FOR STUDENTS. Solutions to Odd-Numbered End-of-Chapter Exercises

Transcription

1 Solutos to Odd-Numbered Ed-of-Chapter Exercses FOR STUDENTS Solutos to Odd-Numbered Ed-of-Chapter Exercses

2 Solutos to Odd-Numbered Ed-of-Chapter Exercses

3 Solutos to Odd-Numbered Ed-of-Chapter Exercses 3 Chapter Revew of Probablty Solutos to Exercses. (a) Probablty dstrbuto fucto for Outcome 0 (umber of heads) probablty (b) Cumulatve probablty dstrbuto fucto for Outcome < 0 0 < < (umber of heads) Probablty (c) µ E ( ) (0 0.5) + ( 0.50) + ( 0.5).00 Usg Key Cocept.3: E E ad var( ) ( ) [ ( )], E ( ) (0 0.5) + ( 0.50) + ( 0.5).50 so that var( ) E ( ) [ E ( )].50 (.00) For the two ew radom varables W 3+ 6 ad V 0 7, we have: (a) EV ( ) E(0 7 ) 0 7 E ( ) , EW ( ) E(3+ 6 ) 3+ 6 E ( ) (b) (c) σ WV σ var (3 + 6 ) 6 σ , W σ var (0 7 ) ( 7) σ V cov (3 + 6, 0 7 ) 6( 7) cov (, ) σwv cor ( W, V ) σ σ W V 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

4 Solutos to Odd-Numbered Ed-of-Chapter Exercses 4 5. Let deote temperature F ad deote temperature C. Recall that 0 whe 3 ad 00 whe ; ths mples (00/80) ( 3) or (5/9). Usg Key Cocept.3, µ 70 F mples that µ (5/9) 70. C, ad σ 7 F mples σ (5/9) C. 7. Usg obvous otato, C M + F ; thus µ µ + µ ad σ σ + σ + cov(, ). Ths 9. mples (a) µ $85,000 per year. (b) (c) C cor M F Cov( M, F) (, ), σmσf C M F so that Cov( M, F) σ σ cor ( M, F). Thus M F C M F MF Cov ( M, F ) , where the uts are squared thousads of dollars per year. σ σ + σ + cov(, ), so that σ , ad C M F MF σ thousad dollars per year. C C (d) Frst you eed to look up the curret Euro/dollar exchage rate the Wall Street Joural, the Federal Reserve web page, or other facal data outlet. Suppose that ths exchage rate s e (say e 0.80 euros per dollar); each $ s therefore wth ee. The mea s therefore eµ C ( uts of thousads of euros per year), ad the stadard devato s eσ C ( uts of thousads of euros per year). The correlato s ut-free, ad s uchaged. Value of Probablty Dstrbuto of Value of Probablty dstrbuto of (a) The probablty dstrbuto s gve the table above. E ( ) E E E σ 4.77 ( ) Var() ( ) [ ( )] 8. (b) Codtoal Probablty of 8 s gve the table below Value of / / / / / Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

5 Solutos to Odd-Numbered Ed-of-Chapter Exercses 5 E ( 8) 4 (0.0/0.39) + (0.03/0.39) + 30 (0.5/0.39) + 40 (0.0/0.39) + 65 (0.09/0.39) 39. E + + ( 8) 4 (0.0/0.39) (0.03/0.39) 30 (0.5/0.39) (0.0/0.39) 65 (0.09/0.39) Var( ) σ (c) E( ) ( 4 0.0) + ( : 0.05) + ( ) 7.7 Cov(, ) E ( ) EE ( ) ( ) Corr(, ) Cov(, )/( σ σ ).0 /( ) (a) 0.90 (b) 0.05 (c) 0.05 (d) Whe ~ χ 0, the /0 ~ F0,. (e) Z, where Z ~ N(0,), thus Pr( ) Pr( Z ) (a) E ( ) Var ( ) + µ + 0 ; EW ( ) Var ( W) + µ W (b) ad W are symmetrc aroud 0, thus skewess s equal to 0; because ther mea s zero, ths meas that the thrd momet s zero. 4 E ( µ ) $ σ (c) The kurtoss of the ormal s 3, so 3 ; solvg yelds yelds the results for W. (d) Frst, codto o 0, so that S W: ES ( 0) 0; ES ( 0) 00, ES ( 0) 0, ES ( 0) 3 00 Smlarly, 3 4. ES ES ES ES 3 4 ( ) 0; ( ), ( ) 0, ( ) 3. From the large of terated expectatos ES ( ) ES ( 0) Pr ( 0) + ES ( ) Pr( ) 0 ES ES ES 4 E( ) 3; a smlar calculato ( ) ( 0) Pr ( 0) + ( ) Pr( ) ES ES ES ( ) ( 0) Pr ( 0) + ( ) Pr( ) 0 ES ES ES ( ) ( 0) Pr ( 0) + ( ) Pr( ) (e) µ ES ( ) 0, thus ES from part d. Thus skewess 0. S 3 3 ( µ S ) ES ( ) 0 Smlarly, σ S ES ( µ S) ES ( ).99, ad ES Thus, kurtoss /(.99 ) ( µ S ) ES ( ) Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

6 Solutos to Odd-Numbered Ed-of-Chapter Exercses 6 5. (a) Pr ( ) Pr 4/ 4/ 4/ Pr Z 4/ 4/ where Z ~ N(0, ). Thus, () 0; Pr Z Pr ( 0.89 Z 0.89) / 4/ () 00; Pr Z Pr(.00 Z.00) / 4/ (b) () 000; As get large Pr Z Pr( 6.3 Z 6.3).000 4/ 4/ c 4/ c 0 c Pr (0 c 0 + c) Pr 4/ 4/ 4/ c c Pr Z. 4/ 4/ gets large, ad the probablty coverges to. (c) Ths follows from (b) ad the defto of covergece probablty gve Key Cocept µ 0.4 ad σ (a) () P( 0.43) () P( 0.37) Pr Pr / 0.4/ 0.4/ Pr Pr / 0.4/ 0.4/ (b) We kow Pr(.96 Z.96) 0.95, thus we wat to satsfy 0.4 >.96 ad <.96. Solvg these equaltes yelds / 0.4/ 9. (a) l Pr ( y ) Pr ( x, y ) j j l Pr ( y x)pr ( x) j 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

7 Solutos to Odd-Numbered Ed-of-Chapter Exercses 7 (b) E( ) y Pr ( y ) y Pr ( y x ) Pr ( x ) j j j j j j l k j j j ypr ( y x) Pr ( x ) l E( x)pr ( x). k k l (c) Whe ad are depedet, so. (a) Pr ( x, y ) Pr ( x )Pr ( y ), σ E[( µ )( µ )] l k j j j j ( x µ )( y µ ) Pr ( x, y ) l k j j ( x µ )( y µ ) Pr ( x) Pr ( y ) j j l k ( x µ ) Pr ( x) ( yj µ ) Pr ( yj j E ( µ ) E ( µ ) 0 0 0, 0 cor (, ) σ 0 σ σ σ σ ( ) [( ) ( )] [ ] E µ E µ µ E µ + µ µ + µ µ (b) E ( ) 3 E ( ) µ + 3 E ( ) µ µ E ( ) 3 E ( ) E ( ) + 3 E ( )[ E ( )] [ E ( )] E ( ) 3 E ( ) E ( ) + E ( ) 3 3 E E ( µ ) [( 3 µ + 3 µ µ )( µ )] E [ 3 µ 3 µ µ µ 3 µ 3 µ µ ] E ( ) 4 E ( ) E ( ) + 6 E ( ) E ( ) 4 EE ( ) ( ) + E ( ) E ( ) 4[ E ( )][ E ( )] + 6[ E ( )] [ E ( )] 3[ E ( )] ad Z are two depedetly dstrbuted stadard ormal radom varables, so µ µ σ σ σ Z 0, Z, Z 0. (a) Because of the depedece betwee ad Z, Pr ( Z z x) Pr ( Z z), ad EZ ( ) EZ ( ) 0. Thus E ( ) E( Z ) E( ) E( Z ) Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

8 Solutos to Odd-Numbered Ed-of-Chapter Exercses 8 (b) (c) (d) E σ µ ( ) +, ad 3 3 µ E + Z E + µ +. ( ) ( ) Z 0 E ( ) E ( + Z) E ( ) + EZ ( ). Usg the fact that the odd momets of a stadard ormal 3 radom varable are all zero, we have E ( ) 0. Usg the depedece betwee ad Z, we 3 have E( Z ) µ µ 0. Thus E ( ) E ( ) + EZ ( ) 0. Z Cov( ) E[( µ )( µ )] E[( 0)( )] E ( ) E ( ) E ( ) σ 0 cor (, ) 0. σ σ σ σ 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

9 Solutos to Odd-Numbered Ed-of-Chapter Exercses 9 Chapter 3 Revew of Statstcs Solutos to Exercses. The cetral lmt theorem suggests that whe the sample sze ( ) s large, the dstrbuto of the sample average ( ) s approxmately σ 43. 0, we have (a) 00, σ 43 σ 0. 43, ad 00, σ wth σ N µ σ. Gve a populato µ 00, Pr ( < 0) Pr < Φ (.55) (b) 64, σ 43 σ , ad Pr(0 < < 03) Pr < < Φ ( ) Φ (. 00) (c) 65, σ 43 σ , ad Pr ( > 98) Pr ( 98) Pr Φ( ) Φ (3. 978) (rouded to four decmal places). 3. Deote each voter s preferece by. f the voter prefers the cumbet ad 0 f the voter prefers the challeger. s a Beroull radom varable wth probablty Pr ( ) p ad Pr ( 0) p. From the soluto to Exercse 3., has mea p ad varace p( p). p ˆ (a) 400 (b) ˆ( ˆ p p) ( ) 4 var( pˆ ) The stadard error s SE ( pˆ) (var( pˆ)) (c) The computed t-statstc s t act p ˆ µ p , SE( pˆ ) Because of the large sample sze ( 400), we ca use Equato (3.4) the text to get the p-value for the test H0 : p 05. vs. H : p 05:. act p-value Φ ( t ) Φ. ( 506) Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

10 Solutos to Odd-Numbered Ed-of-Chapter Exercses 0 (d) Usg Equato (3.7) the text, the p-value for the test H0 : p 05. vs. H : p> 05. s act p-value Φ ( t ) Φ (. 506) (e) Part (c) s a two-sded test ad the p-value s the area the tals of the stadard ormal dstrbuto outsde ± (calculated t-statstc). Part (d) s a oe-sded test ad the p-value s the area uder the stadard ormal dstrbuto to the rght of the calculated t-statstc. (f) For the test H0 : p 05. vs. H : p> 0. 5, we caot reject the ull hypothess at the 5% sgfcace level. The p-value s larger tha Equvaletly the calculated t-statstc. 506 s less tha the crtcal value.645 for a oe-sded test wth a 5% sgfcace level. The test suggests that the survey dd ot cota statstcally sgfcat evdece that the cumbet was ahead of the challeger at the tme of the survey. 5. (a) () The sze s gve by Pr( pˆ 0.5 >.0), where the probablty s computed assumg that p 0.5. (b) () Pr( pˆ 0.5 >.0) Pr( 0.0 pˆ 0.5.0) 0.0 pˆ Pr.5.5/ / /055 pˆ 0.5 Pr / where the fal equalty usg the cetral lmt theorem approxmato () The power s gve by Pr( pˆ 0.5 >.0), where the probablty s computed assumg that p Pr( pˆ 0.5 >.0) Pr( 0.0 pˆ 0.5.0) 0.0 pˆ Pr.53.47/ / / pˆ Pr.53.47/ / /055 pˆ 0.53 Pr / where the fal equalty usg the cetral lmt theorem approxmato t.6, Pr( t >.6).0, so that the ull s rejected at the 5% level /055 () Pr( t >.6).004, so that the ull s rejected at the 5% level. () 0.54 ± / ± 0.03, or 0.5 to (v) 0.54 ± / ± 0.04, or 0.50 to Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

11 Solutos to Odd-Numbered Ed-of-Chapter Exercses (v) 0.54 ± / ± 0.0, or 0.53 to (c) () The probablty s 0.95 s ay sgle survey, there are 0 depedet surveys, so the 0 probablty f () 95% of the 0 cofdece tervals or 9. (d) The relevat equato s.96 SE( pˆ ) <.0 or.96 p( p) / <.0. Thus must be chose so.96 p( p) that >, so that the aswer depeds o the value of p. Note that the largest value that.0 p( p) ca take o s 0.5 (that s, p 0.5 makes p( p) as large as possble). Thus f > 9604, the the marg of error s less tha 0.0 for all values of p The ull hypothess that the survey s a radom draw from a populato wth p 0.. The pˆ 0. t-statstc s t, SE( pˆ where ˆ ˆ ˆ ) SE( p) p( p)/. (A alteratve formula for SE( ˆp ) s 0. ( 0.) /, whch s vald uder the ull hypothess that p 0.). The value of the t-statstc s.7, whch has a p-value of that s less tha 0.0. Thus the ull hypothess p 0.(the survey s ubased) ca be rejected at the % level. 9. Deote the lfe of a lght bulb from the ew process by. The mea of s µ ad the stadard devato of s σ 00 hours. s the sample mea wth a sample sze 00. The stadard σ 00 devato of the samplg dstrbuto of s σ 0 hours. The hypothess test s H : µ 000 vs. 0 H : µ > 000. The maager wll accept the alteratve hypothess f > 00 hours. (a) The sze of a test s the probablty of erroeously rejectg a ull hypothess whe t s vald. The sze of the maager s test s sze Pr( > 00 µ 000) Pr( 00 µ 000) Pr µ Φ (5) Pr( > 00 µ 000) meas the probablty that the sample mea s greater tha 00 hours whe the ew process has a mea of 000 hours. (b) The power of a test s the probablty of correctly rejectg a ull hypothess whe t s vald. We calculate frst the probablty of the maager erroeously acceptg the ull hypothess whe t s vald: β Pr( 00 µ 50) Pr µ Φ(. 5) Φ (. 5) The power of the maager s testg s β (c) For a test wth 5%, the rejecto rego for the ull hypothess cotas those values of the t-statstc exceedg Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

12 Solutos to Odd-Numbered Ed-of-Chapter Exercses t act act act >. > The maager should beleve the vetor s clam f the sample mea lfe of the ew product s greater tha 03.9 hours f she wats the sze of the test to be 5%.. Assume that s a eve umber. The s costructed by applyg a weght of to the odd observatos ad a weght of 3 to the remag observatos. 3 3 E ( ) E ( ) + E ( ) + E ( ) + E ( ) 3 µ + µ µ 9 9 var( ) var( ) var( ) var( ) var( ) σ 5 σ + σ (a) Sample sze 40, sample average 654., sample stadard devato s The 9 5 stadard error of s SE ( ) s The 95% cofdece terval for the mea test score the populato s 40 µ ±. 96SE( ) 654. ± (65. 34, ). (b) The data are: sample sze for small classes 38, sample average , sample stadard devato s 9. 4; sample sze for large classes 8, sample average , sample stadard devato s The stadard error of s s s SE( ) The hypothess tests for hgher average scores smaller classes s The t-statstc s The p-value for the oe-sded test s: H : µ µ 0 vs. H : µ µ > 0. 0 act t SE( ) p-value ( ) (4 0479) act 5 Φ t Φ.... Wth the small p-value, the ull hypothess ca be rejected wth a hgh degree of cofdece. There s statstcally sgfcat evdece that the dstrcts wth smaller classes have hgher average test scores. 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

13 Solutos to Odd-Numbered Ed-of-Chapter Exercses 3 5. Let p deote the fracto of the populato that preferred Bush. (a) pˆ 405/ ; SE ( pˆ).08; 95% cofdece terval s pˆ ±.96 SE ( pˆ) or ±.036 (b) pˆ 378/ ; SE( pˆ).08; 95% cofdece terval s pˆ ±.96 SE( pˆ) or ± 0.36 (c) pˆ pˆ 0.036; SE( pˆ pˆ ) + (because the surveys are depedet. Sep Oct Sep Oct 0.536( 0.536) 0.5( 0.5) The 95% cofdece terval for the chage p s ( pˆ pˆ ) ±.96 SE( pˆ pˆ ) or Sep Oct Sep Oct ±.050. The cofdece terval cludes ( p p ) 0.0, so there s ot statstcally sgfcace evdece of a chage voters prefereces. Sep Oct 7. (a) The 95% cofdece terval s m, 004 m,99 ±.96 SE( m, 004 m,99 ) where Sm, 004 Sm, m, 004 m,99 m, 004 m, SE( ) ; the 95% cofdece terval s ( ) ± 0.63 or.66 ± (b) The 95% cofdece terval s w, 004 w, 99 ±.96 SE( w, 004 w,99 ) where Sw, 004 Sw, w, 004 w,99 w, 004 w, SE( ) ; the 95% cofdece terval s ( ) ± 0.53 or 0.87 ± (c) The 95% cofdece terval s ( ) ( ) ±.96 SE[( ) ( )], where m, 004 m,99 w, 004 w,99 m, 004 m,99 w, 004 w,99 Sm, 004 Sm, 99 Sw, 004 Sw, m, 004 m,99 w, 004 w,99 m, 004 m, 99 w, 004 w, SE[( ) ( )] The 95% cofdece terval s ( ) ( ) ± or 0.79 ± (a) No. E + E jthus ( ) σ µ ad ( j) µ for. + j j E ( ) E E ( ) E ( ) µ + σ (b) es. If gets arbtrarly close to µ wth probablty approachg as gets large, the gets arbtrarly close to µ wth probablty approachg as gets large. (As t turs out, ths s a example of the cotuous mappg theorem dscussed Chapter 7.). Set m w, ad use equato (3.9) wrte the squared SE of m w as 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

14 Solutos to Odd-Numbered Ed-of-Chapter Exercses 4 ( ) ( ) m m w w ( ) ( ) m w + [ SE( )] ( m m ) + ( w w ). ( ) Smlary, usg equato (3.3) pooled m w [ SE ( )] ( m m ) + ( w w ) ( ) ( ) ( m m ) + ( w w ). ( ) 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

15 Solutos to Odd-Numbered Ed-of-Chapter Exercses 5 Chapter 4 Lear Regresso wth Oe Regressor Solutos to Exercses. (a) The predcted average test score s TestScore (b) The predcted chage the classroom average test score s TestScore. ( 5 8 9). ( 5 8 3) 3. 8 (c) Usg the formula for ˆβ Equato (4.8), we kow the sample average of the test scores across 0 the 00 classrooms s TestScore ˆ β + ˆ β CS (d) Use the formula for the stadard error of the regresso (SER) Equato (4.9) to get the sum of squared resduals: Use the formula for SSR SER ( ) (00 ) R Equato (4.6) to get the total sum of squares: SSR 96 TSS R The sample varace s s TSS Thus, stadard devato s 99 s s (a) The coeffcet 9.6 shows the margal effect of Age o AWE; that s, AWE s expected to crease by $9.6 for each addtoal year of age s the tercept of the regresso le. It determes the overall level of the le. (b) SER s the same uts as the depedet varable (, or AWE ths example). Thus SER s measures dollars per week. (c) R s ut free. (d) () $936.7; () $,8.7 (e) No. The oldest worker the sample s 65 years old. 99 years s far outsde the rage of the sample data. (f) No. The dstrbuto of earg s postvely skewed ad has kurtoss larger tha the ormal. (g) ˆ β ˆ, 0 β so that ˆ β ˆ. 0 + β Thus the sample mea of AWE s $, Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

16 Solutos to Odd-Numbered Ed-of-Chapter Exercses 6 5. (a) u represets factors other tha tme that fluece the studet s performace o the exam cludg amout of tme studyg, apttude for the materal, ad so forth. Some studets wll have studed more tha average, other less; some studets wll have hgher tha average apttude for the subject, others lower, ad so forth. (b) Because of radom assgmet u s depedet of. Sce u represets devatos from average E(u ) 0. Because u ad are depedet E(u ) E(u ) 0. (c) () s satsfed f ths year s class s typcal of other classes, that s, studets ths year s class ca be vewed as radom draws from the populato of studets that eroll the class. (3) s satsfed because 0 00 ad ca take o oly two values (90 ad 0). (d) () ; ; () The expectato of ˆβ 0 s obtaed by takg expectatos of both sdes of Equato (4.8): ˆ ˆ ( ˆ Eβ0) E ( β) E β0 + β+ u β ˆ β + E( β β ) + Eu ( ) β, 0 0 where the thrd equalty the above equato has used the facts that ˆβ s ubased so E( β ˆ β) 0 ad Eu ( ) (a) Wth ˆ β ˆ 0, β0, ad ˆ ˆ β0. Thus ESS 0 ad R 0. (b) If R 0, the ESS 0, so that ˆ for all. But ˆ ˆ ˆ β0 + β, so that ˆ for all, whch mples that ˆ β 0, or that s costat for all. If s costat for all, the ( ) 0 ad ˆβ s udefed (see equato (4.7)).. (a) The least squares objectve fucto s ( ). b Dfferetatg wth respect to b yelds ( ) b ( b ). b Settg ths zero, ad solvg for the least squares estmator yelds ˆ β. (b) Followg the same steps (a) yelds ˆ β ( 4) 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

17 Solutos to Odd-Numbered Ed-of-Chapter Exercses 7 Chapter 5 Regresso wth a Sgle Regressor: Hypothess Tests ad Cofdece Itervals Solutos to Exercses (a) The 95% cofdece terval for β s {. 5 8 ±. 96. }, that s 0. 5 β (b) Calculate the t-statstc: act t ˆ β SE( ˆ β ) The p-value for the test H0: β 0 vs. H: β 0 s act p-value Φ ( t ) Φ. ( 6335) The p-value s less tha 0.0, so we ca reject the ull hypothess at the 5% sgfcace level, ad also at the % sgfcace level. (c) The t-statstc s act t ˆ β ( 5.6) 0. SE( ˆ β ). 0.0 The p-value for the test H0: β 5.6 vs. H: β 5.6 s act p-value Φ ( t ) Φ ( 0.0) 0.9 The p-value s larger tha 0.0, so we caot reject the ull hypothess at the 0%, 5% or % sgfcace level. Because β 5.6 s ot rejected at the 5% level, ths value s cotaed the 95% cofdece terval. (d) The 99% cofdece terval for β 0 s {50.4 ± }, that s, β The 99% cofdece terval s.5 {3.94 ± ) or 4.7 lbs WeghtGa 7. lbs. 5 (a) The estmated ga from beg a small class s 3.9 pots. Ths s equal to approxmately /5 of the stadard devato test scores, a moderate crease. act 3.9 (b) The t-statstc s t 5.56, whch has a p-value of Thus the ull hypothess s.5 rejected at the 5% (ad %) level. (c) 3.9 ± ± (a) The t-statstc s 3..3 wth a p-value of 0.03; sce the p-value s less tha 0.05, the ull.5 hypothess s rejected at the 5% level. (b) 3. ± ± Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

18 Solutos to Odd-Numbered Ed-of-Chapter Exercses 8 (c) es. If ad are depedet, the β 0; but ths ull hypothess was rejected at the 5% level part (a). (d) β would be rejected at the 5% level 5% of the samples; 95% of the cofdece tervals would cota the value β (a) β ( ) so that t s lear fucto of,,,. (b) E(,, ) β, thus E( β,, ) E ( ),, ) β( + + ) β. Usg the results from 5.0, ˆ β 0 m ad ˆ Sm β w m. From Chapter 3, SE( ) ad SE( ) +. Pluggg the umbers ˆ β ad SE( ˆ β ˆ 0) 6.; β 38.0 ad sm sw w m m w SE( ˆ β ) (a) es (b) es (c) They would be uchaged (d) (a) s uchaged; (b) s o loger true as the errors are ot codtoally homosckesdastc. 5. Because the samples are depedet, ˆm β, ad ˆ β w, are depedet. Thus var( ˆ β ˆ ˆ ˆ m, βw, ) var( βm, ) + var( βw,). ˆ Var ( β m,) s cosstetly estmated as [ SE( ˆ β m,)] ad ˆ Var( β w,) s cosstetly estmated as [ SE( ˆ β w,)], so that var( ˆ β ˆ m, βw,) s cosstetly estmated by [ SE( ˆ β ˆ m, )] + [ SE( βw,)], ad the result follows by otg the SE s the square root of the estmated varace. m m 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

19 Solutos to Odd-Numbered Ed-of-Chapter Exercses 9 Chapter 6 Lear Regresso wth Multple Regressors Solutos to Exercses. By equato (6.5) the text, we kow Thus, that values of R ( R ). k R are 0.75, 0.89, ad 0.93 for colums () (3). 3. (a) O average, a worker ears $0.9/hour more for each year he ages. (b) Sally s eargs predcto s dollars per hour. Betsy s eargs predcto s dollars per hour. The dfferece s (a) $3,400 (recall that Prce s measured $000s). (b) I ths case BDR ad Hsze 00. The resultg expected chage prce s thousad dollars or $39,000. (c) The loss s $48,800. (d) From the text R ( R ), so R k ( R ), thus, R k 7. (a) The proposed research assessg the presece of geder bas settg wages s too lmted. There mght be some potetally mportat determats of salares: type of egeer, amout of work experece of the employee, ad educato level. The geder wth the lower wages could reflect the type of egeer amog the geder, the amout of work experece of the employee, or the educato level of the employee. The research pla could be mproved wth the collecto of addtoal data as dcated ad a approprate statstcal techque for aalyzg the data would be a multple regresso whch the depedet varable s wages ad the depedet varables would clude a dummy varable for geder, dummy varables for type of egeer, work experece (tme uts), ad educato level (hghest grade level completed). The potetal mportace of the suggested omtted varables makes a dfferece meas test approprate for assessg the presece of geder bas settg wages. (b) The descrpto suggests that the research goes a log way towards cotrollg for potetal omtted varable bas. et, there stll may be problems. Omtted from the aalyss are characterstcs assocated wth behavor that led to carcerato (excessve drug or alcohol use, gag actvty, ad so forth), that mght be correlated wth future eargs. Ideally, data o these varables should be cluded the aalyss as addtoal cotrol varables. 9. For omtted varable bas to occur, two codtos must be true: (the cluded regressor) s correlated wth the omtted varable, ad the omtted varable s a determat of the depedet varable. Sce ad are ucorrelated, the estmator of β does ot suffer from omtted varable bas. 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

20 Solutos to Odd-Numbered Ed-of-Chapter Exercses 0. (a) (b) ( b b ) b ( b b ) b ( b b ) ( b b ) ( b b ) (c) From (b), ˆβ satsfes or ad the result follows mmedately. (d) Followg aalyss as (c) ( ˆ β ˆ β ) 0 ˆ β ˆ β ˆ β ˆ ad substtutg ths to the expresso for ˆβ (c) yelds Solvg for ˆβ yelds: β ˆ β. ˆ β ˆ β ( ) (e) The least squares objectve fucto s ( b 0 b b ) ad the partal dervatve wth respect to b 0 s ( b0 b b ) ( b0 b b ). b 0 Settg ths to zero ad solvg for ˆβ yelds: ˆ 0 β ˆ β ˆ β Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

21 Solutos to Odd-Numbered Ed-of-Chapter Exercses (f) ˆ β 0 ˆ ˆ + β + β 0 β ˆ β ˆ β β (g) ˆ β 0 ˆ ˆ + β + β 0 β ˆ β ˆ β β + β + β + u 0 β u ( β + β + β ) 0 0 u [ ( β + β + β )] 0 [( β β β )( β β β )] 0 0 [ β β β β + β + ββ + ββ β + β β + β + β β 0 0 [ β β β+ β + ββ + ββ ββ + β + β ] β β β+ β + ββ+ ββ ββ + β + β + ˆ β + ˆ β + ˆ β 0 ˆ β ˆ β ˆ β β ˆ β ˆ β ˆ β β + β β + β β + β ] 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

22 Solutos to Odd-Numbered Ed-of-Chapter Exercses Chapter 7 Hypothess Tests ad Cofdece Itervals Multple Regresso Solutos to Exercses. Regressor () () (3) College ( ) 5.46** (0.) 5.48** (0.) 5.44** (0.) Female ( ).64** (0.0).6** (0.0).6** (0.0) Age ( 3) 0.9** (0.04) 0.9** (0.04) Ntheast ( 4) 0.69* (0.30) Mdwest ( 5) 0.60* (0.8) South ( 6) 0.7 (0.6) Itercept.69** (0.4) 4.40** (.05) 3.75** (.06) (a) The t-statstc s 5.46/ >.96, so the coeffcet s statstcally sgfcat at the 5% level. The 95% cofdece terval s 5.46 ± (b) t-statstc s.64/0.0 3., ad 3. >.96, so the coeffcet s statstcally sgfcat at the 5% level. The 95% cofdece terval s.64 ± (a) es, age s a mportat determat of eargs. Usg a t-test, the t-statstc s , 0.04 wth a p-value of , mplyg that the coeffcet o age s statstcally sgfcat at the % level. The 95% cofdece terval s 0.9 ± (b) Age [0.9 ± ] 5 [0.9 ± ].45 ± $.06 to $ Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

23 Solutos to Odd-Numbered Ed-of-Chapter Exercses 3 5. The t-statstc for the dfferece the college coeffcets s t ( ˆ β ˆ ˆ ˆ college,998 βcollege,99 )/ SE( βcollege,998 βcollege,99). Because ˆcollege β ad,998 ˆcollege β are computed,99 from depedet samples, they are depedet, whch meas that cov( ˆ β ˆ college,998, β college,99) 0 var( ˆ β ˆ β ) var( ˆ β ) + var( ˆ β ). Ths mples that Thus, college,998 college,99 college,998 college,998 ˆ ˆ act SE( β β ) ( ). Thus, t There s o sgfcat college,998 college,99 ( ) chage sce the calculated t-statstc s less tha.96, the 5% crtcal value. 7. (a) The t-statstc s < Therefore, the coeffcet o BDR s ot statstcally sgfcatly dfferet from zero. (b) The coeffcet o BDR measures the partal effect of the umber of bedrooms holdg house sze (Hsze) costat. et, the typcal 5-bedroom house s much larger tha the typcal -bedroom house. Thus, the results (a) says lttle about the covetoal wsdom. (c) The 99% cofdece terval for effect of lot sze o prce s 000 [.00 ± ] or.5 to 6.48 ( thousads of dollars). (d) Choosg the scale of the varables should be doe to make the regresso results easy to read ad to terpret. If the lot sze were measured thousads of square feet, the estmate coeffcet would be stead of (e) The 0% crtcal value from the F, dstrbuto s.30. Because 0.08 <.30, the coeffcets are ot jotly sgfcat at the 0% level. 9. (a) Estmate β + γ + β ( + ) + u 0 ad test whether γ 0. (b) Estmate ad test whether γ 0. (c) Estmate β + γ + β ( a ) + u 0 β + γ + β ( ) + u 0 ad test whether γ Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

24 Solutos to Odd-Numbered Ed-of-Chapter Exercses 4 Chapter 8 Nolear Regresso Fuctos Solutos to Exercses (a) The percetage crease sales s %. The approxmato s 00 [l (98) l (96)].05%. 96 (b) Whe Sales 00 05, the percetage crease s % ad the approxmato s 00 [l (05) l (96)] %. Whe Sales 00 50, the percetage crease s % ad the approxmato s 00 [l (50) l (96)] 4.335%. Whe Sales , the percetage crease s % ad the approxmato s 00 [l (500) l (96)] %. (c) The approxmato works well whe the chage s small. The qualty of the approxmato deterorates as the percetage chage creases. 3 (a) The regresso fuctos for hypothetcal values of the regresso coeffcets that are cosstet wth the educator s statemet are: β > 0 ad β < 0. Whe TestScore s plotted agast STR the regresso wll show three horzotal segmets. The frst segmet wll be for values of STR < 0; the ext segmet for 0 STR 5; the fal segmet for STR > 5. The frst segmet wll be hgher tha the secod, ad the secod segmet wll be hgher tha the thrd. (b) It happes because of perfect multcollearty. Wth all three class sze bary varables cluded the regresso, t s mpossble to compute the OLS estmates because the tercept s a perfect lear fucto of the three class sze regressors. 5. (a) () The demad for older jourals s less elastc tha for youger jourals because the teracto term betwee the log of joural age ad prce per ctato s postve. () There s a lear relatoshp betwee log prce ad log of quatty follows because the estmated coeffcets o log prce squared ad log prce cubed are both sgfcat. (3) The demad s greater for jourals wth more characters follows from the postve ad statstcally sgfcat coeffcet estmate o the log of characters. (b) () The effect of l(prce per ctato) s gve by [ l(age)] l(prce per ctato). Usg Age 80, the elastcty s [ l(80)] 0.8. () As descrbed equato (8.8) ad the footote o page 63, the stadard error ca be foud by dvdg 0.8, the absolute value of the estmate, by the square root of the F-statstc testg β l(prce per ctato) + l(80) β l(age) l(prce per ctato) 0. Characters (c) l ( ) l( Characters) l( a ) for ay costat a. Thus, estmated parameter o a Characters wll ot chage ad the costat (tercept) wll chage. 7. (a) () l(eargs) for females are, o average, 0.44 lower for me tha for wome. () The error term has a stadard devato of.65 (measured log-pots) Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

25 Solutos to Odd-Numbered Ed-of-Chapter Exercses 5 () es. But the regresso does ot cotrol for may factors (sze of frm, dustry, proftablty, experece ad so forth). (v) No. I solato, these results do mply geder dscrmato. Geder dscrmato meas that two workers, detcal every way but geder, are pad dfferet wages. Thus, t s also mportat to cotrol for characterstcs of the workers that may affect ther productvty (educato, years of experece, etc.) If these characterstcs are systematcally dfferet betwee me ad wome, the they may be resposble for the dfferece mea wages. (If ths were true, t would rase a terestg ad mportat questo of why wome ted to have less educato or less experece tha me, but that s a questo about somethg other tha geder dscrmato.) These are potetally mportat omtted varables the regresso that wll lead to bas the OLS coeffcet estmator for Female. Sce these characterstcs were ot cotrolled for the statstcal aalyss, t s premature to reach a cocluso about geder dscrmato. (b) () If MarketValue creases by %, eargs crease by 0.37% () Female s correlated wth the two ew cluded varables ad at least oe of the varables s mportat for explag l(eargs). Thus the regresso part (a) suffered from omtted varable bas. (c) Forgettg about the effect or Retur, whose effects seems small ad statstcally sgfcat, the omtted varable bas formula (see equato (6.)) suggests that Female s egatvely correlated wth l(marketvalue). 9. Note that Defe a ew depedet varable The cofdece terval s ˆ γ ±. ( ˆ γ) β + β + β β + β + β + β 0 0 ( ) ( ). Z, ad estmate β + γ + β Z + u. 96 SE Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

26 Solutos to Odd-Numbered Ed-of-Chapter Exercses 6 Chapter 9 Assessg Studes Based o Multple Regresso Solutos to Exercses. As explaed the text, potetal threats to exteral valdty arse from dffereces betwee the populato ad settg studed ad the populato ad settg of terest. The statstcal results based o New ork the 970 s are lkely to apply to Bosto the 970 s but ot to Los Ageles the 970 s. I 970, New ork ad Bosto had large ad wdely used publc trasportato systems. Atttudes about smokg were roughly the same New ork ad Bosto the 970s. I cotrast, Los Ageles had a cosderably smaller publc trasportato system 970. Most resdets of Los Ageles reled o ther cars to commute to work, school, ad so forth. The results from New ork the 970 s are ulkely to apply to New ork 00. Atttudes towards smokg chaged sgfcatly from 970 to The key s that the selected sample cotas oly employed wome. Cosder two wome, Beth ad Jule. Beth has o chldre; Jule has oe chld. Beth ad Jule are otherwse detcal. Both ca ear $5,000 per year the labor market. Each must compare the $5,000 beeft to the costs of workg. For Beth, the cost of workg s forgoe lesure. For Jule, t s forgoe lesure ad the costs (pecuary ad other) of chld care. If Beth s just o the marg betwee workg the labor market or ot, the Jule, who has a hgher opportuty cost, wll decde ot to work the labor market. Istead, Jule wll work home producto, carg for chldre, ad so forth. Thus, o average, wome wth chldre who decde to work are wome who ear hgher wages the labor market. 5 (a) ad (b) (c) γβ 0 γβ 0 γu βv Q +. γ β γ β β0 γ0 u v P +. γ β γ β γβ 0 γβ 0 EQ ( ), γ β EP ( ) β0 γ 0 γ β Var( Q) ( γσ u + βσ v), Var( P) ( σu + σv), ad γ β γ β (d) () Cov( P, Q) ( γσ u + βσ V) γ β ˆ p Cov( Q, P) γσ u + βσ V β, Var( P) σ + σ u V ˆ p (, ) β 0 EQ ( ) EP Cov P Q ( ) Var ( P ) 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

27 Solutos to Odd-Numbered Ed-of-Chapter Exercses 7 () ˆ β β p > 0, usg the fact that γ > 0 (supply curves slope up) ad β < 0 σu ( γ β) σu + σv (demad curves slope dow). 7. (a) True. Correlato betwee regressors ad error terms meas that the OLS estmator s cosstet. (b) True. 9. Both regressos suffer from omtted varable bas so that they wll ot provde relable estmates of the causal effect of come o test scores. However, the olear regresso (8.8) fts the data well, so that t could be used for forecastg.. Aga, there are reasos for cocer. Here are a few. Iteral cosstecy: To the extet that prce s affected by demad, there may be smultaeous equato bas. Exteral cosstecy: The teret ad troducto of E-jourals may duce mportat chages the market for academc jourals so that the results for 000 may ot be relevat Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

28 Solutos to Odd-Numbered Ed-of-Chapter Exercses 8 Chapter 0 Regresso wth Pael Data Solutos to Exercses. (a) Wth a $ crease the beer tax, the expected umber of lves that would be saved s 0.45 per 0,000 people. Sce New Jersey has a populato of 8. mllo, the expected umber of lves saved s The 95% cofdece terval s (0.45 ±.96 0.) 80 [5.8, 73.77]. (b) Whe New Jersey lowers ts drkg age from to 8, the expected fatalty rate creases by 0.08 deaths per 0,000. The 95% cofdece terval for the chage death rate s 0.08 ± [ 0.04, 0.574]. Wth a populato of 8. mllo, the umber of fataltes wll crease by wth a 95% cofdece terval [ 0.04, 0.574] 80 [ 8.34, 7.49]. (c) Whe real come per capta ew Jersey creases by %, the expected fatalty rate creases by.8 deaths per 0,000. The 90% cofdece terval for the chage death rate s.8 ± [.04,.58]. Wth a populato of 8. mllo, the umber of fataltes wll crease by wth a 90% cofdece terval [.04,.58] 80 [840, 09]. (d) The low p-value (or hgh F-statstc) assocated wth the F-test o the assumpto that tme effects are zero suggests that the tme effects should be cluded the regresso. (e) The dfferece the sgfcace levels arses prmarly because the estmated coeffcet s hgher (5) tha (4). However, (5) leaves out two varables (uemploymet rate ad real come per capta) that are statstcally sgfcat. Thus, the estmated coeffcet o Beer Tax (5) may suffer from omtted varable bas. The results from (4) seem more relable. I geeral, statstcal sgfcace should be used to measure relablty oly f the regresso s wellspecfed (o mportat omtted varable bas, correct fuctoal form, o smultaeous causalty or selecto bas, ad so forth.) (f) Defe a bary varable west whch equals for the wester states ad 0 for the other states. Iclude the teracto term betwee the bary varable west ad the uemploymet rate, west (uemploymet rate), the regresso equato correspodg to colum (4). Suppose the coeffcet assocated wth uemploymet rate s β, ad the coeffcet assocated wth west (uemploymet rate) s γ. The β captures the effect of the uemploymet rate the easter states, ad β + γ captures the effect of the uemploymet rate the wester states. The dfferece the effect of the uemploymet rate the wester ad easter states s γ. Usg the coeffcet estmate ( ˆ γ ) ad the stadard error SE( ˆ γ ), you ca calculate the t-statstc to test whether γ s statstcally sgfcat at a gve sgfcace level. 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

29 Solutos to Odd-Numbered Ed-of-Chapter Exercses 9 3. The fve potetal threats to the teral valdty of a regresso study are: omtted varables, msspecfcato of the fuctoal form, mprecse measuremet of the depedet varables, sample selecto, ad smultaeous causalty. ou should thk about these threats oe-by-oe. Are there mportat omtted varables that affect traffc fataltes ad that may be correlated wth the other varables cluded the regresso? The most obvous caddates are the safety of roads, weather, ad so forth. These varables are essetally costat over the sample perod, so ther effect s captured by the state fxed effects. ou may thk of somethg that we mssed. Sce most of the varables are bary varables, the largest fuctoal form choce volves the Beer Tax varable. A lear specfcato s used the text, whch seems geerally cosstet wth the data Fgure 8.. To check the relablty of the lear specfcato, t would be useful to cosder a log specfcato or a quadratc. Measuremet error does ot appear to a problem, as varables lke traffc fataltes ad taxes are accurately measured. Smlarly, sample selecto s a ot a problem because data were used from all of the states. Smultaeous causalty could be a potetal problem. That s, states wth hgh fatalty rates mght decde to crease taxes to reduce cosumpto. Expert kowledge s requred to determe f ths s a problem. 5. Let D f ad 0 otherwse; D3 f 3 ad 0 otherwse D f ad 0 otherwse. Let B t f t ad 0 otherwse; B3 t f t 3 ad 0 otherwse BT t f t T ad 0 otherwse. Let β 0 α + µ ; γ α α ad δ t µ t µ. 7. (a) Average sow fall does ot vary over tme, ad thus wll be perfectly collear wth the state fxed effect. (b) Sow t does vary wth tme, ad so ths method ca be used alog wth state fxed effects. 9. Ths assumpto s ecessary for the usual formula for SEs to be correct. If t s correct, errors are correlated, the usual formula for SEs are wrog ad ferece s faulty. The appedx cludes a dscusso of more geeral formulae for the SEs whe Assumpto #5 s volated.. No, oe of the regressors s. t Ths depeds o. Ths meas that assumpto () s volated. t 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

30 Solutos to Odd-Numbered Ed-of-Chapter Exercses 30 Chapter Regresso wth a Bary Depedet Varable Solutos to Exercses. (a) The t-statstc for the coeffcet o Experece s 0.03/ , whch s sgfcat at the % level. (b) z Matthew ; Φ (.0) (c) z ; Φ (0.7) 0.76 Chrstopher (d) z ; Φ (3.9) 0.999, ths s ulkely to be accurate because the Jed sample dd ot clude ayoe wth more that 40 years of drvg experece. 3. (a) The t-statstc for the coeffcet o Experece s t 0.006/0.00 3, whch s sgfcat a the % level. (b) Prob Matther (c) Prob Chrstopher (d) The probabltes are smlar except whe experece large (>40 years). I ths case the LPM model produces osescal results (probabltes greater tha.0). 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

31 Solutos to Odd-Numbered Ed-of-Chapter Exercses 3 5. (a) Φ( ) 0.84 (b) Φ( ) 0.83 (c) The t-stat o the teracto term s 0.05/ , whch s sgfcat at the 0% level. 7. (a) For a black applcat havg a P/I rato of 0.35, the probablty that the applcato wll be deed F( ) 7.8%. s e (b) Wth the P/I rato reduced to 0.30, the probablty of beg deed s F ( ).49.9%. The dfferece deal probabltes compared to + e (a) s 4.99 percetage pots lower. (c) For a whte applcat havg a P/I rato of 0.35, the probablty that the applcato wll be deed s F( ) %. If the P/I rato s reduced to 0.30, the probablty of + e beg deed s F( ) %. The dfferece deal probabltes s + e.08 percetage pots lower. (d) From the results parts (a) (c), we ca see that the margal effect of the P/I rato o the probablty of mortgage deal depeds o race. I the logt regresso fuctoal form, the margal effect depeds o the level of probablty whch tur depeds o the race of the applcat. The coeffcet o black s statstcally sgfcat at the % level. The logt ad probt results are smlar. 9. (a) The coeffcet o black s 0.084, dcatg a estmated deal probablty that s 8.4 percetage pots hgher for the black applcat. (b) The 95% cofdece terval s ± [3.89%,.9%]. (c) The aswer (a) wll be based f there are omtted varables whch are race-related ad have mpacts o mortgage deal. Such varables would have to be related wth race ad also be related wth the probablty of default o the mortgage (whch tur would lead to deal of the mortgage applcato). Stadard measures of default probablty (past credt hstory ad employmet varables) are cluded the regressos show Table 9., so these omtted varables are ulkely to bas the aswer (a). Other varables such as educato, martal status, ad occupato may also be related the probablty of default, ad these varables are omtted from the regresso colum. Addg these varables (see colums (4) (6)) have lttle effect o the estmated effect of black o the probablty of mortgage deal.. (a) Ths s a cesored or trucated regresso model (ote the depedet varable mght be zero). (b) Ths s a ordered respose model. (c) Ths s the dscrete choce (or multple choce) model. (d) Ths s a model wth cout data. 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

32 Solutos to Odd-Numbered Ed-of-Chapter Exercses 3 Chapter Istrumetal Varables Regresso Solutos to Exercses cgarettes cgarettes. (a) The chage the regressor, l( P ) l( P ), from a $0.0 per pack crease the, 995, 985 retal prce s l.0 l The expected percetage chage cgarette demad s % 4.587%. The 95% cofdece terval s ( 0.94 ±.96 0.) % [ 6.60%,.58%]. (b) Wth a % reducto come, the expected percetage chage cgarette demad s 0.53 ( 0.0) 00%.06%. (c) The regresso colum () wll ot provde a relable aswer to the questo (b) whe recessos last less tha year. The regresso colum () studes the log-ru prce ad come elastcty. Cgarettes are addctve. The respose of demad to a come decrease wll be smaller the short ru tha the log ru. (d) The strumetal varable would be too weak (rrelevat) f the F-statstc colum () was 3.6 stead of 33.6, ad we caot rely o the stadard methods for statstcal ferece. Thus the regresso would ot provde a relable aswer to the questo posed (a). 3. (a) The estmator ˆTSLS 0 ˆTSLS ˆ σ ( ˆ a β β ) s ot cosstet. Wrte ths as ˆ TSLS ˆ σ ( ˆ ˆ a ( )), u β where ˆTSLS ˆTSLS uˆ β0 β. Replacg ˆ TSLS β wth β, as suggested the questo, wrte ths as σ β ˆ ˆ ( ( )) + [ β ˆ ( ) + β ˆ a u u u ( )]. The frst term o the rght had sde of the equato coverges to ˆ σ u, but the secod term coverges to somethg that s o-zero. Thus ˆ σ a s ot cosstet. (b) The estmator σ Σ ˆTSLS β ˆTSLS ˆ β b ( 0 ) s cosstet. Usg the same otato as (a), we ca wrte ˆ σ, b Σ u ad ths estmator coverges probablty to σ u. 5. (a) Istrumet relevace. Z does ot eter the populato regresso for * (b) Z s ot a vald strumet. ˆ wll be perfectly collear wth W. (Alteratvely, the frst stage regresso suffers from perfect multcollearty.) (c) W s perfectly collear wth the costat term. (d) Z s ot a vald strumet because t s correlated wth the error term. 7. (a) Uder the ull hypothess of strumet exogeety, the J statstc s dstrbuted as a χ radom varable, wth a % crtcal value of Thus the statstc s sgfcat, ad strumet exogeety E(u Z, Z ) 0 s rejected. (b) The J test suggests that E(u Z, Z ) 0, but does t provde evdece about whether the problem s wth Z or Z or both. 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

33 Solutos to Odd-Numbered Ed-of-Chapter Exercses (a) There are other factors that could affect both the choce to serve the mltary ad aual eargs. Oe example could be educato, although ths could be cluded the regresso as a cotrol varable. Aother varable s ablty whch s dffcult to measure, ad thus dffcult to cotrol for the regresso. (b) The draft was determed by a atoal lottery so the choce of servg the mltary was radom. Because t was radomly selected, the lottery umber s ucorrelated wth dvdual characterstcs that may affect earg ad hece the strumet s exogeous. Because t affected the probablty of servg the mltary, the lottery umber s relevat. 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

34 Solutos to Odd-Numbered Ed-of-Chapter Exercses 34 Chapter 3 Expermets ad Quas-Expermets Solutos to Exercses. For studets kdergarte, the estmated small class treatmet effect relatve to beg a regular class s a crease of 3.90 pots o the test wth a stadard error.45. The 95% cofdece terval s 3.90 ± [9.098, 8.70]. For studets grade, the estmated small class treatmet effect relatve to beg a regular class s a crease of 9.78 pots o the test wth a stadard error.83. The 95% cofdece terval s 9.78 ± [4.33, 35.37]. For studets grade, the estmated small class treatmet effect relatve to beg a regular class s a crease of 9.39 pots o the test wth a stadard error.7. The 95% cofdece terval s 9.39 ±.96.7 [4.078, 4.70]. For studets grade 3, the estmated small class treatmet effect relatve to beg a regular class s a crease of 5.59 pots o the test wth a stadard error.40. The 95% cofdece terval s 5.59 ± [0.886, 0.94]. 3. (a) The estmated average treatmet effect s TreatmetGroup Cotrol pots. (b) There would be oradom assgmet f me (or wome) had dfferet probabltes of beg assged to the treatmet ad cotrol groups. Let p Me deote the probablty that a male s assged to the treatmet group. Radom assgmet meas p Me 0.5. Testg ths ull hypothess results a t-statstc of p ˆ Me t , so that the ull of Me pˆme ( pˆme ) 0.55( 45) me 00 radom assgmet caot be rejected at the 0% level. A smlar result s foud for wome. 5. (a) Ths s a example of attrto, whch poses a threat to teral valdty. After the male athletes leave the expermet, the remag subjects are represetatve of a populato that excludes male athletes. If the average causal effect for ths populato s the same as the average causal effect for the populato that cludes the male athletes, the the attrto does ot affect the teral valdty of the expermet. O the other had, f the average causal effect for male athletes dffers from the rest of populato, teral valdty has bee compromsed. (b) Ths s a example of partal complace whch s a threat to teral valdty. The local area etwork s a falure to follow treatmet protocol, ad ths leads to bas the OLS estmator of the average causal effect. (c) Ths poses o threat to teral valdty. As stated, the study s focused o the effect of dorm room Iteret coectos. The treatmet s makg the coectos avalable the room; the treatmet s ot the use of the Iteret. Thus, the art majors receved the treatmet (although they chose ot to use the Iteret). 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

35 Solutos to Odd-Numbered Ed-of-Chapter Exercses 35 (d) As part (b) ths s a example of partal complace. Falure to follow treatmet protocol leads to bas the OLS estmator. 7. From the populato regresso we have α + β + β ( D W) + β D + v, t t t 0 t t β ( ) + β [( D D) W] + β ( D D) + ( v v ). 0 By defg, (a bary treatmet varable) ad u v v, ad usg D 0 ad D, we ca rewrte ths equato as + + W + u β0 β β, whch s Equato (3.5) the case of a sgle W regressor. 9. The covarace betwee β ad s cov( β, ) E{[ β E( β )][ E( )]} Eβ Eβ β E + Eβ E E( β ) E( β ) E( ) { ( ) ( ) ( ) ( )} Because s radomly assged, s dstrbuted depedetly of β. The depedece meas E( β ) E( β ) E ( ) ad E( β ) E( β ) E ( ). Thus cov( β, ) ca be further smplfed: So cov( β, ) E( β )[ E ( ) E( )] E( β ) σ. cov( β, ) E( β ) σ E( β ). σ σ. Followg the otato used Chapter 3, let π deote the coeffcet o state sales tax the frst stage IV regresso, ad let β deote cgarette demad elastcty. (I both cases, suppose that come has bee cotrolled for the aalyss.) From (3.) p ˆ TSLS E( β π ) Cov( β, π ) Cov( β, π ) β E( β ) + Average Treatmet Effect +, E( π ) E( π ) E( π ) 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.

36 Solutos to Odd-Numbered Ed-of-Chapter Exercses 36 where the frst equalty uses the uses propertes of covaraces (equato (.34)), ad the secod equalty uses the defto of the average treatmet effect. Evdetly, the local average treatmet effect wll devate from the average treatmet effect whe Cov( β, π ) 0. As dscussed Secto 3.7, ths covarace s zero whe β or π are costat. Ths seems lkely. But, for the sake of argumet, suppose that they are ot costat; that s, suppose the demad elastcty dffers from state to state (β s ot costat) as does the effect of sales taxes o cgarette prces (π s ot costat). Are β ad π related? Mcroecoomcs suggests that mght be. Recall from your mcroecoomcs class that the lower s the demad elastcty, the larger fracto of a sales tax s passed alog to cosumers terms of hgher prces. Ths suggests that β ad π are postvely related, so that Cov( β, π ) > 0. Because E(π ) > 0, ths suggests that the local average treatmet effect s greater tha the average treatmet effect whe β vares from state to state. 05 Pearso Educato, Ic. Publshg as Addso Wesley Reprted wth permsso from Pearso Educato, Ic.