hw_answers5 - E c o n 3 2 1 A n s w e rs H o m e w o rk 0 5...

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Unformatted text preview: E c o n 3 2 1 A n s w e rs H o m e w o rk 0 5 M a rtin S c h m id t S u m m e r 2 0 1 0 R e v ie w E x e r c is e 1 0 .8 ( m e a n t to te s t y o u r u n d e rs ta n d in g o f th e re g re s s io n e ffe c t ) In th e s p e c ific s e ttin g h e re th e re g r e s s io n e ffe c t p re d ic ts th a t a g r o u p o f p e o p le th a t h a d th e s a m e , h ig h e r th a n a v e ra g e b lo o d p re s s u re o n th e fir s t m e a s u re m e n t w ill s till h a v e h ig h e r th a n a v e r a g e b lo o d p r e s s u r e o n th e s e c o n d m e a s u re m e n t, b u t th e g ro u p a v e ra g e w ill b e c lo s e r to th e to ta l a v e ra g e o f th e s e c o n d m e a s u re m e n ts ( h o w m u c h c lo s e r is d e te rm in e d b y th e c o rr e la tio n c o e ffic ie n t) . A n a lo g o u s ly , a g ro u p o f p e o p le th a t h a d th e s a m e , lo w e r th a n a v e ra g e b lo o d p re s s u re o n th e fir s t m e a s u r e m e n t w ill s till h a v e lo w e r th a n a v e ra g e b lo o d p re s s u re o n th e s e c o n d m e a s u r e m e n t, b u t th e g ro u p a v e ra g e w ill b e c lo s e r to th e to ta l a v e r a g e o f th e s e c o n d m e a s u re m e n ts . T h e re g re s s io n e ffe c t s a y s n o th in g a b o u t th e c h a n g e in to ta l a v e ra g e s fr o m firs t to s e c o n d m e a s u re m e n ts a n d , th e r e fo re , c a n ’t e x p la in a c h a n g e in th e a v e ra g e fo r th e w h o le p o p u la tio n . T h e d a ta g iv e s o m e e v id e n c e th a t p a tie n ts a re m o r e re la x e d th e s e c o n d tim e . R e v ie w E x e r c is e 1 0 .1 0 ( m e a n t to p ro v id e a p e rs p e c tiv e o n th e re g r e s s io n e ffe c t fro m a p e r c e n tile s p o in t o f v ie w ) F a ls e . T h e c o rre la tio n c o e ffic ie n t r w ill b e p o s itiv e (h ig h e r G P A i n o n e y e a r w ill b e a s s o c ia te d w ith G P A in a s u b s e q u e n t ). E v e n if n o c o rre la tio n c o e ffic ie n t a n d s ta n d a rd d e v ia tio n s a re g iv e n , w e k n o w th a t 0 < r < 1 a n d th a t th e re g r e s s io n e ffe c t w ill m o v e th e a v e ra g e o f a g ro u p w ith b e lo w a v e ra g e G P A ( 4 0 t h p e rc e n tile ) t o a n a v e ra g e g ro u p G P A c lo s e to th e to ta l a v e ra g e ( 5 0 t h p e rc e n tile ) . A s th e a v e ra g e is th e b e s t p re d ic to r (in te rm s o f m in im iz in g th e R M S o f p re d ic tio n e rro rs ) fo r a g r o u p o f in d iv id u a ls if n o th in g e ls e is k n o w n a b o u t in d iv id u a l d iffe re n c e s a n d a s th e r e g r e s s io n lin e a p p ro x im a te s th e g ra p h o f a v e ra g e s o f a s c a tte r d ia g ra m it is c le a r th a t th e b e s t p re d ic tio n fo r a n in d iv id u a l a t th e 4 0 t h G P A p e rc e n tile is a h ig h e r G P A p e rc e n tile ( s o m e th in g b e tw e e n 4 0 t h a n d 5 0 t h p e rc e n tile , th e h ig h e r th e c o rr e la tio n c o e ffic ie n t th e c lo s e r to th e 4 0 t h p e rc e n tile ). R e v ie w E x e r c is e 1 1 .1 ( m e a n t a s a re m in d e r fo r th e re la tio n s h ip b e tw e e n R M S o f re s id u a ls , c o r re la tio n c o e ffic ie n t a n d s ta n d a r d d e v ia tio n s ) T h e fo rm u la fo r th e R M S o f re s id u a ls fo r p re d ic tin g y fr o m x is g iv e n b y th e s q u a r e ro o t o f 1 ‐ r 2 tim e s S D y . R e v ie w E x e r c is e 1 1 .3 ( m e a n t to re m in d y o u w h ic h s ta n d a rd d e v ia tio n s to u s e in th e c a lc u la tio n o f th e R M S o f re s id u a ls ) ( a ) S q rt( 1 ‐ r 2 ) S D h e i g h t a t 1 8 = S q rt( 1 − 0 .8 2 ) × 2 .5 = 1 .5 in c h e s (s ta n d a rd d e v ia tio n o f th e p re d ic te d v a ria b le ) (b ) S q r t( 1 ‐ r 2 ) S D h e i g h t a t 6 = S q rt( 1 − 0 .8 2 × 1 .7 ) ≈ 1 .0 in c h e s (s ta n d a rd d e v ia tio n o f th e p re d ic te d v a ria b le ) R e v ie w E x e r c is e 1 1 .4 ( m e a n t to illu s tr a te th e m e a n in g o f R M S o f re s id u a ls fo r p re d ic tio n s ) 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 (i.e . a p p r o x im a te ly b iv a r ia te n o r m a l) . T h e re fo re th e R M S o f re s id u a ls is th e s a m e in e v e ry b in / c la s s in te rv a l. (a ) R M S o f re s id u a ls = S q rt( 1 − r 2 ) × S D f i n a l s c o r e s = 1 2 ; th e ty p ic a l p r e d ic tio n e r ro r / re s id u a l w ill b e 1 2 m e a n in g th a t a b o u t 2 / 3 o f th e s tu d e n ts w ill fa ll w ith in a 2 lin e s p a ra lle l a n d 1 S D a b o v e / b e lo w th e r e g re s s io n lin e (b ) 6 5 .8 . T h is s tu d e n t is 3 0 / 2 5 = 1 .2 S D s a b o v e a v e ra g e o n th e m id te rm , a n d s h o u ld b e a b o v e a v e ra g e o n th e fi n a l b y r × 1 .2 = 0 .7 2 S D s . T h a t is 1 0 .8 p o in ts a n d c o rre s p o n d s to th e p re d ic te d a v e ra g e o f h is 8 0 p o in ts m id te rm g ro u p (c ) 1 2 , th e R M S o f re s id u a ls : s e e p a rt (a ) R e v ie w E x e r c is e 1 1 .5 ( m e a n t to te s t if y o u th o r o u g h ly u n d e rs to o d th e re la tio n s h ip s a n d d iff e re n c e s b e tw e e n a v e ra g e s , s ta n d a rd d e v ia tio n s , c o n d itio n a l a v e ra g e s a n d R M S o f re s id u a ls ) 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 (i.e . a p p r o x im a te ly b iv a r ia te n o r m a l) . T h e re fo re , o n e c a n u s e th e n o r m a l a p p ro x im a tio n to c a lc u la te p e rc e n ta g e s fo r a rb itr a r y b in s / c la s s in te rv a ls . (a ) T h e a n s w e r is a b o u t 5 % . 8 0 p o in ts c o rre s p o n d to 1 .6 7 s ta n d a rd d e v ia tio n s a b o v e th e a v e ra g e . T h e a re a u n d e r th e s ta n d a r d n o rm a l c u r v e a n d a b o v e 1 .6 7 e q u a ls c a . 5 % . (b ) A b o u t 1 2 % . T h e r e g re s s io n lin e p re d ic ts a n a v e r a g e fin a l s c o re o f 6 5 .8 fo r th e g ro u p th a t s c o r e d 8 0 p o in ts o n th e ir m id te rm . T o c a lc u la te th e p e rc e n ta g e o f s tu d e n ts w h o s c o re d 8 0 p o in ts o n th e m id te r m a n d m o re th a n 8 0 p o in ts o n th e fin a l e x a m o n e c a n u s e th e n o r m a l a p p ro x im a tio n fo r th e b in / c la s s in te rv a l o f 8 0 m id te rm p o in ts . T h e s ta n d a rd d e v ia tio n o f th e n o r m a l c u rv e in e a c h b in / c la s s in te rv a l e q u a ls th e R M S o f r e s id u a ls , i.e . 1 2 p o in ts . 8 0 p o in ts in th e g ro u p w ith 8 0 m id te rm p o in ts c o r re s p o n d s to (8 0 − 6 5 .8 )/ 1 2 ≈ 1 .1 8 s ta n d a r d d e v ia tio n s . T h e a re a a b o v e 1 .2 8 a n d b e lo w th e n o r m a l c u rv e c a n th e n b e c a lc u la te d fro m th e ta b le o f th e s ta n d a r d n o r m a l c u r v e . R e v ie w E x e r c is e 1 1 .6 ( m e a n t to illu s tr a te a g a in th e d iffe re n c e b e tw e e n a s s o c ia tio n a n d c a u s a lity , a n d th e im p o rta n t ro le o f c o n fo u n d in g fa c to rs ) N o . T h e c o n c lu s io n s e e m s re a s o n a b le , b u t d o e s n o t fo llo w fr o m th e d a ta . A n a s s o c ia tio n d o e s n o t im p ly c a u s a lity a n d v a r io u s c o n fo u n d in g fa c to r s c a n p r o d u c e a n a s s o c ia tio n w ith o u t a n y c a u s a l e ffe c t. F o r e x a m p le , b e tte r s tu d e n ts m ig h t e n jo y d o in g h o m e w o rk m o re th a n w e a k e r s tu d e n ts a n d , th e r e fo re , w o u ld s p e n d m o re tim e d o in g h o m e w o rk a n y w a y . R e v ie w E x e r c is e 1 3 .4 ( m e a n t to illu s tr a te th e m u ltip lic a tio n ru le in th e p r e s e n c e o f e v e n t d e p e n d e n c e ) T o c a lc u la t e th e p ro b a b ility o f 2 e v e n ts th a t o c c u r to g e th e r o n e u s e s th e m u ltip lic a tio n fo rm u la : p ( A a n d B ) = p (A )p (B |A ) = p ( B )p ( A |B ). A c a r d s e t h a s 5 2 c a rd s . T h e re a re 4 p o s s ib ilitie s o u t o f 5 2 to p ic k 1 a c e o u t th e w h o le s e t o f c a rd s . W h e n th e 1 a c e is k e p t th e r e r e m a in 5 1 c a rd s a n d 3 a c e s , th e n 5 0 c a rd s a n d 2 c a r d s , fin a lly 4 9 c a r d s a n d 1 a c e . In th e la s t d ra w th e r e w o u ld b e 4 p o s s ib ilitie s to d ra w 1 k in g o u t o f th e r e m a in in g 4 8 c a r d s . If o n e a s s u m e s e q u a l p ro b a b il itie s fo r a ll p o s s ib le o u tc o m e s in e a c h s in g le d ra w o n e c a n c a lc u la te th e p ro b a b ility w ith th e m u ltip lic a tio n ru le . (i.e . 4 / 5 2 × 3 / 5 1 × 2 / 5 0 × 1 / 4 9 × 4 / 4 8 ≈ 1 / 3 ,0 0 0 ,0 0 0 . R e v ie w E x e r c is e 1 3 .7 ( m e a n t to illu s tr a te a c o m m o n e rro r in p ro b a b ility a s s e s s m e n ts m a d e p o p u la r b y K a h n e m a n a n d T v e r s k y ) If o n e a s s u m e s e q u a l p ro b a b ility fo r e a c h p o s s ib le s e r ie s o f 6 c o in to s s e s , b o th s e q u e n c e s a re e q u a lly lik e ly a n d h a v e a c h a n c e o f 1 / 6 4 . O f c o u rs e , th e r e a re a lo t m o re s e q u e n c e s w ith 3 h e a d s a n d 3 ta ils , b u t th a t w a s n o t th e q u e s tio n . (a ) F a ls e (b ) F a ls e (c ) T ru e R e v ie w E x e r c is e 1 3 .8 ( m e a n t a s a n a p p lic a tio n o f th e m u ltip lic a tio n ru le u n d e r in d e p e n d e n c e ) (a ) O n o n e ro ll, th e p ro b a b ility o f g e ttin g 3 o r m o re s p o ts is 4 / 6 ≈ 0 .6 7 ; th e p r o b a b ility o f g e ttin g 3 o r m o re s p o ts o n 4 ro lls is (0 .6 7 ) 4 ≈ 0 .2 0 , o r 2 0 % . T h e p r o b a b ilitie s c a n b e m u ltip lie d d ire c tly b e c a u s e th e d ie r o lls a re a s s u m e d a s in d e p e n d e n t. (b ) N o n e o f th e ro lls s h o w 3 o r m o re s p o ts if a ll s h o w 2 o r fe w e r . T h e re fo re , th e p r o b a b ility o f 4 d ie ro lls w ith 1 o r 2 s p o ts is (2 / 6 ) 4 ≈ 0 .0 1 2 , o r 1 .2 % . A g a in in d e p e n d e n c e o f d ie ro lls is a s s u m e d . (c ) T h is e v e n t is th e o p p o s ite o f th e e v e n t in (a ) . T h e re fo re , th e p ro b a b ility is 1 0 0 % − 2 0 % = 8 0 % . R e v ie w E x e r c is e 1 4 .2 ( m e a n t to a s a n o th e r a p p lic a tio n o f th e ru le fo r p r o b a b ility o f e v e n ts in s itu a tio n s w ith s y m m e tric o u tc o m e s ) T h e p r o b a b ility is 2 / 3 6 . If a ll p o s s ib le o u tc o m e s (in a n e x p e r im e n t w ith a fin ite n u m b e r o f p o s s ib le o u tc o m e s ) h a v e th e s a m e p ro b a b ility , e .g . o f b e c a u s e o f a n a p p a re n t p h y s ic a l s y m m e try o f th e c h a n c e m e c h a n is m , o n e c a n c a lc u la te e v e n t p ro b a b ilitie s b y c o u n tin g a ll o u tc o m e s th a t r e s u lt in th e o c c u rre n c e o f th e e v e n t a n d d iv id e th is n u m b e r b y th e n u m b e r o f a ll p o s s ib le o u tc o m e s , i.e . p ( A ) = N A / N . F r o m fig u re 1 in c h a p te r 1 4 o n e c a n e a s ily s e e th a t 2 o u tc o m e s re s u lt in th e o c c u r r e n c e o f th e e v e n t a n d th a t 3 6 o u tc o m e s a re p o s s ib le . R e v ie w E x e r c is e 1 4 .6 ( m e a n t to c la rify th e d is tin c tio n b e tw e e n 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 e v e n ts ) If y o u w a n t to fin d th e c h a n c e th a t a t le a s t o n e o f th e tw o e v e n ts w ill h a p p e n , c h e c k to s e e if th e y a re m u tu a lly e x c lu s iv e ; if s o , y o u c a n a d d th e chances. If y o u w a n t to fin d th e c h a n c e th a t b o th e v e n ts w ill h a p p e n , c h e c k to s e e if th e y a re in d e p e n d e n t; if s o , y o u c a n m u ltip ly th e c h a n c e s . R e v ie w E x e r c is e 1 5 .1 ( m e a n t to illu s tr a te a n a p p lic a tio n o f th e b in o m ia l d is trib u tio n ) T h e p r o b a b ility is 4 0 % . T o c a lc u la te th e p ro b a b ility o f th e e v e n t o n e h a s to c o u n t a ll p o s s ib le o u tc o m e s i n w h ic h th e e v e n t c a n o c c u r ( e .g . (A c e ,N ,N ,N ,N ,N ), ( N , A c e , N , N , N , N ) , (N , N , A c e , N , N , N ). A s th e d ie r o lls a r e in d e p e n d e n t th e p ro b a b ility o f e a c h o u tc o m e e q u a ls p (a c e ) p (n o a c e )p (n o a c e )p (n o a c e ) p ( n o a c e ) p ( n o a c e ) = p ( a c e )p (n o a c e ) 5 . A s th e o u tc o m e s a re m u tu a lly e x c lu s iv e o n e c a n a d d th e p ro b a b ilitie s a n d a r riv e a t p ( e x a c tly 1 a c e ) = 6 x 1 / 6 x ( 5 / 6 ) 5 . K n o w le d g e o f th e b in o m ia l d is trib u tio n c o n s id e r a b ly s im p lifie s th e c a lc u la tio n o f p ro b a b ilitie s th a t in v o lv e a n u m b e r o f o c c u rre n c e s a n d n o n ‐ o c c u r re n c e s o f in d e p e n d e n t s p e c i fic e v e n ts . H e re th e fo rm u la fo r th e b in o m ia l d is tr ib u tio n r e d u c e s to : 5 n k 6! 1 5 p (1 − p) n − k = ≈ 40% 1!5! 6 6 k € R e v ie w E x e r c is e 1 5 .2 ( m e a n t to illu s tra te th e tric k o f u s in g th e p ro b a b ility fo r th e o p p o s ite e v e n t if c a lc u la tio n o f th e p ro b a b ility fo r th e e v e n t o f in te re s t is d iffic u lt) T h e c o r re c t a n s w e r is ( iii). T h e p ro b a b ility o f n o t g e ttin g a s ix in o n e ro ll is 1 ‐ ( 1 / 6 ) = 5 / 6 . A s th e p ro b a b ilitie s d ie r o lls a re in d e p e n d e n t o n e c a n fin d th e p r o b a b ility o f th e e v e n t o f g e ttin g n o s ix in 1 0 ro lls c a n b e c a lc u la te d b y s im p le m u ltip lic a tio n . A lte rn a tiv e ly th e b in o m ia l fo rm u la c a n b e u s e d . ...
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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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