STA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #1

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1 STA 08 Appled Lear Models: Regresso Aalyss Sprg 0 Soluto for Homework #. Let Y the dollar cost per year, X the umber of vsts per year. The the mathematcal relato betwee X ad Y s: Y X. Ths s a fuctoal relatoshp, ot a statstcal oe. If the umber of yearly spa vsts for a member s kow, the the eact dollar amout of that member s aual dues ca be calculated..6 (a) See Fg..6 o p.. Your pcture should look smlar. Plot E(Y) X Evaluate E(Y) at X 0, 0 ad 40, ad plot these pots. X 0 : E(Y) (0) 50 X 0 : E(Y) (0) 300 X 40 : E(Y) (40) 400 Sketch a ormal curve aroud each of these mea values to represet the dstrbuto of Y at each of the gve X values. Note that the varace of each probablty dstrbuto s the same (σ 6). (b) β 0 : the value of E(Y) at X 0 s β β : for each ut crease X, E(Y) creases by β 5 uts..7 (a) No, sce the dstrbuto of Y s ukow. (b) Yes. Now Y has a ormal dstrbuto wth E(Y) (5) 00, Var(Y) 5..e., Y ~ N(00, 5) P[95 Y 05] P Z P[ - Z ] Over the specfed rage for X, from 40 to 00, there s a crease producto output after a employee takes a trag program. Ths s because the y-tercept b 0 s equal to 0. For eample, f the producto output was 40 before the trag, t wll be 58 after the trag. Also f the producto output was 00 before the trag, t wll be 5 after. However, lookg outsde the rage of X, f the producto was 000 before the trag, t would oly be 970 after the trag. There s oly a crease wth the rage of X..3 (a) The data are observatoal there was o cotrolled epermet. (b) The cocluso s ot vald. Oe caot make fereces about a causal relatoshp based o observatoal data. There could be cofoudg varables that are related to the creased employee productvty ad creased class preparato tme.

2 .3 (c) Eample : Taleted employees do t eed to sped much tme class preparato but stll have hgher productvty levels tha others. Eample : Readg techcal papers or searchg the web may decrease oe s class preparato tme. However, oe may stll have hgher productvty levels tha others. (d) A epermeter mght take a represetatve sample of good employees who do t sped much tme class for preparg. The partcpats would be radomly assged to oe of two groups. Group would be asked to sped several hours for preparato (say 4 hours per day) ad Group would be asked ot to eceed 4 hours of preparg per day. The amout of tme for preparato ad the productvty level would be recorded for each dvdual.. (a) Yˆ X Arfreght breakage Y X R-Sq 90. % 0 amp tras A lear regresso fucto appears to ft the data well. (b) Whe X, Yˆ () 4. (c) The crease the umber of trasfers (X) s. So, the crease the epected umber of broke ampules, E(Y), s estmated by b 4. (d) Calculate: X, Y 4. As we have see part (b), Yˆ () 4.. The X, Y. ftted regresso le does pass through the pot ( )

3 .5 MINITAB regresso for arfreght breakage data: The regresso equato s y broke trasfers Predctor Coef StDev T P Costat trasf S.483 R-Sq 90.% R-Sq(adj) 88.9% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total Obs trasf y broke Ft StDev Ft Resdual St Resd (a) Ŷ () 4. e Y Ŷ e s a estmate of ε, the vertcal devato of Y from the ukow true regresso le (b) Σe SSE 7.6 MSE.0 MSE estmates σ Assumg X 0 wth the scope of the model, the mplcato of the regresso fucto f β 0 s othg but we epect Y 0 ad the regresso fucto plot passes through the org a) For b 0 9 ad b 3, the crtero s: b) Smlarly, for b 0 ad b 5 0 ( (9 + 3 )) 76 Q y 0 ( ( + 5 )) 60 Q y YES! The crtera Q for these estmates, as epected, are larger tha for the least squares estmates..4 a) The least squares estmator of β s obtaed by mmzg the least square crtera, Q. Hece we eed to mmze:

4 Q ( y β ) To get the estmator, we take the dervatve of Q wth respect to β ad equate t to zero. dq dβ d dβ ( y β ) ( y β ) 0 Thus, solvg the above equato for β, evaluated at b, we get the least square estmator to be: b y b) Frst, the desty of a observato Y for the ormal error model, utlzg the fact that E{ Y } β ad σ { Y } σ s gve by: y β f ep πσ σ The lkelhood fucto for observatos Y, Y,..., Y s the product of the above dvdual destes. Sce the varace of the error termσ s kow ths problem, the lkelhood fucto s a fucto of β oly. Hece, L( β ) ep y β / (πσ ) σ ep / (πσ ) σ ( ) ( y β ) The mamum lkelhood estmator (MLE) of β s obtaed by mamzg the above lkelhood. Sce the value of β that mamze the above lkelhood also mamze LogL (β ), we fd the MLE from: LogL( β ) log πσ ( y β ) σ Net, takg the dervatve wth respect to β ad equatg to zero gve the MLE (Check also secod dervatve). That s, d LogL( β ) β dβ σ ( y ) 0

5 Solvg for β evaluated at b gve the MLE to be: b y c) A estmator b of β s ubased f E {b} β. From the regresso equato y β + ε, we deduce E{ y } β Thus, t follow E { b} E{ y } β β β Therefore, the MLE b s a ubased estmator of β..43 a) Let Y be the umber of actve physcas CDI X be The Total Populato X be Number of Hosptal Beds ad X 3 be Total Persoal Icome The estmated regresso fuctos of the umber of actve physcas o each of the predctors are gve by: Yˆ X Yˆ X Yˆ X b) Plot of regresso fuctos ad data. 3 Number of Actve Physcas Number of Actve Physcas *0^6 4*0^6 6*0^6 8*0^6 Total Populato Number of Hosptal Beds

6 Number of Actve Physcas Yes, the lear regresso relato appears to provde a good ft for each of the three predctor varables. However, the two data pots whch are out of the scatter should be see wth cauto Total Persoal Icome c) The mea square error for each predctor varables ca be obtaed from the aalyss of varace table. Thus, Predctors MSE X 3704 X 309 X So based o the MSE, we ca deduce that regresso model that cota X (the umber of hosptal beds) has the smallest varablty aroud the ftted regresso le..44 Let Y stads for per capta come ad also let X represet the percetage of dvduals havg at least bachelor s degree. a) The estmated regresso fucto for each rego s gve by: Rego NE NC S W Estmated MSE Regresso Fucto Yˆ X 7,335,008 Yˆ X 4,4,34 Yˆ X 7,474,349 Yˆ ,4,38 X

7 b) As to the smlarty of the regresso fuctos, terms of the drecto of relatoshp betwee per capta come ad percetage of dvdual havg at least a bachelor s degree t s the same all regos. A ut crease percetage of bachelor s degree result a crease the per capta come. The rate of cremet, however, dffers amog the regos. For stace, the relatve rate of cremet per capta come for a ut crease percetage of dvduals s hgher NE ad smaller NC. c) The MSE for each rego s show colum 3 above. There s a dfferece the varablty aroud the ftted regresso le amog the groups. The varablty s relatvely hgher W ad smaller NC.

12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model

12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model 1. Estmatg Model parameters Assumptos: ox ad y are related accordg to the smple lear regresso model (The lear regresso model s the model that says that x ad y are related a lear fasho, but the observed