Statistics. Chapter 10 Two-Sample Tests. Copyright 2013 Pearson Education, Inc. publishing as Prentice Hall. Chap 10-1

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1 Statistics Chapter 0 Two-Sample Tests Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-

2 Learig Objectives I this chapter, you lear How to use hypothesis testig for comparig the differece betwee: The meas of two idepedet populatios The meas of two related populatios The proportios of two idepedet populatios The variaces of two idepedet populatios The meas of more tha two populatios Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-

3 Two-Sample Tests Two-Sample Tests Populatio Meas, Idepedet Samples Populatio Meas, Related Samples Populatio Proportios Populatio Variaces Examples: Group vs. Group Same group before vs. after treatmet Proportio vs. Proportio Variace vs. Variace Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-3

4 Differece Betwee Two Meas Populatio meas, idepedet samples * σ ad σ ukow, assumed equal σ ad σ ukow, ot assumed equal 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 Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-4

5 Differece Betwee Two Meas: Populatio meas, idepedet samples * Idepedet Samples Differet data sources Urelated Idepedet Sample selected from oe populatio has o effect o the sample selected from the other populatio σ ad σ ukow, assumed equal Use S p to estimate ukow σ. Use a Pooled-Variace t test. σ ad σ ukow, ot assumed equal Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Use S ad S to estimate ukow σ ad σ. Use a Separate-Variace t test. Chap 0-5

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 Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-6

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 α α α/ α/ -t α t α -t α/ t α/ Reject H 0 if t STAT < -t α Reject H 0 if t STAT > t α Reject H 0 if t STAT < -t α/ or t STAT > t α/ Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-7

8 Hypothesis tests for µ - µ with σ ad σ ukow ad assumed equal Populatio meas, idepedet samples σ ad σ ukow, assumed equal σ ad σ ukow, ot assumed equal * 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 Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-8

9 Hypothesis tests for µ - µ with σ ad σ ukow ad assumed equal Populatio meas, idepedet samples The pooled variace is: S p ( ) ( ) S + ( ) + ( S ) (cotiued) σ ad σ ukow, assumed equal σ ad σ ukow, ot assumed equal * The test statistic is: t STAT ( X ) ( ) X µ µ S Where t STAT has d.f. ( + ) p + Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-9

10 Cofidece iterval for µ - µ with σ ad σ ukow ad assumed equal Populatio meas, idepedet samples σ ad σ ukow, assumed equal σ ad σ ukow, ot assumed equal * The cofidece iterval for µ µ is: ( ) X X ± t + α/ Sp Where t α/ has d.f. + Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-0

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 Sample Size 5 Sample mea Sample std dev.30.6 Assumig both populatios are approximately ormal with equal variaces, is there a differece i mea yield (α 0.05)? Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-

12 Pooled-Variace t Test Example: The test statistic is: Calculatig the Test Statistic H 0 : µ -µ 0 i.e. (µ µ ) H : µ -µ 0 i.e. (µ µ ) (cotiued) t ( X X ) ( µ µ ) ( ) STAT S p S P ( ) S + ( ) S ( ).30 + ( 5 ) ( ) + ( ).6 (-) + (5 ).50 Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-

13 Pooled-Variace t Test Example: Hypothesis Test Solutio H 0 : µ -µ 0 i.e. (µ µ ) H : µ -µ 0 i.e. (µ µ ) α 0.05 df Critical Values: t ±.054 Test Statistic: t STAT Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall.040 Reject H 0 Reject H Decisio: Reject H 0 at α 0.05 Coclusio: There is evidece of a differece i meas. t.040 Chap 0-3

14 Excel Pooled-Variace t test Comparig NYSE & NASDAQ Pooled-Variace t Test for the Differece Betwee Two Meas (assumes equal populatio variaces) Data Hypothesized Differece 0 Level of Sigificace 0.05 Populatio Sample Sample Size COUNT(DATA!$A:$A) Sample Mea 3.7 AVERAGE(DATA!$A:$A) Sample Stadard Deviatio.3 STDEV(DATA!$A:$A) Populatio Sample Sample Size 5 COUNT(DATA!$B:$B) Sample Mea.53 AVERAGE(DATA!$B:$B) Sample Stadard Deviatio.6 STDEV(DATA!$B:$B) Itermediate Calculatios Populatio Sample Degrees of Freedom 0B7 - Populatio Sample Degrees of Freedom 4 B - Total Degrees of Freedom 44 B6 + B7 Pooled Variace Stadard Error Differece i Sample Meas 0.74 B8 - B t Test Statistic.040(B - B4) / B0.50((B6 * B9^) + (B7 * B3^)) / B SQRT(B9 * (/B7 + /B)) Two-Tail Test Lower Critical Value TINV(B5, B8) Upper Critical Value.05 TINV(B5, B8) p-value TDIST(ABS(B),B8,) Reject the ull hypothesis IF(B7<B5,"Reject the ull hypothesis", "Do ot reject the ull hypothesis") Decisio: Reject H 0 at α 0.05 Coclusio: There is evidece of a differece i meas. Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-4

15 Pooled-Variace t Test Example: Cofidece Iterval for µ - µ 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 S 0.74 ± ± α/ p + (0.009,.47) Sice 0 is less tha the etire iterval, we ca be 95% cofidet that µ NYSE > µ NASDAQ Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-5

16 Hypothesis tests for µ - µ with σ ad σ ukow, ot assumed equal Populatio meas, idepedet samples Assumptios: Samples are radomly ad idepedetly draw σ ad σ ukow, assumed equal σ ad σ ukow, ot assumed equal * Populatios are ormally distributed or both sample sizes are at least 30 Populatio variaces are ukow ad caot be assumed to be equal Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-6

17 Hypothesis tests for µ - µ with σ ad σ ukow ad ot assumed equal Populatio meas, idepedet samples σ ad σ ukow, assumed equal σ ad σ ukow, ot assumed equal * (cotiued) The formulae for this test are ot covered i this book. See referece 8 from this chapter for more details. This test utilizes two separate sample variaces to estimate the degrees of freedom for the t test Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-7

18 Related samples Related Populatios The Paired Differece Test Tests Meas of Related Populatios 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 Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-8

19 Related samples Related Populatios The Paired Differece Test The i th paired differece is D i, where D i X i - X i The poit estimate for the paired differece populatio mea µ D is D : D i D (cotiued) i The sample stadard deviatio is S D is the umber of pairs i the paired sample S D i (D i D) Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-9

20 The Paired Differece Test: Fidig t STAT Paired samples The test statistic for µ D is: t STAT D µ S D D Where t STAT has - d.f. Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-0

21 The Paired Differece Test: Possible Hypotheses Paired Samples Lower-tail test: H 0 : µ D 0 H : µ D < 0 Upper-tail test: H 0 : µ D 0 H : µ D > 0 Two-tail test: H 0 : µ D 0 H : µ D 0 α α α/ α/ -t α t α -t α/ t α/ Reject H 0 if t STAT < -t α Reject H 0 if t STAT > t α Reject H 0 if t STAT < -t α/ or t STAT > t α/ Where t STAT has - d.f. Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-

22 The Paired Differece Cofidece Iterval Paired samples The cofidece iterval for µ D is D ± t α / S D where S D i (D i D) Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-

23 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 C.B T.F M.H. 3 - R.K M.O S D D Σ D i (Di D) Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-3

24 Paired Differece Test: Solutio t Has the traiig made a differece i the umber of complaits (at the 0.0 level)? STAT α.0 µ D D S / D H 0 : µ D 0 H : µ D 0 D - 4. t ± d.f. - 4 Test Statistic: / Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall 5.66 Reject α/ Reject α/ Decisio: Do ot reject H 0 (t stat is ot i the reject regio) Coclusio: There is isufficiet evidece there is sigificat chage i the umber of complaits. Chap 0-4

25 Paired t Test I Excel Paired t Test Data Hypothesized Mea Diff. 0 Level of Sigificace 0.05 Itermediate Calculatios Sample Size 5 COUNT(I:I6) Dbar -4. AVERAGE(I:I6) Degrees of Freedom 4B8 - S D 5.67STDEV(I:I6) Stadard Error.54 B/SQRT(B8) t-test Statistic -.66 (B9 - B4)/B Two-Tail Test Lower Critical Value Upper Critical Value p-value Do ot reject the ull Hypothesis Data ot show is i colum I TINV(B5,B0).776 TINV(B5,B0) 0.73 TDIST(ABS(B3),B0,) IF(B8<B5,"Reject the ull hypothesis", "Do ot reject the ull hypothesis") Sice < -.66 <.776 we do ot reject the ull hypothesis. Or Sice p-value 0.73 > 0.05 we do ot reject the ull hypothesis. Thus we coclude that there is Isufficiet evidece to coclude there is a differece i the average umber of complaits. Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-5

26 Populatio proportios Two 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 Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-6

27 Populatio proportios Two Populatio Proportios The pooled estimate for the overall proportio is: I the ull hypothesis we assume the ull hypothesis is true, so we assume π π ad pool the two sample estimates p X + + X where X ad X are the umber of items of iterest i samples ad Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-7

28 Populatio proportios Two Populatio Proportios The test statistic for is a Z statistic: π π (cotiued) Z STAT ( ) ( ) p p p( p) π + π where X + X p, p, p + X X Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-8

29 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 Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-9

30 Lower-tail test: H 0 : π π 0 H : π π < 0 Hypothesis Tests for Two Populatio Proportios Populatio proportios Upper-tail test: H 0 : π π 0 H : π π > 0 (cotiued) Two-tail test: H 0 : π π 0 H : π π 0 α α α/ α/ -z α z α -z α/ z α/ Reject H 0 if Z STAT < -Z α Reject H 0 if Z STAT > Z α Reject H 0 if Z STAT < -Z α/ or Z STAT > Z α/ Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-30

31 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 Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-3

32 Hypothesis Test Example: Two populatio Proportios (cotiued) 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/ Wome: p 35/ The pooled estimate for the overall proportio is: X + X p Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-3

33 Hypothesis Test Example: Two populatio Proportios (cotiued) The test statistic for π π z STAT ( p p ) ( π π ) p( p) ( ) ( 0). 58(. 58 ) 7 50 Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall + Critical Values ±.96 For α.05 + is:. 0 Reject H 0 Reject H Decisio: Reject H 0.05 Coclusio: There is evidece of a differece i proportios who will vote yes betwee me ad wome. Chap 0-33

34 Z Test for Differeces i Two Proportios Two Proportio Test I Excel Data Hypothesized Differece 0 Level of Sigificace 0.05 Group Number of items of iterest 36 Sample Size 7 Group Number of items of iterest 35 Sample Size 50 Itermediate Calculatios Group Proportio 0.5 B7/B8 Group Proportio 0.7 B0/B Differece i Two Proportios -0. B4 - B5 Average Proportio 0.58 (B7 + B0)/(B8 + B) Z Test Statistic -.0 (B6-B4)/SQRT((B7*(-B7))*(/B8+/B)) Two-Tail Test Lower Critical Value Upper Critical Value p-value Reject the ull hypothesis -.96 NORMSINV(B5/).96 NORMSINV( - B5/) 0.08 *( - NORMSDIST(ABS(B8))) IF(B3 < B5,"Reject the ull hypothesis", "Do ot reject the ull hypothesis") Sice -.0 < -.96 Or Decisio: Reject H 0 Sice p-value 0.08 < 0.05 We reject the ull hypothesis Coclusio: There is evidece of a differece i proportios who will vote yes betwee me ad wome. Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-34

35 Cofidece Iterval for Two Populatio Proportios Populatio proportios The cofidece iterval for π π is: ( ) p p ± Z α/ p ( p ) + p ( p ) Copyright 03 Pearso Educatio, Ic. publishig as Pretice Hall Chap 0-35

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