exam2_answers - E c o n 3 2 1 E x a m 2 A n s w e r s M a...

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Unformatted text preview: E c o n 3 2 1 E x a m 2 A n s w e r s M a rtin S c h m id t S u m m e r 2 0 1 0 1 . A s tu d y c o lle c te d d a ta o n fa th e r ’s w e ig h t a n d s o n ’s w e ig h t . 3 8 9 5 fa th e rs to g e th e r w ith th e ir s o n s w e r e p a r t o f th e s tu d y . T h e s c a tte r d ia g ra m is fo o tb a ll s h a p e d a n d th e s u m m a ry s ta tis tic s fo r th is b iv a ria te d a ta s e t are: • A v e ( fa th e r’s w e ig h t ) = 1 5 0 p o u n d s • S D ( fa th e r ’s w e ig h t ) = 1 5 p o u n d s • A v e (s o n ’s w e ig h t ) = 1 8 0 p o u n d s • S D ( s o n ’s w e ig h t) = 2 0 p o u n d s • C o rre la tio n c o e ffic ie n t r = 0 . 5 a . P le a s e w rite d o w n th e re g re s s io n e q u a tio n fo r th e re g re s s io n o f s o n ’s w e ig h t o n fa th e r’s w e ig h t (u s e th e c o n c re te n u m b e rs a n d s o lv e th e e q u a tio n fo r y ) . Answer a general expression for the regression equation is : y−y x−x =r SDy SDx SD SD ⇒ y = y−r y x+r y x SDx SDx 20 pounds 20 pounds ⇒ son' s weight = 180 pounds − 0.5 150 pounds + 0.5 x= 15 pounds 15 pounds 2 son' s weight = 80 pounds + x 3 b . P le a s e c a lc u l a te th e R M S o f e rro rs / re s id u a ls a n d w rite d o w n th e g e n e ra l fo rm u la . € Answer a general expression for the RMS of regression errors/residuals is : RMS(errors) = (1 − r 2 ) SDy 2 ⇒ RMS(errors) = (1 − 0.5 ) ⋅ 20 pounds ≈ 17.3 pounds € c . P le a s e s k e tc h th e r e g re s s io n lin e , la b e l th e a x e s a n d a t le a s t 2 p o in ts th ro u g h w h ic h th e re g re s s io n lin e p a s s e s (e .g . th e o r ig in w o u ld b e la b e le d (0 ,0 )). Answer d . P le a s e p re d ic t th e w e ig h t o f a s o n th a t is ra n d o m ly p ic k e d o u t o f th e g ro u p o f fa th e rs w h e re e a c h fa th e r h a s a w e ig h t o f 1 4 0 p o u n d s . E x p la in w h y y o u r p re d ic tio n m a k e s s e n s e . Answer T h e b e s t p r e d ic to r o f s o n ’s w e ig h t fo r th e g r o u p o f fa th e r s w e ig h in g 1 4 0 p o u n d s w o u ld b e th e a v e r a g e w e ig h t o f th e so n s o f th e se fa th e r s. T h is a v e ra g e is n o t k n o w n , b u t it is k n o w n th a t th e r e g r e ssio n e q u a tio n a p p r o x im a te s th is a v e r a g e . T h e r e fo re , o n e c a n u se th e r e g r e ssio n e q u a tio n to p re d ic t th e w e ig h t o f a so n w h o s e fa th e r w e ig h s 1 4 0 pounds: 2 son' s weight = 80 pounds + x 3 ⇒ predicted son ' s weight | father' s weight is 140 pounds = 2 80 pounds + ⋅ 140 pounds ≈ 173.3 pounds 3 e . P le a s e s k e tc h th e tw o h is to g r a m s fo r s o n s w h o h a v e fa th e r s w ith a w e ig h t o f 1 4 0 p o u n d s a n d 8 0 0 p o u n d s re s p e c tiv e ly . L a b e l a ll a x e s a n d e x p la in w h y y o u r h is to g ra m s m a k e s e n s e . W h a t is th e a v e ra g e a n d w h a t is th e s ta n d a rd d e v ia tio n o f e a c h h is to g ra m ? Answer T h e s c a tte r d ia g r a m is fo o tb a ll s h a p e d . T h is m e a n s th a t i t is h o m o s k e d a s tic a n d a p p ro x im a te s a b iv a r ia te n o rm a l d is tr ib u tio n . T h e r e fo re , o n e k n o w s th a t th e u n iv a r ia te d a ta se rie s y |x , i.e . so n ’s w e ig h t g iv e n a s p e c ific fa th e r ’s w e ig h t, a p p ro x im a te s a u n iv a ria te n o rm a l d is tr ib u tio n . T h e a p p ro x im a te a v e ra g e o f e a c h u n iv a ria te d a ta s e r ie s y |x is g iv e n b y th e re g re s sio n e q u a tio n y (x ) a n d , b e c a u se o f th e h o m o s k e d a s tic ity o f th e sc a tte r d ia g r a m , th e sta n d a r d d e v ia tio n o f th e h is to g ra m is a p p r o x im a te d b y th e R M S (e rro rs). If fa th e rs w e ig h 1 4 0 p o u n d s th e a p p r o x im a te a v e ra g e o f th e h isto g r a m o f so n ’s h e ig h t is 1 4 0 p o u n d s a n d th e a p p r o x im a te sta n d a r d d e v ia tio n is 1 7 .3 p o u n d s . A p o ss ib le d e n sity h is to g ra m lo o k s lik e th is: € A h is to g ra m w h e r e fa th e r s w e ig h 8 0 0 p o u n d s d o e s n o t m a k e se n se . A fa th e r w h o w e ig h s 8 0 0 p o u n d s w o u ld b e c a . 4 3 sta n d a r d d e v ia tio n s a b o v e th e a v e ra g e . A s fa th e r’s w e ig h t is n o rm a lly d is tr ib u te d th e r e w ill v e ry lik e ly b e n o o b se rv a tio n w ith th is w e ig h t. T h e r e fo r e , a h is to g ra m fo r s o n s o f v a r io u s fa th e rs m a k e s e v e n le ss se n s e . E x tr a p o la tio n b e y o n d th e sa m p le is o fte n risk y a n d h e re c o m p le te ly n o n s e n s ic a l. f. Answer A re g r e s s io n c a p tu re s o n ly a lin e a r a sso c ia tio n b e tw e e n x a n d y v a ria b le s . It c a n n o t b e u se d to p re d ic t th e e ffe c t o f a n in te rv e n tio n a s lo n g a s th e d e sig n o f a stu d y m a k e s it p o ssib le to g iv e a c a u s a l in te r p r e ta tio n to th e lin e a r a sso c ia tio n (e .g . in r a n d o m iz e d tr ia ls ). T h e r e fo re , o n e c a n sa y n o th in g m e a n in g fu l a b o u t th e e ffe c t th a t a w e ig h t d e c re a s e o f a fa th e r w o u ld h a v e o n its s o n . g . P le a s e u s e th e d a ta o n fa th e r’s w e ig h t a n d s o n ’s w e ig h t to e x p la in th e re g re s s io n e ffe c t. Answer T h e r e g re s s io n e ffe c t is e m b o d ie d in th e re g re s sio n e q u a tio n in p ro b le m 1 a . It p r e d ic ts th a t, in fo o tb a ll sh a p e d sc a tte r d ia g r a m s , x o b s e r v a tio n s th a t a r e z sta n d a r d d e v ia tio n s a b o v e o r b e lo w th e x a v e ra g e w ill, o n a v e r a g e , c o rre s p o n d to y v a lu e s th a t a re le ss th a n z , n a m e ly r tim e s z , sta n d a rd d e v ia tio n s a b o v e o r b e lo w th e y a v e r a g e . In o u r c a s e th a t m e a n s th a t a g r o u p o f fa th e rs w e ig h in g 0 .6 7 s ta n d a r d d e v ia tio n s , i.e . 1 0 p o u n d s , le ss th a n th e x a v e ra g e o f 1 5 0 p o u n d s w ill, o n a v e r a g e , h a v e so n s th a t w e ig h o n ly 0 .3 3 s ta n d a rd d e v ia tio n s , i.e . 6 .5 p o u n d s, le ss th a n th e y a v e r a g e o f 1 8 0 p o u n d s . T h is is a n e c e ssa r y e ffe c t w h e n e v e r a sc a tte r d ia g r a m fo llo w s a p p r o x im a t e ly a b iv a ria te n o rm a l d istrib u tio n a n d th e lin e a r a s s o c ia tio n b e tw e e n y a n d x v a ria b le s is n o t p e rfe c t, i.e . r ≠ 1 a n d r ≠ ­ 1 . (7 p o in ts ) W h a t e ffe c t c a n b e p re d ic te d fo r th e s o n if h is fa th e r d e c re a s e s h is w e ig h t fr o m 1 5 0 p o u n d s to 1 2 0 p o u n d s a fte r th e s tu d y w a s ru n ? Why? 2 . P le a s e s h o w th a t th e c o e ffic i e n ts a ,b o f th e re g re s s io n e q u a tio n o f a b iv a ria te d a ta s e rie s y = a + b x c a n b e d e te rm in e d th ro u g h m in im iz a t io n o f th e r o o t m e a n s q u a re o f re s id u a ls . T o d o th is p le a s e w rite d o w n th e d e fin itio n o f th e re s i d u a l e i a n d d e r iv e th e e x a c t fo rm o f th e c o e ffic ie n ts a ,b in te rm s o f a v e (x ) , a v e (y ), S D ( x ), S D (y ) , a n d c o rr e la tio n c o e ffic ie n t r . P le a s e e x p la in y o u r s te p s . (6 p o in ts ) Answer the error/residual is defined as the difference between the actual yi value and the yi value that is predicted from the form of the linear equation and the actual xi value ei := y i − a − bx i n 1 RMS(errors) = ∑ ( y i − a − bx i )2 n i=1 to see which coefficients a,b minimize the RMS(errors) one applies the standard calculus procedure and uses the mathematical fact that monotonously increasing transformations of the minimand do not change the minimizing coefficients : of the : 1n 1n ∑ ( y i − a − bx i )2 ⇔ min n ∑ ( y i − a − bx i )2 min a ,b a ,b n i=1 i= 1 1n ∂a : ∑ −2( y i − a − bx i ) = 0 (FOCa ) n i=1 n 1 ∂b : ∑ −2 x i ( y i − a − bx i ) = 0 (FOCb ) n i=1 solving FOCa for a one finds an expression of the optimal a in terms of b : n n 1 1 ⇒ a = ∑ y i − b ∑ x i = y − bx n i=1 n i=1 plugging this expression into FOCb and multiplying terms one finds : 1n ⇒ ∑ x i y i − x i y + bx i x − bx i2 = 0 n i=1 splitting the sum into sub ‐ sums and moving constants in front of the sum one finds : 1n 1n 1n 1n 2 ⇒ ∑ x i y i − y ∑ x i + bx ∑ x i − b ∑ x i = 0 n i=1 n i=1 n i=1 n i=1 using the definition of the average one can rewrite this equation as : ⇒ b( x 2 − x 2 ) = xy − x y the left hand side is equivalent to b times the variance of x and the right hand side is equivalent to the covariance of x and y; therefore : Cov ( x, y ) ⇒b= SDx2 Cov(x, y) finally using the equivalence between r and the standardized covariance, i.e. r = SD x SD y one sees that : SD ⇒b=r y SDx 3 . A s tu d y o f y o u n g c h ild re n fo u n d th a t th o s e w ith m o re b o d y fa t te n d e d to h a v e m o re “c o n tro llin g ” m o th e rs ; th e S a n F ra n c is c o C h r o n ic le c o n c lu d e d th a t “ P a re n ts o f F a t K id s S h o u ld L ig h te n U p .” a . W a s th is a n o b s e rv a tio n a l s tu d y o r a r a n d o m iz e d c o n tr o lle d e x p e rim e n t? W h y ? b . D id th e s tu d y fin d a n a s s o c ia tio n b e tw e e n m o th e r’s b e h a v io r a n d h e r c h ild ’s le v e l o f b o d y fa t? W h y o r w h y n o t? c . I f c o n tro llin g b e h a v io r b y th e m o th e r c a u s e s c h ild re n to e a t m o re , w o u ld th a t e x p la in a n a s s o c ia tio n b e tw e e n c o n tro llin g b e h a v io r b y th e m o th e r a n d h e r c h ild ’s le v e l o f b o d y fa t? W h y ? d . S u p p o s e th e re is a g e n e , w h ic h c a u s e s o b e s ity . W o u ld th a t e x p la in th e a s s o c ia tio n ? W h y ? e . C a n y o u th in k o f a n o th e r w a y to e x p la in th e a s s o c ia tio n ? f. D o th e d a ta s u p p o rt th e C h ro n ic le ’s a d v ic e o n c h ild r e a rin g ? W h y o r w h y n o t? (6 p o in ts ) Answer (a ) T h is is a n o b se r v a tio n a l stu d y . T h e su b je c ts w e r e n o t a ssig n e d b y in v e stig a to r s to tre a tm e n t a n d c o n tr o l g ro u p s, i.e . c h ild re n w e re n o t a ssig n e d b y in v e s tig a to rs to th e tr e a tm e n t g r o u p w ith d o m in a n t m o th e rs n o r to th e c o n tro l g r o u p w ith n o n ­ d o m in a n t m o th e r s. (b ) Y e s, th e s tu d y fo u n d th a t m o r e c o n tr o llin g m o th e rs te n d to b e o b se r v e d to g e th e r w ith c h ild r e n th a t h a v e h ig h e r le v e ls o f b o d y fa t. T h is is w h a t sta tistic ia n s c a ll a n a s s o c ia tio n . (c ) Y e s, it w o u ld if o n e a s s u m e s th a t n o o th e r c o n fo u n d in g fa c to rs a re a b le to o ffse t o r r e v e rs e th e e ffe c t. F o r e x a m p le , if d o m in a n t m o th e r s a r e m o r e p re se n t in h ig h ­ in c o m e fa m ilie s a n d if lo w ­ in c o m e c a u se s h ig h e r le v e ls o f b o d y fa t (e .g . th ro u g h lo w e r q u a lit y fo o d ) th e n th e 2 e ffe c ts a re c o n fo u n d e d a n d w o rk a g a in s t e a c h o th e r . If th e m a g n itu d e o f th e se e ffe c ts is ro u g h ly th e sa m e th e n th e y c a n c e l a n d th e re w ill b e n o a p p a re n t a sso c ia tio n . If th e m a g n itu d e o f th e in c o m e e ffe c t is h ig h e r th a n th e d o m in a n t m o th e r e ffe c t th e n a ss o c ia tio n is r e v e rs e d . (d ) N o . T h e g e n e w o u ld a ls o h a v e to b e a sso c ia te d w ith c o n tr o llin g b e h a v io r b y th e m o th e r ( s e e F r e e d m a n p . 2 0 ). F o r e x a m p le , if th e o b e sity g e n e a ls o fa v o rs a d o m in a n t c h a r a c te r th e n it is c le a r th a t d o m in a n t m o th e r s a r e lik e ly to p a ss o n th e ir o b e s ity a n d th e ir d o m in a n t c h a ra c te r to th e ir c h ild re n . If o n e ju st lo o k s a t th e re la tio n s h ip b e t w e e n d o m in a n t m o th e rs a n d o b e sity o n e m ig h t c o m e b e lie v e th a t a c a u sa l re la tio n sh ip e x ists b e tw e e n d o m in a n c e o f m o th e r s a n d o b e s ity o f c h ild r e n . If th e o b e sity g e n e is u n c o rr e la te d w ith th e p re v a le n c e o f d o m in a n t m o th e r s th a n o n e c a n a s su m e th a t th e o b e s ity g e n e is sim ila rly d is tr ib u te d b e tw e e n c h ild re n o f d o m in a n t a n d n o n ­ d o m in a n t m o th e r s . E v e n if it h a s a n e ffe c t th e e ffe c t o n th e g ro u p o f c h ild r e n w ith d o m in a n t m o th e r s w o u ld b e th e s a m e a s th e e ffe c t o n th e o th e r g ro u p o f c h ild r e n . T h e e ffe c t o f d o m in a n t m o th e rs w o u ld n o t b e c o n fo u n d e d b y th e e ffe c t o f th e o b e sity g e n e . (e ) A p o s s ib ility is th a t a m o th e r w h o se e s h e r c h ild e a t to o m u c h m ig h t re sp o n d in a w a y th a t p s y c h o lo g ists w o u ld in te r p re t a s “c o n tro llin g ” — Jo h n n y , sto p e a tin g ! (f) N o . T h e C h ro n ic le b a s e s its a d v ic e o n a p e r c e iv e d c a u sa l re la tio n sh ip . T h is re la tio n sh ip m ig h t e x is t, b u t c a n n o t b e e sta b lish e d fr o m o n e o b se rv a tio n a l stu d y . If n o c a u sa l r e la tio n s h ip e x ists th e n m o th e r s w h o lig h te n u p w o u ld n o t b e tte r th e o b e s ity o f th e ir c h ild r e n . M o re o v e r, it is n o t c le a r w h y p a r e n ts in c lu d in g fa th e rs s h o u ld lig h te n u p if th e stu d y c o lle c te d d a ta o n ly o n m o th e r s . 4 . A r o u le tte w h e e l is ro ta te d o n c e (a ro u le tte w h e e l h a s 3 6 fie ld s : th e n u m b e r s 1 ,2 ,..,3 6 , a n d 2 m o re s p e c ia l fie ld s : 0 a n d 0 0 ); y o u a r e in te re s te d in th e e v e n t " o n e o f th e 1 2 n u m b e rs 1 , 2 ,..,1 2 " a n d k n o w th a t th is e v e n t h a s p ro b a b ility 1 2 / 3 8 . a . P le a s e w rite d o w n th e o u tc o m e s s p a c e S Answer S = {0, 00,1, 2, 3, .., 36} b . P le a s e w rite d o w n th e re le v a n t e v e n t s p a c e E (n o n e e d to e x p lic itly s h o w th a t c o m p le m e n ts , in te rs e c tio n s a n d u n io n s a re in th e e v e n t s p a c e ; lis tin g a ll e le m e n ts o f th e re le v a n t e v e n t s p a c e is € enough). Answer E = {∅, S,{1, 2, ..,12},{0, 00,13,14, .., 36}} c . P le a s e d e fin e th e p r o b a b ility fu n c tio n . € Answer p : E →[0,1] ∅→0 S →1 {1, 2, ..,12} → 12 38 {0, 00,13,14, .., 36} → 26 38 € d . P le a s e d e fin e a r a n d o m v a ria b le X th a t m a p s th e o u tc o m e s 1 , 2 , ,.., 1 2 in to th e v a lu e 1 , a n d th e re m a in in g o u tc o m e s in to th e v a lu e 0 Answer X : S →R 1→1 2 →1 .. 12 → 1 0→0 00 → 0 13 → 0 14 → 0 .. 36 → 0 € e . P le a s e d e r iv e th e p ro b a b ility d e n s ity fu n c tio n f X (x ) fo r th e ra n d o m v a r ia b le d e fin e d in (d ). Answer to define the probability density function fX ( x ) one has to find the various set of outcomes that are mapped into the same number and then look up the probability from the probability function X ‐1 (1) = {1, 2, ..,12} X ‐1 (0) = {0, 00,13,14, .., 36} p({1, 2, ..,12}) = 12 38 p({0, 00,13,14, .., 36}) = 26 38 p( X −1 (1)) = 12 if x = 1 38 ⇒ fX ( x) = −1 p( X (0)) = 26 if x = 0 38 f. P le a s e p lo t th e h is to g ra m o f th e p ro b a b ility d e n s ity fu n c tio n f X (x ) € Answer T h e s a m e r o u le tte w h e e l is n o w ro ta te d 2 tim e s in d e p e n d e n tly a n d a n e w ra n d o m v a ria b le X 2 = X 1 + X 2 is d e fin e d g . P le a s e c a lc u la te th e e x p e c ta tio n o f th e n e w ra n d o m v a ria b le X 2 Answer as this is an i.i.d. process 12 12 E ( X 2 ) = X1 + X 2 = 2 X i = 2 ⋅ = 38 19 € h . P le a s e c a lc u la te th e s ta n d a rd e rro r o f th e n e w ra n d o m v a r ia b le X 2 Answer as this is an i.i.d. process : 12 12 2 2 SE ( X 2 ) = SDX 1 + SDX 2 = 2 SDX i = 2 (1 − ) ≈ 0.66 38 38 i. P le a s e d e riv e th e p r o b a b ility d e n s ity fu n c tio n fo r th e n e w ra n d o m v a r ia b le X 2 € Answer independence and identity (i.i.d. process) implie that the joint distribution function f(x1 , x2 )look like this : f ( x1, x 2 ) = f X 1 ( x 2 ) f X 2 ( x 2 ) = f ( x 2 ) f ( x 2 ) possible values of the random variable X2 are : 0 if x1 = 0 and x2 = 0 1 if x1 = 0 and x2 = 1 or if x1 = 1 and x2 = 0 2 if x1 = 1 and x2 = 1 f ( x1 = 0) f ( x 2 = 0) ≈ 0.46 if x2 = 0 ⇒ fX 2 ( x 2 ) = f ( x1 = 1) f ( x 2 = 0) + f ( x1 = 0) f ( x 2 = 1) ≈ 0.44 if x2 = 1 2 f ( x1 = 1) f ( x 2 = 1) ≈ 0.1 if x = 2 ( 9 p o in ts ) € 5 . P le a s e a n s w e r a ll o f th e fo llo w in g 8 q u e s tio n s ( n o e x p la n a tio n s needed) a . A s tu d e n t w h o is a t th e 7 0 t h p e rc e n tile o f firs t ‐ y e a r G P A is lik e ly t o b e i. b e lo w th e 7 0 t h p e rc e n tile o f s e c o n d ‐ y e a r G P A ii. a t th e 7 0 t h p e rc e n tile o f s e c o n d ‐ y e a r G P A iii. a b o v e th e 7 0 t h p e rc e n tile o f s e c o n d ‐ y e a r G P A iv . i m p o s s ib le to s a y w ith th e d a ta g iv e n b . S ta tis tic a l a n a ly s is w a s m a d e fo r m id te rm a n d fin a l s c o r e s in a la rg e c o u rs e , w ith th e fo llo w in g re s u lts : a v e m i d t e r m = 1 4 , a v e f i n a l = 4 0 , S D m i d t e r m = 5 , S D f i n a l = 1 5 , r = ‐ 0 .7 5 . T h e s c a t te r d ia g ra m w a s fo o tb a ll s h a p e d . F o r e a c h s tu d e n t, th e fin a l s c o r e w a s p re d ic te d fro m th e m id te rm s c o re u s in g th e r e g re s s io n lin e . F o r a b o u t 1 / 3 o f th e s tu d e n ts , th e p re d ic tio n fo r th e fin a l s c o r e w a s o ff b y m o r e th a n i. 5 p o in ts ii. 1 0 p o in ts iii. 1 5 p o in ts iv . 2 .5 p o in ts c . A d ie w ill b e ro lle d 1 0 tim e s . T h e c h a n c e it n e v e r la n d s s ix c a n b e fo u n d b y o n e o f th e fo llo w in g c a lc u la tio n s . W h ic h o n e ? i. ( 1 ) 6 10 ii. 1 − ( 1 )10 6 iii. ( 5 ) 6 10 10 5 iv. 1 − ( 6 ) d . T h e u n c o n d itio n a l p ro b a b ility o f e v e n t A is 0 .3 ; th e u n c o n d itio n a l € p ro b a b ility o f e v e n t B is 0 . 7 . W h ic h o n e o f th e fo llo w in g s ta te m e n ts is tru e : i. T h e p ro b a b ility th a t A a n d B b o th h a p p e n m u s t b e 0 .3 x 0 .7 . ii. I f A a n d B a re in d e p e n d e n t th e p ro b a b ility th a t b o th e v e n ts h a p p e n m u s t b e 0 .3 x 0 .7 iii. I f A a n d B a re m u tu a lly e x c lu s iv e th e p ro b a b ility th a t b o th e v e n ts h a p p e n s m u s t b e 0 .3 x 0 .7 iv . I f A a n d B a r e in d e p e n d e n t a n d m u tu a lly e x c lu s iv e th e p ro b a b ility th a t a t le a s t o n e o f th e m h a p p e n s = 1 – 0 .3 x 0 .7 e . 1 0 0 d ra w s a re m o d e a t ra n d o m w ith re p l a c e m e n t fro m th e b o x | ‐ 5 | ‐ 3 | ‐ 1 |0 |1 |3 | 5 | . T h e s u m o f th e d ra w s w ill b e a ro u n d ______, g iv e o r ta k e ______ o r s o . W h ic h o f th e fo llo w in g o p tio n s is th e rig h t one? i. 0 , 1 0 0 √ 1 0 ii. 1 0 0 ,1 0 √ 1 0 iii. 0 ,1 0 √ 1 0 iv . 0 ,1 0 0 0 √ 1 0 f. I n th e d ia g r a m b e lo w o n e lin e is s h o w n . It is k n o w n th a t th is lin e is e ith e r th e S D lin e , o r th e re g re s s io n lin e fo r y o n x , o r th e re g re s s io n lin e o f x o n y . i. ii. iii. iv . T h e lin e is th e S D lin e T h e lin e is th e re g re s s io n lin e fo r y o n x T h e lin e is th e re g re s s io n lin e fo r x o n y I t is im p o s s ib le to s a y w h ic h o f th e th re e lin e s th e lin e is . (6 p o in ts ) Answer a . ( i) is c o rre c t; th e 7 0 t h p e rc e n tile c o rre sp o n d s to c a . 0 .5 sta n d a r d d e v ia tio n s a b o v e a v e r a g e ; th e re g r e ssio n e ffe c t p r e d ic ts a r e r e g r e s s io n to w a r d s th e a v e r a g e a s lo n g a s th e c o rr e la tio n is n o t p e rfe c t (s e e p r o b le m 1 ) b . ( ii) is c o r re c t; th e R M S (e rr o rs ) is 1 0 a n d a s th e sc a tte r d ia g ra m is fo o tb a ll s h a p e d c a . 2 / 3 o f a ll o b se rv a tio n s w ill fa ll in th e a r e a b e tw e e n th e re g re s sio n lin e sh ifte d 1 0 u p a n d th e re g r e ssio n lin e s h ifte d 1 0 d o w n c . ( iii) is c o r re c t; th e p ro b a b ility th a t a (fa ir) d ie d o e s n o t la n d 6 in o n e r o ll is 5 / 6 ; r e p e a te d th e ro ll 1 0 tim e s re su lts in (5 / 6 ) 1 0 d . ( ii) is c o rr e c t; 2 e v e n ts c a n n e v e r b e m u tu a lly e x c lu siv e a n d in d e p e n d e n t a t th e sa m e tim e ; th e m u ltip lic a tio n ru le o f e v e n t p ro b a b ilitie s h o ld s o n ly if th e e v e n ts a re in d e p e n d e n t – th e re f o r e , ( i) is n o t c o r re c t e . ( ii) is c o r r e c t; th e a v e r a g e o f th e b o x is 0 , th e sta n d a rd d e v ia tio n is √ 1 0 ; th e re fo r e , th e e x p e c ta tio n o f th e s u m is 0 a n d th e s ta n d a r d e rro r is 1 0 √ 1 0 f. ( ii) is c o rr e c t; S D lin e a n d b o th r e g r e ssio n lin e s c r o ss a t th e p o in t o f a v e ra g e s ; if o n e p lo ts th e S D lin e o n e c a n se e th a t th e g iv e n lin e h a s a s lo p e th a t is le s s ste e p th a n th e slo p e o f th e S D lin e – th e re fo re , th e re g re s s io n lin e h a s to b e fo r y o n x ...
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This note was uploaded on 03/08/2011 for the course ECON 602 taught by Professor Chugh during the Spring '11 term at Johns Hopkins.

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