1 pertemuan 14 peubah acak normal matakuliah: i0134-metode statistika tahun: 2007

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Pertemuan 14Peubah Acak Normal

Matakuliah : I0134-Metode StatistikaTahun : 2007

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Outline Materi:• Sebaran rata-rata sampling• Sebaran proporsi sampling

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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

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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

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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)

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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)

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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

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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

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Unbiasedness ( )

BiasedUnbiased

X X

X f X

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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

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Effect of Large Sample

Larger sample size

Smaller sample size

X

f X

For sampling with replacement:As increases, decreasesXn

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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

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Central Limit TheoremAs Sample Size Gets Large Enough

Sampling Distribution Becomes Almost Normal Regardless of Shape of Population X

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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

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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

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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

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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

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Standardizing Sampling Distribution of Proportion

1S

S

S p S

p

p p pZp pn

Sampling Distribution

Standardized Normal

DistributionSp

1Z

Sp Sp Z0Z

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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