UNIVERSITY OF TORONTO Faculty of Arts and Science MAY 2006 EXAMINATIONS ECO220Y1Y PART 1 OF 2. Duration - 3 hours

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1 UNIVERSITY OF TORONTO Faculy of Ar ad Sciece MAY 6 EXAMINATIONS ECOYY PART OF Duraio - hour Eamiaio Aid: Calculaor, wo piece of paper wih ay yped or hadwrie oe (ma. ize: 8.5 ; boh ide of paper ca be ued) SOLUTIONS () If a chariy i eleced a radom, wha i he probabiliy ha more ha 55% of i operaig budge i provided by he federal goverme? (d) () If 5 chariie are eleced a radom, wha i he probabiliy ha o average more ha 55% of he operaig budge of he ampled chariie i provided by he federal goverme? (a) () Uig he adard 8 hour day, ad aumig ha half of he hour are ille relaed ad half of he hour are vacaio relaed, wha i he ample mea ad adard deviaio of mied work due o ille meaured i day? (b) (4) Baed o he give iformaio, wha could you reaoably ifer abou he hape of he populaio? (c) (5) Coider he umber of cuomer makig purchae i a ore uig a gif cerificae durig a weeked ouide of he peak holiday period. If ypically 8% of raacio ivolve a gif cerificae, wha i he probabiliy ha o a paricular weeked wih 8 cuomer, le ha ue a gif cerificae? (e) (6) Wha i he probabiliy ha a employee wih o game acce would make or more ale? (e) () Wha i he epeced umber of ale for a employee wih game acce? (b) (8) Wha i P ( X > μ + σ )? (a) (9) Which rage of value iclude P ( X + ) 9 () Suppoe ( X ) i > μ? (b) X. We wih o e : μ 5 veru : μ 5 uig a 5% < i igificace level. For wha value of he ample mea hould he ull hypohei be rejeced? (c) () For a radom ample ake from a ormal populaio, wha i he cofidece level of he poi eimae ( X ) of μ? (a) () Which of he followig decribe eperimeal daa o compare wo populaio? (b) PART OF Page of

2 () The 99% cofidece ierval eimae of a populaio proporio i (.55,.68). Baed o hi ierval eimae, which of he followig reearch hypohee are uppored by he daa a he give igificace level? (a) (4) If he 95% cofidece ierval of a populaio proporio i (.5,.5), wha i he ample ize? (c) (5) A reearcher i iereed i udyig he differece i voig behavior i local elecio bewee people who have moved o a area wihi he la year veru hoe ha have lived here for more ha year. The reearcher coed ha log-ime local reide are more likely o voe i local elecio. Suppoe he reearcher obai a p-value of. for he implied hypohei e. Which of he followig i a plauible cocluio? (c) (6) Suppoe : p. 45 i o be eed agai : p >. 45 a he α.4 level of igificace where p i he probabiliy of geig a ye voe i a populaio. If he ample ize i, wha i he malle umber of ye voe ha will caue o be rejeced? (d) () Coider he reearch (aleraive) hypohei ha le ha 5 perce of U of T ude wach he eleviio how America Idol. Suppoe a radom ample of 8 U of T ude i eleced ad aked if hey wach he how. Wih a 5% igificace level, wha i he probabiliy of failig o rejec he ull hypohei if i fac oly perce of U of T ude wach he how? (e) (8) Wha i he p-value i he e of he hypohei ha a larger fracio of youg people ow ipod? (b) (9) Wha i he 99.8% cofidece ierval eimae of he differece i he facio ha ow a ipod? (e) () If he lope eimae, b, i zero, he wha i he iercep eimae, b? (b) () Wha would he eimae be if iead of CANCIT he reearcher iclude NCANCIT? (a) () Coider wheher here i a aiically igifica differece i icome bewee Caadia ad o-caadia ciize. Which i he mo plauible cocluio abou he differece? (c) () ow much of a price differeial i ypically oberved bewee a 4-Sar dowow hoel ad a 4- Sar o-dowow hoel (oher hig equal, which mea ha oher idepede variable i he model are held coa)? (e) (4) ow much of a price differeial i ypically oberved bewee a hoel wih air codiioig bu o pool ad oe wihou air codiioig bu wih wo pool (oher hig equal, which mea ha oher idepede variable i he model are held coa)? (d) (5) Baed o he regreio reul, which of he followig cocluio abou he eiece of a liear relaiohip i uppored by hee daa? (e) PART OF Page of

3 UNIVERSITY OF TORONTO Faculy of Ar ad Sciece MAY 6 EXAMINATIONS ECOYY PART OF Duraio - hour S O L U T I O N S () (a) Soluio: Thi reearch queio call for a eimae of he populaio proporio of ude ha have a workig lapop compuer. Boh a poi eimae ad a ierval eimae would be eeded for a complee aiical aalyi. The poi eimae will ju be he ample proporio of ude ha have a lapop. The ierval eimae ca be obaied uig he followig formula ad a igificace level (coveioal α.5): p ± z α / pˆ( pˆ) / ˆ (b) Soluio: Reearcher ha propoed a cluer amplig mehod (cluer i a cla). There i o bia creaed by hi choice. owever, becaue cla aedace i by o mea perfec, here may be a ubaial bia (o-repoe bia) if here i a yemaic differece bewee ude who aed cla ad hoe ha do o aed cla. For eample, i i poible ha ude who purchae lapop are paricularly eriou abou heir udie ad may be more likely o be i cla. I hi eample, hi could lead o our poi ad cofidece ierval eimae of he proporio o yemaically overae he rue proporio of ude ha have lapop. O he oher had, if ude who purchae lapop are le likely o aed cla, he he eimae could be dowwardly biaed. Aoher bia migh be creaed by he way he iformaio i colleced: a how of had. Some ude may raie heir had eve if hey do o have a lapop becaue hey may fear embarrame if hey ed up beig oe of he few ha doe o raie heir had. Oher ude may o be payig aeio o he urveyor ad o boher o raie heir had. Thee biae would affec boh he ierval ad poi eimae. (c) Soluio: We could recommed uig a coveioal igificace level (α.5). We do o currely have a eimae of he proporio of ude ha ue a lapop, o we could be coervaive ad ue p-ha.5 for he purpoe of figurig ou he miimum ample ize. The reearcher ha o pecified a deired olerace bu i oud like a rough idea i wha i called for. We ca repor a rage of olerace: of, 4, ad 6 perceage poi (τ., τ.4, τ.6). The formula i: z.96 α / pˆ( pˆ) τ.5(.5). 4;.96.5(.5).4 6;.96.5(.5).6 6 PART OF Page of 5

4 () (a) Soluio: : μ : μ < 8.8 a.886 The rejecio crieria: rejec if a < Cocluio: a which i le ha he criical value of.658, o we rejec (Give he large ample ize, a z e i alo accepable.) (b) Soluio: Sice he ample ize i large we ca look a he adard ormal diribuio (Z). The P-value i approimaely equal o P(Z <.89).94. The p-value i he probabiliy of geig a e aiic a far or furher from he ull hypohei (i he direcio of he reearch hypohei) a he oe we go if he ull hypohei were acually rue. (Aoher way o ay hi: The probabiliy of geig a e aiic a ereme a he oe we go if he ull hypohei i acually rue.) We ca rejec he ull hypohei for ay igificace level larger ha he p-value. ece, we ca rejec he ull hypohei for ay igificace level larger ha approimaely.94%. (c) Soluio: l μ.645 l l β prob( l > 8.95 μ 8) a.498 β prob( >.498).668 a Power β.9 (d) Soluio: * μ P Z <.5 a * β prob( >.645 μ 8) Z a a.645 Z a (.645 ) *.645 PART OF Page of 5

5 () (a) Soluio: S y [ XY XY ] [ ] S [ X X ] [ ] b b Y b X.4 (.86) 8.8. The regreio equaio i: Price-ha..86Age. The price of a ecod had car ed o decreae by $86 per addiioal year of age of he car. (The ample doe o iclude ew car o cao coclude ha ew car ell o average for $,.) (b) Soluio: : β b y : β.86 SE( b ) 4 SE( b ) ( ) SSE ( ) b y ( ) ε y ( ) 8.99 ( b ) 4 ( 8.99 (.86)( 5.69) ) y y 45.9 Rejec he ull hypohei ad ifer he reearch (aleraive) hypohei i rue. Therefore, we coclude ha here i a aiically igifica liear relaiohip bewee ellig price ad age. Alerae oluio (equally correc): SST Yi ( Y i ) ( 6.) i b i i i 5 i i ( ) ( ) (.86) 4 ( ) X X b X X SSR SSE SST SSR 45.6 The ANOVA able i SS df MS F PART OF Page of 5

6 Regreio Error Toal Rejec if F 4. 6 ad fail o rejec if F < The e aiic i F 49.4, which fall i he rejecio regio. Rejec ad coclude ha he model i aiically igifica: here i a aiically igifica liear relaiohip bewee ellig price ad age. (c) Soluio: Thi i a oe-ided hypohei e. b : β.5 : β <.5.86 (.5).6.66 SE( b ).65 The ull hypohei cao be rejeced. Therefore cao coclude ha a oe year addiio o he age of he car ed o reduce he ellig price by more ha $5. The claim i o uppored by he daa. (d) Soluio: ) y ±.5, ε * ( X X ) + + ( ) 4. ± ± (5 8.8) (e) Soluio: No, we hould o predic beyod he rage of daa uig he regreio lie. I he cae of year old car he price i egaive if we aemp o ue he regreio for predicio, which of coure i o poible. (4) (a) Soluio: Le Y be he ale of coo fabric. The eimaed regreio equaio i Y ha 886 4X 6X +.5X 6X 4 The regreio coefficie for X i -6. I mea ha he value of Y ed o decreae (icreae) by 6 ui whe he value of X icreae (decreae) by ui, whe all oher eplaaory variable are kep a fied level. The regreio coefficie for X i.5. I mea ha he value of Y ed o icreae (decreae) by.5 ui whe he value of X icreae (decreae) by ui, whe all oher eplaaory variable are kep a fied level. (b) Soluio: β β β β : 4 : No all β, i,,, 4 i PART OF Page 4 of 5

7 A he.5 igificace level, rejec if F. 8, do o rejec if F <. 8. The ANOVA able i Source SS df MS F Regreio Error 46 6 Toal 56 F SSR / k 8 / 4 85 SSE /( k ) 46 /(8 4 ) F 85 fall io he rejecio regio. Rejec ad coclude ha he model i aiically igifica. (c) Soluio: : β, i,,, 4 ; : β, i,,, 4. i A he.5 igificace level, rejec.69 < <.69. i Predicor Coeff SdDev Coa X X X.5..5 X if. 69 or. 69. Do o rejec if From he above able, regreio coefficie for X ad X are aiically igifica. The regreio coefficie for X ad X 4 are o aiically igifica. (d) Soluio: ave wo omial (dummy; idicaor) variable: X W, if warm, if o X C, if cool, if o Sice here are oly wo caegorie, oe mu be ecluded from he regreio model. Y X 4 4 W W OR β + β X + β X + β X + β X + β + ε Y X 4 4 C C β + β X + β X + β X + β X + β + ε To e if weaher ha a aiically igifica effec o ale we would e wheher he coefficie o he icluded weaher dummy i equal o zero. For he fir model we would e : β W ad : β W. For he ecod model we would e : β C ad : β C. Of coure i doe maer which of he wo model we pecified: he oucome of he aiical e will be he ame. PART OF Page 5 of 5