ENGI 441 Prbbly nd Sscs Fculy f Engneerng nd Appled Scence Prblem Se 10 Sluns Ch-Squre Tess; Smple Lner Regressn 1. Is he fllwng se f bservns f bjecs n egh dfferen drecns cnssen wh unfrm dsrbun? Drecn Number f bjecs bserved N 4 NE 6 E 13 SE 11 S 1 SW 18 W 11 NW 5 There re egh cegres, cnnng l f O = 80 80 E = = 10 > 5 8 Tes : d re frm unfrm dsrbun vs. : n. Exendng he ble, Drecn O E O E ( O ( O E E E N 4 10 6 36 3.6 NE 6 10 4 16 1.6 E 13 10 3 9 0.9 SE 11 10 1 1 0.1 S 1 10 4 0.4 SW 18 10 8 64 6.4 W 11 10 1 1 0.1 NW 5 10 5 5.5 Tl: 80 80 0 15.6
ENGI 441 Prblem Se 10 Sluns Pge f 14 1 (cnnued The ch-squre es ssc s χ bs = 15.6. The number f degrees f freedm s ν = n 1= 7 The crcl vlue s χ α,7. A α =.05, c = χ = 14.067, bu α =.01, c = χ = 18.475.05, 7.01, 7 χ bs = 15.6 The decsn depends n he level f sgnfcnce α chsen. A α =.05, χ > c rejec, bu bs bs A α =.01, χ < c d n rejec. Ths resul s mrgnlly sgnfcn. Frm he ssced Excel fle, he p-vlue s.09, s h.01 < p <.05. The nswer s NO YES f α =.05 f α =.01. A mdel f chemcl prcess predcs he numbers f runs f prcess h shuld fll n ech f fve me nervls. Are he vlues shwn n he ble belw cnssen wh he vlues expeced frm he mdel 5% level f sgnfcnce? Tme nervl bserved expeced 0-5 7 10 5-10 19 5 10-15 7 30 15-0 31 5 > 0 16 10 O = E = 100 nd every expeced vlue s > 5. Tes : d re frm he mdel dsrbun vs. Exendng he ble, : n, α =.05.
ENGI 441 Prblem Se 10 Sluns Pge 3 f 14 (cnnued Tme nervl O E O E O ( E ( O E E 0-5 7 10 3 9 0.90 5-10 19 5 6 36 1.44 10-15 7 30 3 9 0.30 15-0 31 5 +6 36 1.44 > 0 16 10 +6 36 3.60 Tl: 100 100 0 7.68 The ch-squre es ssc s χ bs = 7.68. The number f degrees f freedm s ν = n 1= 4 The crcl vlue s c = χα,4 = χ.05,4 = 9.48. A α =.05, χ < c d n rejec. bs Frm he ssced Excel fle, he p-vlue s.104, s h p >.05. The nswer s YES 3. Rndm smples f cbles re ken frm shpmens frm fur mnufcurers. Ech cble n ech smple s grded ccrdng he ld ppled when h cble breks. The resuls re summrzed n hs cnngency ble. Mnufcurer Quly f cble A B C D X 8 14 6 Y 14 61 19 6 Z 10 1 10 9 Are hese d cnssen wh ndependence beween mnufcurer nd quly f cble? Tes : he fcrs mnufcurer nd quly re ndependen vs. ν = I 1 J 1 = 3 = 6 ( ( : n.
ENGI 441 Prblem Se 10 Sluns Pge 4 f 14 3 (cnnued Exendng he ble, Mnufcurer Quly f cble A B C D Tl X 8 14 6 50 Y 14 61 19 6 100 Z 10 1 10 9 50 Tl 3 104 43 1 00 The ble f vlues expeced when s rue s clculed by ej = 3 100 43 Fr exmple, e3 = = = 1.5. 00 Expeced vlues: Quly f cble A B C D Tl X 8 6 10.75 5.5 50 Mnufcurer Y 16 5 1.50 10.50 100 Z 8 6 10.75 5.5 50 Tl 3 104 43 1 00 ( j ej All expeced vlues re > 5. The vlues f ej re: Quly f cble A B C D X 0.000 0.615... 0.98... 0.107... Mnufcurer Y 0.50 1.557... 0.90... 1.98... Z 0.500 0.961... 0.05....678... j. 3 4 = 1 j= 1 ( j ej χbs = = e j 9.94 c= χ = χ = 1.59 r χ = 16.81 χ < c,6.05,6.01,6 bs α Frm he ssced Excel fle, he p-vlue s.17, s h p >.05. D n rejec. Therefre YES These d re cnssen wh ndependence beween mnufcurer nd quly f cble.
ENGI 441 Prblem Se 10 Sluns Pge 5 f 14 4. A prculr ype f mr s knwn hve n upu rque whse rnge n nrml pern fllws nrml dsrbun. Seven mrs re chsen rndm nd re esed wh he ld nd new mehds f cnrllng he rnge f rque vlues. The resuls f he ess re s fllws: Mr: 1 3 4 5 6 7 New mehd: 5.5 3.16 4.43 6.1 5.75.1 6.01 Old mehd: 7.83 6. 7.46 8.83 8.19 5.64 8.88 ( Jusfy yur chce f mehd n (b belw. The d re prs f mesuremens n sngle se f ndvduls (he seven mrs. Therefre he pprpre mehd s pred w-smple -es (b Cnduc n pprpre hyphess es deermne wheher here s suffcen evdence cnclude h he rnge f rques wh he new mehd s les uns less hn wh he ld mehd. Le X = rque (new mehd Y = rque (ld mehd nd D = X Y Tes : µ D = vs. : µ D < Free chce f α. I chse α =.01. Mr: 1 3 4 5 6 7 Sum SSq x = New Mehd: 5.5 3.16 4.43 6.1 5.75.1 6.01 y = Old Mehd: 7.83 6. 7.46 8.83 8.19 5.64 8.88 d = Dfference:.58 3.06 3.03.71.44 3.43.87 0.1 58.5004 D: n= 7, d = 0.1, d = 58.5004 4.688 4 d =.874 85, sd = = 0.1116, sd = 0.334108 4
ENGI 441 Prblem Se 10 Sluns Pge 6 f 14 4 (b (cnnued Mehd 1 s c = µ D.01, 6 n 0.334 = 3.14 7 = 3.14 0.16 = 0.396 =.396 OR OR d =.87 < c Rejec n fvur f. Mehd d µ D.87 ( bs = = = 6.9 s 0.16 n = = c bs <.01, 6 3.14 c Rejec n fvur f. Mehd 3 bs = = 6.9 Clerly P[T < 6.9] <<.01 [Frm sfwre, P[T < 6.9] < 0.0003] Rejec n fvur f ny resnble vlue f α. An Excel spredshee fle s ls vlble llusre hs slun. (c Use he smple lner regressn mdel n hese d fnd he equn f he lne f bes f hese d. Ne h he lne f bes f depends n whch vrble s he predcr nd whch s he respnse. The rles f X nd Y re nerchnged f he new mehd s he respnse, (whch s he mre nurl ssgnmen.
ENGI 441 Prblem Se 10 Sluns Pge 7 f 14 4 (c (cnnued Summry sscs (respnse = Y = ld mehd: n= 7 x= 3.93 x = 168.6941 nsxx = n x ( x = 96.4738 n S xy = n xy ( x( y = 78.4941 nsyy = n y ( y = 65.08 xy= y= y = 60.7758 53.05 411.3579 S ˆ xy 78.4941 β1 = = = 0.813631656 S 96.4738 xx nd β 1 0 ( y β1 x ˆ = ˆ = n 3.751017 490 Therefre, (crrec 3 s.f. n ech ceffcen, OR y = 0.814 x + 3.75 Summry sscs (respnse = Y = new mehd: n= 7 x= 53.05 x = 411.3579 xy= y= y = 60.7758 3.93 168.6941 xx ( 65.08 ( ( 78.4941 ( 96.4738 ns = n x x = n S = n xy x y = xy ns = n y y = yy S ˆ xy 78.4941 β1 = = = 1.03845540 S 65.08 xx nd β 1 0 ( y β1 x ˆ = ˆ = n 4.41914370 Therefre, (crrec 3 s.f. n ech ceffcen, y = 1.0 x 4.4 An Excel spredshee fle s vlble llusre he secnd versn f hs slun.
ENGI 441 Prblem Se 10 Sluns Pge 8 f 14 4 (d Fnd he ceffcen f deermnn R nd use cmmen n yur nswer pr ( bve. Irrespecve f whch f ld r new mehds s he regressr, ( nsxy ( nsxx ( nsyy ( 78.4941 r = = =.979 49 96.4738 65.08 r = 97.9% (crrec 3 s.f. Very hgh crreln cnn use unpred -es Chce n ( s crrec. A Mnb prjec fle nd Mnb Repr Pd RTF fle re ls vlble llusre hs slun. 5. A sudy ws cnduced nlyze he relnshp beween dversng expendure nd sles. The fllwng d were recrded: X Y Adversng ($ Sles ($ 0 310 4 340 30 400 3 40 35 490 Assume smple lner regressn beween sles Y nd dversng X. Clcule he ceffcens β 0 nd β 1 f he lne f bes f hese d nd esme he sles when $8 re spen n dversng. Is here sgnfcn lner sscn beween Y nd X? Exendng he ble: x y x x y y 0 310 400 600 96100 4 340 576 8160 115600 30 400 900 1000 160000 3 40 104 13440 176400 35 490 15 17150 40100 Sum: 141 1960 415 56950 78800
ENGI 441 Prblem Se 10 Sluns Pge 9 f 14 5 (cnnued n S xx = n Σ x (Σ x = 5 415 (141 = 065 19881 = 744 n S xy = n Σ xy (Σ x Σ y = 5 56950 141 1960 = 84750 76360 = 8390 n S yy = n Σ y (Σ y = 5 78800 (1960 = 3941000 3841600 = 99400 S ˆ xy 8390 β1 = = = 11.7688170 S 744 xx 1 nd β0 ( y β1 x ( ˆ ˆ 1960 11. 141 = = = 73.991935 483 n 5 The regressn lne ( 4 s.f. s y = 11.8 x + 73.99 x = 8 y = 11.77... 8 + 73.99... = 389.744 63 7... When $8 s spen n dversng, we predc $390 f sles (crrec he neres dllr. There re mny chces f mehd fr n hyphess es fr lner sscn. Tes : ρ = 0 vs. : ρ 0 r, equvlenly, es : β 1 = 0 vs. : β1 0 ( nsxy ( nsxx ( nsyy 8390 r = = = 744 99 400.95184 jus ver 95% f ll vrn n y s explned by he lner regressn. The crreln (.975... s very srng, (whch suggess h here s lner sscn, bu he smple sze s very smll, s we shuld prceed wh frml hyphess es. r n.95 3 = = 1 r 1.95 7.700 r
ENGI 441 Prblem Se 10 Sluns Pge 10 f 14 5 (cnnued ( nsxy ( n ( nsxx ( nsyy ( nsxy 8390 3 = = 744 99 440 8390 7.700 r clcule he enres n he ANOVA ble s fllws: ( nsxy n( nsxx 8390 SSR = ( yˆ y = = = 189.607 56 5 744 nsyy SST = y y = S yy = = = n 5 ( 99 440 19880 SSE = SST SSR = 957.39 473... MSR = SSR / ν R = SSR = 18 9.607 56... MSE = SSE / ν E = SSE / 3 = 319.130 84 37... f = MSR / MSE = 59.94... d.f. SS MS f R 1 18 9.607 5... 18 9.607 5... 59.94... E 3 957. 39 48... 319.130 86 6... T 4 19 880 = + f = 7.700 r MSE n MSE 5 319.13 sb = = = = S 744 xx ( nsxx ˆ β 0 11.7 s.14 1 = = = b 7.700.144969 Cmpre he bserved = 7.700....005, 3 = 5.841... > α/, 3 fr ny resnble chce f α. [The p-vlue s less hn.0046.] Rejec n fvur f YES, here s sgnfcn lner sscn beween Y nd X. [See ls he ssced Excel, Mnb prjec nd Mnb RTF fles.]
ENGI 441 Prblem Se 10 Sluns Pge 11 f 14 6. [Ths s n exensn f Exmple 1.06 frm he lecure nes.] A smple f 10 desel rucks were run bh h nd cld esme he dfference n fuel ecnmy. The resuls, n mles per glln, re presened n he fllwng ble. (frm In-use Emssns frm evy-duy Desel Vehcles, J. Ynwz, Ph.D. hess, Clrd Schl f Mnes, 001. Truck Cld 1 4.56 4.6 4.46 4.08 3 6.49 5.83 4 5.37 4.96 5 6.5 5.87 6 5.90 5.3 7 4.1 3.9 8 3.85 3.69 9 4.15 3.74 10 4.69 4.19 ( Use he smple lner regressn mdel n hese d fnd he equn f he lne f bes f (fr s he respnse nd Cld s he regressr hese d mnully. Summry sscs: n= 10, x= 45.86, y = 49.84 ns ( xx = n x x = 63.5604 nsxy = n xy ( x( y = 71.356 ns ( yy = n y y = 81.764 x = 16.6700, xy= 35.6988, y = 56.530 S ˆ xy 71.356 β1 = = = 1.1170 Sxx 63.5604 1 ˆ β0 = ( y ˆ β1 x = n 0.16 73 Therefre, crrec hree sgnfcn fgures n ech ceffcen, he regressn lne s y = 1.1 x 0.16
ENGI 441 Prblem Se 10 Sluns Pge 1 f 14 6 (b Clcule he 95% predcn nervl fr sngle fuure bservn f fuel effcency when run h, fr ruck whse fuel effcency when run cld s 4.00 mles per glln. The frmul fr he predcn nervl s ( ˆ ˆ 1 n x β0 + β1x ± α /, ( n s 1 + + n ( nsxx ( nsyy ( nsxy ( ( ns ( x ( ns xx ( 63.5604 81.764 71.356 s = MSE = = n n 10 8 63.5604 = 0.015 461 s = 0.14344 α =.05 = =.306 01 α /, n x 45.86 x = = = 4.586 n 10 xx.05, 8 ( ( x = 4.00 x x = 0.586 ˆ = ˆ + ˆ = 0.16 + 1.1 4 = 4.36 408 Therefre he 95% PI x = 4 s nd y β0 β1x 1 10 0.343396 4.36 408 ±.306 01 0.015 461 1 + + 10 63.5604 = 4.36 408 ± 0.30806 = ( 4.018, 4.634] ( 3 d.p.
ENGI 441 Prblem Se 10 Sluns Pge 13 f 14 6 (c Use Mnb (r nher sfwre pckge check h he dsrbun f he resduls s cnssen wh nrml dsrbun. Ths nrml prbbly pl clerly demnsres cnssency wh nrml dsrbun. (d Use Mnb (r nher sfwre pckge shw he ANOVA ble, he equn f he lne f bes f nd dsply he lne f bes f, he 95% cnfdence nervls nd he 95% predcn nervls n sngle dgrm. Regressn Anlyss: versus Cld The regressn equn s = - 0.163 + 1.1 Cld S = 0.14344 R-Sq = 98.5% R-Sq(dj = 98.3% Anlyss f Vrnce Surce DF SS MS F P Regressn 1 8.00395 8.00395 517.67 0.000 Errr 8 0.1369 0.01546 Tl 9 8.1764
ENGI 441 Prblem Se 10 Sluns Pge 14 f 14 6 (d (cnnued 7.0 6.5 6.0 ENGI 441 Prblem Se 10 Quesn 6 = - 0.163 + 1.1 Cld Regressn 95% CI 95% PI S 0.14344 R-Sq 98.5% R-Sq(dj 98.3% 5.5 5.0 4.5 4.0 3.5 4.0 4.5 Cld 5.0 5.5 6.0 (e Fnd he smple crreln ceffcen r beween h nd cld fuel effcences, crrec hree sgnfcn fgures. SSR 8.00395 r = = =.984 =.99 SST 8.17 64 Crrec hree sgnfcn fgures, r =.99 (whch verfes he clm f srng crreln mde n Exmple 1.06. An Excel fle, Mnb prjec fle nd Mnb RTF fle re vlble fr hs quesn. Bck he ndex f sluns