
B C
D
5
1
2
1
18 20 22 24 214
2.236
N
ii
N
ii
X
N
X
N
.3
.2
.1 0
A B C D (18) (20) (22) (24) Uniform Distribution
P(X)
X
Developing Sampling Distributions(continued)
Summary Measures for the Population Distribution
6
1st 2nd Observation Obs 18 20 22 24 18 18,18 18,20 18,22 18,24 20 20,18 20,20 20,22 20,24
22 22,18 22,20 22,22 22,24
24 24,18 24,20 24,22 24,24
All Possible Samples of Size n=2
16 Samples Taken with Replacement
16 Sample Means1st 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
Developing Sampling Distributions
(continued)
7
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
Sampling Distribution of All Sample Means
18 19 20 21 22 23 240
.1
.2
.3
X
Sample Means
Distribution
16 Sample Means
_
Developing Sampling Distributions
(continued)
P X
8
1
2
1
2 2 2
18 19 19 24 2116
18 21 19 21 24 211.58
16
N
ii
X
N
i Xi
X
X
N
X
N
Summary Measures of Sampling Distribution
Developing Sampling Distributions
(continued)
9
Comparing the Population with Its Sampling Distribution
18 19 20 21 22 23 240
.1
.2
.3
X
Sample Means Distribution
n = 2
A B C D (18) (20) (22) (24)
0
.1
.2
.3
PopulationN = 4
X_
21 2.236 21 1.58X X P X P X
10
Properties of Summary Measures
• – I.e., is unbiased
• Standard Error (Standard Deviation) of the Sampling Distribution is Less Than the Standard Error of Other Unbiased Estimators
• For Sampling with Replacement or without Replacement from Large or Infinite Populations:
– As n increases, decreases
X
X
X n
X
X
11
Unbiasedness ( )
BiasedUnbiased
X X
X f X
12
Less Variability
Sampling Distribution of Median Sampling
Distribution of Mean
X
f X
Standard Error (Standard Deviation) of the Sampling Distribution is Less Than the Standard Error of Other Unbiased Estimators
X
13
Effect of Large Sample
Larger sample size
Smaller sample size
X
f X
For sampling with replacement:As increases, decreasesXn
14
When the Population is Normal
Central Tendency
Variation
Population Distribution
Sampling Distributions
X
X n
X50X
45X
n
162.5X
n
50
10
15
When the Population isNot Normal
Central Tendency
Variation
Population Distribution
Sampling Distributions
X
X n
X50X
45X
n
301.8X
n
50
10
16
Central Limit TheoremAs Sample Size Gets Large Enough
Sampling Distribution Becomes Almost Normal Regardless of Shape of Population X
17
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
18
Example:
Sampling Distribution
Standardized Normal
Distribution2 .425X 1Z
8X 8.2 Z0Z
0.5
7.8 8 8.2 87.8 8.22 / 25 2 / 25
.5 .5 .3830
X
X
XP X P
P Z
7.8 0.5
.1915
X
19
Population Proportions• Categorical Variable
– E.g., Gender, Voted for Bush, College Degree• Proportion of Population Having a Characteristic• Sample Proportion Provides an Estimate
– • If Two Outcomes, X Has a Binomial Distribution
– Possess or do not possess characteristic number of successes
sample sizeSXpn
p
p
20
Sampling Distribution ofSample Proportion
• Approximated by Normal Distribution–
– Mean:•
– Standard error: •
p = population proportion
Sampling Distributionf(ps)
.3
.2
.1 0
0 . 2 .4 .6 8 1ps
5np
1 5n p
Spp
1Sp
p pn
21
Standardizing Sampling Distribution of Proportion
1S
S
S p S
p
p p pZp pn
Sampling Distribution
Standardized Normal
DistributionSp
1Z
Sp Sp Z0Z
22
Example: 200 .4 .43 ?Sn p P p
.43 .4.43 .87 .8078.4 1 .4
200
S
S
S pS
p
pP p P P Z
Sampling Distribution
Standardized Normal
DistributionSp
1Z
Sp
Sp Z0.43 .87
