Applied Statistics and Probability for Engineers, 6 th edition Xy

Transcription

1 Aled Statstcs and Probablty or Engneers, 6 th edton CHAPTER Sectons a) XX Xy b) ˆ c) (8).6(43) a) ˆ (XX) X y.9 ˆ so b).9.93().53(5) a) Model Model y 8 y ( ) 4 y y 8 y y 4( 8) y ( 8) 6 y 3 y MODEL y = y = MODEL

2 Aled Statstcs and Probablty or Engneers, 6 th edton The nteracton term n model aects the sloe o the regresson equaton. That s, t modes the amount o change er unt o on y. b) 6 y ˆ 4(6) 4 Then, s the eected change on y er unt o. No, t does not deend on the value o, because there s no relatonsh or nteracton between these two varables n model. c) Change er unt o (6) (6) Yes, the result does deend on the value o, because nteracts wth. -4 a) There are two regressor varables n ths model based on the sze o the (X X) matr. SS b) The estmate o s the MS Resdual. The MS Resdual = Resdual 37 = DF c) Standard error o ˆ ˆ C (5.583)(.339) a) The results rom comuter sotware ollow. The model can be eressed as Satsacton = Age Severty +.3 Anety n et t SS E 39.9 b) ˆ n ˆ) n c) cov( ( X X ) C, se( ˆ) ˆ C rom the Mntab outut.

3 Aled Statstcs and Probablty or Engneers, 6 th edton d) Because the regresson coecents have derent standard errors the arameters estmators do not have smlar recson o estmaton. Regresson Analyss: Satsacton versus Age, Severty, Anety The regresson equaton s Satsacton = Age Severty +.3 Anety Predctor Coe SE Coe T P Constant Age Severty Anety S = 7.37 R-Sq = 9.4% R-Sq(ad) = 89.% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total Source DF Seq SS Age Severty 97. Anety 74.6 Unusual Observatons Obs Age Satsacton Ft SE Ft Resdual St Resd R -6 Regresson Analyss: y versus,, 3, 4 The regresson equaton s y = Predctor Coe SE Coe T P Constant X X X X S = R-Sq = 7.% R-Sq(ad) = 54.5% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total a) 3 4 ˆ 7.9 b) c) se( ˆ ).8, se( ˆ ).6774, se( ˆ ) 9.68, se( ˆ 3).45, and se( ˆ 4).965 Because the regresson coecents have derent standard errors the arameters estmators do not have smlar recson o estmaton. d) 5.79(4) 4.93(4).78(9).46(98) 9.37

4 Aled Statstcs and Probablty or Engneers, 6 th edton -7 The regresson equaton s mg = cd -. rh -.34 etw +.9 cm ale +.9 n/v Predctor Coe SE Coe T P Constant cd rh etw cm ale n/v S =.83 R-Sq = 89.3% R-Sq(ad) = 84.8% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total a) where cd rh etw cm ale n / v b) ˆ se ( ˆ ) 9.67, se ( ˆ ). 338, se ( ˆ ). 63, se ( ˆ 3 ). 9459, se ( ˆ 4 ).765, se( ˆ 5 ). 39 and se ( ˆ 6 ). 73 c) (5).(53).3(45).9(9.9) 3.855(3.7).897(3.9) = The regresson equaton s y = Predctor Coe StDev T P Constant S =.887 R-Sq = 67.% R-Sq(ad) = 57.8% Analyss o Varance Source DF SS MS F P Regresson Error Total a) b) ˆ. 779 c) se ( ˆ ) 7. 6, se ( ˆ ). 633, se ( ˆ 3 ). 359, se( ˆ 4 ). 5339and se ( ˆ 5 ). 395 y ˆ y ˆ d) ().55(3).8(9).66(.)

5 Aled Statstcs and Probablty or Engneers, 6 th edton -9 Regresson Analyss: E-9y versus E-9, E-9, E-93 The regresson equaton s E-9y = E E E-93 Predctor Coe SE Coe T P Constant E E E S = R-Sq = 99.4% R-Sq(ad) = 99.% Analyss o Varance Source DF SS MS F P Regresson Resdual Error 6 97 Total a) y hat = b) ˆ c) The estmated standard errors o the coecent estmators are rovded n the above table (SE Coe). Because the regresson coecents have derent standard errors the arameters estmators do not have smlar recson o estmaton. d) y hat = (5.) +.45(3) + 8.3(7) = Predctor Coe SE Coe T P Constant tem soaktme soakct dtme dct S =.96 R-Sq = 96.8% R-Sq(ad) = 96.% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total a) where TEMP SOAKTIME SOAKPCT DFTIME DIFFPCT b) ˆ c) The standard errors are lsted under the StDev column above. d).33.9(65).38().39(.).8476().363(.8) y ˆ.39

6 Aled Statstcs and Probablty or Engneers, 6 th edton - The regresson equaton s rads = mams eosure tme Predctor Coe SE Coe T P Constant mams eosure tme S = R-Sq = 84.3% R-Sq(ad) = 83.5% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total a) where mams Eosure Tme b) ˆ se ( ˆ ) 94., se ( ˆ ) 3. 46, and se ( ˆ ) 5. 4 ˆ c) y () 68.8(5) The regresson equaton s ARSNAILS = AGE -.7 DRINKUSE -.45 COOKUSE + 3. ARSWATER Predctor Coe SE Coe T P Constant AGE DRINKUSE COOKUSE ARSWATER S =.36 R-Sq = 8.% R-Sq(ad) = 76.5% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total a) 3 4 where AGE DrnkUse 3 CookUse 4 ARSWater b) ˆ. 557 se ( ˆ ).47, se ( ˆ ). 358, se ( ˆ ). 4747, se ( ˆ 3 ). 848 se ( ˆ 4 ).679 c) y ˆ (55).74(5).45(5) 3.4(.65) , and

7 Aled Statstcs and Probablty or Engneers, 6 th edton -3 The regresson equaton s densty = delectrc constant +. loss actor Predctor Coe SE Coe T P Constant delectrc constant loss actor S =.8834 R-Sq = 99.7% R-Sq(ad) = 99.7% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total LossFactor a) where DelectrcConst b) ˆ. 8 se ( ˆ ).5, se ( ˆ ). 68, and se ( ˆ ) c) y ˆ.5.47(.3).8(.3) The regresson equaton s y = Predctor Coe SE Coe T P Constant S = R-Sq = 93.7% R-Sq(ad) = 9.6% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total ˆ a) y ˆ 9. se ( ˆ ) 8.4, se ( ˆ ). 539, and se ( ˆ ). 539 b) 48 ˆ c) y 7 7.3(4.5).7(.5) = The regresson equaton s Useul range (ng) = Brghtness (%) -.7 Contrast (%) Predctor Coe SE Coe T P Constant Brghtness (%)

8 Aled Statstcs and Probablty or Engneers, 6 th edton Contrast (%) S = R-Sq = 75.6% R-Sq(ad) = 67.4% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total Contrast a) where % Brghtness % b) ˆ 3 se ˆ, se ( ˆ ). 6763, and se ( ˆ ) c) ( ) d) y ˆ (9).767(8) The regresson equaton s Stack Loss(y) = X +.3 X -.5 X3 Predctor Coe SE Coe T P Constant X X X S = R-Sq = 9.4% R-Sq(ad) = 89.8% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total a) 3 b) ˆ. 5 se ˆ, se ( ˆ ). 349, se ( ˆ ). 368, and se ( ˆ 3 ). 563 c) ( ). 9 d) y ˆ (65).953(8).5(9) a) The model can be eressed as: Ratng Pts = Pct Com Pct TD Pct Int Regresson Analyss: Ratng Pts versus Pct Com, Pct TD, Pct Int The regresson equaton s Ratng Pts = Pct Com Pct TD Pct Int

9 Aled Statstcs and Probablty or Engneers, 6 th edton Predctor Coe SE Coe T P Constant Pct Com Pct TD Pct Int S =.3479 R-Sq = 95.3% R-Sq(ad) = 94.8% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total Source DF Seq SS Pct Com Pct TD Pct Int n et t SS E 5.93 b) ˆ 4. 4 n ˆ) n 8 c) cov( ( X X ) C, se( ˆ) rom the SE Coe column n the comuter outut. ˆ C d) Ratng Pts =.99 +.* *5-3.8*4 = Regresson Analyss: W versus GF, GA,... The regresson equaton s W = GF -.83 GA -.54 ADV +.9 PPGF -.4 PCTG -.63 PEN -.8 BMI + 3. AVG +.9 SHT -.6 PPGA PKPCT +.6 SHGF +.6 SHGA +.5 FG Predctor Coe SE Coe T P Constant GF GA ADV PPGF PCTG PEN BMI AVG SHT PPGA PKPCT SHGF SHGA FG S = R-Sq = 9.9% R-Sq(ad) = 86.3% Analyss o Varance

10 Aled Statstcs and Probablty or Engneers, 6 th edton Source DF SS MS F P Regresson Resdual Error Total where GF AVG GA 9 3 SHT ADV 4 PPGA PPGF 5 PKPCT PCTG 6 PEN SHGF ˆ 7.46 The standard errors o the coecents are lsted under the SE Coe column above. -9 Predctor Coe SE Coe T P Constant Xl X S =.674 R-Sq = 98.4% R-Sq(ad) = 97.3% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total a) b) ˆ 59, se( ˆ ) 37.8, se( ˆ ).557, and se( ˆ ).446 c) (5).843() 8.95 d) Predctor Coe SE Coe T P Constant Xl X X*X S =.5443 R-Sq = 99.% R-Sq(ad) = 98.% Analyss o Varance Source DF SS MS F P Regresson Resdual Error.4. Total e) ˆ., se( ˆ ) 8.68, se( ˆ ) 3.89, se( ˆ ).9 and se( ˆ ).3 ) (5).().48(5)() 7.75 The redcted value s smaller 3 7 BMI SHGA 4 4 FG 7

11 Aled Statstcs and Probablty or Engneers, 6 th edton - a) ' ' )] ( ) ( [ ),, ( y ) )]( ( ) ( [ )] ( ) ( [ ' ' ' y y ) )]( ( ) ( [ ' y Settng the dervatves equal to zero yelds ) ( ) ( ) )( ( ) ( ) )( ( ) ( ' ' ' y n y n y n b) From the rst normal equaton, y ' ˆ. c) Substtutng y y or y n the rst normal equaton yelds ˆ '. Sectons - - a) ) ˆ ( ˆ se t, null hyothess ˆ s reected at level n t t, / E R E R MS MS n SS k SS F ) /( /, regresson s sgncant at level n k,, The mssng quanttes are as ollows: Predctor Coe SE Coe T P Constant Source DF SS MS F P Regresson Resdual Error Total 4 39 R-Squared = 784/39 =.9867 b) From the P-value rom the F test (F = ) or regresson s sgncant. c) Each ndvdual regressor s sgncant to the model that contans the other regressors. - a).83 R T SS R SS b) SS E = SS T SS R = =

12 Aled Statstcs and Probablty or Engneers, 6 th edton SSE /( n ) /( 3) Rad.837 SS /( n ) /( ) T SSRegresson c) MS Regresson = 5 k SS MS Error = E.765 n 7 7 F = MS MS Regresson Error The ANOVA table Source DF SS MS F P Regresson <. Resdual Error Total 9 For the F test the P-value <.. Thereore the F test reects the null hyothess at =.5 and also reects at =.. d) The ANOVA table ater addng a thrd regressor Source DF SS MS Regresson Resdual Error Total 9 SSRegresson ( 3,, ) / = MSError 5.94 Because.5,,6 = 4.49, we al to reect H and conclude that the thrd regressor does not contrbute sgncantly to the model. -3 a) n =, k =, = 3, =.5 H :... k H : or at least one (96) S yy y 96 X ' y y y ˆ ' X ' y SSR SS S SS E yy R SS R k 446.3/ 84.5 SSE 83.7 / 7.5,,7 n ,,7 Reect H and conclude that the regresson model s sgncant at =.5. P-value =.

13 Aled Statstcs and Probablty or Engneers, 6 th edton b) ˆ SSE MS E.957 n se( ˆ ) ˆ c.957(.439).9 se( ˆ ) ˆ c.957(.87).99 H : H: ˆ ˆ t t se( ˆ ) se( ˆ ) t /,7 t.5,7.365 Reect H, P-value <. Reect H, P-value <. Both regresson coecents sgncant -4 S 738. yy a) H : H : or at least one =. SS SS R E ( ˆ' X ' y S yy SS R y ) SSR k / SSE 5.45/ 7.,,7 n 9.55 n n ,,7 Reect H and conclude that the regresson model s sgncant at =.. SS 5.45 ˆ E MSE n 7 se( ˆ ) 7.493(7.9799E 5).45 b) H :

14 Aled Statstcs and Probablty or Engneers, 6 th edton H : ˆ t se( ˆ ) t t ,7 t.5,7 Reect H and conclude that s sgncant n the model at =. P-value = ( P( t t )) = ( ) =.675 c) s useul as a regressor n the model. -5 a) Degrees o reedom = 4 = 6 : t 4.8 P-value = (9.44 E-5) =.88 E-4 : t 8. P-value = (.978 E-7) = 3.96 E-7 3 : t.98 P-value = (.7) =.34 b) H : 3 H : 3 =.5 t, P-value = (.7) = Because the P-value > =.5, al to reect H. We conclude that 3 does not contrbute sgncantly to the model. -6 a) H : 3 4 H at least one =..,4, ,4,7 Reect H P-value =.5 b) =. H : H : t.7 t. 87 t t. t t.895 /, n.5,7 t t.5,7 or Reect H or., 3 and 4-7 a) H : H : at least one

15 Aled Statstcs and Probablty or Engneers, 6 th edton 9.53,6,4.,6,4.4.,6,4 Reect H and conclude regresson model s sgncant at =. b) The t-test statstcs or β through β 6 are -.45, -.7, -3.4,.7, -.9,.69. Because t.5,4 =.76, the regressors that contrbute to the model at α =. are etw and ale. -8 a) : H or all : H or at least one 7.6.,4,4.39.,4,4 Reect H and conclude that the regresson s sgncant at =.. P-value =.3 b) ˆ. 779 =. t t. 76 /, n.5,4 H : H : t. 3 t t. 9 t. 9 t t /, 4 t t /, 4 t t /, 4 t t /, 4 Fal to reect H Reect H Reect H Fal to reect H X and X 5 do not contrbute to the model. -9 a) H : 3 H : or at least one.,3, ,3,6 Reect H and conclude regresson s sgncant at =. b) ˆ. 856 =. t /, n t.5, H : 3 H : 3 t.58 t. 84 t 3. 8 t t.5,6 t t.5, 6 t t.5, 6 Reect H Reect H Reect H -3 ARSNAILS = AGE -. DRINKUSE +. COOKUSE Predctor Coe SE Coe T P Constant AGE DRINKUSE COOKUSE

16 Aled Statstcs and Probablty or Engneers, 6 th edton S =.5697 R-Sq = 8.% R-Sq(ad) =.% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total H : a) 3 H : or at least one ; k = 4..5.,3,7 5.8.,3,7 Do not reect H. There s nsucent evdence to conclude that the model s sgncant at α =.. The P-value =.687. b) H : H : =. ˆ.858 t. se( ˆ ).783 t.5,7.898 t t /,7. Fal to reect H, there s not enough evdence to conclude that s sgncant n the model at =.. H : H : =. t ˆ.8 se( ˆ ).8 t.5, t t /,7. Fal to reect H, there s not enough evdence to conclude that s sgncant n the model at =.. H : 3 H : 3 =. ˆ 3.97 t.5 se( ˆ ).798 t 3.5,7.898 t t /,7. Fal to reect H, there s not enough evdence to conclude that 3 s sgncant n the model at =.. -3 a) H : H : or at least one =.5

17 Aled Statstcs and Probablty or Engneers, 6 th edton ,, ,,37 The regresson equaton s rads = mams eosure tme Predctor Coe SE Coe T P Constant mams eosure tme S = R-Sq = 84.3% R-Sq(ad) = 83.5% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total Reect H and conclude regresson model s sgncant at =.5. P-value <. ˆ b) MS t se( ˆ ) H : H : =.5 E ˆ c ˆ se( ˆ ) t.6 t.5,43.5,37 t t /,37, Reect H and conclude that s sgncant n the model at =.5 ˆ se( ) ˆ c 5.4 H : H : =.5 ˆ t se( ˆ )

18 Aled Statstcs and Probablty or Engneers, 6 th edton t.5,43 t.5,37 t t /,37,.6 Reect H conclude that s sgncant n the model at =.5-3 The regresson equaton s y = Predctor Coe SE Coe T P Constant S = R-Sq = 93.7% R-Sq(ad) = 9.6% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total a) H : H : or at least one =.5 SSR k SSE n.5,, ,, / / 6 Reect H and conclude regresson model s sgncant at =.5 P-value b) ˆ MS E t se( ˆ ) H : H : =.5 ˆ c ˆ se( ˆ ) t.447 t.5,93.5,6 t t /,6, Reect H, s sgncant n the model at =.5

19 Aled Statstcs and Probablty or Engneers, 6 th edton t ˆ se( ) ˆ c.539 H : H : =.5 ˆ se( ˆ ) t.447 t.5,93.5,6 t t /,6, Reect H conclude that s sgncant n the model at =.5 c) Wth a smaller samle sze, the derence n the estmate rom the hyotheszed value needs to be greater to be sgncant. -33 Useul range (ng) = Brghtness (%) -.7 Contrast (%) Predctor Coe SE Coe T P Constant Brghtness (%) Contrast (%) S = R-Sq = 75.6% R-Sq(ad) = 67.4% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total H : a) H : or at least one =. SSR 458/ k SSE 798/ 6 n.,,6.9.,,6 9.8 Fal to reect H and conclude that the regresson model s not sgncant at =. P-value =.5 b) ˆ MS E 3 se ˆ ˆ c ( ).6763 H : H : =.

20 Aled Statstcs and Probablty or Engneers, 6 th edton t ˆ se( ˆ ) t.5,93 t.5, t t /,6, Fal to reect H, there s no enough evdence to conclude that s sgncant n the model at =. se ˆ ˆ c ( ).6887 H : H : =. ˆ t se( ˆ ) t.5,93 t.5, t t /,6, Reect H conclude that s sgncant n the model at =. -34 The regresson equaton s Stack Loss(y) = X +.3 X -.5 X3 Predctor Coe SE Coe T P Constant X X X S = R-Sq = 9.4% R-Sq(ad) = 89.8% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total 69.4 a) H : 3 : H or at least one =. SSR k SSE n.,3, / / ,3,7 Reect H and conclude that the regresson model s sgncant at =. P-value <. b) ˆ MSE. 5 se( ˆ ) ˆ c.349

21 Aled Statstcs and Probablty or Engneers, 6 th edton t H : H : =. ˆ se( ˆ ) t.898 t.5, 4.5,7 t t /,7. Reect H and conclude that s sgncant n the model at =.. ˆ se( ) ˆ c.368 t t H : H : =. ˆ se( ˆ ) t.898.5, 4.5,7 t /,7 t t. Reect H and conclude that ˆ se( ) ˆ 3 c.563 H : 3 H : 3 =. ˆ 3 se( ˆ ) t.898 t.5, 4.5,7 t t /,7. s sgncant n the model at =.. Fal to reect H, there s not enough evdence to conclude that 3 s sgncant n the model at = a) Comuter outut ollows. The test statstc s F = 9.9. Because the P-value s near zero, the regresson s sgncant at.. b) t ˆ se( ˆ ), null hyothess ˆ s reected at level t t /, n or the P-value <

22 Aled Statstcs and Probablty or Engneers, 6 th edton The P-values o all regressors are less than.. Thereore, all ndvdual varables n the model are sgncant. c) The comuter outut or three regressors s ollowed by the comuter outut or two regressors. From the regresson SS R ( ) / r sum o squares n each model the F test or s F MS E 4.4 The F-test P-value s near zero. Thereore the regressor (TD ercentage) s sgncant to the model. Ths s the equvalent to the t test on the coecent o. The F statstc = 4.66 =.94, ecet or some round-o error. Results o regresson on three varables and on two varables are shown below. Regresson Analyss: Ratng Pts versus Pct Com, Pct TD, Pct Int The regresson equaton s Ratng Pts = Pct Com Pct TD Pct Int Predctor Coe SE Coe T P Constant Pct Com Pct TD Pct Int S =.3479 R-Sq = 95.3% R-Sq(ad) = 94.8% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total Source DF Seq SS Pct Com Pct TD Pct Int Unusual Observatons Obs Pct Com Ratng Pts Ft SE Ft Resdual St Resd R R R R R denotes an observaton wth a large standardzed resdual. Regresson Analyss: Ratng Pts versus Pct Com, Pct Int The regresson equaton s Ratng Pts = Pct Com Pct Int Predctor Coe SE Coe T P Constant Pct Com Pct Int S = R-Sq = 7.6% R-Sq(ad) = 69.7% Analyss o Varance Source DF SS MS F P

23 Aled Statstcs and Probablty or Engneers, 6 th edton Regresson Resdual Error Total a) H : or all H : or at least one ,5,6,5,6 Reect H and conclude regresson s sgncant at =.5. P-value <. b) =.5 t/, n t. 5, H : : H 3 3 t.83 t. 5 t. 5 t t. 9 Fal to reect H Reect H Fal to reect H Reect H Fal to reect H c) d) H : or all H : or at least one ,, ,,9 Reect H and conclude regresson s sgncant at =.5 =.5 t t. 45 H : : /, n.5,9 H t 8.3 t t t /, 9 t t /, 9 Reect H or each regressor varable and conclude that both varables are sgncant at =.5 ˆ art( d ) e) 6.7E 6. Part c) s smaller, suggestng a better model. ŷ ˆ ˆ ˆ, Assume no nteracton model. -37 H : a) H at least one ,,3 3.8.,,3 Reect H P-value =

24 Aled Statstcs and Probablty or Engneers, 6 th edton b) H : H : H: H: t 6.4 t.57 t /,3 t.5, t /,3 t.5, t t t t.5,3.5,3 Reect H or regressor. Do not reect H or regressor. SSR(, ) H : c) H : =. 6.69,,3.,,3.5,,3 Do not reect H d) 34. H : H at least one =. 7.74,3,.,3,.,3, Do not reect H e) H : H : 99.7 =. SSR(, ) ,, SSR MS 47 E 98.5.,, Do not reect H ) ˆ. ˆ (no nteracton term) = 59 MS ˆ E ( ) was reduced n the model wth the nteracton term. -38 a) : H or all : H or at least one From the comuter outut 4.9.,4, ,4,5 Reect H and conclude that the regresson model s sgncant at =.

25 Aled Statstcs and Probablty or Engneers, 6 th edton b) H : : H t.5,5.947 GF : t 46 Reect H PPGF :. 8 PCTG :. 4 GA : t 83 Reect H ADV : t 5 Fal to reect H t Fal to reect H t Fal to reect H PEN : t 54 Fal to reect H.. BMI : t 45 Fal to reect H AVG : t 53 Fal to reect H. SHT : t 9 Fal to reect H... PPGA : t 5 Fal to reect H PKPCT : t 54 Fal to reect H SHGF : t. 54 Fal to reect H SHGA : t 34 Fal to reect H. FG :. t Fal to reect H It does not seem that all regressors are mortant. Only the regressors "GF" ( ) and GA ( ) are sgncant at =. c) The comuter result s shown below. Regresson Analyss: W versus GF, PPGF The regresson equaton s W = GF -.6 PPGF Predctor Coe SE Coe T P Constant GF PPGF S = R-Sq = 5.8% R-Sq(ad) = 49.3% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total Because PPGF had a t statstc near zero n art (b) there s a concern that t s not an mortant redctor. We wll evaluate ts role n the smaller model wth GF ,, Because >.5,,7, we reect the null hyothess that the coecent o GF and PPGF are both zero.

26 Aled Statstcs and Probablty or Engneers, 6 th edton H : : 4 H 4 t 3.98 t. 4 Reect H Fal to reect H Based on the t-test, ower lay goals or (PPGF) s not a logcal choce to add to the model that already contans GF. -39 a) The comuter outut ollows. The P-value or the F-test s near zero. Thereore, the regresson s sgncant at both.5 or. ˆ b) t se( ˆ ). Because the P-values or Age and Severty are <.5 both regressors are sgncant to the model. Because the P-value or Anety s.33, t s not sgncant to the model at level. 5. Regresson Analyss: Satsacton versus Age, Severty, Anety The regresson equaton s Satsacton = Age Severty +.3 Anety Predctor Coe SE Coe T P Constant Age Severty Anety S = 7.37 R-Sq = 9.4% R-Sq(ad) = 89.% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total Source DF Seq SS Age Severty 97. Anety a) Regresson Analyss: Satsacton versus Age, Severty The regresson equaton s Satsacton = Age Severty Predctor Coe SE Coe T P Constant Age Severty S = R-Sq = 89.7% R-Sq(ad) = 88.7% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total

27 Aled Statstcs and Probablty or Engneers, 6 th edton Because the P-value o the F test s less than =.5 and =., we reect the H and conclude that at least one regressor contrbutes sgncantly to the model at ether level. b) Because the P-values rom the t-test or both age and severty regressors are less than =.5, we reect the H and conclude that both age and severty regressors contrbute sgncantly to the model. c) From MS Resdual, the estmate o the varance = 5.7. From the comuter outut below, the thrd varable anety s added to the model, the estmate o the varance s reduced to The varance changed very slghtly here so t s unlkely that the varable contrbutes sgncantly to the model. Regresson Analyss: Satsacton versus Age, Severty, Anety The regresson equaton s Satsacton = Age Severty +.3 Anety Predctor Coe SE Coe T P Constant Age Severty Anety S = 7.37 R-Sq = 9.4% R-Sq(ad) = 89.% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total Sectons -3 and -4-4 a) ˆ t ˆ /, n c 7.55 t se( ˆ ).5, (.365)(5.7) ˆ t ˆ /, n 3.73 t.5, (.365)(.556) c se( ˆ ) ˆ t ˆ /, n.6 t.5,7.6 (.365)(.693) c se( ˆ )

28 Aled Statstcs and Probablty or Engneers, 6 th edton b) 8 ' X ( X ' X ) X (.365) c) = ' t Y X ( X ' X ) /, n (.365) 6.64 y X (.3565) ˆ ( X ' a) ˆ t ˆ /, n c.9 t se( ˆ ).5,7.9 (.365)(.55) ˆ t ˆ /, n.93 t.5,7.93 (.365)(.87) c ( X ' X ) X (.3565) se( ˆ ) ) ˆ t ˆ /, n.53 t.5,7.53 (.365)(.998) c se( ˆ ) b) ' X ( X ' X ) 9.37 (.365) X Y (.88) 39.49

29 Aled Statstcs and Probablty or Engneers, 6 th edton c) =.5 5 ' t 9.37 X ( X ' X ) /, n 9.37 (.365) (.88) y X.88 ˆ ( X ' ( X ' X ) X ) -43 Analyss o Varance Source DF Sum o Squares Mean Square F Value Pr > F Model <. Error Corrected Total Root MSE R-Square.9937 Deendent Mean 9.6 Ad R-Sq.995 Coe Var Parameter Estmates Varable DF Parameter Estmate Standard Error t Value Pr > t 99% Condence Lmts Intercet < Outut Statstcs Obs Deendent Varable Predcted Value Std Error Mean Predct 99% CL Mean 99% CL Predct Resdual a)

30 Aled Statstcs and Probablty or Engneers, 6 th edton b) y ˆ y 8.93 c) ˆ.957 Y Y -44 a) 95 % CI on coecents b) ˆ 9.65 c) ˆ Y t se( ˆ ) 4.49 t.5,7.365 Y Y /, n Y Y se( ˆ ) 9.65 (.365)(4.49) t ˆ ( X ( X X ) X /, n (.949) 4.74 y a) b) These art b) ntervals are much wder. Yes, the addton o ths term ncreased the standard error o the regresson coecent estmators. -46 Parameter Estmates Varable DF Parameter Estmate Standard Error t Value Pr > t 95% Condence Lmts Intercet

31 Aled Statstcs and Probablty or Engneers, 6 th edton Outut Statstcs Obs Deendent Varable Predcted Value Std Error Mean Predct 95% CL Mean 95% CL Predct Resdual a) b) ˆ Y.7836 Y c) y ˆ y Parameter Estmates Varable DF Parameter Estmate Standard Error t Value Pr > t 95% Condence Lmts Intercet < mams < EosureTme < Outut Statstcs Obs Deendent Varable Predcted Value Std Error Mean Predct 99% CL Mean 99% CL Predct Resdual a) ˆ t ˆ /, n 9.47 t.5, (.6)(3.46) c se( ˆ ) ˆ t ˆ /, n 68.8 t.5, (.6)(5.4) c se( ˆ )

32 Aled Statstcs and Probablty or Engneers, 6 th edton b) ˆ ˆ Y Y t /, n.8 c) y ˆ se( ˆ Y Y ) y Parameter Estmates Varable DF Parameter Estmate Standard Error t Value Pr > t 99% Condence Lmts Intercet Age DrnkUse CookUse Outut Statstcs Obs Deendent Varable Predcted Value Std Error Mean Predct 99% CL Mean 99% CL Predct Resdual a) t , b) ˆ. 488 ˆ Y Y t.4 3 /, n c) y ˆ. 488 Y se( ˆ Y y.3-49 a) t.5, β b) ˆ.8787 Y ) se( ˆ ).96 t.5,6.86 Y

33 Aled Statstcs and Probablty or Engneers, 6 th edton ˆ t Y /, n Y Y se( ˆ ).8787 (.86)(.96) c).8787 se( ) (.34).8549 y.95-5 The regresson equaton s y = Predctor Coe SE Coe T P Constant S = R-Sq = 93.7% R-Sq(ad) = 9.6% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total ˆ t ˆ c a) /, n 7.3 t se( ˆ ).5,6 7.3 (.943)(.539) ˆ t ˆ /, n.5,6 c.7 t se( ˆ ).7 (.943)(.539) b) New Obs Ft SE Ft 9% CI 9% PI (7.899, 53.74) (6.58, 55.6)XX ˆ 4.8 ˆ Y t se( ˆ ) 6.65 t.5,6.447 Y Y /, n Y Y se( ˆ ) 4.8 (.943)(6.65) c) 4.8 se( ) (7.33) 6.58 y 55.6

34 Aled Statstcs and Probablty or Engneers, 6 th edton d) The smaller the samle sze, the wder the nterval -5 The regresson equaton s Useul range (ng) = Brghtness (%) -.7 Contrast (%) Predctor Coe SE Coe T P Constant Brghtness (%) Contrast (%) S = R-Sq = 75.6% R-Sq(ad) = 67.4% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total a) t.5, b) Predcted Values or New Observatons New Obs Ft SE Ft 99% CI 99% PI (-36.7, 5.8) (-.8,.) Values o Predctors or New Observatons New Contrast Obs Brghtness (%) (%) ˆ 44.6 ˆ Y t se( ˆ ).9 t.5, Y Y /, n Y Y se( ˆ ) 44.6 (3.77)(.9) c) 44.6 se( ) (4.44).8 y. d) Predcted Values or New Observatons New Obs Ft SE Ft 99% CI 99% PI (7.4, 67.) (3.7, 344.) Values o Predctors or New Observatons New Contrast Obs Brghtness (%) (%) 5. 5.

35 Aled Statstcs and Probablty or Engneers, 6 th edton CI: Y PI: 3.7 y 344. These ntervals are wder because the regressors are set at etreme values n the sace and the standard errors are greater. -5 Parameter Estmates Varable DF Parameter Estmate Standard Error t Value Pr > t 95% Condence Lmts Intercet < a) t..5, b) Outut Statstcs Obs Deendent Varable Predcted Value Std Error Mean Predct 95% CL Mean 95% CL Predct Resdual Predcton at = 8, =, 3 = 85 s ˆ 3.37 Y 3.77 Y c) y ˆ y d) Outut Statstcs Obs Deendent Varable Predcted Value Std Error Mean Predct 95% CL Mean 95% CL Predct Resdual Predcton at = 8, = 9, 3 = 93 s Y CI:.3 Y PI: 8.73 y

36 Aled Statstcs and Probablty or Engneers, 6 th edton -53 a) The comuter outut ollows. The outut s used to obtan estmates o the coecents and standard errors. The condence ntervals or the coecents are comuted rom ˆ t ( ˆ) ˆ.5,8se t.5,8se( ˆ). Parameter Estmates Varable DF Parameter Estmate Standard Error t Value Pr > t 95% Condence Lmts Intercet PctCom < PctTD < PctInt < From the t table, t.5, 8 =.48. The condence ntervals or the s are b) Outut Statstcs Obs Deendent Varable Predcted Value Std Error Mean Predct 95% CL Mean 95% CL Predct Resdual ' From the comuter outut ˆ 68.3, ( ˆ ) ( ˆ ) ( ' ) Y se Y V Y X X. 9 ˆ ˆ c) t se( ) t se( ), Y.5,8 Y Y Y.5,8 Y Y a) b) ˆ.466 Y 5 Y 3 Y.86 (.76)(.595).99.3 c) ˆ.7 Y Y Y.7 (.699)(.548).6.8 ˆ se( ˆ ).595 t.5,6.76 se( ˆ ).548 t.5,9.699 ˆ

37 Aled Statstcs and Probablty or Engneers, 6 th edton d) : wdth =.8 : wdth =. The nteracton model has a shorter condence nterval. Yes, ths suggests the nteracton model s reerable. -55 a) t.5,4 = b) ˆ 9.7 ˆ Y t Y.5,4 Y 7 Y se( ˆ ).395 Y se( ˆ ) 9.7 (.45)(.395) c) 7 9 t.5, d) The ntervals n art c) are narrower. All o the regressors used n art c) are sgncant, but not all o those used n art a) are sgncant. The model used n art c) s reerable. -56 a) From the Mntab outut n Eercse -8 the estmate, standard error, t statstc and P-value or the coecent o GF are: Predctor Coe SE Coe T P GF The 95% CI on the regresson coecent o GF s ˆ t se( ˆ ) ˆ ˆ t se( ˆ ) /, n /, n ˆ t se( ˆ ) ˆ ˆ t se( ˆ ).5,5.5, (.947)(.3673) ˆ.6374 (.947)(.3673) ˆ.7983 b) The Mntab result s shown below. Regresson Analyss: W versus GF The regresson equaton s W = GF Predctor Coe SE Coe T P Constant GF

38 Score Aled Statstcs and Probablty or Engneers, 6 th edton S = 5.39 R-Sq = 5.8% R-Sq(ad) = 5.% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total c) The 95% CI on the regresson coecent o GF s ˆ t se( ˆ ) ˆ ˆ t se( ˆ ) /, n /, n ˆ t se( ˆ ) ˆ ˆ t se( ˆ ).5,8.5,8.8 (.763)(.3795) ˆ.8 (.763)(.3795).4856 ˆ.3736 d) The smle lnear regresson model has the narrower nterval. Obvously there are etraneous varables n the model rom art a). The shorter nterval s an ntal ndcator that the orgnal model wth all varables mght be mroved. One mght eect there are other good redctors n the model rom art a), only one o whch s ncluded n the model o art b). Secton a) The regresson equaton s mg = cd -. rh -.34 etw +.9 cm ale +.9 n/v Predctor Coe SE Coe T P Constant cd rh etw cm ale n/v S =.83 R-Sq = 89.3% R-Sq(ad) = 84.8% b) There aears to be an outler. Otherwse, the normalty assumton s not volated. Normal Probablty Plot o the Resduals (resonse s mg) Standardzed Resdual 3 c) The lots do not show any volatons o the assumtons.

39 Standardzed Resdual Standardzed Resdual Standardzed Resdual Standardzed Resdual Aled Statstcs and Probablty or Engneers, 6 th edton Resduals Versus the Ftted Values (resonse s mg) Ftted Value 35 4 Resduals Versus cd (resonse s mg) cd 4 5 Resduals Versus etw (resonse s mg) etw Resduals Versus cm (resonse s mg) cm

40 Standardzed Resdual Standardzed Resdual Aled Statstcs and Probablty or Engneers, 6 th edton Resduals Versus ale (resonse s mg) ale Resduals Versus n/v (resonse s mg) n/v 35 4 d).366,.67,.4684,.858,.6788,.4384,.336,.96794,.67746,.659,.756,.69,.464,.735,.8565,.5335,.83,.935,.8,.9845, None o the values s greater than so none o the observatons are nluental. -58 a) R.7 b) The resdual lots look reasonable. There s some ncrease n varablty at the mddle o the redcted values. c) Normalty assumton s reasonable. The resdual lots aear reasonable too.

41 Aled Statstcs and Probablty or Engneers, 6 th edton

42 Aled Statstcs and Probablty or Engneers, 6 th edton -59 a) The comuter outut ollows. The roorton o total varablty elaned by ths model s: R SS SS R T b) Normal Probablty Plot: Some moderate, but not severe, deartures rom normalty are ndcated. c) Plot the resduals versus tted value and versus each regressor. There s no obvous model alure n the lot o tted values versus resduals. There s a modest ncrease n varablty n the mddle range o tted values. The resdual versus PctTD shows some non-random atterns. Possbly a non-lnear term would benet the model.

43 Resdual Resdual Aled Statstcs and Probablty or Engneers, 6 th edton 5. Resduals Versus Pct Com (resonse s Ratng Pts) Pct Com Resduals Versus Pct TD (resonse s Ratng Pts) Pct TD 6 7

44 COOK Resdual Aled Statstcs and Probablty or Engneers, 6 th edton 5. Resduals Versus Pct Int (resonse s Ratng Pts) Pct Int 4 5 d) A lot o Cook s dstance measures ollows. Although no onts eceed the usual crteron o dstance greater than, two onts are derent and mght be urther studed or nluence..5 Tme Seres Plot o COOK Inde Regresson Analyss: Ratng Pts versus Pct Com, Pct TD, Pct Int The regresson equaton s Ratng Pts = Pct Com Pct TD Pct Int Predctor Coe SE Coe T P Constant Pct Com Pct TD Pct Int S =.3479 R-Sq = 95.3% R-Sq(ad) = 94.8% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total Source DF Seq SS Pct Com Pct TD Pct Int Unusual Observatons

45 Percent Resdual Resdual Aled Statstcs and Probablty or Engneers, 6 th edton Obs Pct Com Ratng Pts Ft SE Ft Resdual St Resd R R R R R denotes an observaton wth a large standardzed resdual. -6 a) R =.969 b) Normalty s accetable Normal Probablty Plot o the Resduals (resonse s PITCH) Normal Score c) Plot s accetable. Resduals Versus the Ftted Values (resonse s PITCH) Ftted Value d) Cook s dstance values * The 9 th observaton s nluental -6 a) R 84. 3% b) Assumton o normalty aears adequate. 99 Normal Probablty Plot o the Resduals (resonse s rads) Resdual

46 Percent Resdual Resdual Resdual Aled Statstcs and Probablty or Engneers, 6 th edton c) There are unnel shaes n the grahs, so the assumton o constant varance s volated. The model s nadequate. 75 Resduals Versus the Ftted Values (resonse s rads) Ftted Value 5 Resduals Versus eosure tme (resonse s rads) Resduals Versus mams (resonse s rads) eosure tme mams d) Cook s dstance values No, none o the observatons has a Cook s dstance greater than. -6 a) R 8. % b) Assumton o normalty s not adequate. 99 Normal Probablty Plot o the Resduals (resonse s ARSNAILS) Resdual..5

47 Resdual Resdual Resdual Standardzed Resdual Aled Statstcs and Probablty or Engneers, 6 th edton c) The grahs ndcate non-constant varance. Thereore, the model s not adequate. 4 Resduals Versus the Ftted Values (resonse s ARSNAILS) Ftted Value Resduals Versus AGE (resonse s ARSNAILS) AGE Resduals Versus DRINKUSE (resonse s ARSNAILS) Resduals Versus COOKUSE (resonse s ARSNAILS) DRINKUSE COOKUSE d) Cook s dstance values nnty There are two nluental onts wth Cook s dstance greater than one. The entry nnty n the lst above ndcate a data ont wth h = and an undened studentzed resdual. -63 a) R % b) Assumton o normalty aears adequate.

48 Resdual Resdual Resdual Percent Aled Statstcs and Probablty or Engneers, 6 th edton 99 Normal Probablty Plot o the Resduals (resonse s densty) Resdual.. c) There s a non-constant varance shown n grahs. Thereore, the model s nadequate.. Resduals Versus the Ftted Values (resonse s densty) Ftted Value.. Resduals Versus delectrc constant (resonse s densty) Resduals Versus loss actor (resonse s densty) delectrc constant loss actor d) Cook s dstance values No, none o the observatons has a Cook s dstance greater than. -64 a) R % b) The normal assumton aears nadequate

49 Resdual Resdual Standardzed Resdual Percent Aled Statstcs and Probablty or Engneers, 6 th edton 99 Normal Probablty Plot o the Resduals (resonse s y) Resdual.5 5. c) The constant varance assumton s not nvald.. Resduals Versus the Ftted Values (resonse s y) Ftted Value 9 Resduals Versus (resonse s y) Resduals Versus (resonse s y) d) Cook s dstance values There are two nluental onts wth Cook s dstances greater than. -65 a) R % b) Assumton o normalty aears adequate.

50 Resdual Resdual Resdual Percent Aled Statstcs and Probablty or Engneers, 6 th edton 99 Normal Probablty Plot o the Resduals (resonse s Useul range (ng)) Resdual c) Assumton o constant varance s a ossble concern. One ont s a concern as a ossble outler. 8 Resduals Versus the Ftted Values (resonse s Useul range (ng)) Ftted Value Resduals Versus Contrast (%) (resonse s Useul range (ng)) 8 Resduals Versus Brghtness (%) (resonse s Useul range (ng)) Contrast (%) Brghtness (%) 9 d) Cook s dstance values No, none o the observatons has a Cook s dstance greater than. -66 a) R 9. 4 % b) Assumton o normalty aears adequate.

51 Resdual Resdual Resdual Resdual Percent Aled Statstcs and Probablty or Engneers, 6 th edton 99 Normal Probablty Plot o the Resduals (resonse s Stack Loss(y)) Resdual c) Assumton o constant varance aears reasonable Resduals Versus the Ftted Values (resonse s Stack Loss(y)) Resduals Versus X (resonse s Stack Loss(y)) Ftted Value X Resduals Versus X (resonse s Stack Loss(y)) Resduals Versus X3 (resonse s Stack Loss(y)) X X d) Cook s dstance values No, none o the observatons has a Cook s dstance greater than. -67 a) R.9835 b) R.99 R ncreases wth addton o nteracton term. No, addng addtonal regressor wll always ncrease r

52 Resduals Resdual Resdual Aled Statstcs and Probablty or Engneers, 6 th edton -68 a) R. 955 regressors.. Yes, the R usng these two regressors s nearly as large as the R rom the model wth ve b) Normalty s accetable, but there s some ndcaton o outlers. Resduals Versus the Ftted Values (resonse s PITCH) Normal Probablty Plot o the Resduals (resonse s PITCH) Ftted Value Normal Score c) Cook s dstance values The last observaton s very nluental -69 a) There s some ndcaton o nonconstant varance snce the resduals aear to an out wth ncreasng values o y. Resdual Plot or y b) Source Sum o Squares DF Mean Square F-Rato P- value Model Error Total (Corr.) R-squared = Stnd. error o est. = R-squared (Ad. or d..) =.9953 Durbn-Watson statstc = R or %; R Ad. 995 or 99.5%; Predcted c) Model ttng results or: log(y)

53 Resduals Resduals Aled Statstcs and Probablty or Engneers, 6 th edton d) Indeendent varable coecent std. error t-value sg.level CONSTANT R-SQ. (ADJ.) =.9574 SE=.7899 MAE= DurbWat=.3 Prevously:.... observatons tted, orecast(s) comuted or mssng val. o de. var Resdual Plot or log(y) Predcted Plot ehbts curvature There s curvature n the lot. The lot does not gve much more normaton as to whch model s reerable. e) Resdual Plot or log(y)

54 Resduals Aled Statstcs and Probablty or Engneers, 6 th edton Plot ehbts curvature Varance does not aear constant. Curvature s evdent. ) Model ttng results or: log(y) Indeendent varable coecent std. error t-value sg.level CONSTANT / R-SQ. (ADJ.) =.9893 SE= MAE=.8896 DurbWat=.869 Analyss o Varance or the Full Regresson Source Sum o Squares DF Mean Square F-Rato P- value Model Error Total (Corr.) R-squared =.996 Stnd. error o est. = R-squared (Ad. or d..) =.9893 Durbn-Watson statstc =.8689 Resdual Plot or log(y) Predcted Usng /3 The resdual lot ndcates better conormance to assumtons. Curvature s removed when usng / 3 as the regressor nstead o 3, and the log o the resonse data. -7 a) Regresson Analyss: W versus GF The regresson equaton s W = GF

55 Resdual Resdual Aled Statstcs and Probablty or Engneers, 6 th edton Predctor Coe SE Coe T P Constant GF S = 5.39 R-Sq = 5.8% R-Sq(ad) = 5.% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total b) R-Sq = 5.8% c) Model aears adequate. Resduals Versus the Ftted Values (resonse s W) Ftted Value 5 55 d) No, the resduals do not seem to be related to PPGF. Because there s no attern evdent n the lot, t does not seem that ths varable would contrbute sgncantly to the model. Resduals Versus PPGF (resonse s W) PPGF a) = k + = + = 3 Average sze = /n = 3/5 =. b) Leverage ont crtera:

56 Normal Score Aled Statstcs and Probablty or Engneers, 6 th edton h h h h h ( / n) (.).4 7,7 8, Ponts 7 and 8 are leverage onts. Sectons -6-7 a) b) H : or all H : or at least one =. 7..,, ,,5 Reect H and conclude that model s sgncant at =. c) H : H : =. t.45 t /, n t t.5.5,83 t.5,5.5 Reect H and conclude sucent evdence to suort value o quadratc term n model at =.. d) One resdual s an outler Normalty assumton aears accetable Resduals aganst tted values s somewhat curved, but the mact o the outler should be consdered. Normal Probablty Plot o the Resduals (resonse s y) Resdual

57 Aled Statstcs and Probablty or Engneers, 6 th edton -73 a) b) = 6.86, >.5,,7, reect H and conclude regresson model s sgncant at =.5 t t c).5,7 t =.66, al to reect H and the regresson model s not sgncant at =.5 d) Model s not accetable. Observaton number 8 and have large leverages.

58 Aled Statstcs and Probablty or Engneers, 6 th edton -74 a) b) H : or all H : or at least one =.5.6.5,, ,,9 Reect H and conclude regresson model s sgncant at =.5 H : c) H : =.5 t.55 t.5,9 t t.6.5,9 Reect H and conclude that s sgncant at =.5 d) Observaton number 9 s an etreme outler.

59 Aled Statstcs and Probablty or Engneers, 6 th edton e) H : 33 H : =.5 t t ,8 t t.36.5,8 3 Do not reect H and conclude that cubc term s not sgncant at =.5-75 a) Predctor Coe SE Coe T P Constant l ^ ^ ^ S =.69 R-Sq = 9.7% R-Sq(ad) = 87.% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total b) H all 3 3 H at least one 9.68.,9, ,9,6 3 3

60 Normal Score Aled Statstcs and Probablty or Engneers, 6 th edton Reect H and conclude that the model s sgncant at =.. c) Assumtons aear to be reasonable. Normal Probablty Plot o the Resduals (resonse s y) Resdual d) H : H at least one.,6,6 SS ( R.,6, MS E 3 ) / r Fal to reect H SS ( R ) SS R 3 ( ) SS 3 R ( ) 3

61 Aled Statstcs and Probablty or Engneers, 6 th edton Reduced Model: y a) Create an ndcator varable or se (e.g. or male, or emale) and nclude ths varable n the model. b) The regresson equaton s ARSNAILS = DRINKUSE +.8 COOKUSE AGE +.67 SEXID Predctor Coe SE Coe T P Constant DRINKUSE COOKUSE AGE SEXID S =.54 R-Sq =.8% R-Sq(ad) =.% where SEXID = or male and or emale c) Because the P-value or testng : se erson s se aects arsenc n the nals. H aganst H : s.495, there s no evdence that the se -77 a) Use ndcator varable or transmsson tye. There are three ossble transmsson tyes: L4, L5 and M6. So, two ndcator varables could be used where 3 = trns=l5, otherwse and 4 = trns=m6, otherwse b) 3 4 c) The P-value or testng H : 3 s.99, whch s not sgncant. However, the P-value or testng H : 4 s., whch s sgncant or values o α >.. Thus, t aears that whether or not the transmsson s manual aects mg, but there s not a sgncant derence between the tyes o automatc transmsson. -78 y where Test o derent sloes: H : H : =.5 t t t.79.5,6 t. or tool tye3 or tool tye46.5,6 Fal to reect H. There s not sucent evdence to conclude that two regresson models are needed. Test o derent ntercets and sloes usng etra sums o squares: H : H at least one s not zero

62 Aled Statstcs and Probablty or Engneers, 6 th edton SS(, ) SS(,, ) SS( ) , ) / / MS E.459 SS( Reect H a) The mn C model s:, C 3. and MS E MS model s the same as the mn C. The mn E b) Same as the model n art (a). c) Same as the model n art (a). d) Same as the model n art (a). e) All methods gve the same model wth ether mn C or mn MS E. -8 The deault settngs or F-to-enter and F-to-remove, equal to 4, n the comuter sotware were used. Derent settngs can change the models generated by the method. a) The mn MS E model s:,, 3 C 3.8 MS E The mn C model s:,, C 3.4 MS E b) Same as the mn C model n art (a) c) Same as art mn MS E model n art (a) d) Same as art mn C model n art (a) e) The mnmum MS E and orward models all are the same. Stewse and backward regressons generate the mnmum C model. The mnmum C model has ewer regressors and t mght be reerred, but MS E has ncreased. -8 a) The mn C model s: C.and MS The mn E MS E model s the same as the mn C. b) Same as model n art (a). c) Same as model n art (a). d) Same as model n art (a). e) All methods gve the same model wth ether mn C or mn MS E. -8 The deault settngs or F-to-enter and F-to-remove or Mntab were used. Derent settngs can change the models generated by the method. a) The mn MS E model s:, 3, 4 C.6 MS The mn C model s: 3, 4 C.6 MS. 737 E E b) Same as the mn C model n art (a) c) Same as the mn C model n art (a) 4 4

63 Aled Statstcs and Probablty or Engneers, 6 th edton d) Same as the mn C model n art (a) e) The mnmum MS E and orward models all are the same. Stewse and backward regressons generate the mnmum C model. The mnmum C model has ewer regressors and t mght be reerred, but MS E has ncreased. -83 a) The mn C model s: C. and MS E MS model s the same as the mn C. The mn E b) Same as model n art (a). c) Same as model n art (a). d) Same as model n art (a). e) All methods gve the same model wth ether mn C or mn MS E. -84 a) The mn C model s:, C 3. and MS E MS model s the same as the mn C. The mn E b) Same as model n art (a). c) Same as model n art (a). d) Same as model n art (a). e) All methods gve the same model wth ether mn C or mn MS E. -85 a) The mn C model s:, C.9 and MS. 49 E MS model s the same as the mn C. The mn E b) Same as model n art (a). c) Same as model n art (a). d) Same as model n art (a). e) All methods gve the same model wth ether mn C or mn MS E. ) There are no observatons wth a Cook s dstance greater than so the results wll be the same. -86 The deault settngs or F-to-enter and F-to-remove or Mntab were used. Derent settngs can change the models generated by the method. a) Best Subsets Regresson: W versus GF, GA,... Resonse s W P P P P K S S A P C P B A S P P H H Mallows G G D G T E M V H G C G G F Vars R-Sq R-Sq(ad) C- S F A V F G N I G T A T F A G X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

64 Aled Statstcs and Probablty or Engneers, 6 th edton X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X From the outut the mnmum CP model and mnmum MSE model are the same. The regressors are GF, GA, ADV, SHT, PPGA, PKPCT, SHGA. The comuter outut or ths model ollows. Regresson Analyss: W versus GF, GA, ADV, SHT, PPGA, PKPCT, SHGA The regresson equaton s W = GF -.87 GA ADV +.56 SHT -.44 PPGA PKPCT SHGA Predctor Coe SE Coe T P Constant GF GA ADV SHT PPGA PKPCT SHGA S =.8935 R-Sq = 9.3% R-Sq(ad) = 89.8% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total The model s 457.8GF.87GA.375ADV.56SHT.44PPGA 4.94PKPCT. 489SHGA b) Stewse Regresson: W versus GF, GA,... Alha-to-Enter:.5 Alha-to-Remove:.5 Resonse s W on 4 redctors, wth N = 3 Ste 3 4 Constant GF T-Value

65 Aled Statstcs and Probablty or Engneers, 6 th edton P-Value.... GA T-Value P-Value... SHGA.7.9 T-Value P-Value.6.9 SHT -.6 T-Value -.5 P-Value.43 S R-Sq R-Sq(ad) Mallows C The selected model rom Stewse Regresson has our regressors GF, GA, SHT, SHGA. The comuter outut or ths model ollows. Regresson Analyss: W versus GF, GA, SHT, SHGA The regresson equaton s W = GF -.67 GA -.59 SHT +.93 SHGA Predctor Coe SE Coe T P Constant GF GA SHT SHGA S =.65 R-Sq = 88.7% R-Sq(ad) = 86.9% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total The model s GF.67GA.59SHT. 93SHGA c) Stewse Regresson: W versus GF, GA,... Forward selecton. Alha-to-Enter:.5 Resonse s W on 4 redctors, wth N = 3 Ste 3 4 Constant GF T-Value P-Value.... GA T-Value

66 Aled Statstcs and Probablty or Engneers, 6 th edton P-Value... SHGA.7.9 T-Value P-Value.6.9 SHT -.6 T-Value -.5 P-Value.43 S R-Sq R-Sq(ad) Mallows C The model selected by Forward Selecton s the same as art (b). d) Stewse Regresson: W versus GF, GA,... Backward elmnaton. Alha-to-Remove:. Resonse s W on 4 redctors, wth N = 3 Ste Constant GF T-Value P-Value GA T-Value P-Value ADV T-Value P-Value PPGF T-Value P-Value PCTG T-Value P-Value PEN T-Value P-Value BMI T-Value P-Value AVG T-Value P-Value SHT T-Value P-Value

67 Aled Statstcs and Probablty or Engneers, 6 th edton PPGA T-Value P-Value PKPCT T-Value P-Value SHGF T-Value P-Value SHGA T-Value P-Value FG. T-Value. P-Value.98 S R-Sq R-Sq(ad) Mallows C Ste Constant GF T-Value P-Value... GA T-Value P-Value... ADV T-Value P-Value.9.4 PPGF T-Value P-Value PCTG T-Value P-Value PEN -.39 T-Value -.94 P-Value.358 BMI T-Value P-Value AVG T-Value P-Value SHT

68 Aled Statstcs and Probablty or Engneers, 6 th edton T-Value P-Value..6.8 PPGA T-Value P-Value PKPCT T-Value P-Value SHGF T-Value P-Value SHGA T-Value P-Value...36 FG T-Value P-Value S R-Sq R-Sq(ad) Mallows C The model selected by Backward Selecton ncludes GF, GA, SHT, PPGA, PKPCT, SHGA. The comuter outut or ths model ollows. Regresson Analyss: W versus GF, GA, SHT, PPGA, PKPCT, SHGA The regresson equaton s W = GF -.87 GA +.38 SHT -.34 PPGA PKPCT SHGA Predctor Coe SE Coe T P Constant GF GA SHT PPGA PKPCT SHGA S =.3776 R-Sq = 9.3% R-Sq(ad) = 89.% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total The model s 48.77GF.87GA.38SHT.34PPGA 4.58PKPCT. 387SHGA e) There are several reasonable choces. The seven-varable model GF, GA, ADV, SHT, PPGA, PKPCT, SHGA wth mnmum C s a good choce. It has MSE not much larger than the MSE n the ull model.

69 Aled Statstcs and Probablty or Engneers, 6 th edton The our-varable model GF, GA, SHT, SHGA rom Stewse Regresson(and Forward Selecton) s a smler model wth C = 4. < =5 and good R-squared. Even the three-varable model GF, GA, SHGA s reasonable. It s stll smler wth a good R-squared. The C = 4. and ths s only slghtly greater than = 4. However, the MSE or ths model s somewhat hgher than or the svarable model. -87 a) The comuter outut ollows. The rst model n the table wth seven varables mnmzes MS E and C. Best Subsets Regresson: Pts versus Att, Com,... PctCom, YdsPerAtt, PctTD, PctInt Resonse s Pts Y d P s c P t e P c C C r c t A o o Y A t L I I t m m d t T T n n n Vars R-Sq R-Sq(ad) Mallows C- S t s t D D g t t X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X The comuter outut or ths model ollows. Regresson Analyss: RatngPts versus Att, PctCom,... The regresson equaton s RatngPts = Att +.87 PctCom -.5 Yds YdsPerAtt +.7 TD PctTD PctInt Predctor Coe SE Coe T P Constant Att PctCom Yds YdsPerAtt TD PctTD PctInt S =.3637 R-Sq =.% R-Sq(ad) =.%

70 Aled Statstcs and Probablty or Engneers, 6 th edton Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total b) Stewse regresson selects the our-varable model YdsPerAtt,PctInt,PctTD,PctCom. Stewse Regresson: Pts versus Att, Com,... Alha-to-Enter:.5 Alha-to-Remove:.5 Resonse s Pts on redctors, wth N = 3 Ste 3 4 Constant YdserAtt T-Value P-Value.... PctInt T-Value P-Value... PctTD T-Value P-Value.. PctCom.884 T-Value 96.4 P-Value. S R-Sq R-Sq(ad) Mallows C The comuter outut or ths model ollows. Regresson Analyss: RatngPts versus YdsPerAtt, PctInt, PctTD, PctCom The regresson equaton s RatngPts = YdsPerAtt PctInt PctTD +.88 PctCom Predctor Coe SE Coe T P Constant YdsPerAtt PctInt PctTD PctCom S =.466 R-Sq =.% R-Sq(ad) =.% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total The model s

71 Aled Statstcs and Probablty or Engneers, 6 th edton.3 4.YdsPerAtt 4.6PctInt 3.3PctTD. 88PctCom c) Forward selecton shown below selects the same our-varable model YdsPerAtt,PctInt,PctTD,PctCom as n art (b). Stewse Regresson: Pts versus Att, Com,... Forward selecton. Alha-to-Enter:.5 Resonse s Pts on redctors, wth N = 3 Ste 3 4 Constant YdserAtt T-Value P-Value.... PctInt T-Value P-Value... PctTD T-Value P-Value.. PctCom.884 T-Value 96.4 P-Value. S R-Sq R-Sq(ad) Mallows C d) Backward elmnaton shown below selects the s-varable model Att, PctCom, Yds, YdsPerAtt, PctTD, PctInt. It s smlar to the model wth mnmum MS E ecet varable TD s ecluded. Stewse Regresson: Pts versus Att, Com,... Backward elmnaton. Alha-to-Remove:. Resonse s Pts on redctors, wth N = 3 Ste Constant Att T-Value P-Value Com. T-Value. P-Value.98 PctCom T-Value P-Value.....

72 Aled Statstcs and Probablty or Engneers, 6 th edton Yds T-Value P-Value YdserAtt T-Value P-Value..... TD T-Value P-Value PctTD T-Value P-Value..... Lng.. T-Value.5.5 P-Value Int... T-Value P-Value PctInt T-Value P-Value..... S R-Sq R-Sq(ad) Mallows C The comuter outut or ths model ollows. Regresson Analyss: RatngPts versus Att, PctCom,... The regresson equaton s RatngPts = Att +.89 PctCom -.86 Yds YdsPerAtt PctTD PctInt Predctor Coe SE Coe T P Constant Att PctCom Yds YdsPerAtt PctTD PctInt S = R-Sq =.% R-Sq(ad) =.% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total The model s

73 Aled Statstcs and Probablty or Engneers, 6 th edton.7.55 Att.889 PctCom.8 Yds YdsPerAtt PctTd PctInt e) The our varable model (PctCom, YdsPerAtt, PctTD, PctInt)has the second mnmum C and also has small MS E and large adusted R-squared. It s also a model wth the ew regressors so t s reerred. -88 a) The mn C model s: C. and MS E The mn MS E model s the same as the mn C model b) The ull model that contans all 3 varables where AGE DrnkUse 3 c) No varables are selected CookUse d) The mn C model has only the ntercet term wth. 5 C and MS. 37 The mn MS E model s the same as the mn C n art (a). e) None o the varables seem to be good redctors o arsenc n nals based on the models above (none o the varables are sgncant). -89 Ths analyss ncludes the emssons varables hc, co, and co. It would be reasonable to consder models wthout these varables as regressors. Best Subsets Regresson: mg versus cd, rh,... Resonse s mg a c r e c n c Mallows h t m l / h c o Vars R-Sq R-Sq(ad) C- S d w e v c o X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X E a) The mnmum C (.3) model s: cd The mnmum MSE (4.8) model s: cd etw.345 etw.9 ale cm 3.84 ale.96 c

74 Aled Statstcs and Probablty or Engneers, 6 th edton b) cd etw ale c c) Same model as the mn MS E equaton n art (a) ˆ d) y etw 4.4ale.385n / v e) The mnmum C model s reerred because t has a very low MSE as well (4.9) ) Only one ndcator varable s used or transmsson to dstngush the automatc rom manual tyes and two ndcator varables are used or drv: trans = or automatc (L4, L5) and or manual (M6) and drv = drv = 4 or R and drv =F; drv = drv = 4 or F and drv = R. The mnmum C (4.) model s the same as the mnmum MSE (.67) model: Stewse: drv Forward selecton:..34 etw etw etw drv cm 5.6 n / v 4.5trns 3.drv. 7 n / v. ale drv drv Backward selecton: same as mnmum C and mnmum MSE. Preer the model gvng the mnmum C and mnmum MSE. 3.4 co trans.8 drv c 7.4 trans -9 ' ( ') '.33( ') ( 97.5).33( 97.5) a) ' 47.66( ' ), where ' S y ˆ (.6) 47.66(.6) s b) At = 85 '. 6 c) y ( 97.5).33( 97.5) d) They are the same. e) ' '.44( y y y' and ' where S y S ')

75 Aled Statstcs and Probablty or Engneers, 6 th edton The roorton o total varablty elaned s the same or both the standardzed and un-standardzed models. Thereore, R s the same or both models. y' y' where ' ( ' ) ' ( ' ) y y y ' and S y ' -9 The deault settngs or F-to-enter and F-to-remove, equal to 4, were used. Derent settngs can change the models generated by the method. a) C MS.4 b) C 4.66 MS.4 E E c) The orward selecton model n art (a) s more arsmonous wth a lower C and equvalent MS E. Thereore, we reer the model n art (a). -9 n = 3, k = 9, = 9 + = n ull model. a) ˆ MS E R.9 SSR SSE R S S SS E yy MS E.9 5 yy ( n ) (3 ) S yy S yy SS S SS R yy E 5 3 SSR 3 MSR k 9 MSR MS E.,9,,9, 3.46 Reect H and conclude at least one s sgncant at =.. b) k = 4 = 5 SSE SSE MS E 88 n 3 5 Yes, MS E s reduced wth new model (k = 4). SSE ( ) c) C n C 3 (5) ˆ Yes, C s reduced rom the ull model. S

76 Aled Statstcs and Probablty or Engneers, 6 th edton -93 n = 3 k = 7 MS E( ull ) a) =3 SS E 3 SS E 3 MS E. n 3 3 SS E C n MS E( ull) 3 3 (3) 6 YES, C > b) = 4 SS E 75 SS E 75 MS E.6 n 3 4 Yes, both MS E and C are reduced. C 75 3 (4) 5.5 Sulemental Eercses -94 a) The mssng quanttes are as ollows: Coe T Constant = SE Coe 7.68 From the t table wth 6 degrees o reedom, P-value Constant < (.5), so P-value Constant <. T = Coe SE Coe Coe , SE Coe =. 343 T P-valule < (.5), so P-valule <. Coe T X = SE Coe.969 P-valule < (.5), so P-valule <. Coe.8565 T 3 = SE Coe.665 P-valule 3 < (.5), so P-valule 3 <. Regresson DF = 9 6 = 3 SS Error = SS SS Total Regresson MSRegresson 5767 F =. 543 MS 5 Error P-value <.. R-Squared = 3473/ =.995 R-Squared Adusted = (683/6)/(348943/9) =.994 b) Because the P-value rom the F-test s less than =.5 and less than =., we reect the H or ether value and conclude that at least one regressor sgncantly contrbutes to the model.

77 Aled Statstcs and Probablty or Engneers, 6 th edton c) Because the P-value rom the t-test or the,, and 3 varables are less than =.5, we reect the H s and conclude that each ndvdual regressor contrbutes sgncantly to the model. -95 a) Because the matr s 3 3 two regressors are n the regresson model. The ntercet s also n the model. b) cov( ˆ ) ( X X ) C, thereore the varances o the two varables regresson coecents are: 5(.339) and 5(.98).4554 c) se( ˆ ) ˆ C Analyss o Varance Source DF Sum o Squares Mean Square F Value Pr > F Model <. Error Corrected Total Root MSE R-Square.993 Deendent Mean 394. Ad R-Sq.995 Coe Var.846 Parameter Estmates Varable DF Parameter Estmate Standard Error t Value Pr > t 99% Condence Lmts Intercet < a) b) H : H : or at least one = ,3, Reect H and conclude that regresson s sgncant. P-value <. c) All at =. t. 5, H : 3 H : 4 H : 5 H : 3 H : 4 H : 5 t.97 t t 3. t t /, 36 t t /, 36 t t /, 36 Fal to reect H Reect H Reect H

78 Normal Score Aled Statstcs and Probablty or Engneers, 6 th edton d) R. 993 Ad. R. 995 e) Normalty assumton aears reasonable. However there s a ga n the lne. Normal Probablty Plot o the Resduals (resonse s y) Resdual ) Plot s satsactory. g) Slght ndcaton that varance ncreases as 3 ncreases.

79 Resdual Aled Statstcs and Probablty or Engneers, 6 th edton Resduals Versus 3 (resonse s y) h) Outut Statstcs Obs Deendent Varable Predcted Value Std Error Mean Predct 99% CL Mean 99% CL Predct Resdual y ˆ 389.5(974).3(7).657(63) Analyss o Varance Source DF Sum o Squares Mean Square F Value Pr > F Model <. Error Corrected Total Root MSE.34 R-Square.99 Deendent Mean 8.68 Ad R-Sq.993 Coe Var.59 Parameter Estmates Varable DF Parameter Estmate Standard Error t Value Pr > t 95% Condence Lmts Intercet * < a) H : 3 4 5

80 Resdual Resdual Percent Aled Statstcs and Probablty or Engneers, 6 th edton H : or at least one =.5.5,3, ,3,36 Reect H and conclude that regresson s sgncant. P-value <. b) =.5 t. 8. 5,36 H : 3 H : 4 H : 5 H : 3 H : 4 H : 5 t.3 t t. 48 t t /, 36 t t /, 36 t t /,36 Fal to reect H Reect H Reect H c) Curvature s evdent n the resduals lots rom ths model. 99 Normal Probablty Plot o the Resduals (resonse s y*) Resdual Resduals Versus the Ftted Values (resonse s y*).3 Resduals Versus 3* (resonse s y*) Ftted Value * a) H : H : at least one =.

81 Aled Statstcs and Probablty or Engneers, 6 th edton ,5,9,5,9 Reect H. P-value =.5 b) =.5 t.5,9.93 H : H : H : 3 H : 4 H : 5 H: H: H: 3 H: 4 H: 5 t.48 t.74 t.4 t.8 t.5 t t t t t t /,9 t /,9 t /,9 t /,9 t /,9 Reect H Reect H Reect H Reect H Do not reect H c) 3 4 H : 3 4 H : = ,4,,4, or at least one Reect H. =.5 t.5,.86 H : H : H : 3 H : 4 H: H: H: 3 H: 4 t.53 t.89 t.49 t 3.6 t t t t t /, t /, t /, t /, Reect H Reect H Reect H Reect H d) The addton o the 5 th regressor results n a loss o one degree o reedom n the denomnator and the reducton n SS E s not enough to comensate or ths loss. e) Observaton s unusually large. Studentzed resduals ) R or model n art (a):.558. R or model n art (c):.557. R or model,, 3, 4 wthout obs. #:.84. R ncreased because observaton was not t well by ether o the revous models. H : g) 3 4 H : = ,4,9 Reect H..9.5,4,9 =.5 t.5,9.93 H : H : H : 3 H : 4 H: H: H: 3 H: 4 t 3.96 t 6.43 t 3.64 t 3.39 t t t t t t t t.5,9.5,9.5,9.5,9

82 Aled Statstcs and Probablty or Engneers, 6 th edton Reect H Reect H Reect H Reect H h) There s some ndcaton o curvature.

83 Aled Statstcs and Probablty or Engneers, 6 th edton -99 Note that data n row are deleted to ollow the nstructons n the eercse. a) The regresson equaton s y* = * +.3 * * -.4 4* Predctor Coe SE Coe T P Constant * * * * S =.8333 R-Sq = 95.8% R-Sq(ad) = 94.9% Analyss o Varance Source DF SS MS F P Regresson Resdual Error Total b) H : 3 4 : at least one H =.5 9., P-value, Reect H at =.5. T tests aear n the revous comuter outut. Because all P-values, all tests reect H c) The resdual lots are more satsactory than the lots n the revous eercse.