1 Pertemuan 18 Pembandingan Dua Populasi-2 Matakuliah: A0064 / Statistik Ekonomi Tahun: 2005 Versi: 1/1.

1

Pertemuan 18Pembandingan Dua Populasi-2

Matakuliah : A0064 / Statistik Ekonomi

Tahun : 2005

Versi : 1/1

2

Learning Outcomes

Pada akhir pertemuan ini, diharapkan mahasiswa

akan mampu :

• Membandingkan pengujian sampel besar untuk perbedaan antara dua proporsi populasi dan pengujian untuk kesamaan dua populasi

3

Outline Materi

• Pengujian Sampel Besar untuk Perbedaan antara Dua Proporsi Populasi

• Sebaran-F dan Uji untuk Kesamaan Dua Ragam Populasi

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8-4

• Hypothesized difference is zero I: Difference between two population proportions is 0

• p1= p2

» H0: p1 -p2 = 0

» H1: p1 -p20

II: Difference between two population proportions is less than 0

• p1p2

» H0: p1 -p2 0

» H1: p1 -p2 > 0

• Hypothesized difference is other than zero: III: Difference between two population proportions is less than D

• p1 p2+D

» H0:p-p2 D

» H1: p1 -p2 > D

8-5 A Large-Sample Test for the Difference between Two Population Proportions

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8-5

A large-sample test statistic for the difference between two population proportions, when the hypothesized difference is zero:

where is the sample proportion in sample 1 and is the sample

proportion in sample 2. The symbol stands for the combined sample proportion in both samples, considered as a single sample. That is:

A large-sample test statistic for the difference between two population proportions, when the hypothesized difference is zero:

where is the sample proportion in sample 1 and is the sample

proportion in sample 2. The symbol stands for the combined sample proportion in both samples, considered as a single sample. That is:

zp p

p pn n

( )

( )

1 2

1 2

0

11 1

pxn1

1

1

When the population proportions are hypothesized to be equal, then a pooled estimator of the proportion ( ) may be used in calculating the test statistic.

pxn1

1

1

p

21

11ˆnn

xxp

p

Comparisons of Two Population Proportions When the Hypothesized Difference Is Zero: Test Statistic

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8-6

Carry out a two-tailed test of the equality of banks’ share of the car loan market in 1980 and 1995.

Population 1: 1980

n1 = 100

x1 = 53

p1 = 0.53

H

H

Critical point: z0.05

= 1.645

H0 may not be rejected even at a 10%

level of significance.

0 1 2 0

1 1 2 0

1 2 0

11

1

1

2

0 53 0 43

48 521

100

1

100

0 10

0 004992

0 10

0 070651 415

:

:

( )

( )

. .

(. )(. )

.

.

.

..

p p

p p

zp p

p pn n

Population 2: 1995

n = 100

x = 43

p = 0.43

x1 + x2

n1 n2

2

2

2

.p

53 43

100 1000 48

Comparisons of Two Population Proportions When the Hypothesized Difference Is Zero: Example 8-8

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0.4

0.3

0.2

0.1

0.0z

f( z)

Standard Normal Distribution

NonrejectionRegion

RejectionRegion

-z0.05=-1.645 z0.05=1.645

Test Statistic=1.415

RejectionRegion

0

Since the value of the test statistic is within the nonrejection region, even at a 10% level of significance, we may conclude that there is no statistically significant difference between banks’ shares of car loans in 1980 and 1995.

Since the value of the test statistic is within the nonrejection region, even at a 10% level of significance, we may conclude that there is no statistically significant difference between banks’ shares of car loans in 1980 and 1995.

Example 8-8: Carrying Out the Test

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8-8

Example 8-8: Using the Template

P-value = P-value = 0.157, so do 0.157, so do not reject Hnot reject H00

at the 5% at the 5% significance significance level.level.

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8-9

Carry out a one-tailed test to determine whether the population proportion of traveler’s check buyers who buy at least $2500 in checks when sweepstakes prizes are offered as at least 10% higher than the proportion of such buyers when no sweepstakes are on.

Population 1: With Sweepstakes

n1 = 300

x1 = 120

p1 = 0.40

H

H

Critical point: z0.001

= 3.09

H0 may be rejected at any common level of significance.

0 1 2 0 10

1 1 2 0 10

1 2

11

1

1

21

2

2

0 40 0 20 0 10

0 40 0 60

300

0 20 80

700

0 10

0 032073 118

: .

: .

( )

( ) ( )

( . . ) .

( . )( . ) ( . )(. )

.

..

p p

p p

zp p D

p p

n

p p

n

Population 2: No Sweepstakes

n = 700

x = 140

p = 0.20

2

2

2

Comparisons of Two Population Proportions When the Hypothesized Difference Is Not Zero: Example 8-9

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8-10

0.4

0.3

0.2

0.1

0.0z

f( z)

Standard Normal Distribution

NonrejectionRegion

RejectionRegion

z0.001=3.09

Test Statistic=3.118

0

Since the value of the test statistic is above the critical point, even for a level of significance as small as 0.001, the null hypothesis may be rejected, and we may conclude that the proportion of customers buying at least $2500 of travelers checks is at least 10% higher when sweepstakes are on.

Since the value of the test statistic is above the critical point, even for a level of significance as small as 0.001, the null hypothesis may be rejected, and we may conclude that the proportion of customers buying at least $2500 of travelers checks is at least 10% higher when sweepstakes are on.

Example 8-9: Carrying Out the Test

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8-11

Example 8-9: Using the Template

P-value = P-value = 0.0009, so 0.0009, so reject Hreject H00 at at

the 5% the 5% significance significance level.level.

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8-12

A (1-) 100% large-sample confidence interval for the difference between two population proportions:

A (1-) 100% large-sample confidence interval for the difference between two population proportions:

A 95% confidence interval using the data in example 8-9:

A 95% confidence interval using the data in example 8-9:

( )

( ) ( )

p p z

p p

n

p p

n1 2

2

11

1

1

21

2

2

( )

( ) ( )

( . . ) .( . )( . ) ( . )( . )

. ( . )( . ) . . [ . , . ]

p p z

p p

n

p p

n1 2

2

11

1

1

21

2

20 4 0 2 1 96

0 4 0 6

300

0 2 0 8

700

0 2 1 96 0 0321 0 2 0 063 0 137 0 263

Confidence Intervals for the Difference between Two Population Proportions

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8-13

Confidence Intervals for the Difference between Two Population Proportions – Using the Template – Using the Data

from Example 8-9

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8-14

The F distribution is the distribution of the ratio of two chi-square random variables that are independent of each other, each of which is divided by its own degrees of freedom.

The F distribution is the distribution of the ratio of two chi-square random variables that are independent of each other, each of which is divided by its own degrees of freedom.

An F random variable with k1 and k2 degrees of freedom:An F random variable with k1 and k2 degrees of freedom:

Fk

k

k k1 2

1

2

1

2

2

2

,

8-6 The F Distribution and a Test for Equality of Two Population Variances

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8-15

• The F random variable cannot be negative, so it is bound by zero on the left.

• The F distribution is skewed to the right.

• The F distribution is identified the number of degrees of freedom in the numerator, k1, and the number of degrees of freedom in the denominator, k2.

• The F random variable cannot be negative, so it is bound by zero on the left.

• The F distribution is skewed to the right.

• The F distribution is identified the number of degrees of freedom in the numerator, k1, and the number of degrees of freedom in the denominator, k2.

543210

1.0

0.5

0.0

F

F Distributions with different Degrees of Freedom

f(F

)

F(5,6)

F(10,15)

F(25,30)

The F Distribution

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8-16

Critical Points of the F Distribution Cutting Off a Right-Tail Area of 0.05

k1 1 2 3 4 5 6 7 8 9

k2

1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.1810 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.0211 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.9012 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.8013 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.7114 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.6515 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59

3.01

543210

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

F0.05=3.01

f(F)

F Distribution with 7 and 11 Degrees of Freedom

F

The left-hand critical point to go along with F(k1,k2) is given by:

Where F(k1,k2) is the right-hand critical point for an F random variable with the reverse number of degrees of freedom.

1

2 1F k k,

Using the Table of the F Distribution

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8-17

The right-hand critical point read directly from the table of the F distribution is:

F(6,9) =3.37

The corresponding left-hand critical point is given by:

The right-hand critical point read directly from the table of the F distribution is:

F(6,9) =3.37

The corresponding left-hand critical point is given by:

1 1410

0 24399 6F , .

. 543210

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

F

f(F)

F Distribution with 6 and 9 Degrees of Freedom

F0.05=3.37F0.95=(1/4.10)=0.2439

0.05

0.05

0.90

Critical Points of the F Distribution: F(6, 9), = 0.10

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8-18

Test statistic for the equality of the variances of two normallydistributed populations:

Fs

sn n1 21 1

1

2

2

2 ,

Test statistic for the equality of the variances of two normallydistributed populations:

Fs

sn n1 21 1

1

2

2

2 ,

I: Two-Tailed Test

• 1 = 2

• H0: 1 = 2

• H1: 2

II: One-Tailed Test

• 12

• H0: 1 2

• H1: 1 2

I: Two-Tailed Test

• 1 = 2

• H0: 1 = 2

• H1: 2

II: One-Tailed Test

• 12

• H0: 1 2

• H1: 1 2

Test Statistic for the Equality of Two Population Variances

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8-19

The economist wants to test whether or not the event (interceptions and prosecution of insider traders) has decreased the variance of prices of stocks.

70.223,24

01.0

01.223,24

05.0

0.322

s

24=2

n

After :2 Population

3.921

s

25=1

n

Before :1 Population

F

F

H

H

H0 may be rejected at a 1% level of significance.

0 1

2

2

2

1

2

1 1

2

2

2

1 1 2 1 24 23

12

22

9 3

3 031

:

:

, ,

.

..

F

n nF

s

s

Example 8-10

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8-20

Distribution with 24 and 23 Degrees of Freedom

543210

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

F0.01=2.7

f(F

)

F

Test Statistic=3.1

Since the value of the test statistic is above the critical point, even for a level of significance as small as 0.01, the null hypothesis may be rejected, and we may conclude that the variance of stock prices is reduced after the interception and prosecution of inside traders.

Since the value of the test statistic is above the critical point, even for a level of significance as small as 0.01, the null hypothesis may be rejected, and we may conclude that the variance of stock prices is reduced after the interception and prosecution of inside traders.

Example 8-10: Solution

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8-21

Example 8-10: Solution Using the Template

Observe that the p-Observe that the p-value for the test is value for the test is 0.0042 which is less 0.0042 which is less than 0.01. Thus the than 0.01. Thus the null hypothesis null hypothesis must be rejected at must be rejected at this level of this level of significance of 0.01.significance of 0.01.

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8-22

Population 1 Population 2

n1

= 14 n2

= 9

s12 s

22

0 122 0 112

0 05

13 83 28

0 10

13 82 50

. .

.

,.

.

,.

F

F

H

H

H0

may not be rejected at the 10% level of significance.

0 12

22

1 12

22

1 1 2 1 13 812

22

0122

0112119

:

:

, ,

.

..

F

n nF

s

s

Example 8-11: Testing the Equality of Variances for Example 8-5

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8-23

Since the value of the test statistic is between the critical points, even for a 20% level of significance, we can not reject the null hypothesis. We conclude the two population variances are equal.

Since the value of the test statistic is between the critical points, even for a 20% level of significance, we can not reject the null hypothesis. We conclude the two population variances are equal.

F Distribution with 13 and 8 Degrees of Freedom

543210

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

F

f(F)

F0.10=3.28F0.90=(1/2.20)=0.4545

0.10

0.10

0.80

Test Statistic=1.19

Example 8-11: Solution

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8-24

Template to test for the Difference between Two Population Variances:

Example 8-11

Observe that the p-Observe that the p-value for the test is value for the test is 0.8304 which is larger 0.8304 which is larger than 0.05. Thus the than 0.05. Thus the null hypothesis null hypothesis cannot be rejected at cannot be rejected at this level of this level of significance of 0.05. significance of 0.05. That is, one can That is, one can assume equal assume equal variance.variance.

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8-25

The F Distribution Template to

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8-26

The Template for Testing Equality of Variances

27

Penutup

• Pembandingan Dua Populasi merupakan bagian dari pengujian Hipotesis dimana populasinya lebih dari satu,hal ini juga merupakan salah satu bentuk inferensial statistik yang berupa pengambilan kesimpulan/ pengambilan keputusan tentang menolak atau tidak menolak (menerima) suatu pernyataan/hipotesis