Statistik Bisnis. Week 11 Two-Sample Tests

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1 Statistik Bisis Week Two-Sample Tests

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, assumed equal ukow, assumed uequal

4 Differece Betwee Two Meas Idepedet Samples ukow, assumed equal ukow, assumed 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, assumed 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, assumed 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

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 assumed equal Idepedet Samples ukow, assumed equal ukow, assumed 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 assumed equal

9 Hypothesis tests for µ - µ with σ ad σ ukow ad assumed equal (cotiued) Idepedet Samples ukow, assumed equal ukow, assumed 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 assumed equal Idepedet Samples ukow, assumed equal ukow, assumed uequal The cofidece iterval for μ μ is: X X t a/ Sp Where t α/ has d.f. = +

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 assumed equal Idepedet Samples ukow, assumed equal ukow, assumed 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 assumed to be equal

16 (cotiued) Hypothesis tests for µ - µ with σ ad σ ukow ad ot assumed 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, assumed equal ukow, assumed uequal

17 EXERCISE

18 0.8 (cot d) Meurut peelitia terbaru, aak-aak yag meoto kartu dega ikla sebuah produk makaa, rata-rata, maka 8,5 gram Goldfish crackers. Semetara itu, aak-aak yag meoto kartu tapa ikla tersebut rata-rata maka 9,7 gram Goldfish crackers. Terdapat 8 aak dalam peelitia ii, yaitu 59 aak utuk setiap kelompok. Simpaga baku sampel utuk aak yag meoto kartu dega ikla adalah 8,6 gram da simpaga baku sampel utuk aak meoto tapa ikla adalah 7,9 gram.

19 0.8 Dega megasumsika variasi populasi sama, apakah terdapat cukup bukti bahwa rata-rata jumlah Goldfish crackers yag dimaka oleh aak-aak yag meoto kartu dega ikla lebih tiggi?

20 0.0 (cot d) 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

21 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?

22 0.6 (cot d) Apakah aak-aak megguaka telepo selular? Sepertiya demikia, meurut peelitia baru-baru ii, peggua telepo selular berusia dibawah tahu rata-rata 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.

23 0.6 (cot d) 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.

24 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)?

26 Related Populatios The Paired Differece Test Tests Meas of Related Populatios Related samples Paired or matched samples Repeated measures (before/after) Use differece betwee paired values: D i = X i - X i Elimiates Variatio Amog Subjects Assumptios: Both Populatios Are Normally Distributed Or, if ot Normal, use large samples

27 Related Populatios The Paired Differece Test (cotiued) The i th paired differece is D i, where Related samples D i = X i - X i The poit estimate for the paired Differece populatio mea μ D is D : D i D i The sample stadard deviatio is S D is the umber of pairs i the paired sample S D i (D i D)

28 The Paired Differece Test: Fidig t STAT The test statistic for μ D is: Related samples t STAT D μ S D D Where t STAT has - d.f.

29 The Paired Differece Test: Possible Hypotheses Paired Samples Lower-tail test: Upper-tail test: Two-tail test: H 0 : μ D 0 H : μ D < 0 H 0 : μ D 0 H : μ D > 0 H 0 : μ D = 0 H : μ D 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/ Where t STAT has - d.f.

30 The Paired Differece Cofidece Iterval The cofidece iterval for μ D is Related samples D t a / S D where S D i (D i D)

31 Paired Differece Test: Example Assume you sed your salespeople to a customer service traiig workshop. Has the traiig made a differece i the umber of complaits? You collect the followig data: Number of Complaits: () - () Salesperso Before () After () Differece, D i D = D i C.B T.F M.H. 3 - R.K M.O S D = (D i D)

32 Paired Differece Test: Solutio t Has the traiig made a differece i the umber of complaits (at the 0.0 level)? STAT a =.0 μ D D S / D H 0 : μ D = 0 H : μ D 0 D = - 4. t = ± d.f. = - = 4 Test Statistic: / Reject a/ Reject Decisio: Do ot reject H 0 (t stat is ot i the reject regio) a/ Coclusio: There is isufficiet evidece there is sigificat chage i the umber of complaits.

33 EXERCISE

34 0.0 (cot d) Sembila ahli memberi peilaia pada dua merek kopi Kolombia dalam sebuah percobaa taste-testig. Sebuah ratig berskala 7 ( = sagat tidak memuaska, 7 = sagat memuaska) diberika utuk empat karakteristik: rasa, aroma, richess, da keasama. Data berikut meujukka total ratig dari keempat karakteristik tersebut. Merek Ahli A B C.C. 4 6 S.E. 7 7 E.G. 9 B.L. 4 7 C.M. 5 C.N. 6 7 G.N. 7 6 R.M. 5 7 P.V. 3

35 0.0 a. Dega tigkat sigifikasi 0,05, adakah bukti yag meujukka bahwa rata-rata ratig atara kedua merek tersebut berbeda? b. Asumsi apa yag diperluka megeai distribusi populasi utuk dapat melakuka uji ii? c. Buat da iterpretasika estimasi selag kepercayaa perbedaa rata-rata ratig atara kedua merek tersebut dega tigkat kepercayaa 95%.

36 0.4 (cot d) Kaker plasma darah (Multiple myeloma), dapat diketahui dari peigkata pembetukka pembuluh darah (agiogeesis) pada sumsug tulag yag merupakah faktor peetu keberlagsuga hidup pederitaya. Salah satu pegobata utuk peyakit ii adala trasplatasi sel iduk (stem cell). Berikut merupaka data kepadata pembuluh darah pada sumsum tulag pada pasie yag telah meyelesaika prosedur trasplatasi sel iduk (diukur dega tes darah da uri).

37 0.4 (cot d) Pasie Sebelum Sesudah

38 0.4 Pegukura dilakuka tepat sebelum trasplatasi sel iduk da ketika seluruh prosedur trasplatasi sel iduk selesai dilaksaaka. Pada tigkat sigifikasi 0,05, apakah terdapat bukti bahwa rata-rata kepadata pembuluh darah pada susum tulag sebelum trasplatasi sel iduk lebih tiggi jika dibadigka dega setelah trasplatasi?

40 Two Populatio Proportios Populatio proportios Goal: test a hypothesis or form a cofidece iterval for the differece betwee two populatio proportios, π π The poit estimate for the differece is p p

41 Two Populatio Proportios Populatio proportios I the ull hypothesis we assume the ull hypothesis is true, so we assume π = π ad pool the two sample estimates The pooled estimate for the overall proportio is: p X X where X ad X are the umber of items of iterest i samples ad

42 Two Populatio Proportios ) ( p p π π p p Z STAT The test statistic for π π is a Z statistic: (cotiued) X, p X, p X X p where Populatio proportios

43 Hypothesis Tests for Two Populatio Proportios Populatio proportios Lower-tail test: Upper-tail test: Two-tail test: H 0 : π π H : π < π i.e., H 0 : π π 0 H : π π < 0 H 0 : π π H : π > π i.e., H 0 : π π 0 H : π π > 0 H 0 : π = π H : π π i.e., H 0 : π π = 0 H : π π 0

44 Hypothesis Tests for Two Populatio Proportios (cotiued) Lower-tail test: H 0 : π π 0 H : π π < 0 Populatio proportios Upper-tail test: H 0 : π π 0 H : π π > 0 Two-tail test: H 0 : π π = 0 H : π π 0 a a a/ a/ -z a z a -z a/ z a/ Reject H 0 if Z STAT < -Z a Reject H 0 if Z STAT > Z a Reject H 0 if Z STAT < -Z a/ or Z STAT > Z a/

45 Hypothesis Test Example: Two populatio Proportios Is there a sigificat differece betwee the proportio of me ad the proportio of wome who will vote Yes o Propositio A? I a radom sample, 36 of 7 me ad 35 of 50 wome idicated they would vote Yes Test at the.05 level of sigificace

46 Hypothesis Test Example: Two populatio Proportios The hypothesis test is: H 0 : π π = 0 (the two proportios are equal) H : π π 0 (there is a sigificat differece betwee proportios) The sample proportios are: Me: p = 36/7 = 0.50 Wome: p = 35/50 = 0.70 The pooled estimate for the overall proportio is: p X X (cotiued)

47 Hypothesis Test Example: Two populatio Proportios (cotiued) Reject H 0 Reject H 0 The test statistic for π π is: z STAT p p π π p( p) (. 58 ) 7 Critical Values = ±.96 For a = Decisio: Reject H 0.05 Coclusio: There is evidece of a differece i proportios who will vote yes betwee me ad wome.

48 Cofidece Iterval for Two Populatio Proportios Populatio proportios The cofidece iterval for π π is: p p Z a/ p ( p ) p ( p )

50 Testig for the Ratio Of Two Populatio Variaces Hypotheses F STAT Tests for Two Populatio Variaces F test statistic H 0 : σ = σ H : σ σ H 0 : σ σ H : σ > σ Where: S = Variace of sample (the larger sample variace) = sample size of sample S = Variace of sample (the smaller sample variace) = sample size of sample = umerator degrees of freedom = deomiator degrees of freedom S S

51 The F Distributio The F critical value is foud from the F table There are two degrees of freedom required: umerator ad deomiator The larger sample variace is always the umerator Whe F STAT I the F table, S S df = ; df = umerator degrees of freedom determie the colum deomiator degrees of freedom determie the row

52 Fidig the Rejectio Regio H 0 : σ = σ H : σ σ H 0 : σ σ H : σ > σ a/ F a 0 Do ot reject H 0 F α/ Reject H 0 0 Do ot reject H 0 F α Reject H 0 F Reject H 0 if F STAT > F α/ Reject H 0 if F STAT > F α

53 F Test: A Example You are a fiacial aalyst for a brokerage firm. You wat to compare divided yields betwee stocks listed o the NYSE & NASDAQ. You collect the followig data: NYSE NASDAQ Number 5 Mea Std dev.30.6 Is there a differece i the variaces betwee the NYSE & NASDAQ at the a = 0.05 level?

54 F Test: Example Solutio Form the hypothesis test: H 0 : σ H : σ σ σ Fid the F critical value for a = 0.05: (there is o differece betwee variaces) (there is a differece betwee variaces) Numerator d.f. = = = 0 Deomiator d.f. = = 5 = 4 F α/ = F.05, 0, 4 =.33

55 F Test: Example Solutio The test statistic is: (cotiued) H 0 : σ = σ H : σ σ S.30 F STAT.56 S.6 a/ =.05 F STAT =.56 is ot i the rejectio regio, so we do ot reject H 0 0 Do ot reject H 0 Reject H 0 F 0.05 =.33 F Coclusio: There is isufficiet evidece of a differece i variaces at a =.05

56 EXERCISE

57 0.30 (cot d) Apakah tahu ii diperluka usaha lebih utuk keluar dari sebuah mailig list dari tahu sebelumya? Sebuah peelitia dari 00 peritel olie besar meujukka data berikut: PERLU TIGA ATAU LEBIH KLIK SEBELUM KELUAR TAHUN Ya Tidak

58 0.30 a. Tetuka hipothesis kosog da hipothesis alteratif utuk megetahui apakah diperluka usaha lebih utuk keluar dari sebuah mailig list jika dibadigka dega tahu sebelumya. b. Lakuka uji hipothesis utuk poi (a), dega megguaka tigkat sigifikasi 0,05. c. Apakah hasil dari poi (b) sesuai dega klaim bahwa diperluka usaha lebih utuk keluar dari sebuah mailig list jika dibadigka dega tahu sebelumya?

59 0.34 (cot d) Bagaimaa perasaa orag Amerika megeai ikla di halama web? Sebuah survey yag dilakuka pada.000 peggua iteret dewasa diketahui bahwa 670 orag meetag adaya ikla di halama web. Misalka sebuah survei lai pada.000 orag peggua iteret berusia 7 tahu meemuka bahwa 50 orag meetag adaya ikla di halama web. Dega megguaka tigkat sigifikasi 0,05, adakah bukti bahwa terdapat perbedaa proporsi atara peggua iteret dewasa da peggua iteret berusia 7 tahu yag meetag ikla?

60 0.46 (cot d) 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 meyebarjab 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 S

61 0.46 a. Dega tigkat sigifikasi 0,05, adakah bukti yag meujukka perbedaa sebara (variability) kecemasa komputer (computer axiety) yag dialami pria da waita? b. Asumsi apa yag ada perluka tetag kedua populasi tersebut utuk dapat megguaka uji F? c. Berdasarka poi (a) da (b), uji t maakah yag seharusya ada guaka utuk meguji apakah ada perbedaa yag sigifika atara kecemasa komputer yag dirasaka oleh waita da pria?

62 THANK YOU