exam1_answers - E c o n 3 2 1 A n s w e rs E x a m 1 M a...

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Unformatted text preview: E c o n 3 2 1 A n s w e rs E x a m 1 M a rtin S c h m id t S u m m e r 2 0 1 0 1 . I n c o m in g s tu d e n ts a t a la w m e d ic a l s c h o o l h a v e a n a v e ra g e L S A T (L a w S c h o o l A p titu d e T e s t) s c o re o f 1 5 8 a n d a n S D o f 1 2 . T o m o rr o w , o n e o f th e s e s tu d e n ts w ill b e p ic k e d a t ra n d o m . Y o u h a v e to g u e s s th e s c o r e n o w ; th e g u e s s w ill b e c o m p a re d w ith th e a c tu a l s c o r e to s e e h o w fa r o ff it is . E a c h p o in t o ff w ill c o s t a d o lla r. (F o r e x a m p le , if th e g u e s s is 1 6 5 a n d th e s c o re is re a lly 1 6 1 y o u w ill h a v e to p a y $ 4 ) . (a ) W h a t is th e b e s t g u e s s a n d w h y is th is th e b e s t g u e s s ? (b ) T h e r e is a b o u t 1 c h a n c e in 2 to lo s e m o r e th a n a c e rta in a m o u n t o f d o lla rs ? W h a t is th is a m o u n t o f d o lla rs a n d h o w d o y o u a rr iv e a t th is a m o u n t? (c ) N o w a w h o le s e rie s o f s tu d e n ts is c h o s e n . T h e r .m .s . s iz e o f y o u r lo s s e s s h o u ld b e a ro u n d w h a t a m o u n t? P le a s e e x p la in h o w y o u a r riv e a t th is a m o u n t. (2 p o in t s ) Answer (a ) th e a v e r a g e o f 1 5 8 w o u ld b e th e b e st g u e ss in th is p ro b le m ; th e a v e ra g e is th e n u m b e r th a t m in im iz e s th e R M S o f d e v ia tio n s g iv e n a g r o u p o f o b se r v a tio n s a n d is in th is s e n se th e b e st g u e s s; b e c a u se d e v ia tio n s a re n o t sq u a r e d b e fo r e th e y g e t tra n s la te d in to lo sse s, b u t a r e ju st a d d e d u p in a b so lu te te rm s th e m e d ia n w o u ld a c tu a lly h a v e b e e n a b e tte r g u e ss , b u t a s it is n o t k n o w n o n e sh o u ld fa ll b a c k to th e a v e ra g e ; m o r e o v e r, te st d istrib u tio n s lik e th e L S A T s c o re a r e a p p r o x im a te ly n o r m a l ly d istrib u te d a n d so th e m e d ia n a n d a v e ra g e a re v e r y c lo s e (b ) t h e r e is a 5 0 % c h a n c e to lo se m o re th a n 1 6 / 3 d o lla rs; g iv e n th e sim ila rity o f th e L S A T te s t s c o re d is trib u tio n a n d th e n o r m a l d istr ib u tio n o n e c a n u se th e n o r m a l a p p ro x im a tio n to fin d th e p e r c e n ta g e o f o b se rv a tio n s th a t fa ll in sid e th e in te r v a l [a v e r a g e – z S D , a v e ra g e + z S D ]. If th e p e rc e n ta g e e q u a ls 5 0 % th e z v a lu e is a b o u t 0 .6 7 . C o n se q u e n tly , 5 0 % (1 0 0 % ­ 5 0 % ) o f a ll te st sc o re s w ill fa ll o u tsid e th e in te rv a l [ 1 5 8 – 1 6 / 3 , 1 5 8 + 1 6 / 3 ]. T h is m e a n s th a t th e r e is a 5 0 % c h a n c e to lo s e m o re th a n 1 6 / 3 d o lla rs. (c ) th e R M S o f lo s s e s s h o u ld a p p r o x im a te ly e q u a l th e sta n d a rd d e v ia tio n o f 8 (if a w h o le s e rie s o f in d iv id u a ls is p ic k e d ra n d o m ly o n e w a n ts to k n o w th e a v e ra g e d e v ia tio n fro m th e m e a n o f 1 5 8 , th e R M S o f d e v ia tio n fro m th e m e a n g iv e s th e ty p ic a l d e v ia tio n fr o m th e m e a n : R M S (X ­ A v e (X )), b u t th is is n o th in g b u t th e d e fin itio n o f th e s ta n d a r d d e v ia tio n ) 2 . P le a s e d e s c rib e th e d o m a in o f d e s c rip tiv e s ta tis tic s a n d p re s e n t a t le a s t 4 to o ls th a t d e s c rip tiv e s ta tis tic s u s e s to e x tra c t in fo rm a tio n o u t o f la rg e d a ta s e r ie s ? W h y a re th e s e to o ls n e c e s s a r y a t a ll? W h a t a r e th e lim ita tio n s o f d e s c rip tiv e s ta tis tic s ? (2 p o in t s ) Answer D e sc rip tiv e s ta tis tic s tr ie s to e x tra c t a n d u n d e rsta n d in fo r m a tio n fr o m a d a ta se t. It u se s to o ls lik e th e h isto g ra m o r sc a tte r d ia g r a m to p re se n t d a ta v isu a lly a n d e a s e th e r e c o g n itio n o f p a tte r n s in d a ta se ts. M o r e o v e r, it re d u c e s th e d a ta s e t to ta b le s o r su m m a r y sta tistic s lik e a v e r a g e (m e a su rin g th e c e n t e r o f u n iv a r ia te d a ta se r ie s), sta n d a rd d e v ia tio n (m e a s u rin g th e sp re a d a ro u n d th e c e n te r o f u n iv a r ia te d a ta se r ie s ) a n d c o r re la tio n c o e ffic ie n t ((m e a s u r in g th e lin e a r a sso c ia tio n o f u n iv a r ia te d a ta se r ie s ). A ll th e se to o ls a re n e e d e d to r e n d e r a la r g e n u m b e r o f o b se r v a tio n s in te llig ib le a n d a llo w c o m p a ris o n a c r o s s g ro u p s o f in d iv id u a l o b se r v a tio n s. A la r g e lis t o f n u m b e r s is in v ir tu a lly in c o m p re h e n sib le to a ll h u m a n b ra in s a n d th a t is w h y h u m a n s h a v e to re ly o n v isu a liz a tio n a n d d a ta r e d u c tio n te c h n iq u e s . D e sc rip tiv e sta ti s tic s c a n p ro v id e in fo rm a tio n e x c lu siv e ly a b o u t th e in d iv id u a l o b s e rv a tio n s in th e a v a ila b le d a ta se t. W ith d e sc rip tiv e sta tis tic s it is im p o s s ib le to g e n e ra liz e fr o m a n a v a ila b le d a ta s e t a n d th e c o r re sp o n d in g in d iv id u a ls to o th e r g r o u p s o f in d iv id u a ls, w h ic h a re n o t c o n ta in e d in th e d a ta se t. T o m a k e g e n e ra l c o n c lu sio n s a b o u t o th e r p o p u la tio n s o r d a ta g e n e ra tin g p r o c e s s e s o n e h a s to u se in fe r e n tia l sta tistic s. 3 . P le a s e d e s c r ib e a g e n e ra l s ta tis tic a l a p p ro a c h to in fe r c a u s a l e ffe c ts . P le a s e e x p la in w h y th is a p p ro a c h m a k e s s e n s e a n d w h y it h a s s h o r tc o m in g s . P le a s e u s e a n e x a m p le to ill u s tra te w h a t y o u a re s a y in g . (2 p o in t s ) Answer T o in fe r th e e ffe c t o f in te r v e n tio n s o n g ro u p s o f in d iv id u a ls sta tistic ia n s c a n c o m p a r e g r o u p s th a t h a v e u n d e rg o n e a n in te r v e n tio n (p o ss ib ly d iffe re n t k in d s o f in te r v e n tio n s) – tr e a tm e n t g ro u p s ­ w ith g r o u p s th a t h a v e n o t – c o n tro l g r o u p s . T o m a k e s u r e th a t d iffe re n c e s b e tw e e n tr e a tm e n t a n d c o n tr o l g ro u p s a re c a u s e d b y th e in te rv e n tio n a n d n o t b y d iffe r e n c e s in g r o u p c h a ra c te r is tic s th e s ta tis tic ia n h a s to m a k e su re th a t g r o u p s a re v e ry sim ila r in a ll c h a r a c te ris tic s th a t c a n in flu e n c e th e r e sp o n se o f in d iv id u a ls in th e tr e a tm e n t a n d c o n tr o l g r o u p s. T h e m o st a c c e p te d w a y to e x c lu d e th e p o ssib ility th a t g r o u p s d iffe r in c h a ra c te ristic s th a t c a n in flu e n c e th e re sp o n s e o f in d iv id u a ls in th e tre a tm e n t a n d c o n tr o l g ro u p s is to a ss ig n in d iv id u a ls to g r o u p s th r o u g h a c h a n c e m e c h a n ism . T h is w a y o f a s sig n in g in d iv id u a ls to tre a tm e n t a n d c o n tro l g r o u p s r e n d e r s s im ila r c h a r a c te ristic s a m o n g g r o u p s v e ry lik e ly e v e n if it c a n n o t e x c lu d e th e p o ssib ility th a t th e c h a n c e m e c h a n is m p ro d u c e s b ia s e d c o m p a r iso n g ro u p s. T h is a p p ro a c h is u su a lly te rm e d ra n d o m iz a tio n a p p ro a c h o r ra n d o m iz e d c o n tro lle d e x p e rim e n t. If it is fo llo w e d o n e is w e ll ju stifie d to a sc rib e d iffe r e n c e s in re sp o n s e s o f g ro u p s to th e in te rv e n tio n . A m a jo r s h o r tc o m in g o f th is a p p ro a c h is th a t it is o fte n n o t a p p lic a b le in so c ia l s e ttin g s b e c a u se o f e th ic a l c o n str a in ts (e .g . ra n d o m iz e d a ssig n m e n t to d e c a d e lo n g s m o k in g a n d n o n ­ sm o k in g g r o u p s is u n a c c e p ta b le ) o r p r a c tic a l c o n stra in ts (re c ru itm e n t a n d m o n ito rin g o f a la r g e n u m b e r o f in d iv id u a ls is u su a lly v e r y c o s tly ). M o re o v e r, th e e sta b lis h m e n t o f a c a u s a l e ffe c t o f th e tr e a tm e n t d o e s n o t r e v e a l th e c a u sa l m e c h a n ism . T h e la st p r o b le m o c c u rs e v e n fo r v e ry s tra ig h tfo r w a r d in te r v e n tio n s lik e te stin g th e e ffic a c y o f a d ru g (o fte n e ffic a c y is s h o w n , b u t it re m a in s u n c le a r w h a t th e p h a rm a c o lo g ic a l m e c h a n ism o f th e d r u g r e a lly i s). A n e x a m p le is a c o n tr o lle d e x p e r im e n t th a t te sts th e e ffe c tiv e n e ss o f fe rtiliz e r. A s a p lo t o f la n d w ill n e a rly n e v e r b e h o m o g e n o u s (d u e to d iffe re n c e s i n s o il, a n im a l a tta c k s, w in d p r o te c tio n , e tc .) it c o u ld b e m isle a d in g to c u t a p lo t o f la n d in 2 p a rts a n d a p p ly fe r tiliz e r to o n e p a rt a n d n o t to th e o th e r . A c o n tr o lle d e x p e rim e n t c o u ld c u t th e p lo t in to m a n y m o r e p a rts (le ts s a y 1 0 0 ) a n d th e n a ssig n e a c h p a rt to tr e a tm e n t w ith fe r tiliz e r o r c o n tr o l w ith o u t fe rtiliz e r b y a c o in to ss . O n a v e ra g e e ffe c ts fro m h e te r o g e n e o u s la n d p lo ts s h o u ld c a n c e l o u t a n d th e d iffe r e n c e b e tw e e n tr e a tm e n t a n d c o n tr o l g r o u p s w o u ld g iv e a re lia b le m e a s u re o f th e c a u s a l e ffe c t o f fe r tiliz e r. Im p r o v e m e n ts o n th e se tu p a re p o s sib le a n d c o m m o n in p ra c tic e . 4 . B re a s t c a n c e r is o n e o f th e m o s t c o m m o n m a lig n a n c ie s a m o n g w o m e n in th e U .S . If it is d e te c te d e a rly e n o u g h b e fo re th e c a n c e r s p re a d s c h a n c e s o f s u c c e s s fu l tre a tm e n t a r e m u c h b e tte r. D o s c re e n in g p r o g ra m s s p e e d u p d e te c tio n b y e n o u g h to m a tte r? T h e firs t la rg e ‐ s c a le tria l w a s ru n b y th e H e a lth In s u ra n c e P la n o f G re a te r N e w Y o rk , s ta rtin g in 1 9 6 3 . T h e s u b je c ts (a ll m e m b e rs o f th e p la n ) w e r e 6 2 ,0 0 0 w o m e n a g e 4 0 to 6 4 . T h e s e w o m e n w e re d iv id e d a t ra n d o m in to tw o e q u a l g ro u p s . In th e tre a tm e n t g r o u p , w o m e n w e r e e n c o u ra g e d to c o m e in fo r a n n u a l s c re e n in g , in c lu d in g e x a m in a tio n b y a d o c to r a n d X ‐ ra y s . A b o u t 2 0 ,2 0 0 w o m e n in th e tre a tm e n t g ro u p d id c o m e in fo r th e s c re e n in g ; b u t 1 0 ,8 0 0 re fu s e d . T h e c o n tro l g r o u p w a s o ffe re d u s u a l h e a lth c a r e . A ll th e w o m e n w e re fo llo w e d fo r m a n y y e a rs . R e s u lts fo r th e fir s t 5 y e a rs a r e s h o w n in th e ta b le b e lo w (“H IP ” is th e u s u a l a b b re v ia tio n fo r th e H e a lth In s u r a n c e P la n ) : T r e a tm e n t group E x a m in e d R e fu s e d T o ta l C o n tr o l group 20200 10800 31000 31000 C a u s e o f D e a th B r e a s t c a n c e r A ll o th e r Number R a te Number R a te 23 16 39 63 1 .1 1 .5 1 .3 2 .0 428 409 836 879 21 38 27 28 E p id e m io lo g is ts w h o w o rk e d o n th e s tu d y fo u n d t h a t ( i) s c re e n in g h a d little im p a c t o n d is e a s e s o th e r th a n b re a s t c a n c e r; (ii) p o o r e r w o m e n w e re le s s lik e ly to a c c e p t s c re e n in g th a n ric h e r o n e s ; a n d (iii) m o s t d is e a s e s fa ll m o re h e a v ily o n th e p o o r th a n th e ric h . P le a s e a n s w e r th e fo llo w in g q u e s tio n s : (a ) D o e s s c re e n in g s a v e liv e s ? W h ic h n u m b e rs in th e ta b le a re e v id e n c e fo r th is p o in t? (b ) W h y is th e d e a th r a te fro m a ll o th e r c a u s e s in t h e w h o le tre a tm e n t g ro u p ( "e x a m in e d " a n d " re fu s e d " c o m b in e d ) a b o u t th e s a m e a s th e r a te in th e c o n tro l g ro u p ? (c ) B re a s t c a n c e r ( lik e p o lio , b u t u n lik e m o s t o th e r d is e a s e s ) a ffe c ts th e ric h m o re th a n th e p o o r. W h ic h n u m b e rs in th e ta b le c o n fir m th is a s s o c ia tio n b e tw e e n b re a s t c a n c e r a n d in c o m e ? (d ) T h e d e a th ra te ( fro m a ll c a u s e s ) a m o n g w o m e n w h o a c c e p te d s c re e n in g is a b o u t h a lf th e d e a th ra te a m o n g w o m e n w h o re fu s e d . D id s c re e n in g c u t th e d e a th ra te in h a lf? If n o t, w h a t e x p la in s th e d iffe r e n c e in d e a th ra te s ? (e ) T o s h o w th a t s c re e n in g re d u c e s th e ris k fro m b r e a s t c a n c e r , s o m e o n e w a n ts to c o m p a r e th e ra te s 1 .1 a n d 1 .5 . Is th is a g o o d c o m p a r is o n ? Is it b ia s e d a g a in s t s c r e e n in g ? F o r s c re e n in g ? (f) S o m e o n e c la im s th a t e n c o u ra g in g w o m e n to c o m e in fo r b re a s t c a n c e r s c re e n in g in c re a s e s th e ir h e a lth c o n s c io u s n e s s , s o th e s e w o m e n ta k e b e tte r c a re o f th e m s e lv e s a n d liv e lo n g e r fo r th a t re a s o n . Is th e ta b le c o n s is te n t o r in c o n s is te n t w ith th e c la im ? (g ) In th e firs t y e a r o f th e H IP tria l, 6 7 b re a s t c a n c e rs w e re d e te c te d in th e " e x a m in e d " g ro u p , 1 2 in th e " re fu s e d " g r o u p , a n d 5 8 in th e c o n tro l g ro u p . T ru e o r fa ls e , a n d e x p la in : s c re e n in g c a u s e s b re a s t c a n c e r . (5 p o in ts ) Answer (a ) Y e s . If o n e c o m p a r e s th e ra te s fo r th e ra n d o m iz e d tr e a tm e n t a n d c o n tro l g ro u p s o n e fin d s th a t th e to ta l b r e a st c a n c e r ra te in th e tr e a tm e n t g r o u p is 1 .3 p e r 1 0 0 0 w h e re a s it is 2 .0 p e r 1 0 0 0 in th e c o n tro l g ro u p . A s th e stu d y is ra n d o m iz e d o n e c a n su b s c rib e th e ra te d iffe re n c e o f 0 .7 p e r 1 0 0 0 to th e e ffe c t o f sc re e n in g (a b stra c tin g fr o m c h a n c e v a ria b ility ). T h e e ffe c t is p ro b a b ly e v e n h ig h e r a s 1 / 3 o f th e in d iv id u a ls in th e tre a tm e n t g r o u p r e fu se d to ta k e p a rt in th e s c re e n in g . (b ) I f th e c la im th a t sc re e n in g h a d little im p a c t o n d ise a se s o th e r th a n b r e a s t c a n c e r h o ld s tr u e , th e n o n e w o u ld e x p e c t v e ry sim ila r d e a th r a te s fo r tre a tm e n t a n d c o n tr o l g ro u p . S c re e n in g sh o u ld h a v e n o e ffe c t o n th e d e a th r a te s fr o m o th e r c a u se s . T h is is e x a c tly w h a t o n e fin d s (a d e a th r a te o f 2 7 p e r 1 0 0 0 in th e tre a tm e n t g ro u p a n d a d e a th r a te o f 2 8 p e r 1 0 0 0 in th e c o n tro l g ro u p ). (c ) I f b r e a s t c a n c e r a ffe c ts th e r ic h m o r e th a n th e p o o r o n e w o u ld e x p e c t th a t a c o m p a ris o n b e tw e e n g ro u p s w ith d iffe re n t n u m b e r s o f r ic h p e o p le a n d th e s a m e le v e l o f tr e a tm e n t sh o u ld s h o w h ig h e r r a te s o f b r e a s t c a n c e r in th e g ro u p w ith r ic h e r p e o p le (th is h o ld s tr u e o n ly if th e d iffe r e n c e in g ro u p s c r e a te d b y th e d iffe r e n t n u m b e r o f r ic h p e o p le d o e s n o t r e s u lt in o th e r re le v a n t g r o u p d iffe r e n c e s a n d c o n fo u n d in g o f fa c to r s ) . T h is is e x a c tly w h a t o n e fin d s if o n e c o m p a re s th e c o n tro l g ro u p (n o tre a tm e n t) a n d th e tr e a tm e n t su b g ro u p th a t r e fu s e d tre a tm e n t. B e c a u se r ic h p e o p le a re le ss lik e ly to r e fu s e tr e a tm e n t th a n p o o r p e o p le th e re la tiv e n u m b e r o f p o o r p e o p le w ill b e h ig h e r in th e s u b g ro u p o f p e o p le w h o r e fu se d tre a tm e n t. A s th e tr e a tm e n t a n d c o n tr o l g r o u p in itia lly h a d s im ila r c o m p o sitio n o n e c a n a s s u m e th a t th e c o n tr o l g ro u p w ill h a v e a h ig h e r fra c tio n o f r ic h p e o p le th a n th e s u b g r o u p o f p e o p le w h o re fu se d tre a tm e n t. T h e b r e a s t c a n c e r d e a th ra te o f th e c o n tr o l g ro u p (2 .0 p e r 1 0 0 0 ) is 0 .5 h ig h e r th a n th e b re a s t c a n c e r in th e su b g ro u p th a t re fu se d tre a tm e n t (1 .5 p e r 1 0 0 0 ). (d ) C o m p o s itio n b ia s c a n e x p la in th e d iffe re n c e s in d e a th r a te s. B e c a u se in d iv id u a ls c a n d e c id e if th e y a c c e p t o r re fu se s c r e e n in g sp littin g o f th e tr e a tm e n t g r o u p in to 2 su b g ro u p s c a n r e su lt in c o m p o sitio n b ia s if th e r e e x is t d iffe r e n c e s b e tw e e n p e o p le w h o a c c e p t o r re fu s e sc re e n in g . If ric h p e o p le a r e m o r e lik e ly to a c c e p t sc re e n in g it fo llo w s th a t th e tr e a tm e n t g r o u p w h o re fu se d sc r e e n in g w ill c o n sist o f re la tiv e ly m o r e p o o r p e o p le . If m o s t d ise a se s fa ll m o re h e a v ily o n p o o r p e o p le o n e w o u ld e x p e c t h ig h e r d e a th ra te s fo r th e su b g r o u p th a t r e fu se d tr e a tm e n t. T h is is w h a t th e d a ta sh o w : 2 1 p e r 1 0 0 0 v s. 3 8 p e r 1 0 0 0 . (e ) T h e c o m p a r is o n is b ia se d a g a in st sc re e n in g . It is g e n e ra lly p ro b le m a tic to c o m p a re 2 g ro u p s th a t c a n d iffe r in im p o rta n t c h a ra c te r is tic s. If o n e c o m p a re s th e su b g r o u p th a t a c c e p te d tr e a tm e n t w ith th e s u b g r o u p th a t r e fu s e d tre a tm e n t th e c o m p o sitio n o f th e s e 2 g ro u p s c a n d iffe r in m a n y w a y s b e c a u se in d iv id u a ls a s s ig n e d th e m s e lv e s to th e su b g ro u p s. If c h a ra c te r istic s th a t in flu e n c e a s s ig n m e n t to s c re e n in g (e .g . in c o m e ) a lso in flu e n c e d e a th ra te s th e n th e c h a r a c te r is tic s a n d th e e ffe c t o f sc r e e n in g a r e c o n fo u n d e d . If r ic h e r p e o p le te n d to a ssig n th e m se lv e s m o re fre q u e n tly to sc re e n in g a n d if r ic h e r p e o p le a re m o r e o fte n su b je c t to b re a st c a n c e r th e n o n e w o u ld e x p e c t th e s c r e e n in g g ro u p to sh o w h ig h e r b re a st c a n c e r d e a th r a te s w ith o u t th e im p ro v e m e n ts fro m sc r e e n in g – c le a r p re se n c e o f a b ia s (f) T h e ta b le d o e s n o t su p p o rt th is s ta te m e n t. A s m e n tio n e d a b o v e s c re e n in g h a s a n e ffe c t o n d e a th r a te s fr o m b re a s t c a n c e r , b u t n o t o n d e a th ra te s fr o m o th e r c a u se s . T a k in g b e tte r c a re o f o n e se lf sh o u ld n o t o n ly d e c re a se th e b re a st c a n c e r d e a th r a te , b u t a lso th e d e a th r a te fr o m a ll o th e r c a u se s. T h is is c le a r ly n o t th e c a se . (g ) F a lse . T h e sta te m e n t m e n tio n s e x p lic itly d e te c tio n o f b re a s t c a n c e r . S c r e e n in g a llo w s e a r lie r d e te c tio n o f b re a st c a n c e r th a n o th e r w is e ( a n d e a r ly d e te c tio n a llo w s m o re e ffe c tiv e tre a tm e n t ) . D e te c tio n m e a n s u n c o v e rin g th e c o n s e q u e n c e o f a p r e v io u s e ffe c t. T h e re fo r e , p a rtic u la r ly in th e first y e a r o f th e stu d y o n e w o u ld e x p e c t h ig h e r d e te c tio n ra te s w ith in th e su b g ro u p th a t a c c e p te d sc re e n in g th a n w ith in th e c o n tro l g ro u p o r th e su b g r o u p th a t re fu se d sc r e e n in g . A p a r t fr o m th e fa c t th a t a b so lu te n u m b e rs a n d n o t r a te s a re re p o rte d , o n e s h o u ld n o t c o n fu se d e te c tio n a n d c a u sa tio n . 5 . T ru e o r fa ls e , a n d e x p la in p le a s e : (a ) If y o u s u b tra c t 2 fro m e a c h e n try o n a lis t, th a t d e c re a s e s th e a v e ra g e b y 2 . (b ) If y o u s u b tra c t 2 fro m e a c h e n try o n a lis t, th a t d e c r e a s e s th e S D b y 2. (c ) If y o u h a lf e a c h e n try o n a lis t, th a t h a lv e s th e a v e r a g e . (d ) If y o u h a lf e a c h e n try o n a lis t, th a t h a lv e s th e S D . (e ) If y o u c h a n g e th e s ig n o f e a c h e n try o n a lis t, th a t c h a n g e s th e s i g n o f th e a v e ra g e . (f) If y o u c h a n g e th e s ig n o f e a c h e n try o n a lis t, th a t c h a n g e s th e s i g n o f th e S D . (2 p o in ts ) Answer A ll a n s w e r s fo llo w fr o m sp e c ific p ro p e r tie s o f th e a v e r a g e a n d th e sta n d a rd d e v ia tio n , n a m e l y A v e ( a + b x ) = a + b A v e (x ) € € n x := 1 ∑ x i n i=1 1n 1n 1n ⇒ a + bx = ∑ a + bx i = ∑ a + b∑ x i = a + bx n i=1 n i=1 n i=1 a n d S D ( a + b x ) = |b |S D (x ) 1n SD := ∑ ( x − x ) 2 x n i=1 i 1n 1n 2 ⇒ SD ∑ (a + bx i − (a + bx )) = n ∑ (a + bx i − a + bx ))2 a + bx = n i=1 i= 1 1n 2 1n = ∑ b ( x i − x ))2 = b2 n ∑ ( x i − x ))2 = bSDx i= 1 n i=1 O th e r m o re in tu itiv e e x p la n a tio n s b a se d o n h is to g r a m s o r v e rb a l re a s o n in g a r e p o s s ib le . A ll th e se im p ly th e sa m e a n sw e rs to th e q u e stio n s : (a ) t r u e ; a v e ( ­ 2 + x ) = ­ 2 + a v e (x ) (b ) f a ls e ; S D ( ­ 2 + x ) = S D (x ) (c ) t r u e ; a v e (½ x ) = ½ a v e (x ) (d ) t r u e ; S D (½ x ) = ½ S D (x ) (e ) t r u e ; a v e ( ­ 1 x ) = ­ 1 a v e (x ) (f) f a ls e ; S D ( ­ 1 x ) = 1 S D (x ) 6 . O n th e M a th S A T , m e n h a v e a d is tin c t e d g e . In 2 0 0 5 , fo r in s ta n c e , th e m e n a v e ra g e d a b o u t 5 3 8 , a n d th e w o m e n a v e ra g e d a b o u t 5 0 4 . Y o u m a y a s s u m e th a t th e h is to g ra m s fo llo w e d th e n o rm a l c u rv e , a n d th a t b o th S D s w e re a b o u t 1 2 0 . (a ) P le a s e e s tim a te th e p e r c e n ta g e o f m e n g e ttin g o v e r 7 0 0 o n th is te s t in 2 0 0 5 , a n d s h o w h o w a n d w h y y o u a rriv e a t th e r e s u lts . (b ) P le a s e e s tim a te th e p e rc e n ta g e o f w o m e n g e ttin g o v e r 7 0 0 o n th is te s t in 2 0 0 5 , a n d s h o w h o w a n d w h y y o u a rriv e a t th e re s u lts (2 p o in ts ) Answer T h e n o rm a l a p p r o x im a tio n a llo w s o n e to c a lc u la te th e p e r c e n ta g e w ith th e h e lp o f th e ta b le fo r th e s ta n d a rd n o r m a l c u rv e . (a ) 9 % ; 7 0 0 p o in ts c o r r e s p o n d to 1 .3 5 s ta n d a rd d e v ia tio n s a b o v e a v e r a g e ; o n e is th e n in te r e s te d in th e a r e a a b o v e 1 .3 5 a n d b e lo w th e s ta n d a r d n o rm a l c u r v e th e ta b le fo r th e s ta n d a r d n o r m a l c u rv e ta b u la te s th e p e rc e n ta g e o f e n tr ie s c o n ta in e d in th e in te r v a l [ ­ 1 .3 5 , 1 .3 5 ] a s 8 2 % ; th e r e fo re , o n e c a n c a lc u la te th e p e rc e n ta g e o u tsid e th e in te r v a l a s 1 0 0 % ­ 8 2 % the defintion of the correlation coefficient is r( x, y ) := u sin g th e sy m m e try o f th e sta n d a rd n o r m a l c u rv e o n e c a n • standard units x SU , y SU imply by construction that x SU = 0, SD x = 1, y SU = 0, SD y = 1 th e n o b ta in th e v a lu e o f t h e a re a to th e st nd and, consequently, 5 rig h t o f 1 .3 by a formula linking standard deviation and 1 and 2 d iv in g b y 2 , i.e . 1 8 % / 2 b y moments : 2 2 2 2 2 2 x SU = SDx + x SU = 1, y SU = SDy + y SU = 1 = 9 % SU SU SU SU 1 n x i − x y i − y 1 n = ∑ x SU i y SU i ∑ n i=1 SDx SDy n i=1 • 0 ≤ 2 1n ∑ ( x SU i + y SU i ) holds always true for any arbitrary values of x SU i and y SU i (a sum of n i=1 squared real numbers is always positive and division by a natural number does not change this) application of the binomial formula, i.e. ( x + y) 2 = x 2 + 2 xy + y 2 , and using the definitions of averages for x, y, and the correlation coefficent then implies : (b ) n 5 % , 2 1 n a n a lo g o u s to (a ) 1 1n 2 1n 2 0 ≤ n ∑ ( x SU + y SU i ) = n ∑ x + n ∑ y +2 n ∑ x SU i y SU i = 1 + 1 + 2 r = 2 + 2r i= 1 i= 1 i= 1 i= 1 i SUi SU i manipulation of the inequality 2 + 2 r ≥ 0 then produces the desired result : r( x SU , y SU ) ≥ −1 7 . P le a s e p ro v e th a t ‐ 1 ≤ r( x ,y ) ≤ 1 (e v e r y e q u a lity o r in e q u a lity n e e d s e x p la n a tio n ) . 2 1n ∑ ( x SU i − y SU i ) implies n i=1 (3 p o in ts ) 0≤ • an analogous argument for the inequality 0 ≤ 2 Answer 1n 1n 1n 1n ∑ ( x SU i − y SU i ) = n ∑ x 2SUi + n ∑ y 2SU i −2 n ∑ x SU i y SU i = 1 + 1 − 2r = 2 n i=1 i= 1 i= 1 i= 1 manipulation of the inequality 2 ‐ 2 r ≥ 0 then produces : ⇒ r( x SU , y SU ) ≤ 1 combining the 2 inequalites for r( x SU , y SU ) then completes the proof : ⇒ −1 ≤ r( x SU , y SU ) = r( x, y ) ≤ 1 the subscripts for standard units can be dropped as the correlation coefficient is calculated in standard units and is invariant to all kinds of linear transformations a + bx, c + dy as long as b,d > 0 8 . P le a s e d ra w th e s c a tte r d ia g ra m a n d c a lc u la te a ll 5 c o m m o n s u m m a r y s ta tis tic s fo r th e d a ta s e t b e lo w . P le a s e w rite d o w n th e g e n e ra l fo rm u la s fo r th e s u m m a ry s ta tis tic s . W h a t a s p e c t o f th e d a ta s e r ie s d o th e 5 s u m m a ry s ta tis tic s try to c a p tu r e ? x 1 1 1 2 2 2 3 4 y 5 3 7 3 1 1 1 1 (5 p o in ts ) Answer The scatter diagram looks like this: 8 0 1 2 y 4 6 3 1 8 6 4 2 0 y x The 5 summary statistics (Ave(x), Ave(y), SD(x), SD(y), r(x,y)) can be calculated through application of their defining formulas: x 2 3 4 1n x = x (measure of the center of the univariate data series x) n∑ i i=1 1n y = y (measure of the center of the univariate data series y) n∑ i i=1 1n 2 SDx = n ∑ ( x i − x ) (measure of the spread around the center of the univariate data series x) i= 1 1n 2 SDy = n ∑ ( y i − y ) (measure of the spread around the center of the univariate data series y) i= 1 n r( x, y ) := 1 ∑ x i − x y i − y (measure of the linear association between the bivariate data series x, y) SD n i=1 SDx y € The corresponding values are: Ave(x) = 2 Ave(y) = 2.75 SD(x) = 1.07 SD(y) = 2.25 r(x,y) = ­0.71 9. W h a t d o e s e c o lo g ic a l c o rre la tio n m e a n ? Is e c o lo g ic a l c o rr e la tio n p re s e n t in th e d a ta o f q u e s tio n 8 a n d , if s o , w h a t is th e lik e ly e ffe c t? D o th e d a ta o f q u e s tio n 8 in d ic a te a c a u s a l e ffe c t o f x o n y ? (2 p o in ts ) Answer E c o lo g ic a l c o rr e la tio n m e a su r e s th e lin e a r a ss o c ia tio n b e tw e e n a g g re g a te d o b s e r v a tio n s . A g g r e g a te d o b se r v a tio n s a re c o n stru c te d th ro u g h a n a v e r a g in g p r o c e s s th a t c o m b in e s v a r io u s in d iv id u a l o b se r v a tio n s in to o n e a g g r e g a te o b se r v a tio n (e .g . c o m b in in g o b se r v a tio n o f in d iv id u a l in c o m e a n d in d iv id u a l e d u c a tio n in to sta te a v e r a g e s o v e r in c o m e a n d e d u c a tio n ). It is im p o ssib le to sa y if e c o lo g ic a l c o rr e la tio n is p re s e n t in th e d a ta o f q u e stio n 8 w ith o u t k n o w in g if th e d a ta r e p r e se n t in d iv id u a l o b s e r v a tio n s o r a g g re g a te d o b se rv a tio n s. T h e n e g a tiv e c o rr e la tio n c o e ffic ie n t o f ­ 0 .7 1 in d ic a te s a stro n g lin e a r a sso c ia tio n b e tw e e n th e x a n d y v a r ia b le s (it sh o u ld b e n o te d th a t th e s c a tte r d ia g r a m d o e s n o t lo o k fo o tb a ll s h a p e d a n d th a t th e a p p lic a b ility o f th e c o rr e la tio n su m m a ry sta tis tic is q u e stio n a b le ) . If th e x a n d y v a r ia b le s r e p r e se n t a g g re g a te s o v e r in d iv id u a l o b se r v a tio n s th e n th e c o r re la tio n c o e ffic ie n t w ill v e ry lik e ly o v e r e s tim a te th e lin e a r a sso c ia tio n a m o n g in d iv id u a l o b se r v a tio n s . In g e n e r a l, a lin e a r a sso c i a tio n o n its o w n a n d w ith o u t fu rth e r k n o w le d g e (e .g . o n e x p e r im e n ta l d e sig n o r su b je c t m a tte r) c a n n e v e r in d ic a te a c a u s a l e ffe c t o f th e in d e p e n d e n t v a r ia b le o n th e d e p e n d e n t v a ria b le . F o r e x a m p le , e x c h a n g in g th e 2 v a ria b le s w o u ld re su lt in th e s a m e th e c o r re la tio n c o e ffic ie n t a n d in d ic a te a re v e rs io n o f c a u s a lity . L in e a r a s s o c ia tio n is o n ly a n e c e s sa r y , b u t n o t su ffic ie n t fo r c a u s a lity (a s s u m in g th a t o n e c a n iso la te th e c a u sa l e ffe c t). 1 0 . T h e fig u re ( a d a p te d fro m a 1 9 9 2 n e w s p a p e r a r tic le ) s h o w s th e d is trib u tio n o f A m e ric a n fa m ilie s b y in c o m e . R a n g e s in c lu d e th e le ft e n d p o in t b u t n o t th e rig h t. F o r e x a m p le , 3 .7 % o f th e fa m ilie s h a d in c o m e s in th e r a n g e $ 0 ‐ $ 4 ,9 9 9 , 5 .8 % h a d in c o m e s in th e ra n g e $ 5 ,0 0 0 ‐ $ 9 ,9 9 9 , a n d s o fo rth . T ru e o r fa ls e , a n d p le a s e e x p la in : (a ) A lth o u g h A m e ric a n fa m ilie s a re n o t s p re a d e v e n ly o v e r th e w h o le in c o m e ra n g e , th e fa m ilie s th a t e a r n b e tw e e n $ 1 0 ,0 0 0 a n d $ 3 5 ,0 0 0 a r e s p re a d fa irly e v e n ly o v e r th a t ra n g e . (b ) T h e fa m ilie s th a t e a r n b e tw e e n $ 3 5 ,0 0 0 a n d $ 7 5 ,0 0 0 a re s p r e a d fa irly e v e n ly o v e r th a t ra n g e . (c ) T h e g ra p h is a h is to g ra m . (2 p o in ts ) Answer (a ) tru e (if o n e d ra w s a d e n sity h isto g ra m th e r e b y a d ju s tin g fo r th e siz e o f c la ss in te r v a ls o n e fin d s th a t th e d e n sity b e tw e e n $ 1 0 k a n d $ 3 5 k is a b o u t th e sa m e ) (b ) fa lse (if o n e d r a w s a d e n sity h isto g r a m th e re b y a d ju stin g fo r th e siz e o f c la ss in te rv a ls o n e fin d s th a t th e d e n sity b e tw e e n $ 3 5 k a n d $ 7 5 k d iffe r s) (c ) fa ls e (a s se e n b e fo r e a d e n s ity h isto g r a m lo o k s d iffe r e n t b e c a u se it a d ju sts fo r siz e o f c la s s in te rv a l s ) ...
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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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