Statistik Bisnis 2. Week 5 Comparing the Means of Two Independent Populations

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1 Statistik Bisis Week 5 Comparig the Meas of Two Idepedet Populatios

2 Learig Objectives The meas of two idepedet populatios The meas of two related populatios I this chapter, you lear how to use hypothesis testig for comparig the differece betwee: The proportios of two idepedet populatios The variaces of two idepedet populatios by testig the ratio of the two variaces

3 Two-Sample Tests Two-Sample Tests Populatio Meas Populatio Proportios Populatio Variaces Idepedet Samples Related Samples ukow, equal ukow, uequal

4 Differece Betwee Two Meas Idepedet Samples ukow, equal ukow, uequal Goal: Test hypothesis or form a cofidece iterval for the differece betwee two populatio meas, μ μ The poit estimate for the differece is X X

5 Differece Betwee Two Meas: Idepedet Samples Idepedet Samples ukow, equal Differet data sources Urelated Idepedet Sample selected from oe populatio has o effect o the sample selected from the other populatio Use S p to estimate ukow σ. Use a Pooled-Variace t test. ukow, uequal Use S ad S to estimate ukow σ ad σ. Use a Separate-Variace t test.

6 Hypothesis Tests for Two Populatio Meas Two Populatio Meas, Idepedet Samples Lower-tail test: Upper-tail test: Two-tail test: H 0 : μ μ H : μ < μ i.e., H 0 : μ μ 0 H : μ μ < 0 H 0 : μ μ H : μ > μ i.e., H 0 : μ μ 0 H : μ μ > 0 H 0 : μ = μ H : μ μ i.e., H 0 : μ μ = 0 H : μ μ 0

Hypothesis tests for μ μ Two Populatio Meas,

a t a -t a/ t a/ Reject H 0 if t STAT < -t a Reject

7 Hypothesis tests for μ μ Two Populatio Meas, Idepedet Samples Lower-tail test: H 0 : μ μ 0 H : μ μ < 0 Upper-tail test: H 0 : μ μ 0 H : μ μ > 0 Two-tail test: H 0 : μ μ = 0 H : μ μ 0 a a a/ a/ -t a t a -t a/ t a/ Reject H 0 if t STAT < -t a Reject H 0 if t STAT > t a Reject H 0 if t STAT < -t a/ or t STAT > t a/

8 Hypothesis tests for µ - µ with σ ad σ ukow ad equal Idepedet Samples ukow, equal ukow, uequal Assumptios: Samples are radomly ad idepedetly draw Populatios are ormally distributed or both sample sizes are at least 30 Populatio variaces are ukow but equal

9 Hypothesis tests for µ - µ with σ ad σ ukow ad equal (cotiued) Idepedet Samples ukow, equal ukow, uequal The pooled variace is: S p S ( ) ( The test statistic is: t STAT S ) Where t STAT has d.f. = ( + ) X X μ μ S p

10 Cofidece iterval for µ - µ with σ ad σ ukow ad equal Idepedet Samples ukow, equal ukow, uequal The cofidece iterval for μ μ is: X X t a/ Sp Where t α/ has d.f. = +

Pooled-Variace t Test Example You are a fiacial aalyst for a brokerage firm. Is there a differece i divided yield betwee stocks listed o the NYSE & NASDAQ?

11 Pooled-Variace t Test Example You are a fiacial aalyst for a brokerage firm. Is there a differece i divided yield betwee stocks listed o the NYSE & NASDAQ? You collect the followig data: NYSE NASDAQ Number 5 Sample mea Sample std dev.30.6 Assumig both populatios are approximately ormal with equal variaces, is there a differece i mea yield (a = 0.05)?

12 Pooled-Variace t Test Example: Calculatig the Test Statistic The test statistic is: H0: μ - μ = 0 i.e. (μ = μ ) H: μ - μ 0 i.e. (μ μ ) (cotiued) t STAT X X μ μ S p S P S S.30 5 ( ) ( ).6 (-) (5 ).50

13 Pooled-Variace t Test Example: Hypothesis Test Solutio H 0 : μ - μ = 0 i.e. (μ = μ ) Reject H 0 Reject H 0 H : μ - μ 0 i.e. (μ μ ) a = 0.05 df = = 44 Critical Values: t = ± t.040 Test Statistic: t STAT Decisio: Reject H 0 at a = 0.05 Coclusio: There is evidece of a differece i meas.

14 Pooled-Variace t Test Example: Cofidece Iterval for µ - µ DCOVA Sice we rejected H 0 ca we be 95% cofidet that µ NYSE > µ NASDAQ? 95% Cofidece Iterval for µ NYSE - µ NASDAQ X X t / Sp a (0.009,.47) Sice 0 is less tha the etire iterval, we ca be 95% cofidet that µ NYSE > µ NASDAQ

15 Hypothesis tests for µ - µ with σ ad σ ukow, ot equal Idepedet Samples ukow, equal ukow, uequal Assumptios: Samples are radomly ad idepedetly draw Populatios are ormally distributed or both sample sizes are at least 30 Populatio variaces are ukow ad caot be to be equal

16 (cotiued) Hypothesis tests for µ - µ with σ ad σ ukow ad ot equal S S S S The test statistic is: STAT S S μ μ X X t t STAT has d.f. ν = Idepedet Samples ukow, equal ukow, uequal

17 EXERCISE

18 0.7 Meurut sebuah peelitia baru-baru ii, ketika berbelaja barag-barag mewah dalam jariga (olie shoppig), pria rata-rata membaelajaka $,40, semetara waita rata-rataya $,57. Misalka peelitia tersebut dilakuka pada 600 orag pria da 700 orag waita, da simpaga baku dari jumlah uag yag dibelajaka tersebut adalah $,00 utuk pria da $,000 utuk waita. a. Tetuka hipothesis kosog da alteratifya jika ada igi meetuka apakah rata-rata uag yag dibelajaka oleh pria lebih bayak daripada waita b. Pada koteks peelitia ii, apakah yag dimaksud dega kesalaha tipe I? c. Pada koteks peelitia ii, apakah yag dimaksud dega kesalaha tipe I? d. Dega tigkat sigifikasi 0.0, apakah terdapat bukti bahwa rata-rata jumlah uag yag dibelajaka pria lebih bayak daripada waita?

19 0.0 () Computer Axiety Ratig Scale (CARS) megukur tigkat kecemasa terhadap komputer (computer axiety), dega skala dari 0 (tidak ada kecemasa) higga 00 (sagat cemas). Peeliti dari Miami Uiversity meyebarka CARS pada 7 mahasiswa bisis. Salah satu tujua dari peelitia tersebut adalah meetuka apakah terdapat perbedaa tigkat kecemasa komputer yag dirasaka oleh mahasiswa bisis pria da waita. Mereka meemuka data berikut: Pria Waita X 40,6 36,85 S 3,35 9,4 00 7

20 0.0 () a. Dega tigkat sigifikasi 0.05, apakah terdapat bukti bahwa kecemasa komputer yag dirasaka oleh mahasiswa bisis waita berbeda dari yag dirasaka oleh mahasiswa bisis pria? b. Apakah asumsi-asumsi yag harus ada buat megeai kedua populasi tersebut utuk dapat megguaka uji t?

21 0.6 () Apakah aak-aak megguaka telepo selular? Sepertiya demikia, meurut peelitia baru-baru ii, peggua telepo selular berusia dibawah tahu ratarata melakuka 37 paggila telepo per bula. Cukup tiggi, jika dibadigka dega 3 paggila telepo per bula yag dilakuka oleh peggua telepo selular berusia 3 higga 7 tahu. Misalka hasil tersebut diambil dari sampel 50 orag peggua telepo selular utuk setiap grup peggua da simpaga baku sampel peggua telepo selular berusia dibawah tahu adalah 5,7 paggila telepo per bula da simpaga baku sampel peggua telepo selular berusia 3 higga 7 tahu adalah 67,6 paggila telepo per bula.

22 0.6 () a. Dega megasumsika bahwa variasi populasi dari peggua telepo selular adalah sama, adakah bukti yag meujukka bahwa terdapat perbedaa rata-rata pegguaa telepo selular atara kelompok usia dibawah tahu da kepompok usia 3 higga 7 tahu? (Guaka tigkat sigifikasi 0,05.) b. Selai kesamaa variasi, sebutka asumsi lai yag diperluka dalam melakuka uji hipotesis pada poi (a)?

23 THANK YOU

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