Pertemuan 04 Baru-Estimasi Dua Populasi

8/10/2019 Pertemuan 04 Baru-Estimasi Dua Populasi

1/40

Pertemuan 4

Estimasi Dua Populasi

Rudi Salam

[email protected] - 1


2/40

Membandingkan Dua Populasi

Previously we looked at techniques to estimate and createconfidence interval for one population:

Population Mean , Population Variance 2, andPopulation Proportion p

Kita tetap menggunakan parameter tersebut untukduapopulasi, di mana perhatiannya sekarang adalah:

The differencebetween two means. The ratioof two variances.

The differencebetween two proportions.



3/40

Estimasi Dua PopulasiMengestimasi nilai

dua populasi

Rata-rata

Populasi,independent

samples

SampelBerpasangan

ProporsiPopulasi

Group 1 vs.independentGroup 2

Same groupbefore vs. aftertreatment

Proportion 1 vs.Proportion 2

Examples:

VariansPopulasi

Varians 1 vs.Varians 2



4/40

Pertemuan 4

Estimasi Dua Rata-rata

Rudi Salam



5/40

Perbedaan Dua Rata-Rata

Dalam mengestimasi dan menguji perbedaan di antara

rata-rata dua populasi, kita mengambil sampel acak darimasing-masing populasi. Kita akan menganggap keduasampel adalah independen, maksudnya, kedua sampelsama sekali tidak berhubungan satu sama lain.

(Likewise, we consider for popn 2)

Sample, size: n1

Population 1

Parameters: Statistics:



6/40

Perbedaan Dua Rata-rata

Population means,independent

samples

1 and 2 known

1 and 2 unknown,

n1 and n2 30

1 and 2 unknown,n1 or n2 < 30

Tujuan: Form a confidenceinterval for the differencebetween two population

means, 1 2

The point estimate for thedifference is

x1 x2

*



7/40

Sampel Independen


samples

1 and 2 known

1 and 2 unknown,

n1 and n2 30


Different data sources

Unrelated Independent

Sample selected fromone population has noeffect on the sampleselected from the otherpopulation

Use the difference between 2sample means

Use z test or pooled variancet test

*



8/40


samples

1 and 2 known

1 and 2 unknown,

n1 and n2 30


1 dan 2 diketahui

Assumptions:

Samples are randomly andindependently drawn

population distributions arenormal or both sample sizes

are

30

Population standarddeviations are known

*



9/40


samples

1 and 2 known

1 and 2 unknown,

n1 and n2 30


and the standard error of

x1 x2 is

When 1 and 2 are known and

both populations are normal orboth sample sizes are at least 30,the test statistic is a z-value

2

2

2

1

2

1

xx n

n

21

(continued)1 dan 2 diketahui

*



10/40


samples

1 and 2 known

1 and 2 unknown,

n1 and n2 30


2

2

2

1

2

1/221

n

n

zxx

Confidence interval untuk1 2 adalah:

1 dan 2 diketahui (continued)

*



11/40


samples

1 and 2 known

1 and 2 unknown,

n1 and n2 30


1 and 2 unknown, large samples

Assumptions:

Samples are randomly andindependently drawn

both sample sizesare 30

Population standarddeviations are unknown

*



12/40


samples

1 and 2 known

1 and 2 unknown,

n1 and n2 30


1 and 2 unknown, large samples

Forming intervalestimates:

use sample standarddeviation s to estimate

the test statistic is a z value

(continued)

*



13/40


samples

1 and 2 known

1 and 2 unknown,

n1 and n2 30


2

2

2

1

2

1/221

n

s

n

szxx

The confidence interval for1 2 is:

1 and 2 unknown, large samples(continued)

*



14/40


samples

1 and 2 known

1 and 2 unknown,

n1 and n2 30


1 = 2 unknown, small samples

Assumptions: populations are normally

distributed

the populations have equalvariances

samples are independent

*



15/40


samples

1 and 2 known

1 and 2 unknown,

n1 and n2 30



Forming interval

estimates:

The population variances

are assumed equal, so usethe two sample standarddeviations and pool them toestimate

the test statistic is a t valuewith (n1 + n2 2) degreesof freedom

(continued)

*



16/40


samples

1 and 2 known

1 and 2 unknown,

n1 and n2 30



Standar deviasigabungannya adalah:

(continued)

2nn

s1ns1ns

21

2

22

2

11p

*



17/40


samples

1 and 2 known

1 and 2 unknown,

n1 and n2 30


21

p/221

n1

n1stxx

The confidence interval for

1 2 is:

1 = 2 unknown , small samples(continued)

Where t/2 has (n1 + n2 2) d.f.,

and

2nn

s1ns1ns

21

2

22

2

11p

*



18/40


samples

1 and 2 known

1 and 2 unknown,

n1 and n2 30


Interval kepercayaan untuk

1 2 jika 1 2 adalah:

1 2 unknown , small samples(continued)

Where t/2 adalah nilai distribusi tdengan derajat bebas v di mana :

*

2 2

1 2

1 2 /21 2

s s

x x n nt

22 2

1 1 2 2

2 22 2

1 1 2 2

1 2

1 1

s n s nv

s n s n

n n



19/40

Sampel BerpasanganTests Means of 2 Related Populations

Paired or matched samples Repeated measures (before/after)

Use difference between paired values:

Eliminates Variation Among Subjects

Assumptions:

Both Populations Are Normally Distributed

Or, if Not Normal, use large samples

Pairedsamples

d = x1- x2



20/40

Paired DifferencesThe ith paired difference is di , where

Pairedsamples

di = x1i - x2i

The point estimate for

the population meanpaired difference is d :

1n

)d(d

s

n

1i

2

i

d

n

d

d

n

1i

i

The sample standarddeviation is

n is the number of pairs in the paired sample



21/40

Paired Differences

The confidence interval for d isPairedsamples

1n

)d(d

s

n

1i

2i

d

n

std d

/2

Where t/2 has

n - 1 d.f. and sd is:

(continued)

n is the number of pairs in the paired sample



22/40

Contoh (varians known)

Suatu ujian kimia yang telah dibakukandiberikan pada 50 siswa wanita dan 75 siswapria. Nilai rata-rata wanita 76, sedangkan muridpria mendapat nilai rata-rata 82. Carilah CI 96%untuk selisih 1-2, bila 1 menyatakan rataannilai semua siswa pria dan 2 rataan nilaisemua siswa wanita yang mungkin akanmengikuti ujian ini. Anggap simpangan bakupopulasi untuk wanita dan pria, masing-masing6 dan 8.



23/40

Contoh (varians known)

Solusi:

Estimasi titik untuk1-2 adalah

Dengan =0,04 diperoleh z0,02=2,054.

Selang kepercayaannya adalah:

1 2 82 76 6x x

1 2

1 2

64 36 64 366 2,054 6 2,054

75 50 75 50

atau

3,43 8,57



24/40

Pertemuan 4

Selisih Dua Proporsi

Rudi Salam



25/40

Two Population ProportionsGoal: Form a confidence interval for

or test a hypothesis about thedifference between two populationproportions, p1 p2

The point estimate forthe difference is p1 p2

ProporsiPopulasi

Assumptions:

n1p1 5 , n1(1-p1) 5

n2p2

5 , n2(1-p2)

5



26/40

Confidence Interval for

Two Population Proportions

ProporsiPopulasi

1 1 2 2

1 2 /21 2

(1 ) (1 )

n n

p p p pp p z

The confidence interval forp1 p2 is:



27/40

Contoh

A consumer packaged goods (CPG) company is test

marketing two new versions of soap packaging. Version one(bright colors) is distributed in one supermarket, whileversion two (simple colors) is in another.

Here is

the summary data:



28/40

Contoh

Create a 95% confidence interval for thedifference between the two proportions ofpackaged soap sales



29/40

Pertemuan 4

Rasio Dua Varians

Rudi Salam



30/40

Rasio Dua Varians

When looking at two population variances, weconsider the ratioof the variances, i.e. theparameter of interest to us is:

Ingat:

2 22 1

1 2 1 1 2 22 2

1 2

, di mana 1 dan 1s

F F v v v n v ns


2

1

2

2


31/40

Rasio Dua Varians

Sehingga

dapat ditulis: 1 2 1 21

2 2, , 1P f v v F f v v

f1-/2 f/2

1-/2 /2



32/40

Rasio Dua Varians

Diuraikan:

1 2 1 212 2

2 2

2 11 2 1 22 2

11 22 2

2 2 2

1 1 1

2 2 2

2 1 2 2 2 1 21

2 2

2 2 2

1 1 12 12 2 2

2 1 2 2 2 2

2

, , 1

, , 1

1 11

, ,

1, 1

,

P f v v F f v v

sP f v v f v v

s

s sP

s f v v s f v v

s sP f v v

s f v v s



33/40

Rasio Dua Varians

Bila S12 dan s2

2 adalah varians dari sampel bebas masing-

masing ukuran n1 dan n2 dari populasi normal makaselang kepercayaan (1-)100% untuk adalah

Di mana f/2(v1,v2) adalah nilai f dengan df v1=n1-1 dan

v2=n2-1 sehingga luas di sebelah kanannya adalah /2,dan f/2(v2,v1) adalah nilai f yang sama dengan dfv2=n2-1 dan v1=n1-1.

2 2 2

1 1 1

2 12 2 22 1 2 2 2 2

2

1

,,

s s

f v vs f v v s


2 2

1 2


34/40

Contoh

A sample of 150 people was randomly drawn. Eachperson was identified as a consumer or a non-consumer of high-fiber cereal. For each person thenumber of calories consumed at lunch was recorded.

The data



35/40

Contoh

If we wanted to determine the 95% confidenceinterval estimate of the ratio of the twopopulation variances, we would proceed asfollows

The confidence interval

estimator for is:2 2

1 2



36/40

Contoh

Dari tabel:

The 95% confidence interval estimate of the ratio of thetwo population variances is:

That is, we estimate that lies between 0.2388

and 0.6614. Note that one (1.00) is not within this interval

2 2

1 2



37/40

Soal 1

Suatu perusahaan taksi ingin menentukan apakahmembeli ban merk A atau merk B untuk armadataksinya. Untuk menaksir perbedaan kedua merk,dilakukan suatu percobaan menggunakan 12 ban

dari tiap merk. Ban dipakai sampai aus. Hasilmerk A: X1=36.300 km, s1=5000 km ; merk B:X2=38.100 km, s2=6100 km. Hitunglah selang

kepercayaan 95% untuk 1-2, kedua populasidiasumsikan normal



38/40

Soal 2

Kembali ke soal 1, hitunglah selang

kepercayaan 99% untuk A-B bila suatu bandari tiap merk dipasang secara acak di rodabelakang delapan taksi dan jarak yang ditempuh(dalam km) adalah

Anggap selisih jarak berdistribusi hampir normal

Taksi 1 2 3 4 5 6 7 8

Merk A 34400 45500 36700 32000 48400 32800 38100 30100

Merk B 36700 46800 37700 31100 47800 36400 38900 31500



39/40

Soal 3

Seorang ahli genetika ingin mengetahui proporsipria dan wanita dalam penduduk yangmempunyai sejenis penyakit darah. Suatusampel acak 1000 pria menunjukkan 250 yangterserang, sedangkan sebanyak 275 dari 1000wanita yang diperiksa juga terserang penyakittersebut. Hitunglah selang kepercayaan 95%untuk selisih kedua proporsi pria dan wanitayang terserang penyakit tersebut.


40/40

Soal 4

Suatu perusahaan taksi ingin menentukan apakah

membeli ban merk A atau merk B untuk armadataksinya. Untuk menaksir perbedaan kedua merk,dilakukan suatu percobaan menggunakan 12 ban

dari tiap merk. Ban dipakai sampai aus. Hasilmerk A: XA=36300 km, sA=5000 km ; merk B:XB=38100 km, sB=6100 km. Buatlah selang

kepercayaan 90% untuk12/ 22. Apakah asumsibahwa 1

2=22 mendapat dukungan dalam

membuat selang kepercayaan untuk1-2.

Pertemuan 04 Baru-Estimasi Dua Populasi

Documents

Transcript of Pertemuan 04 Baru-Estimasi Dua Populasi