11: The Analysis of Variance

Transcription

1 : The Aalysis of Variace. I comparig 6 populaios, here are ANOVA able is show below. Source df Treames 5 Error 5 Toal 59 k degrees of freedom for reames ad ( ) = 60 = 60. The. a Refer o Eercise.. The give sums of squares are isered ad missig eries foud by subracio. The mea squares are foud as MS = SS df. Source df SS MS F Treames Error Toal 59. b The F saisic, F = MST MSE, has df = 5 ad df = 5 degrees of freedom. c Wih α =.05 ad degrees of freedom from b, H 0 is rejeced if F > F d Sice F =.67 falls i he rejecio regio, he ull hypohesis is rejeced. There is a differece amog he meas. e The criical values of F wih df = 5 ad df 60 (Table 6) for boudig he p-value for his oe-ailed es are show below. α F α Sice he observed value F =.67 falls bewee F.0 ad F.005,.005 < p-value <.0 ad H 0 is rejeced as i par d.. Refer o Eercise.. MSE. a ± ± ±.9 0 or.7 < μ <.09. ±.05 MSE (.07.5) ± ±.80 or.07 < μ μ <.0 b ( ). Similar o Eercise.. Wih 6 ( ) show below. = = ad k =, he sources of variaio ad associaed df are Source df Treames Error 0 Toal.5 a Refer o Eercise.. The give sums of squares are isered ad missig eries foud by subracio. The mea squares are foud as MS = SS df. 86

2 Source df SS MS F Treames Error Toal 7. b The F saisic, F = MST MSE, has df = ad df = 0 degrees of freedom. c Wih α =.05 ad degrees of freedom from b, H 0 is rejeced if F > F.05 =.0. d Sice F = 6.98 falls i he rejecio regio, he ull hypohesis is rejeced. There is a differece amog he meas. e The criical values of F wih df = ad df = 0 (Table 6) for boudig he p-value for his oe-ailed es are show below. α F α Sice he observed value F = 6.98 is greaer ha F.005, p -value <.005 ad H 0 is rejeced as i par d..6 Refer o Eercise.5. MSE 6.67 a ± ± ±.89 6 or 86.8 < μ < ±.05 MSE ( ) ± ±.57 or.58 < μ μ < 6.67 b ( ).7 The followig prelimiary calculaios are ecessary: T = T = 9 T = 5 G = 8 ( ij ) ( 8) a CM = = = Toal SS = ij CM = CM = = b Ti 9 5 SST = CM = + + CM = =.507 i 5 5 SST.507 ad MST = = = 7.56 k c By subracio, SSE = Toal SS SST = =.500 ad he degrees of freedom, by subracio, are =. The SSE.500 MSE = = =.7 d The iformaio obaied i pars a-c is cosolidaed i a ANOVA able. Source df SS MS Treames Error Toal e The hypohesis o be esed is H : μ = μ = μ versus H : a leas oe pair of meas are differe 0 a 87

3 f MST 7.56 The rejecio regio for he es saisic F = = = 6.6 is based o a F-disribuio wih MSE.7 ad degrees of freedom. The criical values of F for boudig he p-value for his oe-ailed es are show below. α F α Sice he observed value F = 6.6 is bewee F.0 ad F.05,.0 < p-value <.05 ad H 0 is rejeced a he 5% level of sigificace. There is a differece amog he meas..8 The hypohesis o be esed is H 0 : μ = μ versus H a : μ μ ad he es saisic is ( ) = = =.59 s.7 5 Noice ha he bes esimaor of σ is s = MSE, which is used i he calculaio. The rejecio regio wih α =.05 ad degrees of freedom is >.05 =.0 ad he ull hypohesis is rejeced. We coclude ha here is a differece bewee he meas..9 a The 90% cofidece ierval for μ is MSE.7 ±.05.8 ± ±.85 5 or.95 < μ <.65. b The 90% cofidece ierval for μ μis ( ) ±.05 MSE (.8.5) ± ±.8 or.7 < μ μ <.8.0 a The followig prelimiary calculaios are ecessary: T = 80 T = 99 T = 6 G = 80 ( ij ) ( 80) CM = = = 6,5.55 Toal SS = ij CM = 65, 86 CM = Ti SST = CM = + + CM = i 5 Calculae MS = SS df ad cosolidae he iformaio i a ANOVA able. Source df SS MS Treames Error Toal b The hypohesis o be esed is H : μ = μ = μ versus H : a leas oe pair of meas are differe 0 a 88

4 ad he F es o deec a differece i mea sude respose is F = MST = 5.5. MSE The rejecio regio wih α =.05 ad ad 8 df is F >.6 ad H 0 is rejeced. There is a sigifica differece i mea respose due o he hree differe mehods.. a The 95% cofidece ierval for μa is MSE 6. A ± ± ± 8. 5 or < μ A < 8.. b The 95% cofidece ierval for A μb is MSE 6. B ± ± ± 0.5 B or 55.8 < μ B < c The 95% cofidece ierval for μa μbis ( A B) ±.05 MSE A B ( ) ± ±.96 or.69 < μa μb <.96 d Noe ha hese hree cofidece iervals cao be joily valid because all hree employ he same value of s = MSE ad are depede.. a From he compuer priou, he es saisic is F = 5.70 ad he p-value is give o he priou as P =.05. Tha is, H 0 ca be rejeced for ay value of α greaer ha.05. The ull hypohesis of equaliy of meas is rejeced a he 5% (bu o a he %) level of sigificace, ad we coclude ha here is a differece i mea assembly imes for he hree programs. b From he priou, MSE =.9 ad he 99% cofidece ierval for μa μbis ( A B) ±.005 MSE A B ( ) ± ± 9.58 or.75 < μa μb < 5. c The 99% cofidece ierval for μ A MSE.9 A ± ± ± 6.7 A or 5. < μ A < d Sice he measuremes represe averages of four assembly imes ad sice ime iself is a coiuous radom variable, he Ceral Limi Theorem assures us ha eve for small values of, he average assembly imes will have a fairly moud-shaped disribuio.. a We would be reasoably cofide ha he daa saisfied he ormaliy assumpio because each measureme represes he average of 0 coiuous measuremes. The Ceral Limi Theorem assures us ha his mea will be approimaely ormally disribued. b We have a compleely radomized desig wih four reames, each coaiig 6 measuremes. The aalysis of variace able is give i he Miiab priou. The F es is 89

5 MST F = = = 57.8 MSE.5 wih p-value =.000 (i he colum marked P ). Sice he p-value is very small (less ha.0), H 0 is rejeced. There is a sigifica differece i he mea leaf legh amog he four locaios wih P <.0 or eve P <.00. c The hypohesis o be esed is H 0 : μ = μ versus H a : μ μ ad he es saisic is = = =.09 MSE The p-value wih 0 df = is P( >.09) p -value < (.005) =.0 is bouded (usig Table ) as ad he ull hypohesis is rejec. We coclude ha here is a differece bewee he meas. d The 99% cofidece ierval for μ μis ( ) ±.005 MSE ( ) ± ±.557 or.80 < μ μ <.9 e Whe coducig he ess, remember ha he saed cofidece coefficies are based o radom samplig. If you looked a he daa ad oly compared he larges ad smalles sample meas, he radomess assumpio would be disurbed.. a The desig is compleely radomized wih four reames. The aalysis of variace able is give o he Miiab priou ad he es saisic is MST.60 F = = = 6.66 MSE.00 wih p-value =.000. H 0 is rejeced ad he resuls are declared highly sigifica. There is a sigifica differece i he mea dissolved oyge coe for he four locaios. b The 95% cofidece ierval for μ μis ( ) ±.05 MSE ( 6..78) ± ±.7 or.87 < μ μ <.9.5 The desig is compleely radomized wih reames ad 5 replicaios per reame. The Miiab priou o he e page shows he aalysis of variace for his eperime. 90

6 Oe-way ANOVA: Calcium versus Mehod Source DF SS MS F P Mehod Error Toal S = R-Sq = 7.9% R-Sq(adj) = 68.7% Idividual 95% CIs For Mea Based o Pooled SDev Level N Mea SDev (------*------) (------*------) (------*------) Pooled SDev = The es saisic, F = 6.8 wih p-value =.000 idicaes he resuls are highly sigifica; here is a differece i he mea calcium coes for he hree mehods. All assumpios appear o have bee saisfied..6 a The followig prelimiary calculaios are ecessary: T =.55 T =.9 T = 0. T =.6 G = 0. ij 0. ( ) ( ) CM = = =.96 Toal SS = ij CM = 5.69 CM = Ti SST = CM = CM =.95 i 8 Calculae MS = SS df ad cosolidae he iformaio i a ANOVA able. Source df SS MS Treames Error Toal The hypohesis o be esed is H 0 : μ = μ = μ = μ versus H a : a leas oe pair of meas are differe ad he F es o deec a differece i average prices is MST F = =.9. MSE The rejecio regio wih α =.05 ad ad df is approimaely F >.9 ad H 0 is o rejeced. There is o eough evidece o idicae a differece i average prices for he four ypes of ua. b The 95% cofidece ierval for μ μis c The 95% cofidece ierval for μ μis ( ) ±.05 MSE (.896.7) ± ±.59 or.6 < μ μ <.08 9

7 ( ) ±.05 MSE (.8.5) ± ±.55 or.9 < μ μ <.60 d The researcher migh be ieresed i he differece bewee whie ad ligh ua!.7 a The desig is a compleely radomized desig (four idepede samples). b The followig prelimiary calculaios are ecessary: T = T = 07 T = 58 T = G = 686 ij 686 ( ) ( ) CM = = =,097,99.8 Toal SS = ij CM =,0,86 CM = 9. 0 Ti SST = CM = CM = 7. i Calculae MS = SS df ad cosolidae he iformaio i a ANOVA able. Source df SS MS Treames Error Toal 9 9. c The hypohesis o be esed is H 0 : μ = μ = μ = μ versus H a : a leas oe pair of meas are differe ad he F es o deec a differece i average prices is MST F = = 6.. MSE The rejecio regio wih α =.05 ad ad 6 df is approimaely F >. ad H 0 is rejeced. There is eough evidece o idicae a differece i he average prices for he four saes..8 a The desig is a compleely radomized desig (four idepede samples). b The followig prelimiary calculaios are ecessary: T = 5 T = 577 T = 97 T = 567 G = 55 ij 55 ( ) ( ) CM = = =,.68 Toal SS = ij CM = 8, 7 CM = Ti SST = CM = CM = i Calculae MS = SS df ad cosolidae he iformaio i a ANOVA able. Source df SS MS Treames Error Toal c The hypohesis o be esed is H 0 : μ = μ = μ = μ versus H a : a leas oe pair of meas are differe ad he F es o deec a differece i average scores is 9

8 MST F = =.76. MSE The rejecio regio wih α =.05 ad ad 5 df is approimaely F >.9 ad H 0 is o rejeced. There is o eough evidece o idicae a differece i he average scores for he four eachig mehods..9 Sample meas mus be idepede ad based upo samples of equal size..0 Use Tables (a) ad (b). a q.05 ( 5, 7) = 5.06 b ( ) c q.0 (,8) = 6.0 d ( ). a q ( ) q.05,0 =.88 q.0 7,5 = 9. s s ω =.05, =.0 =.878s 5 5 s s ω = q.0 6, = 6.0 =.567s 8 8 b ( ) ω =.05 6,8 =.9 = 6.78 b The raked meas are show below. A lie uder wo or more meas idicaes a differece less ha ω ad hece o differeces bewee ha group of meas a q ( ). Wih k =, df = 0, = 6, ω = q.0 (, 0) MSE.5 = 5.0 =.69 6 The raked meas are show below The Miiab priou for pairwise comparisos is reproduced below. Tukey 95% Simulaeous Cofidece Iervals All Pairwise Comparisos amog Levels of Mehod Idividual cofidece level = 97.9% Mehod = subraced from: Mehod Lower Ceer Upper Mehod (-----*-----) (-----*-----) Mehod = subraced from: Mehod Lower Ceer Upper Mehod (-----*-----)

9 Miiab adjuss he differeces i he sample meas, calculaig a ierval ( i j) ± ω for all pairs of reames. If he ierval coais zero, he wo meas are o judged o be sigificaly differe. Hece, mehods ad are o sigificaly differe. The raked meas are show below The desig is compleely radomized wih reames ad 5 replicaios per reame. The Miiab priou below shows he aalysis of variace for his eperime. Oe-way ANOVA: mg/dl versus Lab Source DF SS MS F P Lab Error.5 5. Toal 65.0 S = 5.9 R-Sq = 9.5% R-Sq(adj) = 0.00% Idividual 95% CIs For Mea Based o Pooled SDev Level N Mea SDev ( * ) ( * ) ( * ) Pooled SDev = 5.9 Tukey 95% Simulaeous Cofidece Iervals All Pairwise Comparisos amog Levels of Lab Idividual cofidece level = 97.9% Lab = subraced from: Lab Lower Ceer Upper ( * ) ( * ) Lab = subraced from: Lab Lower Ceer Upper ( * ) a The aalysis of variace F es for H 0 :μ = μ = μis F =.60 wih p -value =.56. The resuls are o sigifica ad H 0 is o rejeced. There is isufficie evidece o idicae a differece i he reame meas. b Sice he reame meas are o sigificaly differe, here is o eed o use Tukey s es o search for he pairwise differeces. Noice ha all hree iervals geeraed by Miiab coai zero, idicaig ha he pairs cao be judged differe..6 Wih k =, df = 6, = 5, ω = q.0 (,6) MSE.5 = 5.9 =.9 5 The raked meas are show below The secod sample mea is differe ha he oher hree; meas ad are differe from each oher. 9

10 .7 a The followig prelimiary calculaios are ecessary: T = 85 T = 00 T = 7 G = 8859 ij 8859 ( ) ( ) CM = = = 55. Toal SS = ij CM = 5, 95, 69 CM = Ti SST = CM = + + CM = 75.8 i Calculae MS = SS df ad cosolidae he iformaio i a ANOVA able. Source df SS MS Treames Error Toal The hypohesis o be esed is H 0 : μ = μ = μ versus H a : a leas oe pair of meas are differe ad he F es o deec a differece i average scores is MST F = = MSE The rejecio regio wih α =.05 ad ad df is approimaely F >.89 ad H 0 is rejeced. There is evidece of a differece i he average scores for he hree graduae programs. b The 95% cofidece ierval for μ μis ( ) ±.05 MSE ± ± 6. or 57. < μ μ < 8.59 c Wih k =, df =, = 5, ω = q.05 (,) MSE 8. =.77 = The raked meas are show below There is o sigifica differece bewee programs ad, bu programs ad,, ad are differe from each oher..8 I comparig reames wihi 6 blocks, here are k = reame degrees of freedom ad b = 5block df. The ANOVA able is show below. Source df Treames Blocks 5 Error 0 Toal 7.9 Refer o Eercise.8. The give sums of squares are isered ad missig eries foud by subracio. The mea squares are foud as MS = SS df. 95

11 Source df SS MS F Treames Blocks Error 0.. Toal To compare he reame meas, he es saisic F = MST MSE =.0 ad he rejecio regio wih ad 0 df is F > F.05 =.0. The ull hypohesis is o rejeced. There is isufficie evidece o idicae a differece bewee reame meas.. The 95% cofidece ierval for μa μbis he ( A B) ±.05 MSE b (.9.) ± ±.5 or.8 < μ μ <.767. To es for differeces amog block meas, he es saisic is F = MSB MSE =.. The criical values of F from Table 6 wih 5 ad 0 df are show below. α F α Sice he observed value F =.is less ha F.0, p -value >.0 ad he ull hypohesis is o rejeced. There is isufficie evide o idicae differeces amog block meas.. Use Miiab o obai a ANOVA priou, or use he followig calculaios: ( ij ) ( ) CM = = = Toal SS = CM = CM = CM= ij j 7 0 T SST = CM = CM = 5.58 B i SSB = CM = CM = ad SSE = Toal SS SST SSB =.6667 Calculae MS = SS df ad cosolidae he iformaio i a ANOVA able. Source df SS MS F Treames Blocks Error Toal a To es he differece amog reame meas, he es saisic is F = MST = 8.58 = 9.9 MSE. ad he rejecio regio wih α =.05 ad ad 6 df is F >.76. There is a sigifica differece amog he reame meas. b To es he differece amog block meas, he es saisic is F = MSB = 60. = 5.75 MSE. 96 A B

12 ad he rejecio regio wih α =.05 ad ad 6 df is F > 5.. There is a sigifica differece amog he block meas. c Wih k =, df = 6, =, MSE. ω = q.0 (, 6) = 7.0 =.7 The raked meas are show below d The 95% cofidece ierval is ( A B) ±.05 MSE b ( 7..) ±.7. ±. or 5. < μa μb <.668 e Sice here is a sigifica differece amog he block meas, blockig has bee effecive. The variaio due o block differeces ca be isolaed usig he radomized block desig.. Similar o Eercise.. Use he Miiab priou o aalyze he eperime. You should oice ha here are sigifica differeces amog reame meas ad ha here are also sigifica differeces amog he block meas. Sice here is a differece amog he block meas, blockig has bee effecive. The variaio due o block differeces ca be isolaed usig he radomized block desig. To deermie where he reame differeces lie, use Tukey s es wih ω = q.05 (,8) MSE.08 =.0 =.0 5 The raked meas are show below The sigifica differece is bewee reame A ad he oher wo reames, B ad C..5 a By subracio, he degrees of freedom for blocks is b = 8= 6. Hece, here are b = 7 blocks. b There are always b = 7 observaios i a reame oal. c There are k = + = 5 observaios i a block oal. d Source df SS MS F Treames Blocks Error Toal.9 e To es he differece amog reame meas, he es saisic is MST.55 F = = = 9.68 MSE.667 ad he rejecio regio wih α =.05 ad ad df is F >.78. There is a sigifica differece amog he reame meas. f To es he differece amog block meas, he es saisic is MSB.5 F = = = 8.59 MSE

13 ad he rejecio regio wih α =.05 ad 6 ad df is F >.5. There is a sigifica differece amog he block meas..6 Use Miiab o obai a ANOVA priou, or use he followig calculaios: ( ij ) ( 5.) CM = = = 88.9 Toal SS = CM = CM = 6.76 ij ( ) T j ( 06.) + ( 0.9) + ( 08.) SS formulaios = CM = CM =.895 b ( ) + ( ) + ( ) + ( ) B SS( auo) = i CM = CM =.50 ad k SSE = Toal SS SST SSB =.5 Calculae MS = SS df ad cosolidae he iformaio i a ANOVA able. Source df SS MS F Treames Blocks Error Toal a To es he ull hypohesis ha here is o differece i mea mileage per gallo for he hree formulaios, he es saisic is F = MST = 6.6 MSE The criical values of F from Table 6 wih ad 6 df are show below. α F α Sice he observed value F = 6.6 is bewee F.05 ad F.05,.05 < p-value <.05 ad he ull hypohesis is rejeced a he 5% level of sigificace. There is a sigifica differece amog he reame meas. b To es he ull hypohesis ha here is o differece i mea mileage for he four auomobiles, he es saisic is F = MSB =.75 MSE ad he p-value wih ad 6 df is.05 < p-value <.0 There is o evidece of a sigifica differece amog he auomobiles. c The 90% cofidece ierval is ( A B) ±.05 MSE b ( ) ± ±.650 or.85 < μa μb <.55 d To deermie where he reame differeces lie, use Tukey s es wih ω = q.05 (, 6) MSE.67 =. =.07 The raked meas are show below A C B 98

14 Oly gasolie formulaios A ad B are sigificaly differe from each oher..7 Similar o previous eercises. The Miiab priou for his radomized block eperime is show below. Two-way ANOVA: Measuremes versus Blocks, Chemicals Source DF SS MS F P Blocks Chemicals Error Toal.9067 S = R-Sq = 95.85% R-Sq(adj) = 9.0% Idividual 95% CIs For Mea Based o Pooled SDev Blocks Mea (----*-----).700 (----*-----).5 (-----*----) Idividual 95% CIs For Mea Based o Pooled SDev Chemicals Mea (-----*-----). (-----*-----).000 (-----*-----).8000 (-----*-----) Boh he reame ad block meas are sigificaly differe. Sice he four chemicals represe he reames i his eperime, Tukey s es ca be used o deermie where he differeces lie: ω = q.05 (, 6) MSE.0897 =.90 =.85 The raked meas are show below The chemical falls io wo sigificaly differe groups A ad C versus B ad D..8 a Use he properies of addiiviy o obai SS. Degrees of freedom are obaied usig b = 0 ad k =. Source df SS MS F Mirrors Drivers Error Toal b To es he equaliy of reame meas, he es saisic is F = MST = 6.96 MSE The criical values of F from Table 6 wih ad 7 ( 0) df are show below. α F α Sice he observed value F = 6.96 is greaer ha F.005, p -value <.005 ad H 0 is rejeced. There is a sigifica differece i mea glare raigs for he four mirrors. c To es he equaliy of block meas, he es saisic is The criical values of F from Table 6 wih 9 ( 0) F = MSB =.7 MSE ad 7 ( 0) 99 df are show below.

15 α F α Sice he observed value F =.7 is greaer ha F.005, p -value <.005. There is a differece amog he drivers..9 The facor of ieres is soil preparaio, ad he blockig facor is locaios. A radomized block desig is used ad he aalysis of variace able ca be obaied usig he compuer priou. a The F saisic o deec a differece due o soil preparaios is F = MST = 0.06 MSE wih p -value =.0. The ull hypohesis ca be rejeced a he 5% level of sigificace; here is a sigifica differece amog he reame meas. b The F saisic o deec a differece due o locaios is F = MSB = 0.88 MSE wih p -value =.008. The ull hypohesis ca be rejeced a he % level of sigificace; here is a highly sigifica differece amog he block meas. c Tukey s es ca be used o deermie where he differeces lie: MSE.8889 ω = q.0 (, 6) = 6. =.5 The raked meas are show below Preparaios ad are he oly wo reames ha ca be declared sigificaly differe. d The 95% cofidece ierval is ( B A) ±.05 MSE b ( 6.5.5) ± ±.8 or. < μ μ < A rearrageme of he resuls gives Dogs Level of Digialis A B C The aalysis of variace able ca be foud o he priou. a There are k b+ = + = 6degrees of freedom associaed wih SSE. b The F saisic o deec a differece i mea upake of calcium for he hree levels of digialis is F = MST = 58. MSE wih p -value =.000. The digialis levels are very sigificaly differe; a leas oe of he levels of digialis causes a differe mea upake of calcium i he hear muscle of dogs. c Tukey s es ca be used o deermie where he differeces lie: MSE 05 ω = q.0 (, 6) = 6. = B A

16 The raked meas are show below A B C Average resposes are differe for all hree levels of digialis. d The F saisic o deec a differece i mea upake i calcium for he four hear muscles is F = MSB = MSE wih p -value =.000. The ull hypohesis of o differece should be rejeced; here is evidece of a sigifica differece amog he hear muscles. e Tukey s es ca be used o deermie where he differeces lie: MSE 05 ω = q.0 (, 6) = 7.0 = 9. The raked meas are show below The hear muscle i he firs dog is sigificaly differe from he oher hree dogs. f The sadard deviaio of he differece bewee he mea calcium upake for wo levels of digialis is s + = MSE = 05 =.5 b b b g The 95% cofidece ierval is ( ) MSE b A B ±.05 ( ) ±.7(.5) 7.5 ± 55. or 9.8 < μ μ < 8.. A radomized block desig has bee used wih esimaors as reames ad cosrucio job as he block facor. The aalysis of variace able is foud i he Miiab priou below. Two-way ANOVA: Cos versus Esimaor, Job Source DF SS MS F P Esimaor Job Error Toal S = R-Sq = 9.6% R-Sq(adj) = 8.% Idividual 95% CIs For Mea Based o Pooled SDev Esimaor Mea A.65 ( * ) B.8875 ( * ) C.875 ( * ) Boh reames ad blocks are sigifica. The reame meas ca be furher compared usig Tukey s es wih ω = q.05 (, 6) MSE.757 =. =.885 The raked meas are show below A C A B B 0

17 Esimaors A ad B show a sigifica differece i average coss.. a A radomized block desig has bee used wih isurace compaies as reames ad locaios as he block facor. The aalysis of variace able is foud i he Miiab priou below. Two-way ANOVA: Cos versus Locaio, Compay Source DF SS MS F P Locaio Compay Error Toal S = 6. R-Sq = 89.8% R-Sq(adj) = 8.5% b-c There is a sigifica differece due o compaies ( F = 7.9 wih p -value =.00 ) ad also due o locaio (blocks) ( F =.0 wih p -value =.000 ). d The reame meas ca be furher compared usig Tukey s es wih ω = q.05 ( 5,) MSE 68, 69 =.5 = The raked meas are show below s Ceury AAA Allsae Firema s Fud Sae Farm The compaies overlap io groups wihi which he meas are o sigificaly differe.. a A radomized block desig has bee used wih sores as reames ad iems as he block facor. b The F saisic o deec a differece i mea prices for he five sores is F = MST =.79 MSE wih p -value =.000. The ull hypohesis of o differece should be rejeced; here is evidece of a sigifica differece i average prices amog he sores. c The F saisic o deec a differece i mea prices for he 8 iems is F = MSB = 9.9 MSE wih p -value =.000. The ull hypohesis of o differece should be rejeced; here is evidece of a sigifica differece i average prices from iem o iem. Tha is, blockig has bee effecive. d If WiCo has specifically chose o lis he iems for which hey kow heir cos is lower ha he oher sores, he eperime will be biased i heir favor. The average cos of iems migh o be less ha he oher sores if he iems o be priced had bee radomly seleced.. a The reame meas ca be furher compared usig Tukey s es. From Table (a), we use a coservaive esimae wih df = ad q.05(5,) =.7. b Calculae q ( ) MSE.865 ω =.05, =.7 =.78 8 c The raked meas are show below. WiCo Food Less Saers Ralphs Albersos The average price a WiCo is o sigificaly lower ha Food Less, bu i is lower ha he oher hree sores..5 a-b There are 5= 0reames ad 5 = 60oal observaios. 0

18 c I a facorial eperime, variaio due o he ieracio A Bis isolaed from SSE. The sources of variaio ad associaed degrees of freedom are give o he e page. Source df A B A B Error 0 Toal 59.6 a The complee ANOVA able is show below. Sice facor A is ru a levels, i mus have df. Oher eries are foud by similar reasoig. Source df SS MS F A B A B Error.5.07 Toal.7 b The es saisic is F = MS( AB) MSE = 0.9 ad he rejecio regio is F >.00. Hece, H 0 is o rejeced. There is isufficie evidece o idicae ieracio bewee A ad B. c The es saisic for esig facor A is F =.0 wih F.05 =.89. The es saisic for facor B is F =.9 wih F.05 =.9. Neiher A or B are sigifica..7 Refer o Eercise.6. The 95% cofidece ierval is ( ) ±.05 MSE r ( 8. 6.) ± ±. or. < μ μ < 5..8 a The ie reame (cell) oals eeded for calculaio are show i he able. Facor A Facor B Toal Toal CM = = 9. 8 Toal SS = 66 CM = SSA = CM =. SSB = CM = SS( AB) = SSA SSB CM = 6. Source df SS MS F A B A B Error Toal

19 F = = ad he rejecio regio is F >.6. There is evidece of a sigifica ieracio. Tha is, he effec of facor A depeds upo he level of facor B a which A is measured. d Sice F = 6.67 lies bewee F.0 ad F.005,.005 < p-value <.0. b-c The es saisic is MS( AB) MSE 6.67 e Sice he ieracio is sigifica, he differeces i he four facor-level combiaios should be eplored idividually, usig a ieracio plo such as he oe geeraed by Miiab below. Facor A Mea Facor B Look a he differeces bewee he hree levels of facor A whe facor B chages from level o level. Levels ad behave very similarly while level behaves quie differely. Whe facor B chages from level o level, levels ad of facor A behave similarly, ad level behaves differely..9 a Similar o Eercise.8. Based o he fac ha he mea respose for he wo levels of facor B behaves very differely depedig o he level of facor A uder ivesigaio, here is a srog ieracio prese bewee facors A ad B. b The es saisic for ieracio is F = MS( AB) MSE = 7.85 wih p -value =.000 from he Miiab priou. There is evidece of a sigifica ieracio. Tha is, he effec of facor A depeds upo he lvel of facor B a which A is measured. c I ligh of his ype of ieracio, he mai effec meas (averaged over he levels of he oher facor) differ oly slighly. Hece, a es of he mai-effec erms produces a o-sigifica resul. d No. A sigifica ieracio idicaes ha he effec of oe facor depeds upo he level of he oher. Each facor-level combiaio should be ivesigaed idividually. e Aswers will vary..50 Use he compuig formulas give i his secio or a compuer sofware package o geerae he ANVOA able for his facorial eperime. The followig priou was geeraed usig Miiab. Two-way ANOVA: Perce Gai versus Markup, Locaio Source DF SS MS F P Markup Locaio Ieracio Error Toal.67 S = 5.90 R-Sq = 85.05% R-Sq(adj) = 7.60% a From he priou, F =.wih p -value =.6. Hece, a he α =.05 level, H 0 is o rejeced. There is isufficie evidece o idicae ieracio. b Sice o ieracio is foud, he effecs of A ad B ca be esed idividually. Boh A ad B are sigifica. 0

20 c The ieracio plo geeraed by Miiab is show o he e page. Noice ha he lies, alhough o eacly parallel, do o idicae a sigifica differece i he behavior of he mea resposes for he wo differe locaios. 0 Locaio 0 Mea Markup d The 95% cofidece ierval is ( ) ±.05 MSE r ( 7 +.5) ± ±.5 or.0 < μ μ <.99.5 a The oal umber of paricipas was siy, wey i each of hree caegories. Hece, he oal degrees of freedom is fify-ie. Facor T was ru a wo levels, facor A a hree levels, resulig i he give degrees of freedom. b MST MSA F = = =.66 F = = = 6.87 MSE 8.05 MSE 8.05 MS( TA).9905 F = = =. MSE 8.05 c Sice ieracio is sigifica, he mai effecs eed o be esed idividually. Aeio should be focused o he idividual cell meas. d The abled values for he approimae df are show below. α F α (,60) F α (,60) For facor T,.005 < p-value <.0. For facor A, p -value <.005, ad for he ieracio A T,.0 < p-value < Aswers will vary from sude o sude. There is o sigifica ieracio, or is he mai effec for ciies sigifica. There is a sigifica differece i he average cos per mile based o he disace raveled, wih he cos per mile decreasig as he disace icreases. Perhaps a sraigh lie may model he coss as a fucio of ime..5 a The desig is a facorial eperime wih r = 5 replicaios. There are wo facors, Geder ad School, oe a wo levels ad oe a four levels. 05

21 b The aalysis of variace able ca be foud usig a compuer priou or he followig calculaios: Schools Geder Toal Male Female Toal CM = = Toal SS = 589 CM = SSG = CM = SS( Sc) = CM = SS( G Sc) = SSG SS( Sc) CM = Source df SS MS F G Sc G Sc Error Toal c The es saisic is F = MS( GSc) MSE =.9 ad he rejecio regio is F >.9 (wih α =.05 ). Aleraely, you ca boud he p -value >.0. Hece, H 0 is o rejeced. There is isufficie evidece o idicae ieracio bewee geder ad schools. d You ca see i he ieracio plo ha here is a small differece bewee he average scores for male ad female sudes a schools ad, bu o differece o speak of a he oher wo schools. The ieracio is o sigifica Geder 650 Mea School e The es saisic for esig geder is F =.09 wih F.05 =.7 (or p -value >.0 ). The es saisic for schools is F = 7.75 wih F.05 =.9 (or p -value <.005 ). There is a sigifica effec due o schools. Usig Tukey s mehod of paired comparisos wih α =.0, calculae MSE 96. ω = q.0 (, ) =.80 =

22 The raked meas are show below a The eperimeal uis are he supervisors. b The wo facors are he raiig mehod (raied or uraied) ad he siuaio (sadard or emergecy). c There are wo levels of each facor. d There are = reames. e The desig is a facorial eperime, wih replicaios per reame..55 a The aalysis of variace able ca be foud usig a compuer priou or he followig calculaios: Traiig (A) Siuaio (B) Traied No Traied Toal Sadard Emergecy Toal CM = = Toal SS = 6660 CM = SSA = CM = = SSB = CM = SS( A B) = SSA SSB CM = 56.5 Source df SS MS F A B A B Error Toal 5 56 b The es saisic is F = MS( A B) MSE =.7 ad he rejecio regio is F >.75 (wih α =.05 ). Aleraely, you ca boud he p -value >.0. Hece, H 0 is o rejeced. The ieracio erm is o sigifica. c The es saisic is F = MSB MSE =.6 ad he rejecio regio is F >.75 (wih α =.05 ). Aleraely, you ca boud he.05 < p-value <.0. Hece, H 0 is o rejeced. Facor B (Siuaio) is o sigifica. d The es saisic is F = MSA MSE = 7.9 ad he rejecio regio is F >.75 (wih α =.05 ). Aleraely, you ca boud he p -value <.005. Hece, H 0 is rejeced. Facor A (Traiig) is highly sigifica. e The ieracio plo is show o he e page. The respose is much higher for he supervisors who have bee raied. You ca see very lile chage i he respose for he wo differe siuaios (sadard or emergecy). The parallel lies idicae ha here is o ieracio bewee he wo facors. 07

23 80 Traiig 70 Mea Siuaio.56 The desig is compleely radomized wih five reames, coaiig four, seve, si, five ad five measuremes, respecively. The aalysis of variace able ca be foud usig he compuer priou or he followig calculaios: ( ij ) ( 0.6) CM = = = Toal SS = CM = CM =.78 ij ( ) ( ) ( ) T.5.7. i SST = CM = CM =. 7 5 i SSE = Toal SS SST =.57 a The F es is F =.67 wih p -value =.000. The resuls are highly sigifica, ad H 0 is rejeced. There is a differece i mea reacio imes due o he five simuli. b The hypohesis o be esed is H 0 : μa = μd versus H a : μa μd ad he es saisic is A D = = =.7 MSE.06 5 A D The rejecio regio wih α =.05 ad degrees of freedom is >.05 =.07 ad he ull hypohesis is rejeced. We coclude ha here is a differece bewee he meas..57 The iervals provided i he Miiab priou allow you o declare a differece bewee a pair of meas oly whe boh edpois have he same sig. Sigifica differeces are observed bewee reames A ad C, B ad C, C ad E ad D ad E. The raked meas are show below. E A B D C The residuals i he upper ail of he ormal probabiliy plo are smaller ha epeced, bu overall, here is o a problem wih ormaliy. The spreads of he residuals whe ploed agais he fied values is relaively cosa..59 The objecive is o deermie wheher or o mea reacio ime differs for he five simuli. The four people used i he eperime ac as blocks, i a aemp o isolae he variaio from perso o perso. A radomized block desig is used, ad he aalysis of variace able is give i he priou. a The F saisic o deec a differece due o simuli is F = MST = 7.78 MSE 08

24 wih p -value =.000. There is a sigifica differece i he effec of he five simuli. b The reame meas ca be furher compared usig Tukey s es wih MSE ω = q.05 ( 5,) =.5 =.90 The raked meas are show below. E A B D C c The F es for blocks produces F = 6.59 wih p -value =.007. The block differeces are sigifica; blockig has bee effecive..60 A compleely radomized desig has bee used. The aalysis of variace able ca be foud usig a compuer program or he followig calculaios: ( ij ) ( 6) CM = = =, Toal SS = ij CM =,70 CM = Ti SST = CM = CM = i Calculae MS = SS df ad cosolidae he iformaio i a ANOVA able. Source df SS MS Treames Error Toal The Miiab compuer priou is show below. Oe-way ANOVA: 0-9, 0-9, 0-59, Source DF SS MS F P Facor Error Toal S = R-Sq = 6.7% R-Sq(adj) = 0.00% Idividual 95% CIs For Mea Based o Pooled SDev Level N Mea SDev ( * ) ( * ) ( * ) ( * ) Pooled SDev = a The F es for reames is MST F = =.87 MSE wih p -value =.68 ad H 0 is o rejeced. There is o evidece o sugges a differece amog he four groups. b The 90% cofidece ierval for μ μis 09

25 ( ) ±.05 MSE ( ) ± ±.750 or.050 < μ μ < 6.50 c The 90% cofidece ierval for μ is MSE ± ± ±.65 or.88 < μ < 0.5. d Wih B =, σ MSE ad, he ecessary iequaliy is MSE MSE = or Samples of size = 6 will be required. I his case, he degrees of freedom associaed wih MSE will be = 00, which is large eough ha is a valid approimaio..6 Aswers will vary from sude o sude. A compleely radomized desig has bee used. The aalysis of variace able is show i he priou. Oe-way ANOVA:,,, Source DF SS MS F P Facor Error Toal S = 6.85 R-Sq = 56.% R-Sq(adj) = 50.50% Idividual 95% CIs For Mea Based o Pooled SDev Level N Mea SDev (------* ) (-----*-----) ( *------) (-----*-----) Pooled SDev = 6.85 Tukey 95% Simulaeous Cofidece Iervals All Pairwise Comparisos Idividual cofidece level = 98.90% subraced from: Lower Ceer Upper (------*------) ( * ) (-----*------) subraced from: Lower Ceer Upper (------* ) (------*-----) subraced from: Lower Ceer Upper (------* )

26 The sude should recogize he sigifica differece i he mea resposes for he four raiig programs, ad should furher ivesigae hese differeces usig Tukey s es wih raked meas show below: This is similar o Eercise.5. a-b There are = 8reames ad r = 8r oal observaios. c The sources of variaio ad associaed degrees of freedom are give below. Source df A B A B Error 8r 8 Toal 8r.6 This is similar o previous eercises. The complee ANOVA able is show below. Source df SS MS F A B A B Error Toal 9 8. a The es saisic is F = MS( AB) MSE =.0 ad he rejecio regio is F >.0. There is isufficie evidece o idicae a ieracio. b Usig Table 6 wih df = ad df =, he followig criical values are obaied. α F α The observed value of F is less ha F.0, so ha p -value >.0. c The es saisic for esig facor A is F = 6.5wih F.05 =.6. There is evidece ha facor A affecs he respose. d The es saisic for facor B is F = 7.7 wih F.05 =.0. Facor B also affecs he respose..6 Refer o Eercise.6. The 95% cofidece ierval is ( ) ±.05 MSE (.7.) ± ±.5 or.985 < μ μ < a The eperime is a facorial ad a wo-way aalysis of variace is geeraed. b Usig he Miiab priou give i he eercise, he F es for ieracio is F =.5 wih p -value =.6. There is isufficie evidece o sugges ha he effec of emperaure is differe depedig o he ype of pla. c The plo of reame meas for coo ad cucumber as a fucio of emperaure is show o he e page. The emperaure appears o have a quadraic effec o he umber of eggs laid i boh coo ad cucumber. However, he reame meas are higher overall for he cucumber plas.

27 60 55 Pla 0 50 Mea Temp 8 d The 95% cofidece ierval for μcoo μcucumber is ( Coo Cucumber ) ±.05 MSE Coo Cucumber ± ± 8.6 or.56 < μ μ < 5.8 Coo Cucumber.66 The compleely radomized desig has bee used. The aalysis of variace able ca be obaied usig a compuer program or he compuig formulas. Oe-way ANOVA: A, B, C, D Source DF SS MS F P Facor Error Toal S = 0.76 R-Sq = 9.7% R-Sq(adj) = 9.87% Idividual 95% CIs For Mea Based o Pooled SDev Level N Mea SDev A ( * ) B ( * ) C ( * ) D ( * ) Pooled SDev = 0.76 MST a To es he differece i reame meas, use F = = 5.0 wih p -value =.0. H 0 is rejeced a MSE he 5% level of sigificace; here is evidece o sugges a differece i mea discharge for he four plas. b The hypohesis o be esed is H 0 : μa =.5 versus H a : μa >.5 ad he es saisic is A μa = = =.88 MSE A The rejecio regio wih α =.05 ad 6 df is >.05 =.76 ad he ull hypohesis is o rejeced. We cao coclude ha he limi is eceeded a pla A. c The 95% cofidece ierval for μa μd is

28 ( A D) ±.05 MSE A D ( ) ± ±. or.579 < μ μ <.7.67 a The desig is a radomized block desig, wih weeks represeig blocks ad sores as reames. b The Miiab compuer priou is show below. Two-way ANOVA: Toal versus Week, Sore Source DF SS MS F P Week Sore Error Toal S =.799 R-Sq = 8.97% R-Sq(adj) = 7.5% c The F es for reames is F = 7. wih p -value =.00. The p-value is small eough o allow rejecio of H 0. There is a sigifica differece i he average weekly oals for he five supermarkes. d Wih k = 5, df =, =, ω = q.05 ( 5,) MSE.0 =.5 = 0.8 The raked meas are show below A D.68 Aswers will vary from sude o sude. The sudes should meio he sigificace of boh block ad reame effecs. There appear o be o violaios of he ormaliy ad commo variace assumpios. Sice he reame meas were sigificaly differe, Tukey s es is used o eplore he differeces wih ω = q.05 ( 5, 0) MSE.965 =. =.9 6 The raked meas are show below. E B A C D a This is a facorial eperime. The Miiab aalysis of variace is show below. Two-way ANOVA: VO versus Geder, Aciviy Source DF SS MS F P Geder Aciviy Ieracio Error Toal S =.88 R-Sq = 9.97% R-Sq(adj) = 9.57% b The F-es for ieracio is F = 7.6wih a p-value of.00. There is evidece of sigifica ieracio bewee geder ad levels of physical aciviy. The F-es for geder is F = wih a p-value of.000. There is sufficie evidece o idicae a differece i he average maimum oyge upake due o geder.

29 The F-es for levels of physical aciviy is F = 59.0 wih a p-value of.000. There is sufficie evidece o idicae a differece i he average maimum oyge upake due o levels of physical aciviy. c Sice he ieracio was sigifica, Tukey s es is used o eplore he differeces i he si facorlevel meas wih MSE. ω = q.05 ( 6,8).9 =.67 The raked meas are show below. FL FS FM ML MS MM The males who have higher levels of physical aciviy have sigificaly higher average maimum oyge upake..70 a-b This is a facorial desig wih r = 5 replicaios. The Miiab aalysis of variace is show below. Two-way ANOVA: Salary versus Geder, School Type Source DF SS MS F P Geder School Type Ieracio Error Toal S = 7.76 R-Sq =.96% R-Sq(adj) =.% Idividual 95% CIs For Mea Based o Pooled SDev Geder Mea ( * ) 6.57 ( * ) Idividual 95% CIs For Mea Based o School Pooled SDev Type Mea ( * ) 7.0 ( * ) 65. ( * ) b The F-es for ieracio is F = 0.0 wih a p-value of.0. There is o evidece of sigifica ieracio. The F-es for geder is F =.89 wih a p-value of.0 ad he F-es for school ype is F =.60 wih a p-value of.00. There is sufficie evidece o idicae a differece i he average salaries due school ype, bu o due o geder. c The 95% cofidece ierval for μm μf is ( M F) ±.05 MSE F M ( ) ± ± 5.8 or.96 < μ μ < 0.00 Sice he value μm μf = 0 falls i he ierval, here is o eough evidece o idicae a differece i he average salaries for males ad females. d Tukey s es is used o eplore he differeces due o school ype wih M F

30 ω = q.0 (, ) MSE = The raked meas are show below. Public Church Privae There is a differece i average salary bewee public ad privae isiuios, bu o bewee he oher wo pairs..7 a The desig is a compleely radomized desig wih hree samples, each havig a differe umber of measuremes. b Use he compuig formulas i Secio.5 or he Miiab priou below. Oe-way ANOVA: Iro versus Sie Source DF SS MS F P Sie Error Toal.7 S = 0.7 R-Sq = 9.6% R-Sq(adj) = 9.6% The F es for reames has a es saisic F = 6.85 wih p-value =.000. The ull hypohesis is rejeced ad we coclude ha here is a sigifica differece i he average perceage of iro oide a he hree sies. c The diagosic plos are show below. There appears o be o violaio of he ormaliy assumpios; he variaces may be uequal, judgig by he differig bar widhs above ad below he ceer lie. 99 Normal Probabiliy Plo of he Residuals (respose is Iro) Residuals Versus he Fied Values (respose is Iro) Perce Residual Residual -.5 Fied Value a The eperime is ru i a radomized block desig, wih elephoe compaies as reames ad ciies as blocks. b Use he compuig formulas i Secio.8 or he Miiab priou below. Two-way ANOVA: Score versus Ciy, Carrier Source DF SS MS F P Ciy Carrier Error Toal S =.87 R-Sq = 88.80% R-Sq(adj) = 8.% c The F es for reames (carriers) has a es saisic F = 9.90 wih p-value =.000. The ull hypohesis is rejeced ad we coclude ha here is a sigifica differece i he average saisfacio scores for he four carriers. 5

31 d The F es for blocks (ciies) has a es saisic F =.88 wih p-value =.09. The ull hypohesis is rejeced ad we coclude ha here is a sigifica differece i he average saisfacio scores for he four ciies..7 There is o evidece of o-ormaliy. There may be a slighly larger error variaio for he smaller values compared o he larger values of y..7 a The eperime is a facorial eperime, wih wo facors (rak ad geder). There are r = 0 replicaios per facor-level combiaio. b Use he compuig formulas i Secio.0 or he Miiab priou below. Two-way ANOVA: Salary versus Geder, Rak Source DF SS MS F P Geder Rak Ieracio Error Toal S =.650 R-Sq = 96.67% R-Sq(adj) = 96.6% c The F es for ieracio has a es saisic F =.78 wih p-value =.07. The ull hypohesis is o rejeced ad we coclude ha here is o sigifica ieracio bewee rak ad geder. d The F es for rak has a es saisic F = 75.0 wih p-value =.000, ad he F es for geder has a es saisic F = wih p-value =.000. Boh facors are highly sigifica. We coclude ha here is a differece i average salary due o boh geder ad rak. e The ieracio plo is show below. Noice he differeces i salary due o boh rak ad geder. Ieracio Plo for Salary Daa Meas 0 00 Rak A ssisa A ssociae Full Mea Female Geder Male Usig Tukey s es is used o eplore he differeces wih MSE. ω = q.0 (, 5).7 =.56 0 The raked meas are show below. All hree of he raks have sigificaly differe average salaries. Assisa Associae Full Case Sudy: A Fie Mess The desig is a wo-way classificaio, wih ype of icke as he reame ad ciies as blocks. The Miiab priou for he radomized block desig is show below. Two-way ANOVA: Fie versus Ciy, Type Source DF SS MS F P 6

32 Ciy Type Error Toal S =.8 R-Sq = 56.58% R-Sq(adj) =.% Idividual 95% CIs For Mea Based o Pooled SDev Type Mea (------*------) 5.08 (------* ).58 (------*------) Sudes should oice he sigifica differece i average icke prices for he hree ypes of ickes, bu o from ciy o ciy. I does o appear ha blockig has bee effecive. To eplore he differeces bewee he hree ypes of ickes, use Tukey s procedure wih ω = q.05 (, ) MSE 6. =.5 =.55 The raked meas are show below. Overime Red Zoe Hydra The red zoe ad fire hydra icke amous are o sigificaly differe, bu he amou for overime parkig appears o be sigificaly less ha he oher wo ypes. Aswers will vary, bu should summarize he above resuls. 7