Solutions to Exercises in Chapter 5

Transcription

1 in 5. (a) The required inerval is b ± se( ) b where b = 4.768, =.4 and se( b ) =.39. Tha is ±.4.39 = ( 4.4, 88.57) We esimae ha β lies beween 4.4 and In repeaed samples 95% of similarly onsrued inervals would onain β. (b) To es H : β = agains H: β we ompue he -value b = = = se( b ).39 β.84 Sine he 5% riial value =.4 exeeds.84, we do no reje H. The daa do no reje he zero-inerep hypohesis. () The p-value.734 represens he sum of he areas under he disribuion o he lef of.84 and o he righ of.84. Sine he disribuion is symmeri, eah of he ail areas will be.734 =.367. Eah of he areas in he ails beyond he riial values ± = ±. is.5. Sine.5 <.367, H is no rejeed. From Figure 5. we an see ha having a p-value >.5 is equivalen o having < <. (d) Tesing H : β = agains H: β >, requires he same -value as in par (b), =.84. Beause i is a one-ailed es, he riial value is hosen suh ha here is a probabiliy of.5 in he righ ail. Tha is, =.686. Sine =.84 > =.69, H is rejeed and we onlude ha he inerep is posiive. In his ase p-value = P( >.84) =.367. We see from Figure 5. ha having he p-value <.5 is equivalen o having > PDF T Figure 5. Criial and Observed Values for Two-Tailed Tes in Quesion 5.()

2 .4.3 PDF T Figure 5. Observed and Criial Vlaues for One-Tailed Tes in Quesion (d) (e) The erm "level of signifiane" is used o desribe he probabiliy of rejeing a rue null hypohesis when arrying ou a hypohesis es. The erm "level of onfidene" refers o he probabiliy of an inerval esimaor yielding an inerval ha inludes he rue parameer. When arrying ou a wo-ailed es of he form H : β k = versus H :, βk nonrejeion of H implies lies wihin he onfidene inerval, and vie versa, providing he level of signifiane is equal o one minus he level of onfidene. (f) False. Srily speaking, we anno make probabiliy saemens abou onsan unknown parameers like β. Thus, if 95% onfiden is regarded as synonymous wih a 95% probabiliy, he saemen is false. However, if we rea he erm "onfiden" more loosely, he saemen ould be regarded as rue. The probabiliy of aeping H: β > when i is false is.5. Thus, afer we have aeped H, in his sense we an say we are 95% onfiden ha β is posiive. 5. (a) The oeffiien of EXPER indiaes ha, on average, a drafsman's qualiy raing goes up by.76 for every addiional year of experiene. (b) The 95% onfidene inerval for β is given by b ± se( b ) =.76± = (.6,.68) We are 95% onfiden ha he proedure we have used for onsruing a onfidene inerval whih yield an inerval ha inludes β. () For esing H : β = agains H: β, he p-value is. I is given as he sum of he areas under he -disribuion o he lef of.7 and o he righ of.7. The area in eah of hese ails is. =.56. We do no reje H beause, for α=.5, p-value >.5. (d) The predied qualiy raing of a drafsman wih 5 years experiene is raing = = 3.58

3 3 The seps required o ompue a prediion inerval will depend on he sofware you are using. Mos sofware will give you a sandard error of he foreas error se( f ), obained as he square roo of var( ˆ yˆ ) ˆ (5 x) y =σ + + T ( x x) Then, a 95% prediion inerval an be obained from yˆ ± se( f) = 3.58 ±.74se( f) 5.3 (a) The esimaed slope oeffiien indiaes ha, on average, a % inrease in real oal expendiure leads o a.3% inrease in real food expendiure. I is he elasiiy of food expendiure wih respe o oal expendiure. (b) For esing H : β =.5 agains he alernaive H: β.5, we ompue he value, assuming H is rue, as b β.34.5 = = = se( b ) The riial value for a wo-ailed es, a. signifiane level and 3 degrees of freedom is =.87. Sine = 3.7 > =.87, we reje H and onlude he elasiiy for food expendiure is no equal o.5. () A 95% onfidene inerval for β is given by (d) b ± se( b ) =.34 ± = (.8,.363) The error erms mus be normally and independenly disribued wih zero mean and onsan variane. This assumpion is neessary for he raio ( b β ) se( b) o have a - disribuion. If he sample size was we ould dispense wih he assumpion of a normally disribued error and rely on a enral limi heorem o show ha ( b β ) se( b ) has an approximae or normal disribuion. (d) Omiing an imporan variable will bias he esimae of β and make he formulas for ompuing he es saisi and onfidene inerval inorre. 5.4 Sine he repored -saisi is given by = b se( b ) and he esimaed variane is var( ˆ b ) = [se( b )], in his ase we have ˆ var( b) = ( b ) = ( ) = 3, (a) For p =.5, he null hypohesis would be rejeed a boh he 5% and % levels of signifiane. (b) For p =.8, he null hypohesis would be rejeed a he 5% level of signifiane, bu no a he % level of signifiane.

4 4 5.6 (a) Hypoheses: H : β = agains H: β Calulaed -value: =.3.8 = 3.78 Criial -value: ± = ±.89 Deision: Reje H beause = 3.78 > =.89. (b) Hypoheses: H : β = agains H: β > Calulaed -value: =.3.8 = 3.78 Criial -value: =.58 Deision: Reje H beause = 3.78 > =.58. () Hypoheses: H : β = agains H: β < Calulaed -value: =.3.8 = 3.78 Criial -value: =.77 Deision: Do no reje H beause = 3.78 > =.77. (d) Hypoheses: H : β =.5 agains H: β.5 Calulaed -value: = (.3.5).8 =.3 Criial -value: ± = ±.74 Deision: Reje H beause =.3 < =.74. (e) A 99% inerval esimae of he slope is given by b ± se( ) b =.3 ±.89.8 = (.79,.54) We esimae β o lie beween.79 and.54 using a proedure ha works 99% of he ime in repeaed samples. 5.7 (a) When esimaing E( y ), we are esimaing he average value of y for all observaional unis wih an x-value of x. When prediing y, we are prediing he value of y for one observaional uni wih an x-value of x. The firs exerise does no involve he random error e ; he seond does. (b) Eb ( + bx ) = Eb ( ) + Eb ( ) x =β + βx var( b + b x ) = var( b ) + x var( b ) + x ov( b, b ) σ x σ x σ x x = + T x x x x x x ( ) ( ) ( ) σ ( ( x x) + Tx ) σ ( x x x) = + T ( x x) ( x x) x xx + x ( x x) =σ + =σ + T ( x x) T ( x x)

5 5 5.8 I is no appropriae o say ha E( yˆ ) = y beause y is a random variable. E( y ) = β + β x β + β x + e = y ˆ We need o inlude y in he expeaion so ha ( ) E( yˆ y ) = E( yˆ ) E( y ) =β + β x β + β x + E( e ) =. 5.9 The esimaed equaion is prie = sqf (56.) (.83) (se) (a) A 95% onfidene inerval for β is b ± se( ) b = 46.5 ± = (4.48, 5.53) (b) To es H : β = agains H: β >, we ompue he -value = = 6.4. A a 5% signifiane level he riial value for a one-ailed es and degrees of freedom is =.65. Sine = 6.4 > =.65, H is rejeed. We onlude here is a posiive relaionship beween house size and prie. () To es H : β = 5 agains H: β 5, we ompue he -value = (46.5 5).83 =.43. (d) A a 5% signifiane level he riial values for a wo-ailed es and degrees of freedom are ± = ±.97. Sine =.43 lies beween.97 and.97, we do no reje H. The daa are no in onfli wih he hypohesis ha says he value of a square foo of housing spae is $5. The poin prediion for house prie for a house wih square fee is prie = = 9,583 A 95% inerval prediion for house prie for a house wih square fee is prie ± se( f ) = 9583 ± = (7544, 774) ˆ = + = + 5= 6 5. y b bx ( x x) (5 ) T ( x x) 5 var( ˆ f ) =σ ˆ + + = = se( f ) = = 3.864

6 6 5. Using appropriae ompuer sofware we find ha b =.4656 v!ar( b ) =.3897 se(b ) =.75 b =.946 v!ar( b ) =.675 se(b ) =.9 (a) The inerval esimaors for β and β are given by b ± se( b) and b ± se( b) where =.6 is he 5% riial value wih 3 degrees of freedom. Therefore, he inerval esimae for β is (b) The inerval esimae for β is.4656 ±.6(.75) = (.8,.795).946 ±.6(.9) = (.645,.34) If we use he inerval esimaors o ompue a large number of inerval esimaes like hese, in repeaed samples, 95% of hese inervals will onain β and β. To es he hypohesis ha β = agains he alernaive i is posiive, we se up he hypoheses H : β = vs H : β >. The es saisi is = b se ( b ). Sine he es is a one-ailed es, a a 5% signifiane level he rejeion region is >.77. The value of he es saisi is =.4656/.75 = Sine = 3.96 > =.77, we reje he null hypohesis indiaing ha he daa are no ompaible wih β = ; hey suppor he hypohesis β >. () The hypoheses are H : β = vs H : β >. The es saisi is = b ( b ) se. For a 5% signifiane level and a one-ailed es, he rejeion region is >.77. The value of he es saisi is =.946/.9 =.68. Sine =.68 > =.77, we reje he null hypohesis and onlude ha he daa are no ompaible wih β = ; hey suppor he alernaive hypohesis ha β is posiive. (d) The marginal produ of he inpu is dy / dx whih is equal o β. Thus, he hypoheses (e) are H : β =.35 vs H : β.35. The es saisi is = ( 35) b. se( b ). A a 5% signifiane level, he rejeion region is >.6. The value of he es saisi is = ( ) /.9 = Sine = 4.45 < =.6, we reje he null hypohesis and onlude ha he daa are no ompaible wih β =.35. The daa do no suppor he hypohesis ha he marginal produ of he inpu is.35. The sampling variabiliy for he inpu level 8 is ( 8 x) ( x x) ( 8 8) v!ar ( y! y )! = = + + σ = The sampling variabiliy for he inpu level 6 is ( 6 x) ( x x) ( 6 8) v!ar ( y! y )! = σ = + + =.658 The prediion error variane is smalles a he sample mean x = 8 and beomes larger he furher x is from x. Sine x = 6 is ouside he sample range, he prediion error variane in his ase is greaer han he squares of all he sandard errors in he able in

7 7 par (b). The variane of he prediion error refers o he variane of (!y y) in repeaed samples, where, for eah sample, we have differen leas squares esimaes b and b, and hene a differen predior!y, as well as a differen realized fuure value y. 5. The leas squares esimaed demand equaion is ln q = ln p (.44) (.4) The figures in parenheses are sandard errors. (a) To es he hypohesis ha he elasiiy of demand is equal o, we se up he hypoheses H : β = versus H : β. The es saisi is = [ b ( ) ] se ( b). Wih degrees of freedom and a 5% signifiane level he rejeion region is >.8. The value of he es saisi is = = Sine = 4.38 <.8, we reje he null hypohesis and onlude ha he elasiiy of demand for hamburgers is no equal o. (b) The predied logarihm of he number of hamburgers sold when prie is $ is ( ) ( ) ln q! = ln = and so a poin prediion for he number of hamburgers is!q = exp(5.868) = Thus, if he prie is $, i is predied ha 336 hamburgers will be sold. To find an inerval prediion for he number of hamburgers, we firs find an inerval prediion for he logarihm of he number of hamburgers. A 95% inerval predior for he logarihm is ln (!q ) ±.8 se( f ) Now, se( ) = = is f, and so a 95% inerval prediion for ln(q ) when ln( p ) = ln() ±.8(.3578) = (5.543, 6.94) Given exp(5.543) = 48 and exp(6.94) = 455, a 95% inerval prediion for he number of hamburgers sold is (48, 455).

8 8 5.3 (a) The linear relaionship beween life insurane and inome is esimaed as (b)!y = x (7.3835) (.) where he numbers in parenheses are orresponding sandard errors. The relaionship in par (a) indiaes ha, as inome inreases, he amoun of life insurane inreases, as is expeed. The value of b = implies ha if a family has no inome, hen hey would purhase $6855 worh of insurane. I is neessary o be areful of his inerpreaion beause here is no daa for families wih an inome lose o zero. Pars (i), (ii) and (iii) disuss he slope oeffiien. (i) If inome inreases by $, hen an esimae of he resuling hange in he amoun of life insurane is $388.. (ii) The sandard error of b is.. To es a hypohesis abou β he es saisi is b β se ~ T ( ) ( b ) [ ] An inerval esimaor for β is b ( b ) b ( b ) se + se,, where is he riial value for wih (T ) degrees of freedom a he α level of signifiane. (iii) To es he laim, he relevan hypoheses are H : β = 5 versus H : β 5. The alernaive β 5 has been hosen beause, before we sample, we have no reason o suspe β > 5 or β < 5. The es saisi is ha given in par (ii) wih β se equal o 5. The rejeion region (8 degrees of freedom) is >.. The value of he es saisi is b = = = 999. se b. ( ) As = <., we reje he null hypohesis and onlude ha he esimaed relaionship does no suppor he laim. (iv) Life insurane ompanies are ineresed in household haraerisis ha influene he amoun of life insurane over ha is purhased by differen households. One likely imporan deerminan of life insurane over is household inome. To see if inome is imporan, and o quanify is effe on insurane, we se up he model y = β + β x + e where y is life insurane over by he -h household, x is household inome, β and β are unknown parameers ha desribe he relaionship, and e is a random unorrelaed error ha is assumed o have zero mean and onsan variane σ. To esimae our hypohesized relaionship, we ake a random sample of households, olle observaions on y and x, and apply he leas-squares esimaion proedure. The esimaed equaion, wih sandard errors in parenheses, is given in par (a). The poin esimae for he response of life-insurane over o an inome inrease of $ is $388 and a 95% inerval esimae for his quaniy is ($3645, $46). This inerval is a relaively narrow one, suggesing we have reliable informaion abou he response. The inerep esimae is no signifianly differen

9 9 () (d) from zero, bu his fa by iself is no a maer for onern; as menioned in par (b), we do no give his value a dire eonomi inerpreaion. The esimaed equaion ould be used o assess likely requess for life insurane and where hanges may our as a resul of inome hanges. To es he hypohesis ha he slope of he relaionship is one, we proeed as we did in par (b)(iii), using insead of 5. Thus, our hypoheses are H : β = versus H : β. The rejeion region is >.. The value of he es saisi is 388. = = Sine = 5.7 > =., we reje he hypohesis ha he amoun of life insurane inreases a he same rae as inome inreases. If inome = $,, hen he predied amoun of life insurane is!y = () = Tha is, he predied life insurane is $394,875 for an inome of $,. 5.4 (a) A 95% inerval esimaor for β is b ±.45 se(b ). Using our sample of daa he orresponding inerval esimae is (b).3857 ± = (.469,.385) If we used he inerval esimaor in repeaed samples, hen 95% of inerval esimaes like he above one would onain β. Thus, β is likely o lie in he range given by he above inerval. We se up he hypoheses H : β = versus H : β <. The alernaive β < is hosen beause we would expe, if here is learning, ha uni oss of produion would deline as umulaive produion inreased. The es saisi, given H is rue, is b = ~ se( b ) (4) The rejeion region is <.76. The value of he es saisi is.3857 = =.7.36 Sine =.7 <.76, we reje H and onlude ha learning does exis. We onlude in his way beause.7 is an unlikely value o have ome from he disribuion whih is valid when here is no learning. () The prediion of he log of uni os when q = is ln( u ˆ) = ln() = The 95% prediion inerval for he uni os of produion is ( u f ) exp ln( ) ± se( ) = exp(3.875 ± ) = (9.63, 4.48) ˆ

10 (d) How quikly workers learn o perform heir asks, and hene he speed wih whih uni oss of produion fall as produion proeeds, are imporan piees of informaion o managers of produion plans. To invesigae his relaionship for he produion of a ianium dioxide by he DuPon Corporaion, we se up he eonomi model u = uq where u is he uni os of produion afer produing q unis, u is he uni os of produion for he firs uni and a is he elasiiy of uni oss wih respe o umulaive produion. A orresponding saisial model is ln( u ) = β + β ln( q ) + e where he subsrip denoes he year for whih observaions u and q were reorded, β = ln(u ), β = a and e is assumed o be an unorrelaed random error wih zero mean and onsan variane. Using 6 observaions from 955 o 97, he esimaed relaionship is ln( u ˆ ) = ln( q ) (.75) (.36) Boh oeffiiens have he expeed signs and are signifianly differen from zero a a. level of signifiane. The esimaed os of he firs uni produed is u ˆ ˆ = exp( β ) = exp(6.9) = 4.. A % inrease in produion dereases uni oss by.386%. Using a 95% inerval esimae o assess he reliabiliy of his poin esimae, we esimae ha he perenage deline in uni oss lies beween.463 and.38. The DuPon managemen an use his informaion o predi fuure uni oss. For example, afer produing unis, he uni os of produion is predied o fall o a value wihin he 95% inerval (9.63, 4.48). 5.5 (a) We se up he hypoheses H : β = versus H : β <. The relevan es saisi, given H is rue, is b = ~ se ( 8) ( b ) (b) The rejeion region is <.658. The value of he es saisi is. 747 = = Sine = 3.33 < =.658, we reje H and onlude ha Mobil Oil's bea is less han. A bea equal o suggess a sok's variaion is he same as he marke variaion. A bea less han implies he sok is less volaile han he marke; i is a defensive sok. The esimaed model is given by!y = x where x is he risk premium of he marke porfolio and y is Mobil's risk premium. Prediing Mobil's premium when x =., we have!y = =.39 When x =., he prediion is

11 !y = =.757 Inerval esimaes for eah value of x are given by y! ± se( f ) where, for a 95% inerval (and 8 degrees of freedom), =.98. Also, for x =., se( f ) =.6434 and for x =., se( f ) = The wo 95% inerval esimaes are: () (d) for x =.:.39 ± = (.6,.388) for x =.:.757 ± = (.57,.4) In he onex of he problem (prediing Mobil's risk premium), hese inervals are very wide and no very informaive. The wo hypoheses are H : β = versus H : β. The es saisi, given H is rue, is b = se ~ ( 8) ( b ) The rejeion region is >.98. The value of he es saisi is. 448 = = Sine =.7 < =.98, we do no reje H. The daa are ompaible wih a zero inerep. Wihou an inerep he esimaed model is!y =.7 x (.85) wih he number in parenheses being he sandard error. Tesing H : β = agains H : β <, he es saisi, given H is rue, is b = ~ se ( 9) ( b ) The rejeion region is <.658. The value of he es saisi is =. 7 = Sine = 3.8 <.658, we reje H and onlude ha Mobil Oil's bea is less han. Prediing Mobil's risk premium for x =. and x =., we have for x =.:!y =.7. =.7 for x =.:!y =.7. =.7 Before urning o inerval prediions for hese wo values of x, noe ha he formula we have been using for he variane of he prediion error is only valid when he model has an inerep. Your ompuer sofware will reognize he hange and give he righ answer. However, i is insruive o derive he orre expression for models wihou an inerep. The prediion error is given by ( β )! f = y y = b x β x e = b x e

12 x σ var = var + var = + σ ( f) x ( b ) ( e ) β x (The ovariane beween (b β ) and e is zero.) To show ha var( b ) noe ha, from Exerise 3.7, b xy x = = x = σ = x and var( b ) x var( y) x Reurning o he sandard error of he prediion error, we have se( f ) = x x x + = +! σ σ x / / When x =., se( f ) =.6395 and he 95% prediion inerval is.7 ± = (.94,.338) When x =., se( f ) =.645 and he 95% prediion inerval is. x = σ, (e).7 ± = (.556,.998). Before invesing on he sok marke, invesors appreiae an indiaion of he riskiness of alernaive soks. Some invesors may be prepared o buy a sok wih a low expeed reurn providing is variane is also low. Ohers may go for risky soks in he hope of a big gain. And, some migh develop a porfolio of soks ha have a variey of risks. Whaever he siuaion, i is imporan o be able o assess he riskiness of differen soks. This riskiness an be examined by looking a he magniude of β j in he model ( ) α β ( ) r r = + r r + e j f j j m f j where r j, r f and r m are he reurn on seuriy j, he risk free rae, and he marke rae, respeively. Values of β j less han sugges sok j is less volaile han he marke and no a risky sok. Values of β j greaer han are an indiaion ha sok j is risky; is variaion is very sensiive o variaion in he marke. To assess he haraerisis of Mobil Oil's sok monhly observaions on r j, r f and r m, for he period 978 o 987, are olleed. The leas-squares esimaed equaion is ( r! j rf ) = ( rm rf ) (.588) (.86) A 95% inerval esimae for Mobil's β j is (.545,.884). Thus, we an onlude ha Mobil's sok is less volaile han he overall marke. I is a good hoie for a risk averse invesor. However, redued volailiy an bring wih i he os of a redued rae of reurn. As we disovered in par (b), when he marke risk premium is %, he predied risk premium

13 3 for Mobil is only 7.57%. Wih a low marke risk premium, suh as %, he prediion for Mobil is omparaively higher (.4%). This higher value is a onsequene of he posiive inerep esimae. In boh ases, i mus be reognized ha our model is no a good one for prediing Mobil's risk premium. The wide prediion inervals mean ha here is a grea deal of unerainy assoiaed wih he realized value of he risk premium. When he marke risk premium is %, we predi ha Mobil's risk premium will lie beween 5.7% and.4%; for a marke risk premium of %, he orresponding prediion is beween 6.6% and 3.88%. Thus, while we have been able o onfidenly onlude ha Mobil's sok is less volaile han he marke, we have no been able o give a reliable prediion of Mobil's risk premium or rae of reurn. 5.6 (a) (a) b = se( b) = =.73 (b) p-value = ( P (.57) ) < = (.896) =.48 () se( b) = b = =.33 var( ˆ b ) = se( b ) =.738 = 4.75 (d) [ ] (b) The esimaed slope b =.8 indiaes ha a % inrease in males 8 and older, who are high shool graduaes, inreases average inome of hose males by $8. The posiive sign is as expeed; more eduaion should lead o higher salaries. () A 99% onfidene inerval for he slope is given by b ± se( ) b =.8 ± = (.96,.64) (d) For esing H : β =. agains H: β., we alulae = (.8.).33 =.634. The riial values for a wo-ailed es wih a 5% signifiane level and 49 degrees of freedom are ± = ±.. Sine =.634 lies in he inerval (.,.), we do no reje H. The null hypohesis suggess ha a % inrease in males 8 or older, who are high shool graduaes, leads o an inrease in average inome for hose males of $. Nonrejeion of H means ha his laim is ompaible wih he sample of daa. (e) The Louisiana residual is (f) ê = =.59. The prediion is MIM ˆ = = 6.4

14 4 5.7 (a) Le y be he quaniy of soda onsumed and x be he maximum emperaure. The linear relaionship beween y and x is y = β + β x + e. Using he daa given, he leas squares esimaes of he equaion are given by (b) y ˆ = x (7.3) (.74) where sandard errors are in parenhesis. R =.9338 To es wheher inreases in emperaure inrease he quaniy onsumed, we es he hypohesis ha H: β = agains H: β >. Given H is rue, he es saisi is = b se( b ). Using a 5% signifiane level, and noing we have 6 degrees of freedom, he rejeion region is >.746. The value of he es saisi is 5.76 = = Sine 5.9 >.746, we reje H and onlude ha here is enough daa evidene o sugges ha higher emperaures do inrease he quaniy onsumed. () A x = 7, he poin prediion for he amoun of soda sold is ŷ = (7) = 3. To ompue a prediion inerval we need he sandard error of he prediion error. Using ompuer sofware, i is found o be se( f ) = A 95% prediion inerval is given by yˆ ± se( f) = 3 ± = (9.7, 6.3) (d) The emperaure for whih we predi zero sodas o be sold is ha value of x whih saisfies he equaion = x or, x = = (a) The relaionship shows how an inrease (or derease) in apprehensions of people enering he U.S. illegally depends on an inrease (or derease) in ime spen poliing he borders. The slope oeffiien gives he elasiiy of ( A A ) wih respe o hanges in ( E E ). Sine he variables are measured in erms of he logs of raios or "log differenes", hey represen relaive hanges raher han original magniudes. To es he signifiane of he esimaed slope, we es H: β = agains H: β. The alulaed -value is =.5.6 = 4.5. Wih suh a large sample size we an ake =.96 as he 5% riial value. Sine 4.5 >.96 we reje H and onlude ha he esimaed slope is signifian.

15 5 (b) This relaionship desribes how he hange in apprehensions of illegal enrans depends on hanges in he Mexian wage rae in is manufauring seor. The slope oeffiien gives he elasiiy of ( A A ) wih respe o hanges in ( MW MW ). Again, noe ha he variables represen hanges in he logs. To es he signifiane of he esimaed slope, we es H: β = agains H: β. The alulaed -value is = = 3.5. Sine 3.5 <.96 we reje H a he 5% level of signifiane. The esimaed slope is signifian. 5.9 (a) The esimaed slopes in he able show how he growh rae of a ounry is expeed o hange when here is a hange in life expeany. Aording o he heory of Swanson and Kopeky, he signs should be posiive. Thus, he sign for he OECD ounries is no wha you would expe. (b) The es resuls for H: β = agains H: β > appear in he following able. Group d of f -value Deision Afria Reje H OECD.7.7 Do no reje H Lain Ameria.34.7 Do no reje H Asia Reje H All Counries Reje H () A 95% inerval esimae of he slope for Lain Ameria is given by b ± se( b ) =. ± = (.6,.86) A 95% inerval esimae of he slope for Asia is b ± se( b ) =.3 ±.3.45 = (.7,.9)