Statistical Inference for Two Samples. Applied Statistics and Probability for Engineers. Chapter 10 Statistical Inference for Two Samples

Transcription

1 4/3/6 Applied Statitic ad Probability for Egieer Sixth Editio Dougla C. Motgomery George C. Ruger Chapter Statitical Iferece for Two Sample Copyright 4 Joh Wiley & So, Ic. All right reerved. CHAPTER OUTLINE - Iferece o the Differece i Mea of Two Normal Ditributio, Variace Kow -. Hypothei tet o the differece i mea, variace kow -. Type II error ad choice of ample ize -.3 Cofidece iterval o the differece i mea, variace kow - Iferece o the Differece i Mea of Two Normal Ditributio, Variace Ukow -. Hypothei tet o the differece i mea, variace ukow -. Type II error ad choice of ample ize -.3 Cofidece iterval o the differece i mea, variace ukow -3 A Noparametric Tet o the Differece i Two Mea -4 Paired t-tet -5 Iferece o the Variace of Two Normal Populatio -5. F ditributio Statitical Iferece for Two Sample Chapter Title ad Outlie Copyright 4 Joh Wiley & So, Ic. All right reerved. -5. Hypothei tet o the ratio of two variace -5.3 Type II error ad choice of ample ize -5.4 Cofidece iterval o the ratio of two variace -6 Iferece o Two Populatio Proportio -6. Large ample tet o the differece i populatio proportio -6. Type II error ad choice of ample ize -6.3 Cofidece iterval o the differece i populatio proportio -7 Summary Table ad Roadmap for Iferece Procedure for Two Sample

2 4/3/6 Learig Objective for Chapter After careful tudy of thi chapter, you hould be able to do the followig:. Structure comparative experimet ivolvig two ample a hypothei tet.. Tet hypothee ad cotruct cofidece iterval o the differece i mea of two ormal ditributio. 3. Tet hypothee ad cotruct cofidece iterval o the ratio of the variace or tadard deviatio of two ormal ditributio. 4. Tet hypothee ad cotruct cofidece iterval o the differece i two populatio proportio. 5. Ue the P-value approach for makig deciio i hypothei tet. 6. Compute power, Type II error probability, ad make ample ize deciio for two-ample tet o mea, variace & proportio. 7. Explai & ue the relatiohip betwee cofidece iterval ad hypothei tet. Chapter Learig Objective Copyright 4 Joh Wiley & So, Ic. All right reerved. 3 -: Iferece o the Differece i Mea of Two Normal Ditributio, Variace Kow Aumptio. Let X X, X be a radom ample from populatio.,,. Let X X, X be a radom ample from populatio.,, 3. The two populatio X ad X are idepedet. 4. Both X ad X are ormal. Sec - Iferece o the Differece i Mea of Two Normal Ditributio, Variace Kow Copyright 4 Joh Wiley & So, Ic. All right reerved. 4

3 4/3/6 -: Iferece o the Differece i Mea of Two Normal Ditributio, Variace Kow The quatity Z X X ( ) (-) ha a N(, ) ditributio. Sec - Iferece o the Differece i Mea of Two Normal Ditributio, Variace Kow Copyright 4 Joh Wiley & So, Ic. All right reerved Hypothei Tet o the Differece i Mea, Variace Kow Null hypothei: H : = Tet tatitic: X X Z (-) Alterative Hypothee P-Value Rejectio Criterio For Fixed-Level Tet H : Probability above z ad probability below z, P = [ ( z )] z z or z z H : > Probability above z, z z P = (z ) H : < Probability below z, P = (z ) z z Sec - Iferece o the Differece i Mea of Two Normal Ditributio, Variace Kow Copyright 4 Joh Wiley & So, Ic. All right reerved. 6 3

4 4/3/6 EXAMPLE - Pait Dryig Time A product developer i itereted i reducig the dryig time of a primer pait. Two formulatio of the pait are teted; formulatio i the tadard chemitry, ad formulatio ha a ew dryig igrediet that hould reduce the dryig time. From experiece, it i kow that the tadard deviatio of dryig time i 8 miute, ad thi iheret variability hould be uaffected by the additio of the ew igrediet. Te pecime are paited with formulatio, ad aother pecime are paited with formulatio ; the pecime are paited i radom order. The two ample average dryig time are x miute ad x miute, repectively. What cocluio ca the product developer draw about the effectivee of the ew igrediet, uig.5? The eve-tep procedure i:. Parameter of iteret: The differece i mea dryig time,, ad.. Null hypothei: H :. 3. Alterative hypothei: H :. Sec - Iferece o the Differece i Mea of Two Normal Ditributio, Variace Kow Copyright 4 Joh Wiley & So, Ic. All right reerved. 7 EXAMPLE - Pait Dryig Time - Cotiued 4. Tet tatitic: The tet tatitic i where (8) 64 ad. z x x 5. Reject H if: Reject H : if the P-value i le tha Computatio: Sice x miute ad x miute, the tet tatitic i z Cocluio: Sice z =.5, the P-value i P (.5).59, o we reject H at the.5 level Iterpretatio: We ca coclude that addig the ew igrediet to the pait igificatly reduce the dryig time. Sec - Iferece o the Differece i Mea of Two Normal Ditributio, Variace Kow Copyright 4 Joh Wiley & So, Ic. All right reerved. 8 4

5 4/3/6 -. Type II Error ad Choice of Sample Size Ue of Operatig Characteritic Curve Two-ided alterative: d Oe-ided alterative: d Sec - Iferece o the Differece i Mea of Two Normal Ditributio, Variace Kow Copyright 4 Joh Wiley & So, Ic. All right reerved Type II Error ad Choice of Sample Size Sample Size Formula Two-ided alterative: For the two-ided alterative hypothei with igificace level, the ample ize required to detect a true differece i mea of with power at leat i ( z ~ / z ) ( ) (-5) ( ) Oe-ided alterative: For a oe-ided alterative hypothei with igificace level, the ample ize required to detect a true differece i mea of ( ) with power at leat i ( ) ( z z ) ( ) (-6) Sec - Iferece o the Differece i Mea of Two Normal Ditributio, Variace Kow Copyright 4 Joh Wiley & So, Ic. All right reerved. 5

6 4/3/6 EXAMPLE -3 Pait Dryig Time Sample Size To illutrate the ue of thee ample ize equatio, coider the ituatio decribed i Example -, ad uppoe that if the true differece i dryig time i a much a miute, we wat to detect thi with probability at leat.9. Uder the ull hypothei,. We have a oe-ided alterative hypothei with,.5 (o z = z.5.645), ad ice the power i.9,. (o z z. =.8). Therefore, we may fid the required ample ize from Equatio -6 a follow: ( z z ) ( (.645.8) [(8) ( ) Thi i exactly the ame a the reult obtaied from uig the OC curve. ) (8) ] Sec - Iferece o the Differece i Mea of Two Normal Ditributio, Variace Kow Copyright 4 Joh Wiley & So, Ic. All right reerved. -.3 Cofidece Iterval o a Differece i Mea, Variace Kow x x If ad are the mea of idepedet radom ample of ize ad from two idepedet ormal populatio with kow variace ad, repectively, a ( ) cofidece iterval for i x x z / x x z/ (-7) where z i the upper percetage poit of the tadard ormal ditributio Sec - Iferece o the Differece i Mea of Two Normal Ditributio, Variace Kow Copyright 4 Joh Wiley & So, Ic. All right reerved. 6

7 4/3/6 EXAMPLE -4 Alumium Teile Stregth Teile tregth tet were performed o two differet grade of alumium par ued i maufacturig the wig of a commercial traport aircraft. From pat experiece with the par maufacturig proce ad the tetig procedure, the tadard deviatio of teile tregth are aumed to be kow. The data obtaied are a follow: =, x 87.6,, =, x 74.5, ad.5. If ad deote the true mea teile tregth for the two grade of par, we may fid a 9% cofidece iterval o the differece i mea tregth a follow: x x z/ () x (.5) ( ) (.5) Copyright 4 Joh Wiley & So, Ic. All right reerved. x 3.98 z / Sec - Iferece o the Differece i Mea of Two Normal Ditributio, Variace Kow 3 -: Iferece for a Differece i Mea of Two Normal Ditributio, Variace Kow Choice of Sample Size z / ( ) (-8) E Remember to roud up if i ot a iteger. Thi eure that the level of cofidece doe ot drop below ( α)%. Sec - Iferece o the Differece i Mea of Two Normal Ditributio, Variace Kow Copyright 4 Joh Wiley & So, Ic. All right reerved. 4 7

8 4/3/6 -: Iferece for a Differece i Mea of Two Normal Ditributio, Variace Kow Oe-Sided Cofidece Boud Upper Cofidece Boud x x z (-9) Lower Cofidece Boud x x z (-) Sec - Iferece o the Differece i Mea of Two Normal Ditributio, Variace Kow Copyright 4 Joh Wiley & So, Ic. All right reerved Hypothee Tet o the Differece i Mea, Variace Ukow Cae : We wih to tet: H : H : The pooled etimator of, deoted by S p, i defied by The quatity S p ( ) S ( ) S (-) T X X S p ( ) (-3) ha a t ditributio with degree of freedom. Sec - Hypothee Tet o the Differece i Mea, Variace Ukow Copyright 4 Joh Wiley & So, Ic. All right reerved. 6 8

9 4/3/6 Tet o the Differece i Mea of Two Normal Ditributio, Variace Ukow ad Equal Null hypothei: H : Tet tatitic: T X X S p (-4) Alterative Hypothei H : P-Value Probability above t ad probability belowt Rejectio Criterio for Fixed-Level Tet t t or /, t t /, H : Probability above t H : Probability below t t t, t t, Sec - Hypothee Tet o the Differece i Mea, Variace Ukow Copyright 4 Joh Wiley & So, Ic. All right reerved. 7 EXAMPLE -5 Yield from a Catalyt Two catalyt are beig aalyzed to determie how they affect the mea yield of a chemical proce. Specifically, catalyt i curretly i ue, but catalyt i acceptable. Sice catalyt i cheaper, it hould be adopted, providig it doe ot chage the proce yield. A tet i ru i the pilot plat ad reult i the data how i Table -. I there ay differece betwee the mea yield? Ue.5, ad aume equal variace. Obervatio Number Catalyt Catalyt x 9.55 x Sec - Hypothee Tet o the Differece i Mea, Variace Ukow Copyright 4 Joh Wiley & So, Ic. All right reerved. 8 9

10 4/3/6 EXAMPLE -5 Yield from a Catalyt - Cotiued The eve-tep hypothei-tetig procedure i a follow:. Parameter of iteret: The parameter of iteret are ad, the mea proce yield uig catalyt ad, repectively.. Null hypothei: H : 3. Alterative hypothei: H : 4. Tet tatitic: The tet tatitic i t x p x 5. Reject H if: Reject H if the P-value i le tha.5. Sec - Hypothee Tet o the Differece i Mea, Variace Ukow Copyright 4 Joh Wiley & So, Ic. All right reerved. 9 EXAMPLE -5 Yield from a Catalyt - Cotiued 6. Computatio: From Table - we have x 9.55, =.39, = 8, x, =.98, ad = Therefore ad ( ) ( ) (7)(.39) 7(.98) p p x x t Cocluio: From Appedix Table V we ca fid t.4,4.58 ad t.5,4.69. Sice, , we coclude that lower ad upper boud o the P-value are.5 P.8. Therefore, ice the P-value exceed.5, the ull hypothei caot be rejected. Iterpretatio: At 5% level of igificace, we do ot have trog evidece to coclude that catalyt reult i a mea yield that differ from the mea yield whe catalyt i ued. Sec - Hypothee Tet o the Differece i Mea, Variace Ukow Copyright 4 Joh Wiley & So, Ic. All right reerved.

11 4/3/6 -. Hypothee Tet o the Differece i Mea, Variace Ukow Cae : If H : i true, the tatitic T * X X S S (-5) i ditributed a t with degree of freedom give by v / / (-6) Sec - Hypothee Tet o the Differece i Mea, Variace Ukow Copyright 4 Joh Wiley & So, Ic. All right reerved. EXAMPLE -6 Areic i Drikig Water Areic cocetratio i public drikig water upplie i a potetial health rik. A article i the Arizoa Republic (May 7, ) reported drikig water areic cocetratio i part per billio (ppb) for metropolita Phoeix commuitie ad commuitie i rural Arizoa. The data follow: Metro Phoeix Rural Arizoa ( x.5, = 7.63) ( x 7.5, = 5.3) Phoeix, 3 Rimrock, 48 Chadler, 7 Goodyear, 44 Gilbert, 5 New River, 4 Gledale, Apache Juctio, 38 Mea, 5 Buckeye, 33 Paradie Valley, 6 Nogale, Peoria, Black Cayo City, Scottdale, 5 Sedoa, Tempe, 5 Payo, Su City, 7 Caa Grade, 8 Determie if there i ay differece i mea areic cocetratio betwee metropolita Phoeix commuitie ad commuitie i rural Arizoa. Sec - Hypothee Tet o the Differece i Mea, Variace Ukow Copyright 4 Joh Wiley & So, Ic. All right reerved.

12 4/3/6 EXAMPLE -6 Areic i Drikig Water - Cotiued The eve-tep procedure i:. Parameter of iteret: The parameter of iteret are the mea areic cocetratio for the two geographic regio, ay, ad, ad we are itereted i determiig whether.. No hypothei: H :, or H : 3. Alterative hypothei: H : 4. Tet tatitic: The tet tatitic i * x x t 5. Reject H if : The degree of freedom o are foud a v / / / 5.3 / 9 9 Sec - Hypothee Tet o the Differece i Mea, Variace Ukow Copyright 4 Joh Wiley & So, Ic. All right reerved. 3. ~ 3 3 EXAMPLE -6 Areic i Drikig Water - Cotiued Therefore, uig.5 ad a fixed-igificace-level tet, we would reject H : if *.6 or if * t t t t. 6..5,3 6. Computatio: Uig the ample data we fid x x * t.5,3.77 * 7. Cocluio: Becaue t.77 t.5, 3.6, we reject the ull hypothei. Iterpretatio: We ca coclude that mea areic cocetratio i the drikig water i rural Arizoa i differet from the mea areic cocetratio i metropolita Phoeix drikig water. Sec - Hypothee Tet o the Differece i Mea, Variace Ukow Copyright 4 Joh Wiley & So, Ic. All right reerved. 4

13 4/3/6 -. Type II Error ad Choice of Sample Size Example -8 Yield from Catalyt Sample Size Coider the catalyt experimet i Example -5. Suppoe that, if catalyt produce a mea yield that differ from the mea yield of catalyt by 4.%, we would like to reject the ull hypothei with probability at leat.85. What ample ize i required? Uig p.7 a a rough etimate of the commo tadard deviatio, we have d Δ /σ = 4. /[()(.7)] =.74. From Appedix Chart VIIe with d =.74 ad β=.5, we fid * =, approximately. Therefore, becaue * = -, * ~.5 (ay) ad we would ue ample ize of = = =. Sec - Hypothee Tet o the Differece i Mea, Variace Ukow Copyright 4 Joh Wiley & So, Ic. All right reerved Cofidece Iterval o the Differece i Mea, Variace Ukow Cae : If x, x,, ad are the ample mea ad variace of two radom ample of ize ad, repectively, from two idepedet ormal populatio with ukow but equal variace, the a ( - α)% cofidece iterval o the differece i mea µ - µ i x x t /, p x x t /, p (-9) where p [( ) ( ) ]/( ) i the pooled etimate of the commo populatio tadard deviatio, ad t/, i the upper α/ percetage poit of the t ditributio with + - degree of freedom. Sec - Hypothee Tet o the Differece i Mea, Variace Ukow Copyright 4 Joh Wiley & So, Ic. All right reerved. 6 3

14 4/3/6 Example -9 Cemet Hydratio Te ample of tadard cemet had a average weight percet calcium of x 9. with a ample tadard deviatio of = 5., ad 5 ample of the lead-doped cemet had a average weight percet calcium of x 87. with a ample tadard deviatio of = 4.. Aume that weight percet calcium i ormally ditributed with ame tadard deviatio. Fid a 95% cofidece iterval o the differece i mea, µ - µ, for the two type of cemet. The pooled etimate of the commo tadard deviatio i foud a follow: ( ) ( ) 9(5.) 4(4.) p p The 95% cofidece iterval i foud uig Equatio -9: x x t.5,3 p x x t.5,3 p Upo ubtitutig the ample value ad uig t.5,3 =.69, which reduce to -.7 m - m 6.7 Sec - Hypothee Tet o the Differece i Mea, Variace Ukow Copyright 4 Joh Wiley & So, Ic. All right reerved Cofidece Iterval o the Differece i Mea, Variace Ukow Cae : If x, x,, ad are the mea ad variace of two radom ample of ize ad, repectively, from two idepedet ormal populatio with ukow ad uequal variace, a approximate ( - α)% cofidece iterval o the differece i mea µ - µ i x x t /, v x x t /, v (-) where v i give by Equatio -6 ad t /,v the upper / percetage poit of the t ditributio with v degree of freedom. Sec - Hypothee Tet o the Differece i Mea, Variace Ukow Copyright 4 Joh Wiley & So, Ic. All right reerved. 8 4

15 4/3/6-4: Paired t-tet Null hypothei: H : D Tet tatitic: D T (-4) S D / Alterative Hypothei P-Value Rejectio Criterio for Fixed-Level Tet H : D Probability above t ad probability belowt t t t t,, or H : D Probability above t H : D Probability below t t t t, t, Sec -4 Paired t-tet Copyright 4 Joh Wiley & So, Ic. All right reerved. 9 Example - Shear Stregth of Steel Girder A article i the Joural of Strai Aalyi [983, Vol. 8()] report a compario of everal method for predictig the hear tregth for teel plate girder. Data for two of thee method, the Karlruhe ad Lehigh procedure, whe applied to ie pecific girder, are how i the table below. Girder Karlruhe Method Lehigh Method Differece d j S/ S/ S3/ S4/ S5/ S/ S/ S/ S/ Determie whether there i ay differece (o the average) for the two method. Sec -4 Paired t-tet Copyright 4 Joh Wiley & So, Ic. All right reerved. 3 5

16 4/3/6 Example - Shear Stregth of Steel Girder The eve-tep procedure i:. Parameter of iteret: The parameter of iteret i the differece i mea hear tregth for the two method.. Null hypothei: H : µ D = 3. Alterative hypothei: H : µ D 4. Tet tatitic: The tet tatitic i d t d / 5. Reject H if: Reject H if the P-value i < Computatio: The ample average ad tadard deviatio of the differece d j are d.769 ad d =.35, ad o the tet tatitic i d.769 t /.35/ 9 d 7. Cocluio: Becaue t.5.8 = 5.4 ad the value of the tet tatitic t = 6.5 exceed thi value, the P-value i le tha (.5) =.. Therefore, we coclude that the tregth predictio method yield differet reult. Iterpretatio: The data idicate that the Karlruhe method produce, o the average, higher tregth predictio tha doe the Lehigh method. Sec -4 Paired t-tet 6.5 Copyright 4 Joh Wiley & So, Ic. All right reerved. 3 A Cofidece Iterval for D from Paired Sample If d ad D are the ample mea ad tadard deviatio of the differece of radom pair of ormally ditributed meauremet, a ( - α)% cofidece iterval o the differece i mea 3 µ D = µ - µ i d t d t (-5) /, D/ D /, D/ where t α/,- i the upper α /% poit of the t ditributio with - degree of freedom. Sec -4 Paired t-tet Copyright 4 Joh Wiley & So, Ic. All right reerved. 3 6

17 4/3/6 Example - Parallel Park Car The joural Huma Factor (96, pp ) reported a tudy i which = 4 ubject were aked to parallel park two car havig very differet wheel bae ad turig radii. The time i ecod for each ubject wa recorded ad i give i Table -4. From the colum of oberved differece, we calculate d =. ad D =.68. Fid the 9% cofidece iterval for µ D = µ - µ. d t D/ D d td/ / D 4.79 D 7. Notice that the cofidece iterval o µ D iclude zero. Thi implie that, at the 9% level of cofidece, the data do ot upport the claim that the two car have differet mea parkig time µ ad µ. That i, the value µ D = µ - µ = i ot icoitet with the oberved data. / 4 Subject (x j ) (x j ) (d j ) Sec -4 Paired t-tet Table -4 Copyright 4 Joh Wiley & So, Ic. All right reerved Iferece o the Variace of Two Normal Populatio -5. The F Ditributio We wih to tet the hypothee: H H : : Let W ad Y be idepedet chi-quare radom variable with u ad v degree of freedom repectively. The the ratio W/ u F Y/ v ha the probability deity fuctio (-8) u/ u v u ( u x v f ( x) u v u x v /) ad i aid to follow the ditributio with u degree of freedom i the umerator ad v degree of freedom i the deomiator. It i uually abbreviated a F u,v. Sec -5 Iferece o the Variace of Two Normal Populatio Copyright 4 Joh Wiley & So, Ic. All right reerved. uv /, x (-9) 34 7

18 4/3/6-5. Hypothei Tet o the Ratio of Two Variace Let X, X,, X be a radom ample from a ormal populatio with mea µ ad variace, ad let X, X,, X be a radom ample from a ecod ormal populatio with mea µ ad variace. Aume that both ormal populatio are idepedet. Let S ad S be the ample variace. The the ratio F S S / / ha a F ditributio with umerator degree of freedom ad deomiator degree of freedom. Sec -5 Iferece o the Variace of Two Normal Populatio Copyright 4 Joh Wiley & So, Ic. All right reerved Hypothei Tet o the Ratio of Two Variace Null hypothei: H : Tet tatitic: F S S (-3) Alterative Hypothee Rejectio Criterio H : H : H : f f f f /,, f,, f,, or f f /,, Sec -5 Iferece o the Variace of Two Normal Populatio Copyright 4 Joh Wiley & So, Ic. All right reerved. 36 8

19 4/3/6 Example -3 Semicoductor Etch Variability Oxide layer o emicoductor wafer are etched i a mixture of gae to achieve the proper thicke. The variability i the thicke of thee oxide layer i a critical characteritic of the wafer, ad low variability i deirable for ubequet proceig tep. Two differet mixture of gae are beig tudied to determie whether oe i uperior i reducig the variability of the oxide thicke. Sixtee wafer are etched i each ga. The ample tadard deviatio of oxide thicke are =.96 agtrom ad =.3 agtrom, repectively. I there ay evidece to idicate that either ga i preferable? Ue a fixed-level tet with α =.5. The eve-tep hypothei-tetig procedure i:. Parameter of iteret: The parameter of iteret are the variace of oxide thicke ad.. Null hypothei: H : 3. Alterative hypothei: H : Sec -5 Iferece o the Variace of Two Normal Populatio Copyright 4 Joh Wiley & So, Ic. All right reerved Iferece o the Variace of Two Normal Populatio 4. Tet tatitic: The tet tatitic i f 5. Reject H if : Becaue = = 6 ad α =.5, we will reject if f > f.5,5,5 =.86 or if f <f.975,5,5 = /f.5,5,5 = /.86 = Computatio: Becaue ad (.96) 3.84, the tet (.3) tatitic i 3.84 f Cocluio: Becaue f..975,5,5 =.35 <.85 < f.5,5,5 =.86, we caot reject the ull hypothei H at the.5 level of igificace. : Iterpretatio: There i o trog evidece to idicate that either ga reult i a maller variace of oxide thicke. Sec -5 Iferece o the Variace of Two Normal Populatio Copyright 4 Joh Wiley & So, Ic. All right reerved. 38 9

20 4/3/6-5.3 Type II Error ad Choice of Sample Size Appedix Chart VIIo, VIIp, VIIq, ad Vllr provide operatig characteritic curve for the F-tet give i Sectio -5. for a =.5 ad a =., aumig that = =. Chart VIIo ad VIIp are ued with the two-ided alterate hypothei. They plot b agait the abcia parameter (-3) for variou l = =. Chart VIIq ad VIIr are ued for the oe-ided alterative hypothee. Sec -5 Iferece o the Variace of Two Normal Populatio Copyright 4 Joh Wiley & So, Ic. All right reerved. 39 Example -4 Semicoductor Etch Variability Sample Size For the emicoductor wafer oxide etchig problem i Example -3, uppoe that oe ga reulted i a tadard deviatio of oxide thicke that i half the tadard deviatio of oxide thicke of the other ga. If we wih to detect uch a ituatio with probability at leat.8, i the ample ize = = adequate? Note that if oe tadard deviatio i half the other, By referrig to Appedix Chart VIIo with = = ad =, we fid that ~.. Therefore, if. ~, the power of the tet (which i the probability that the differece i tadard deviatio will be detected by the tet) i.8, ad we coclude that the ample ize = = are adequate. Sec -5 Iferece o the Variace of Two Normal Populatio Copyright 4 Joh Wiley & So, Ic. All right reerved. 4

21 4/3/6-5.4 Cofidece Iterval o the Ratio of Two Variace If ad are the ample variace of radom ample of ize ad, repectively, from two idepedet ormal populatio with ukow variace ad, the a ( )% cofidece iterval o the ratio / i /,, f /,, f (-33) f where f/, ad are the upper ad lower /, /,, percetage poit of the F ditributio with umerator ad deomiator degree of freedom, repectively..a cofidece iterval o the ratio of the tadard deviatio ca be obtaied by takig quare root i Equatio -33. Sec -5 Iferece o the Variace of Two Normal Populatio Copyright 4 Joh Wiley & So, Ic. All right reerved. 4 Example -5 Surface Fiih for Titaium Alloy A compay maufacture impeller for ue i jet-turbie egie. Oe of the operatio ivolve gridig a particular urface fiih o a titaium alloy compoet. Two differet gridig procee ca be ued, ad both procee ca produce part at idetical mea urface roughe. The maufacturig egieer would like to elect the proce havig the leat variability i urface roughe. A radom ample of = part from the firt proce reult i a ample tadard deviatio =5. microiche, ad a radom ample of = 6 part from the ecod proce reult i a ample tadard deviatio of = 4.7 microiche. Fid a 9% cofidece iterval o the ratio of the two tadard deviatio, /. Aumig that the two procee are idepedet ad that urface roughe i ormally ditributed, we ca ue Equatio -33 a follow: f.95,5, f.5,5, Sec -5 Iferece o the Variace of Two Normal Populatio Copyright 4 Joh Wiley & So, Ic. All right reerved. 4

22 4/3/6 Example -5 Surface Fiih for Titaium Alloy - Cotiued (5.) (4.7).39 (5.) (4.7).85 or upo completig the implied calculatio ad takig quare root, Notice that we have ued Equatio -3 to fid f.95,5, = /f.5,,5 = /.54=.39. Iterpretatio: Sice thi cofidece iterval iclude uity, we caot claim that the tadard deviatio of urface roughe for the two procee are differet at the 9% level of cofidece. Sec -5 Iferece o the Variace of Two Normal Populatio Copyright 4 Joh Wiley & So, Ic. All right reerved Large-Sample Tet o the Differece i Populatio Proportio We wih to tet the hypothee: H H : : p p p p The followig tet tatitic i ditributed approximately a tadard ormal ad i the bai of the tet: Z Pˆ Pˆ p( p ( p ) p ) p( p ) (-34) Sec -6 Iferece o Two Populatio Proportio Copyright 4 Joh Wiley & So, Ic. All right reerved. 44

23 4/3/6-6. Large-Sample Tet o the Differece i Populatio Proportio Null hypothei: H : p p Tet tatitic: Z Pˆ Pˆ Pˆ( Pˆ) (-35) Alterative Hypothei H : p p P-Value Probability above z ad probability below - z. P = [ - ( z )] Rejectio Criterio for Fixed-Level Tet z > z / or z < -z / H : p > p Probability above z. P = - (z ) z > z H : p < p Probability below z. P = (z ) Sec -6 Iferece o Two Populatio Proportio z < -z Copyright 4 Joh Wiley & So, Ic. All right reerved. 45 Example -6 St. Joh' Wort Extract of St. Joh' Wort are widely ued to treat depreio. A article i the April 8,, iue of the Joural of the America Medical Aociatio compared the efficacy of a tadard extract of St. Joh' Wort with a placebo i outpatiet diagoed with major depreio. Patiet were radomly aiged to two group; oe group received the St. Joh' Wort, ad the other received the placebo. After eight week, 9 of the placebotreated patiet howed improvemet, ad 7 of thoe treated with St. Joh' Wort improved. I there ay reao to believe that St. Joh' Wort i effective i treatig major depreio? Ue a =.5. The eve-tep hypothei tetig procedure lead to the followig reult:. Parameter of iteret: The parameter of iteret are p ad p, the proportio of patiet who improve followig treatmet with St. Joh' Wort (p ) or the placebo (p ).. Null hypothei: H : p = p 3. Alterative hypothei: H : p p 4. Tet Statitic: The tet tatitic i z pˆ pˆ pˆ( pˆ) Sec -6 Iferece o Two Populatio Proportio Copyright 4 Joh Wiley & So, Ic. All right reerved. 46 3

24 4/3/6 Example -6 St. Joh' Wort - Cotiued where p 7/.7, p 9/. 9, = =, ad ˆ ˆ x x pˆ Reject H if: Reject H : p = p if the P-value i le tha Computatio: The value of the tet tatitic i.7.9 z.34.3(.77) 7. Cocluio: Sice z =.34, the P-value i P = [ (.34)] =.8, we caot reject the ull hypothei. Iterpretatio: There i iufficiet evidece to upport the claim that St. Joh' Wort i effective i treatig major depreio. Sec -6 Iferece o Two Populatio Proportio Copyright 4 Joh Wiley & So, Ic. All right reerved Type II Error ad Choice of Sample Size If the alterative hypothei i two ided, the -error i z / z pq(/ / / Pˆ Pˆ pq(/ ) ( p / Pˆ Pˆ p ) ( p ) p ) (-37) Sec -6 Iferece o Two Populatio Proportio Copyright 4 Joh Wiley & So, Ic. All right reerved. 48 4

25 4/3/6-6. Type II Error ad Choice of Sample Size If the alterative hypothei i H : p p, z pq(/ / ) ( p p) Pˆ ˆ P (-38) ad if the alterative hypothei i H : p p, z pq(/ / ) ( p p) Pˆ ˆ P (-39) Sec -6 Iferece o Two Populatio Proportio Copyright 4 Joh Wiley & So, Ic. All right reerved Type II Error ad Choice of Sample Size For the two-ided alterative, the commo ample ize i [ z / ( p p )( q q )/ z ( p p ) p q p q ] (-4) where q = p ad q = p. Sec -6 Iferece o Two Populatio Proportio Copyright 4 Joh Wiley & So, Ic. All right reerved. 5 5

26 4/3/6-6.3 Cofidece Iterval o the Differece i the Populatio Proportio If ˆp ad ˆp are the ample proportio of obervatio i two idepedet radom ample of ize ad that belog to a cla of iteret, a approximate two-ided ( )% cofidece iterval o the differece i the true proportio p p i pˆ pˆ z / p p pˆ ( pˆ ) pˆ pˆ pˆ ( pˆ ) z / pˆ ( pˆ ) pˆ ( pˆ ) (-4) where z / i the upper / percetage poit of the tadard ormal ditributio. Sec -6 Iferece o Two Populatio Proportio Copyright 4 Joh Wiley & So, Ic. All right reerved. 5 Example -7 Defective Bearig Coider the proce of maufacturig crakhaft hearig decribed i Example 8-8. Suppoe that a modificatio i made i the urface fiihig proce ad that, ubequetly, a ecod radom ample of 85 bearig i obtaied. The umber of defective bearig i thi ecod ample i 8. Therefore, becaue = 85, pˆ /85.76, = 85, ad pˆ 8/ Obtai a approximate 95% cofidece iterval o the differece i the proportio of detective bearig produced uder the two procee. pˆ pˆ z p p pˆ ( pˆ ) pˆ pˆ pˆ z ( pˆ ) pˆ ( pˆ ) pˆ ( pˆ ) Sec -6 Iferece o Two Populatio Proportio Copyright 4 Joh Wiley & So, Ic. All right reerved. 5 6

27 4/3/6 Example -7 Defective Bearig - Cotiued (.884) 85.94(.959) 85.76(.884) p p (.959) 85 Thi implifie to.685 p p.55 Iterpretatio: Thi cofidece iterval iclude zero. Baed o the ample data, it eem ulikely that the chage made i the urface fiih proce have reduced the proportio of defective crakhaft bearig beig produced. Sec -6 Iferece o Two Populatio Proportio Copyright 4 Joh Wiley & So, Ic. All right reerved. 53-7: Summary Table ad Road Map for Iferece Procedure for Two Sample Sec -7 Summary Table ad Roadmap for Iferece Procedure for Two Sample Copyright 4 Joh Wiley & So, Ic. All right reerved. 54 7

28 4/3/6-7: Summary Table ad Road Map for Iferece Procedure for Two Sample Sec -7 Summary Table ad Roadmap for Iferece Procedure for Two Sample Copyright 4 Joh Wiley & So, Ic. All right reerved. 55 Importat Term & Cocept of Chapter Comparative experimet Cofidece iterval o: Differece Ratio Critical regio for a tet tatitic Idetifyig caue ad effect Null ad alterative hypothee & -ided alterative hypothee Operatig Characteritic (OC) curve Chapter Summary Paired t-tet Pooled t-tet P-value Referece ditributio for a tet tatitic Sample ize determiatio for: Hypothei tet Cofidece iterval Statitical hypothee Tet tatitic Wilcoxo rak-um tet Copyright 4 Joh Wiley & So, Ic. All right reerved. 56 8