Pertemuan 17 Pembandingan Dua Populasi-1

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Pertemuan 17Pembandingan Dua Populasi-1

Matakuliah : A0064 / Statistik EkonomiTahun : 2005 Versi : 1/1

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Learning Outcomes

Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu :• Membandingkan dua observasi yang

berpasangan dan pengujian perbedaan antara dua rata-rata populasi

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Outline Materi

• Pembandingan Observasi yang Berpasangan

• Pengujian Perbedaan antara Dua Rata-rata Populasi

COMPLETE 5 t h e d i t i o nBUSINESS STATISTICS

Aczel/SounderpandianMcGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002

8-4

• Using Statistics• Paired-Observation Comparisons• A Test for the Difference between Two Population

Means Using Independent Random Samples• A Large-Sample Test for the Difference between

Two Population Proportions• The F Distribution and a Test for the Equality of

Two Population Variances• Summary and Review of Terms

The Comparison of Two Populations8



8-5

• Inferences about differences between parameters of two populationsPaired-ObservationsObserve the samesame group of persons or things

– At two different times: “before” and “after”– Under two different sets of circumstances or “treatments”

Independent Samples• Observe differentdifferent groups of persons or things

– At different times or under different sets of circumstances

8-1 Using Statistics



8-6

• Population parameters may differ at two different times or under two different sets of circumstances or treatments because:The circumstances differ between times or treatmentsThe people or things in the different groups are

themselves different

• By looking at paired-observations, we are able to minimize the “between group” , extraneous variation.

8-2 Paired-Observation Comparisons



8-7

freedom. of degrees 1)-(on with distributi t a has statistic the, is differencemean population theand trueis hypothesis

null When the.hypothesis null under the differencemean population theis symbol The ns.observatio of pairs of

number theis , size, sample theand s,difference theseofdeviation standard sample theis s ns,observatio ofpair

eachbetween difference average sample theis D where

: t testnsobservatio-paired for the statisticTest

0

0

0

D

n

n

ns

Dt

D

D

D

D

Paired-Observation Comparisons of Means



8-8

A random sample of 16 viewers of Home Shopping Network was selected for an experiment. All viewers in the sample had recorded the amount of money they spent shopping during the holiday season of the previous year. The next year, these people were given access to the cable network and were asked to keep a record of their total purchases during the holiday season. Home Shopping Network managers want to test the null hypothesis that their service does not increase shopping volume, versus the alternative hypothesis that it does.

Shopper Previous Current Diff 1 334 405 71 2 150 125 -25 3 520 540 20 4 95 100 5 5 212 200 -12 6 30 30 0 7 1055 1200 145 8 300 265 -35 9 85 90 510 129 206 7711 40 18 -2212 440 489 4913 610 590 -2014 208 310 10215 880 995 11516 25 75 50

H0: D 0H1: D > 0

df = (n-1) = (16-1) = 15

Test Statistic:

Critical Value: t0.05 = 1.753

Do not reject H0 if : t 1.753 Reject H0 if: t > 1.753

tD D

sD

n

0

Example 8-1



8-9

tD D

sDn

0 32 81 0

55 75

16

2 354.

..

2.131= t0.025

2.602= t0.01

1.753= t0.05

2.354=test statistic

50-5

0.4

0.3

0.2

0.1

0.0t

f(t)

t Distribution: df=15

Nonrejection Region

Rejection Region

t = 2.354 > 1.753, so H0 is rejected and we conclude that there is evidence that shopping volume by network viewers has increased, with a p-value between 0.01 an 0.025. The Template output gives a more exact p-value of 0.0163. See the next slide for the output.

Example 8-1: Solution



8-10

Example 8-1: Template for Testing Paired Differences



8-11

It has recently been asserted that returns on stocks may change once a story about a company appears in The Wall Street Journal column “Heard on the Street.” An investments analyst collects a random sample of 50 stocks that were recommended as winners by the editor of “Heard on the Street,” and proceeds to conduct a two-tailed test of whether or not the annualized return on stocks recommended in the column differs between the month before and the month after the recommendation. For each stock the analysts computes the return before and the return after the event, and computes the difference in the two return figures. He then computes the average and standard deviation of the differences.

H0: D 0H1: D > 0

n = 50D = 0.1%sD = 0.05%

Test Statistic:

zD D

sD

n

0

This test result is highly significant,and H0 may be rejected at any reasonablelevel of significance.

p - value:

zD D

sDn

p z

0 0 1 0

0 05

50

14 14

14 14 0

.

..

( . )

Example 8-2



8-12

instead. .2

z usemay welarge, is size sample When theright, its to2

of area

an off cuts that freedom of degrees 1)-(non with distributi t theof value theis 2

twhere

2tD

:D

differencemean for the interval confidence 100% )-(1A

nD

s

Confidence Intervals for Paired Observations



8-13

95% confidence interval for the data in Example 8 2 :

D z 0.1 1.96 0.0550

Note that this confidence interval does not include the value 0.

2

01 196 0071

01 0 014 0 086 0114

sDn

. ( . )(. )

. . [ . , . ]

Confidence Intervals for Paired Observations – Example 8-2



8-14

Confidence Intervals for Paired Observations – Example 8-2 Using the

Template



8-15

• When paired data cannot be obtained, use independentindependent random samples drawn at different times or under different circumstances.Large sample test if:

• Both n130 and n230 (Central Limit Theorem), or

• Both populations are normal and 1 and 2 are both known

Small sample test if:• Both populations are normal and 1 and 2 are unknown

8-3 A Test for the Difference between Two Population Means Using Independent Random Samples



8-16

• I: Difference between two population means is 0 1= 2

• H0: 1 -2 = 0

• H1: 1 -2 0

• II: Difference between two population means is less than 0 12

• H0: 1 -2 0

• H1: 1 -2 0

• III: Difference between two population means is less than D 1 2+D

• H0: 1 -2 D

• H1: 1 -2 D

Comparisons of Two Population Means: Testing Situations



8-17

Large-sample test statistic for the difference between two population means:

The term (1- 2)0 is the difference between 1 an 2 under the null hypothesis. Is is equal to zero in situations I and II, and it is equal to the prespecified value D in situation III. The term in the denominator is the standard deviation of the difference between the two sample means (it relies on the assumption that the two samples are independent).

2

2

2

1

2

1

02121)()(

nn

xxz

Comparisons of Two Population Means: Test Statistic



8-18

212 = 452 =x1200=n

Visa Preferred :1 Population

1

1

1

185 = 523 =x800=n

Card Gold :2 Population

2

2

2

Is there evidence to conclude that the average monthly charge in the entire population of American Express Gold Card members is different from the average monthly charge in the entire population of Preferred Visa cardholders?

cesignifican of levelcommon any at rejected is 0

H

0 -7.926)<p(z :value-p

926.796.8

71

2346.80

71

800

2185

1200

2212

0)523452(

2

22

1

21

0)

21()

21(

021

:1

H

021

:0

H

nn

xxz

Two-Tailed Test for Equality of Two Population Means: Example 8-3



8-19

0.4

0.3

0.2

0.1

0.0z

f( z)

Standard Normal Distribution

NonrejectionRegion

RejectionRegion

-z0.01=-2.576 z0.01=2.576

Test Statistic=-7.926

RejectionRegion

0

Since the value of the test statistic is far below the lower critical point, the null hypothesis may be rejected, and we may conclude that there is a statistically significant difference between the average monthly charges of Gold Card and Preferred Visa cardholders.

Example 8-3: Carrying Out the Test



8-20

Example 8-3: Using the Template



8-21

84= 308=x100=n

Duracell :1 Population

1

1

1

67= 254=x100=n

Energizer :2 Population

2

2

2

Is there evidence to substantiate Duracell’s claim that their batteries last, on average, at least 45 minutes longer than Energizer batteries of the same size?

cesignifican of level

common any at rejected benot may 0

H

0.201=0.838)>p(z :value-p

838.075.10

9

45.115

9

100

267

100

284

45)254308(

2

22

1

21

0)

21()

21(

4521

:1

H

4521

:0

H

nn

xxz

Two-Tailed Test for Difference Between Two Population Means: Example 8-4



8-22

Is there evidence to substantiate Duracell’s claim that their batteries last, on average, at least 45 minutes longer than Energizer batteries of the same size?

Two-Tailed Test for Difference Between Two Population Means: Example 8-4 –

Using the Template



8-23

A large-sample (1-)100% confidence interval for the difference between two population means, 1- 2 , using independent random samples:

2

22

1

21

2

)21

(nn

zxx

A 95% confidence interval using the data in example 8-3:

]56.88,44.53[800

21851200

221296.1)452523(

2

22

1

21

2

)21

( nn

zxx

Confidence Intervals for the Difference between Two Population Means



8-24

• If we might assume that the population variances 12 and 2

2 are equal (even though unknown), then the two sample variances, s1

2 and s22,

provide two separate estimators of the common population variance. Combining the two separate estimates into a pooled estimate should give us a better estimate than either sample variance by itself.

x1

** ** ** * ** * ** * *

}Deviation from the mean. One for each sample data point.

Sample 1

From sample 1 we get the estimate s12 with

(n1-1) degrees of freedom.

Deviation from the mean. One for each sample data point.

* * ** ** * * * * ** * *

x2

}

Sample 2

From sample 2 we get the estimate s22 with

(n2-1) degrees of freedom.

From both samples together we get a pooled estimate, sp2 , with (n1-1) + (n2-1) = (n1+ n2 -2)

total degrees of freedom.

8-4 A Test for the Difference between Two Population Means: Assuming Equal Population Variances



8-25

A pooled estimate of the common population variance, based on a sample variance s1

2 from a sample of size n1 and a sample variance s22 from a sample

of size n2 is given by:

The degrees of freedom associated with this estimator is:df = (n1+ n2-2)

s n s n sn np

2 1 12

2 22

1 2

1 12

( ) ( )

The pooled estimate of the variance is a weighted average of the two individual sample variances, with weights proportional to the sizes of the two samples. That is, larger weight is given to the variance from the larger sample.

Pooled Estimate of the Population Variance



8-26

The estimate of the standard deviation of (x1 x2 is given by: sp2

)1

1

1

2n n

Test statistic for the difference between two population means, assuming equal population variances:

t =(x1 x2 1 2

sp2

where 1 2 is the difference between the two population means under the null hypothesis (zero or some other number D).

The number of degrees of freedom of the test statistic is df = ( 1 (the

number of degrees of freedom associated with sp2 , the pooled estimate of the

population variance.

) ( )

( )

)

0

1

1

1

20

2 2

n n

n n

Using the Pooled Estimate of the Population Variance



8-27

Population 1: Oil price = $27.50n1 = 14x1 = 0.317%s1 = 0.12%

Do the data provide sufficient evidence to conclude that average percentage increase in the CPI differs when oil sells at these two different prices?

HH

Critical point: t0.025

= 2.080

H0 may be rejected at the 5% level of significance

0 1 2 0

1 1 2 0

1 2 1 2 0

1 1 12

2 1 22

1 2 2

1

1

1

20 107

0 00247

0 107

0 04972 154

::

( ) ( )

( ) ( )

.

.

.

..

tx x

n s n s

n n n nPopulation 2: Oil price = $20.00n = 9x = 0.21%s = 0.11%

df = (n1

2

2

2

n2

2 14 9 2 21) ( )

Example 8-5



8-28

Do the data provide sufficient evidence to conclude that average percentage increase in the CPI differs when oil sells at these two different prices?


P-value = P-value = 0.0430, so 0.0430, so reject Hreject H00 at at the 5% the 5% significance significance level.level.



8-29

Population 1: Before Reductionn1 = 15

x1 = $6598

s1 = $844

The manufacturers of compact disk players want to test whether a small price reduction is enough to increase sales of their product. Is there evidence that the small price reduction is enough to increase sales of compact disk players?

cesignifican of level 10% at theeven rejected benot may 0

H

1.316=0.10

t:point Critical

91.096.298

272

25.89375

272

12

1

15

1

21215

2669)11(2844)14(

0)65986870(

2

1

1

1

221

22

)12

(21

)11

(

0)

12()

12(

012

:1

H

012

:0

H

nnnn

snsn

xxt

Population 2: After Reductionn = 12x = $6870s = $669

df = (n1

2

2

2

n2

2 15 12 2 25) ( )

Example 8-6



8-30


P-value = P-value = 0.1858, so do 0.1858, so do not reject Hnot reject H00 at the 5% at the 5% significance significance level.level.



8-31

543210-1-2-3-4-5

0.4

0.3

0.2

0.1

0.0t

f(t)

t Distribution: df = 25

NonrejectionRegion

RejectionRegion

t0.10=1.316

Test Statistic=0.91

Since the test statistic is less than t0.10, the null hypothesis cannot be rejected at any reasonable level of significance. We conclude that the price reduction does not significantly affect sales.

Example 8-6: Continued



8-32

A (1-) 100% confidence interval for the difference between two population means, 1- 2 , using independent random samples and assuming equal population variances:

( )x x t sn np1 2

2

2 1

1

1

2

A 95% confidence interval using the data in Example 8-6:

( ) ( ) . ( )( . ) [ . , . ]x x t s p n n1 2

2

2 1

1

1

26870 6598 2 06 595835 0 15 343 85 887 85

Confidence Intervals Using the Pooled Variance



8-33

Confidence Intervals Using the Pooled Variance and the Template-

Example 8-6

Confidence IntervalConfidence Interval

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Penutup

• Pembahasan materi dilanjutkan dengan Materi Pokok 18 (Pembandingan Dua Populasi-2)

Pertemuan 17 Pembandingan Dua Populasi-1

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