M227 Chapter 9 Section 1 Testing Two Parameters: Means, Variances, Proportions

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1 M7 Chapter 9 Sectio 1 OBJECTIVES Tet two mea with idepedet ample whe populatio variace are kow. Tet two variace with idepedet ample. Tet two mea with idepedet ample whe populatio variace are equal Tet two mea with idepedet ample whe populatio variace are ot equal Tet two proportio with idepedet ample. INTRODUCTION There are may itace that require the compario betwee two mea or two proportio: o Compare the average tread wear betwee two et of tire. o Compare the effectivee of two differet fertilizer. o Compare the proportio of me who exercie regularly to the proportio of wome who exercie regularly. The baic tep of hypothei tetig are ued whe we compare two mea or two proportio. The cae that we will examie i thi chapter are: o Two mea with idepedet ample whe the two populatio variace are kow (z tet with all three method of hypothei tetig: Traditioal, P-value, Cofidece Iterval.) o Two variace with idepedet ample - F tet. o Two mea with idepedet ample whe the populatio variace are ot kow (t tet with all three method of hypothei tetig: Traditioal, P-value, Cofidece Iterval.) Whe variace are ot kow, we ditiguih betwee two cae: The two populatio variace are equal The two populatio variace are ot equal o Two populatio with idepedet ample (z tet with all three method of hypothei tetig: Traditioal, P-value, Cofidece Iterval.) The BIG differece i i the defiitio ad computatio of the TEST VALUE. NOTATION: We ue the ame otatio we have bee uig o far for the variou tatitic ad parameter with the additio of two ubcript: Subcript 1 deote parameter for Populatio 1 ad tatitic for ample derived from Populatio 1. Subcript deote parameter for Populatio ad tatitic for ample derived from Populatio. Sectio /0/007

2 M7 Chapter 9 Sectio TWO MEANS INDEPENDENT (large) SAMPLES σ 1 & σ KNOWN or 1, 30 Tet Value: for the Traditioal or P-value method, ue the z-ditributio with the followig Tet Value: Tet Value z = ( X 1 X ) ( µ 1 µ ) σ 1 σ + Null ad Alterative Hypothee: Two-Tailed: H 0 : µ 1 = µ (or µ 1 µ = 0) H 1: µ 1 µ (or µ 1 µ 0) Right-Tailed: H 0 : µ 1 = µ (or µ 1 µ = 0) H 1: µ 1 > µ (or µ 1 µ > 0) Left-Tailed: H 0 : µ 1 = µ (or µ 1 µ = 0) H 1: µ 1 < µ (or µ 1 µ < 0 ) Cofidece Iterval ca be cotructed a follow: Error 1 σ σ E = zα + σ1 σ σ1 σ α Cofidece Iterval: ( ) α µ µ ( ) X X z + < < X X + z + Example 9-1, 9-, 9-3. Excel: Ue MEGASTAT > Hypothei Tet > Compare Two Idepedet Group, z-tet. Sectio 9-4/0/007

3 M7 Chapter 9 Sectio 3 TWO VARIANCES INDEPENDENT SAMPLES Ue F tet. F ditributio : i a amplig ditributio of the ratio 1 of the variace of two ample elected from two ormally ditributed populatio with equal populatio variace. Propertie of F Ditributio o The value of F caot be egative, becaue variace are alway poitive or zero o The ditributio i poitively kewed. o The mea value of F i approximately equal to 1. o The F ditributio i a family of curve baed o the degree of freedom of the variace of the umerator ad the degree of freedom of the variace of the deomiator F Ditributio F tet: F = 1, where 1 i the larget of the two variace i quetio. F Tet ha two degree of freedom, oe for the umerator, ad oe for the deomiator, deoted repectively a: d.f.n = 1-1. ad d.f.d = -1. Ue Table H, i Appedix C, or the Excel fuctio: FDIST ad FINV The table ue oe-tailed value for the α. Thu, if a two-tailed tet i eeded, ay with α = 0.05, the ue the α = 0.05 table etry. Hypothee for variace tetig: Right-Tailed Left-Tailed Two-Tailed 0 : 1 = 1 : 1 > 0 : 1 = 1 : 1 Sectio /0/007

4 M7 Chapter 9 Sectio 3 Note o the Ue of the F Tet The larget variace hould alway be placed i the umerator For a two=tailed tet, the α value mut be divided by ad the critical value placed o the right ide of the F curve. If the tadard deviatio itead f the variace are give i the problem, they mut be quared for the formula for the F tet. If the degree of freedom caot be foud i the table, ue the cloet value o the maller ide I Hypothei Tetig, for the tet value ue the F value The aumptio that the larger variace i i the umerator, make ueceary to fid the left tail of the critical regio; i additio, left-tailed tet doe ot have a meaig. Aumptio for Tetig the Differece Betwee two Variace The populatio from which the ample were obtaied mut be ormally ditributed The ample mut be idepedet of each other Example 9-4, 9 5; alo ue Excel. Example 9-6. NOTE: Calculatig P-value through the table i a rather tediou tak. Ue the EXCEL fuctio FDIST itead, if there i a eed to ue the P-value method. Sectio /0/007

5 M7 Chapter 9 Sectio 4 TWO MEANS INDEPENDENT (Small) SAMPLES σ 1 & σ NOT Kow AND Oe or Both Sample Size Are Le Tha 30 Ue t ditributio Two cae: o Populatio variace are equal o Populatio variace ot equal. Whe problem doe ot tate whether the populatio variace are equal or ot, there are two optio: o Aume that they are NOT equal o Ue F tet firt. Ue F tet oly if the problem pecifically ak you to ue it! Otherwie, aume that the two populatio variace are ot equal. Tet Value for Uequal Variace: t = ( X 1 X ) ( µ 1 µ ) 1 +, d. f. = mi ( 1, 1) Tet Value for Equal Variace: t = (Sometime called pooled variace) ( X 1 X ) ( µ 1 µ ) ( 1) ( 1) d. f. = +, Example 9-9, 9-10 Ue MEGASTAT > Hypothei Tet > Compare Two Idepedet Group ad chooig the appropriate t-tet. X X E < < X X + E, I order to calculate Cofidece Iterval: ( ) µ µ ( ) ue the followig formula for the error: Uequal Variace: 1 E = tα + with d. f. = mi ( 1 1, 1) Equal Variace: ( ) + ( ) E = tα + + with d. f. = 1 + Sectio /0/007

6 M7 Chapter 9 Sectio 6 TWO PROPORTIONS INDEPENDENT SAMPLES Hypothee: Two-Tailed: H0 : p1 = p (or p1 p = 0 ) H1: p1 p (or p1 p 0 ) Right-Tailed: H0 : p1 = p (or p1 p = 0 ) H1: p1 > p (or p1 p > 0 ) Left-Tailed: H0 : p1 = p (or p1 p = 0 ) H1: p1 < p (or p1 p < 0) Tet Value for Traditioal ad P-value method ( pˆ Tet Value: 1 pˆ ) ( p1 p) z =, where: X 1 + p = X, pˆ 1 = X, q = 1 p, pˆ = X pq + 1 Example Do ame Example uig MEGASTAT Cofidece Iterval pˆ Error 1qˆ 1 pˆ qˆ E = z α + pˆ Iterval: 1qˆ 1 pˆ qˆ ( ˆ pˆ p1 pˆ ) zα + < p 1 p < 1qˆ 1 pˆ qˆ ( pˆ 1 pˆ ) + zα + Sectio /0/007