Homework Solution (#5)

Size: px

Start display at page:

Download "Homework Solution (#5)"

Berenice Harper
6 years ago
Views:

1 Homework Soluto (# Chapter : #6,, 8(b, 3, 4, 44, 49, 3, 9 ad 7 Chapter. Smple Lear Regresso ad Correlato.6 (6 th edto 7, old edto Page 9 Rafall volume ( vs Ruoff volume ( : : a. Yes, the scatter-plot shows a strog lear relatoshp betwee rafall volume ad ruoff volume, thus t supports the use of the smple lear regresso model. ( 798 b. 3., , S 634,86.4, ( 643 ( 798( 643 S 4,999 4,43.7, ad S,3 7,4.4. S 7,4.4 ˆ.8697 S,86.4 ad ˆ ( c ( 4. 7 µ. d. S ˆ 4,43.7 (.8697( 7,34.4 S SSE. SSE 37.7 s ˆ σ.4. 3 SSE 37.7 e. r So 97.3% of the observed varato SST 4,43.7 ruoff volume ca be attrbuted to the smple lear regresso relatoshp betwee ruoff ad rafall.. (6 th edto 9, old edto Page a. Verfcato 49.4 b. ˆ , ˆ c. Predcted: ˆ + ˆ ( 7. 7

2 ˆ ˆ µ Y d. + ( (6 th edto, old edto Page ˆ b. We wsh b ad b to mmze f(b, b Σ [ ( b + b ( ] f b to f b to elds b + b Σ( Σ, b Σ ( + b Σ( Σ( Σ(. Sce Σ(, b Σ ( Σ( ( [ Because ( Σ( Thus ˆ * Y ad ˆ * ˆ.. Equatg, ad sce Σ ], b ˆ..3 (old edto Page 9 a. ˆ t s :.49 ± ,.7. The terval cotas ± ( ( α /, ˆ the value, dcatg s a possble value for, ad showg lttle usefuless of the model. b. The p-value assocated wth the model s., whch eceeds stadard levels of α. Ths dcates least square s ot a good wa to predct age from trasparet dete cotet..33 ( 6 th edto 8 a ˆ t s :.748 ±.6(.8.8,.34. The terval cotas ± ( α /, ˆ the value, dcatg s a possble value for, ad showg lttle usefuless of the model. b H., H :. : a > P-value ˆ.748. P( >. H P( T > P( T > >., so we.8 caot reject the ull hpothess at sgfcace level...4 (6 th edto 9, old edto Page 9 E. ( ( Σ ˆ Σ E ˆ We use the fact that ˆ s ubased for E( Σ ( ˆ E ( ΣY Σ( + E +.44 (6 th edto 36, old edto Page 6 a. The mea of the data Eercse. s 4.. Sce 4 s closer to 4. tha s 6, the quatt ( 4 must be smaller tha ( 6. Therefore, sce these quattes are the ol oes that are dfferet the two s ˆ. values, the value for 4 must ecessarl be smaller tha the for 6. Sad brefl, the closer s to, the smaller the value of s ˆ. s ˆ s ˆ

3 b. From the prtout Eercse., the error degrees of freedom s df. t.,.6, so the terval estmate whe 4 s : 7.9 ± (.6( ±.369 ( 7.3, We estmate, wth a hgh degree of cofdece, that the true average stregth for all beams whose MoE s 4 GPa s betwee 7.3 MPa ad 7.96 MPa. c. From the prtout Eercse., s.867, so the 9% predcto terval s ˆ ± t ˆ ( ( (., s + s 7.9 ± ±.8 (.77, Note that the predcto terval s almost tmes as wde as the cofdece terval. d. For two 9% tervals, the smultaeous cofdece level s at least ( (. 9%.49 (6 th edto 37, old edto Page 7 9% CI: (46., 97.7; mdpot 9.9; t. 36 ; ( 7., (.36 ˆ ˆ ˆ ( 97. ˆ + ˆ ˆ + 99% CI: 9.9 ± ( 3.3( 9.4 ( 43.3,68. s, so s ( (6 th edto 38, old edto Page 7 Choce a wll be the smallest, wth d beg largest. a s less tha b ad c (obvousl, ad b ad c are both smaller tha d. Nothg ca be sad about the relatoshp betwee b ad c..9 (6 th edto 47, old edto Page 37 ( 9 ( 47.9 a. S,97 4,7, S , ad 8 8 ( 9( 47.9 S , so r.966. There s a ver strog postve correlato 4, betwee the two varables. b. Because the assocato betwee the varables s postve, the specme wth the larger shear force wll ted to have a larger percet dr fber weght. c. Chagg the uts of measuremet o ether (or both varables wll have o effect o the calculated value of r, because a chage uts wll affect both the umerator ad deomator of r b eactl the same multplcatve costat. r.966. d. ( 933 r e. H o : ρ vs H a : ρ >. t ; Reject H o at level. f r t t.,6.83. t , so H o should be rejected. The.966 data dcates a postve lear relatoshp betwee the two varables.

4 .7 (6 th edto 49, old edto Page 39 a. Sample sze 8 b. ˆ ( Whe 3., ˆ c. Yes, the model utlt test s statstcall sgfcat at the level.. d. Notce the sg of the slope s egatve, hece r r e. Frst check to see f the value 4 falls wth the rage of values used to geerate the least-squares regresso equato. If t does ot, ths equato should ot be used. Furthermore, for ths partcular model a value of 4 elds a g value of 9.8, whch s a mpossble value for. Chapter 3: #9, 9 ad 9 Chapter 3. Nolear ad Multple Regresso 3.9 (6 th edto 63, old edto Page Both a scatter plot ad resdual plot (based o the smple lear regresso model for the frst data set suggest that a smple lear regresso model s reasoable, wth o patter or fluetal data pots whch would dcate that the model should be modfed. However, scatter plots for the other three data sets reveal dffcultes. Scatter Plot for Data Set # Scatter Plot for Data Set #

5 Scatter Plot for Data Set #3 Scatter Plot for Data Set # For data set #, a quadratc fucto would clearl provde a much better ft. For data set #3, the relatoshp s perfectl lear ecept oe outler, whch has obvousl greatl flueced the ft eve though ts value s ot uusuall large or small. The sgs of the resduals here (correspodg to creasg are , ad a resdual plot would reflect ths patter ad suggest a careful look at the chose model. For data set #4 t s clear that the slope of the least squares le has bee determed etrel b the outler, so ths pot s etremel fluetal (ad ts value does le far from the remag oes. 3.9 (6 th edto 7, old edto Page 6 a. No, there s defte curvature the plot. b. Y + ( + ε where ad l(lfetme. Plottg vs. temp gves a plot, whch has a proouced lear appearace (ad fact r.94 for the straght le ft. c..873, 3. 64,. 3783, , Σ Σ Σ Σ.79, from whch ˆ ad ˆ. 4 (values read from computer output. Wth,.44 so ˆ ˆ ( ad thus ˆ e 87.. d. For the trasformed data, SSE.3987, ad 6, Σ , ,. 389, from whch SSPE.3694, SSLF /.993, f. 33. Comparg ths to F.,, 8. 68, t s clear that.3694 / H o caot be rejected. 3.9 (6 th edto 8, old edto Page 7 a. From computer output: ŷ : ŷ :

6 3.37 SSE, s. 69, s 7.9. ( Σ 3.37 b. SST Σ 63, so R c. H wll be rejected favor of H : f ether t t or Thus ( ( :.84 f t Wth t 3. 83, H o caot be rejected; the data does ot.48 argue strogl for the cluso of the quadratc term. d. To obta jot cofdece of at least 9%, we compute a 98% C.I. for each coeffcet usg t., For the C.I. s 8.6 ± ( 6.96( 4. ( 9.87,3. 99 ( a etremel wde terval, ad for the C.I. s.84 ± ( 6.96(.48 (.8,.. e. t.9 ad ˆ + 4 ˆ + 6 ˆ 4. 7, so the C.I. s 4.7±.9.., a, ( ( (.8, ± 4.63 f. If we kew ˆ ˆ ˆ,,, the value of whch mamzes ˆ + ˆ + ˆ would be obtaed b settg the dervatve of ths to ad solvg: ˆ +. The estmate of ths s.9. ˆ Etra Problems: A & B A. There are ma problems wth cocludg that ths data teds to cofrm the clam. I ll epla some here. a. The regresso effect or regresso to the mea. Ths pheomeo tells us that sce we re eamg ol the lowest schools we should epect o average that the wll all move closer to the mea. Ths s a well documeted pheomeo gog all the wa back to Galto s data o the heght of sos/daughters ad predctg these heghts as a fucto of the heghts of parets. b. Ths data does t sa athg about what mprovemets, f a, were made b the other schools. If those schools also creased ther scores the we should adjust the gas made b the lower to reflect the overall tred of scorg hgher o the tests. A better wa (though stll ot perfect to coduct ths stud would be to make t a trul radom test. Take the test scores from the frst ear ad radoml choose some schools to tell ther scores (ad hece ther rak overall ad wthhold the scores from the remag schools. I ths maer ou would be able to eame the effect that kowledge of the eamato score has o the et ear s eamato. Wth the curret aalss there s o evdece to suggest that

7 kowledge of the score had a effect o crease scores. The stud orgazers dd t attempt to measure a dfferece betwee schools wth kowledge ad schools wthout kowledge of ther scores from the prevous ear. B. a. 3 Respose Salar ($g Whole Model Actual b Predcted Plot Salar ($g Actual Salar ($g Predcted P<. RSq.79 RMSE4.764 Summar of Ft RSquare.7866 RSquare Adj Root Mea Square Error Mea of Respose 3.94 Observatos (or Sum Wgts Aalss of Varace Source DF Sum of Squares Mea Square F Rato Model Error Prob > F C. Total <. Parameter Estmates Term Estmate Std Error t Rato Prob> t Itercept <. Pck # <. Resdual b Predcted Plot Salar ($g Resdual Salar ($g Predcted b. I $, s: tpcal pck: ($9.97, $3.49 dvdual pck: ($.9, $.

8 c. Both the Y b predcted ad the resdual plot show a sstematc varato. We ca hopefull accout for ths volato of the regresso requremets b trasformg the data. d. I have omtted the regresso tables for space cocers but here are the approprate cofdece ad predcto tervals ( $, s: e , e.3773 $7.9, $.76 A.: Tpcal pck: ( ( Idvdual pck: ( e , e ( $4., $. A.: Tpcal pck: ($6.64, $.6 Idvdual pck: ($.4, $4.3 A.3: Tpcal pck: ($8.43, $.76 Idvdual pck: ($4.3, $.7 e. Because we kow that salares wll, geeral, ol cotue to decrease t seems ulkel that usg a quadratc regresso wll be accurate ear the rght tal of our data set. A rse salares s t what we epect to happe ad so the predctve power of A. ear the later pcks the draft should t be trusted.

Objectives of Multiple Regression

Objectives of Multiple Regression Obectves of Multple Regresso Establsh the lear equato that best predcts values of a depedet varable Y usg more tha oe eplaator varable from a large set of potetal predctors {,,... k }. Fd that subset of