Chapter Chapter 10 Two-Sample Tests X 1 X 2. Difference Between Two Means: Different data sources Unrelated. Learning Objectives

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1 Chaper 0 0- Learig Objecives I his chaper, you lear how o use hypohesis esig for comparig he differece bewee: Chaper 0 Two-ample Tess The meas of wo idepede populaios The meas of wo relaed populaios The proporios of wo idepede populaios The variaces of wo idepede populaios by esig he raio of he wo variaces Chap 0- Chap 0- Two-ample Tess ifferece Bewee Two Meas Populaio Meas, Idepede amples Examples: Group vs. Group Two-ample Tess Populaio Meas, Relaed amples ame group before vs. afer reame Populaio Proporios Proporio vs. Proporio Populaio Variaces Variace vs. Variace Populaio meas, idepede σ ad σ ukow, σ ad σ ukow, o Goal: Tes hypohesis or form a cofidece ierval for he differece bewee wo populaio meas, µ µ The poi esimae for he differece is X X Chap 0-3 Chap 0-4 ifferece Bewee Two Meas: Idepede amples Hypohesis Tess for Two Populaio Meas iffere daa sources Urelaed Idepede Populaio meas, idepede ample seleced from oe populaio has o effec o he sample seleced from he oher populaio u σ ad σ ukow, σ ad σ ukow, o Use p o esimae ukow σ. Use a Pooled-Variace es. Use ad o esimae ukow σ ad σ. Use a eparae-variace es. Chap 0-5 Two Populaio Meas, Idepede amples Lower-ail es: H 0 : µ µ H : µ < µ H 0 : µ µ 0 H : µ µ < 0 Upper-ail es: H 0 : µ µ H : µ > µ H 0 : µ µ 0 H : µ µ > 0 Two-ail es: H 0 : µ µ H : µ µ H 0 : µ µ 0 H : µ µ 0 Chap 0-6

2 Chaper 0 0- Hypohesis ess for µ µ Hypohesis ess for µ - µ wih σ ad σ ukow ad Two Populaio Meas, Idepede amples Lower-ail es: H 0 : µ µ 0 H : µ µ < 0 Upper-ail es: H 0 : µ µ 0 H : µ µ > 0 Two-ail es: H 0 : µ µ 0 H : µ µ 0 / / - - / / Rejec H 0 if TAT < - Rejec H 0 if TAT > Rejec H 0 if TAT < - / or TAT > / Populaio meas, idepede σ ad σ ukow, σ ad σ ukow, o Assumpios: amples are radomly ad idepedely draw Populaios are ormally disribued or boh sample sizes are a leas 30 Populaio variaces are ukow bu Chap 0-7 Chap 0-8 Hypohesis ess for µ - µ wih σ ad σ ukow ad Cofidece ierval for µ - µ wih σ ad σ ukow ad Populaio meas, idepede σ ad σ ukow, σ ad σ ukow, o The pooled variace is: p ( ) ( ) ( ) ( ) The es saisic is: TAT Where TAT has d.f. ( ) ( X X ) ( µ µ ) p Populaio meas, idepede σ ad σ ukow, σ ad σ ukow, o The cofidece ierval for µ µ is: ( ) X X ± / p Where / has d.f. Chap 0-9 Chap 0-0 Pooled-Variace Tes Example Pooled-Variace Tes Example: Calculaig he Tes aisic You are a fiacial aalys for a brokerage firm. Is here a differece i divided yield bewee socks lised o he NYE & NAAQ? You collec he followig daa: NYE NAAQ Number 5 ample mea ample sd dev.30.6 Assumig boh populaios are approximaely ormal wih equal variaces, is here a differece i mea yield ( 0.05)? The es saisic is: P ( X X ) ( µ µ ) ( ) ( ) ( ) ( ).30 ( 5 ) p ( ) ( ) H0: µ - µ 0 i.e. (µ µ ) H: µ - µ 0 i.e. (µ µ ) (-) (5 ) 0 TAT Chap 0- Chap 0-

3 Chaper Pooled-Variace Tes Example: Hypohesis Tes oluio H 0 : µ - µ 0 i.e. (µ µ ) H : µ - µ 0 i.e. (µ µ ) 0.05 df 5-44 Criical Values: ±.054 Rejec H 0 Rejec H Tes aisic:.040 ecisio: TAT.040 Rejec H 0 a Coclusio: 5 There is evidece of a differece i meas. Excel Pooled-Variace es Comparig NYE & NAAQ Pooled- Variace Tes for he ifferece Bewee Two Meas (assumes equal populao variaces) aa Hypohesized ifferece 0 Level of igificace 0.05 Populao ample ample ize COUNT(ATA!$A:$A) ample Mea 3.7 AVERAGE(ATA!$A:$A) ample adard eviao.3 TEV(ATA!$A:$A) Populao ample ample ize 5 COUNT(ATA!$B:$B) ample Mea.53 AVERAGE(ATA!$B:$B) ample adard eviao.6 TEV(ATA!$B:$B) Iermediae Calculaos Populao ample egrees of Freedom 0 B7 - Populao ample egrees of Freedom 4 B - Toal egrees of Freedom 44 B6 B7 Pooled Variace.50 ((B6 B9^) (B7 B3^)) / B8 adard Error QRT(B9 (/B7 /B)) ifferece i ample Meas 0.74 B8 - B Tes asc.040 (B - B4) / B0 Two- Tail Tes Lower Crical Value TINV(B5, B8) Upper Crical Value.05 TINV(B5, B8) p- value TIT(AB(B),B8,) Rejec he ull hypohesis IF(B7<B5,"Rejec he ull hypohesis", "o o rejec he ull hypohesis") ecisio: Rejec H 0 a 0.05 Coclusio: There is evidece of a differece i meas. Chap 0-3 Chap 0-4 Miiab Pooled-Variace es Comparig NYE & NAAQ Pooled-Variace Tes Example: Cofidece Ierval for µ - µ Two-ample T-Tes ad CI ample N Mea ev E Mea ifferece mu () - mu () Esimae for differece: % CI for differece: (0.009,.47) T-Tes of differece 0 (vs o ): T-Value.04 P-Value F 44 Boh use Pooled ev.56 ecisio: Rejec H 0 a 0.05 Coclusio: There is evidece of a differece i meas. Chap 0-5 ice we rejeced H 0 ca we be 95% cofide ha µ NYE > µ NAAQ? 95% Cofidece Ierval for µ NYE - µ NAAQ ( X X ) ± p 0.74 ± (0.009,.47) / ice 0 is less ha he eire ierval, we ca be 95% cofide ha µ NYE > µ NAAQ Chap 0-6 Hypohesis ess for µ - µ wih σ ad σ ukow, o Hypohesis ess for µ - µ wih σ ad σ ukow ad o Populaio meas, idepede σ ad σ ukow, Assumpios: amples are radomly ad idepedely draw Populaios are ormally disribued or boh sample sizes are a leas 30 Populaio meas, idepede σ ad σ ukow, The es saisic is: ν TAT TAT has d.f. ν ( X X ) ( µ µ ) σ ad σ ukow, o Populaio variaces are ukow ad cao be assumed o be equal σ ad σ ukow, o Chap 0-7 Chap 0-8

4 Chaper Relaed Populaios The Paired ifferece Tes Relaed Tess Meas of Relaed Populaios Paired or mached Repeaed measures (before/afer) Use differece bewee paired values: i X i - X i Elimiaes Variaio Amog ubjecs Assumpios: Boh Populaios Are Normally isribued Or, if o Normal, use large Relaed Populaios The Paired ifferece Tes The i h paired differece is i, where Relaed i X i - X i The poi esimae for he paired differece populaio mea µ is : The sample sadard deviaio is is he umber of pairs i he paired sample i i i ( ) i Chap 0-9 Chap 0-0 The Paired ifferece Tes: Fidig TAT The Paired ifferece Tes: Possible Hypoheses The es saisic for µ is: Paired amples Paired TAT µ Lower-ail es: H 0 : µ 0 H : µ < 0 Upper-ail es: H 0 : µ 0 H : µ > 0 Two-ail es: H 0 : µ 0 H : µ 0 / / Where TAT has - d.f. - - / / Rejec H 0 if TAT < - Rejec H 0 if TAT > Rejec H 0 if TAT < - / Where TAT has - d.f. or TAT > / Chap 0- Chap 0- The Paired ifferece Cofidece Ierval The cofidece ierval for µ is Paired ± / where i ( ) i Paired ifferece Tes: Example Assume you sed your salespeople o a cusomer service raiig workshop. Has he raiig made a differece i he umber of complais? You collec he followig daa: Number of Complais: () - () alesperso Before () Afer () ifferece, i C.B T.F M.H. 3 - R.K M.O Σ i -4. (i ) 5.67 Chap 0-3 Chap 0-4

5 Chaper Has he raiig made a differece i he umber of complais (a he 0.0 level)? TAT.0-4. µ / 5.67/ 5 H 0 : µ 0 H : µ ± d.f. - 4 Paired ifferece Tes: oluio Tes aisic: Rejec / Rejec / ecisio: o o rejec H 0 ( sa is o i he rejec regio) Coclusio: There is isufficie evidece here is sigifica chage i he umber of complais. Chap 0-5 Paired Tes I Excel Paired Tes aa Hypohesized Mea iff. 0 Level of igificace 0.05 Iermediae Calculaos ample ize 5 COUNT(I:I6) bar - 4. AVERAGE(I:I6) egrees of Freedom 4 B TEV(I:I6) adard Error.54 B/QRT(B8) - Tes asc -.66 (B9 - B4)/B Two- Tail Tes Lower Crical Value Upper Crical Value p- value o o rejec he ull Hypohesis aa o show is i colum I TINV(B5,B0).776 TINV(B5,B0) 0.73 TIT(AB(B3),B0,) IF(B8<B5,"Rejec he ull hypohesis", "o o rejec he ull hypohesis") ice < -.66 <.776 we do o rejec he ull hypohesis. Or ice p-value 0.73 > 0.05 we do o rejec he ull hypohesis. Thus we coclude ha here is Isufficie evidece o coclude here is a differece i he average umber of complais. Chap 0-6 Paired Tes I Miiab Yields The ame Coclusios Two Populaio Proporios Paired T-Tes ad CI: Afer, Before Paired T for Afer - Before N Mea ev E Mea Afer Before ifferece % CI for mea differece: (-.5,.85) T-Tes of mea differece 0 (vs o 0): T-Value -.66 P-Value 0.73 Populaio proporios Goal: es a hypohesis or form a cofidece ierval for he differece bewee wo populaio proporios, π π The poi esimae for he differece is p p Chap 0-7 Chap 0-8 Two Populaio Proporios Two Populaio Proporios Populaio proporios I he ull hypohesis we assume he ull hypohesis is rue, so we assume π π ad pool he wo sample esimaes The pooled esimae for he overall proporio is: X X p Populaio proporios The es saisic for π π is a Z saisic: Z TAT ( p p ) ( π π ) p( p) where X ad X are he umber of iems of ieres i ad where X X p, p, p X X Chap 0-9 Chap 0-30

6 Chaper Hypohesis Tess for Two Populaio Proporios Hypohesis Tess for Two Populaio Proporios Lower-ail es: H 0 : π π H : π < π H 0 : π π 0 H : π π < 0 Populaio proporios Upper-ail es: H 0 : π π H : π > π H 0 : π π 0 H : π π > 0 Two-ail es: H 0 : π π H : π π H 0 : π π 0 H : π π 0 Lower-ail es: H 0 : π π 0 H : π π < 0 Populaio proporios Upper-ail es: H 0 : π π 0 H : π π > 0 Two-ail es: H 0 : π π 0 H : π π 0 / / -z z -z / z / Rejec H 0 if Z TAT < -Z Rejec H 0 if Z TAT > Z Rejec H 0 if Z TAT < -Z / or Z TAT > Z / Chap 0-3 Chap 0-3 Hypohesis Tes Example: Two populaio Proporios Is here a sigifica differece bewee he proporio of me ad he proporio of wome who will voe Yes o Proposiio A? I a radom sample, 36 of 7 me ad 35 of 50 wome idicaed hey would voe Yes Tes a he.05 level of sigificace The hypohesis es is: H 0 : π π 0 (he wo proporios are equal) H : π π 0 (here is a sigifica differece bewee proporios) Hypohesis Tes Example: Two populaio Proporios The sample proporios are: Me: p 36/ Wome: p 35/ The pooled esimae for he overall proporio is: X X p Chap 0-33 Chap 0-34 Hypohesis Tes Example: Two populaio Proporios The es saisic for π π is: z TAT ( p p ) ( π π ) p( p) ( ) ( 0). 58(. 58 ) 7 50 Criical Values ±.96 For Rejec H 0 Rejec H ecisio: Rejec H 0.05 Coclusio: There is evidece of a differece i proporios who will voe yes bewee me ad wome. Chap 0-35 Two Proporio Tes I Excel Z Tes for iffereces i Two Proporos aa ice -.0 < -.96 Hypohesized ifferece 0 Level of igificace 0.05 Group Or Number of iems of ieres 36 ample ize 7 Group ice p-value 0.08 < 0.05 Number of iems of ieres 35 ample ize 50 We rejec he ull hypohesis Iermediae Calculaos Group Proporo 0.5 B7/B8 Group Proporo 0.7 B0/B ifferece i Two Proporos - 0. B4 - B5 ecisio: Rejec H 0 Average Proporo 0.58 (B7 B0)/(B8 B) Z Tes asc -.0 (B6- B4)/QRT((B7(- B7))(/B8/B)) Two- Tail Tes Lower Crical Value Upper Crical Value p- value Rejec he ull hypohesis Coclusio: There is -.96 NORMINV(B5/) evidece of a differece i.96 NORMINV( - B5/) 0.08 ( - NORMIT(AB(B8))) proporios who will voe IF(B3 < B5,"Rejec he ull hypohesis", yes bewee me ad "o o rejec he ull hypohesis") wome. Chap 0-36

7 Chaper Two Proporio Tes I Miiab hows The ame Coclusios Cofidece Ierval for Two Populaio Proporios Tes ad CI for Two Proporios ample X N ample p Populaio proporios The cofidece ierval for π π is: ifferece p () - p () Esimae for differece: % CI for differece: ( , ) Tes for differece 0 (vs o 0): Z -.8 P-Value 0.0 ( p p ) ± Z / p( p) p( p) Chap 0-37 Chap 0-38 Tesig for he Raio Of Two Populaio Variaces Tess for Two H 0 : σ σ Hypoheses F TAT Populaio H : σ σ Variaces H 0 : σ σ H : σ > σ F es saisic Where: Variace of sample (he larger sample variace) sample size of sample Variace of sample (he smaller sample variace) sample size of sample umeraor degrees of freedom deomiaor degrees of freedom The F isribuio The F criical value is foud from he F able There are wo degrees of freedom required: umeraor ad deomiaor The larger sample variace is always he umeraor Whe F TAT I he F able, df ; df umeraor degrees of freedom deermie he colum deomiaor degrees of freedom deermie he row Chap 0-39 Chap 0-40 Fidig he Rejecio Regio F Tes: A Example 0 H 0 : σ σ H : σ σ / o o Rejec H 0 rejec H 0 F / Rejec H 0 if F TAT > F / F 0 H 0 : σ σ H : σ > σ o o rejec H 0 F Rejec H 0 Rejec H 0 if F TAT > F F You are a fiacial aalys for a brokerage firm. You wa o compare divided yields bewee socks lised o he NYE & NAAQ. You collec he followig daa: NYE NAAQ Number 5 Mea d dev.30.6 Is here a differece i he bewee he NYE NAAQ a he 0.05 level? variaces & Chap 0-4 Chap 0-4

8 Chaper F Tes: Example oluio Form he hypohesis es: H : σ σ 0 H : σ σ (here is o differece bewee variaces) (here is a differece bewee variaces) Fid he F criical value for 0.05: F Tes: Example oluio The es saisic is: H 0 : σ σ H : σ σ.30 F TAT.56.6 /.05 Numeraor d.f. 0 eomiaor d.f. 5 4 F TAT.56 is o i he rejecio regio, so we do o rejec H 0 0 o o rejec H 0 Rejec H 0 F F F / F.05, 0, 4.33 Coclusio: There is isufficie evidece of a differece i variaces a.05 Chap 0-43 Chap 0-44 Two Variace F Tes I Excel Two Variace F Tes I Miiab Yields The ame Coclusio F Tes for iffereces i Two Variables aa Level of igificace 0.05 Larger- Variace ample ample ize ample Variace ^ maller- Variace ample ample ize 5 ample Variace ^ Iermediae Calculaos F Tes asc Populao ample egrees of Freedom 0 B6 - Populao ample egrees of Freedom 4 B9 - Two- Tail Tes Upper Crical Value p- value o o rejec he ull hypohesis.37 FINV(B4/,B4,B5) FIT(B3,B4,B5) IF(B9<B4,"Rejec he ull hypohesis", "o o rejec he ull hypohesis") Coclusio: There is isufficie evidece of a differece i variaces a.05 because: F saisic.56 <.37 F / or p-value > Tes ad CI for Two Variaces Null hypohesis igma() / igma() Aleraive hypohesis igma() / igma() o igificace level Alpha 0.05 aisics ample N ev Variace Raio of sadard deviaios. Raio of variaces.56 95% Cofidece Iervals CI for isribuio CI for ev Variace of aa Raio Raio Normal (0.735,.739) (0.540, 3.04) Tess Tes Mehod F F aisic P-Value F Tes (ormal) Chap 0-45 Chap 0-46 Chaper ummary Chaper ummary Compared wo idepede Performed pooled-variace es for he differece i wo meas Performed separae-variace es for differece i wo meas Formed cofidece iervals for he differece bewee wo meas Compared wo relaed (paired ) Performed paired es for he mea differece Formed cofidece iervals for he mea Compared wo populaio proporios Formed cofidece iervals for he differece bewee wo populaio proporios Performed Z-es for wo populaio proporios Performed F es for he raio of wo populaio variaces Chap 0-47 differece Chap 0-48