hw_answers4 - E c o n 3 2 1 A n s w e rs H o m e w o rk 0 4...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: E c o n 3 2 1 A n s w e rs H o m e w o rk 0 4 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 .1 ( m e a n t to d e v e lo p y o u r s k ill to in te rp re t re g io n s in a a s c a tte r d ia g ra m ) A & (i); a r e a b e lo w lin e th a t c o n n e c ts p o in ts (2 0 0 ,8 0 0 ) a n d (8 0 0 ,2 0 0 ) ; e q u a tio n o f lin e : V e rb a l = 1 0 0 0 ‐ M a th B & (iii); a re a s lig h tly a b o v e / b e lo w lin e th a t c o n n e c ts p o in ts (2 0 0 , 2 0 0 ) a n d (8 0 0 , 8 0 0 ) ; e q u a tio n o f lin e : V e rb a l = M a th C & (ii); a r e a s lig h tly a b o v e / b e lo w lin e th a t c o n n e c ts p o in ts (2 0 0 , 8 0 0 ) a n d (8 0 0 , 2 0 0 ) , e q u a tio n o f lin e : V e rb a l = 1 0 0 0 ‐ M a th R e v ie w E x e r c is e 1 0 .2 ( m e a n t to s h o w th e c o n n e c tio n b e tw e e n g ro u p e s tim a te s a n d in d iv id u a l p r e d ic tio n s ) (a ) 1 1 2 ; th e id e a o f re g re s s io n s a y s th a t in d iv id u a ls w h o s c o re d 1 1 5 (= a v e ra g e 1 8 + 1 S D 1 8 ) w ill o n h a v e a n a v e ra g e s c o re a t a g e 3 5 th a t e q u a ls a v e ra g e 3 5 + r x 1 S D 3 5 = 1 0 0 + 0 .8 0 x 1 5 = 1 1 2 ; th is v a lu e is n o t e x a c t b e c a u s e th e r e g re s s io n lin e d o e s o n ly a p p ro x im a te th e a v e r a g e f o r d iffe re n t c la s s in te rv a ls o f th e in d e p e n d e n t v a ria b le (b ) 1 1 2 ; th e b e s t p re d ic to r ( in te rm s o f m in im a l R M S ) fo r a n in d iv id u a l th a t is c h o s e n r a n d o m ly fro m a g ro u p w ith k n o w n c h a r a c te r is tic s e q u a ls th e a v e ra g e o f th a t g ro u p ; th e b e s t p re d ic to r fo r a n in d iv id u a l th a t is c h o s e n ra n d o m ly fr o m th e g ro u p th a t c o m p ris e s in d iv id u a ls w h o s c o re d 1 1 5 a t th e a g e o f 1 8 w o u ld b e th e a v e ra g e s c o r e o f th e s e in d iv id u a ls a t a g e 3 5 ; th e e x a c t a v e r a g e s c o re is n o t k n o w n , b u t th e re g re s s io n lin e a p p ro x im a te s th is a v e ra g e c l o s e ly ; th e re fo r e , th e s c o r e p re d ic te d b y th e re g re s s io n lin e is th e b e s t p o s s ib le e s tim a te ( s a m e c a lc u la tio n a s in ( a )) R e v ie w E x e r c is e 1 0 .5 ( m e a n t to illu s tra te p itfa lls in th e in te rp re ta tio n o f c o rr e la tio n a n d re g re s s io n ) (a ) g e n e ra lly fa ls e ; th e s ta t e m e n t w o u ld b e tr u e o n ly if th e p o in t o f a v e ra g e s is ( A v e ( w e ig h t a t h l e t e ), 0 .6 x A v e (w e ig h t a t h l e t e )) a n d th e re g re s s io n lin e h a s a s lo p e o f 1 ( th is im p lie s th a t r x S D (w e ig h t l i f t )/ S D (w e ig h t a t h l e t e ) = 1 ; th e r e fo re , S D ( w e ig h t l i f t ) = 1 / 0 .6 x S D (w e ig h t a t h l e t e ) = 5 / 3 x S D (w e ig h t a t h l e t e ) ) (b ) fa ls e ; th e s ta te m e n t w o u ld b e tru e o n ly if S D (w e ig h t a t h l e t e ) = S D (w e ig h t l i f t ) b e c a u s e th e n : c h a n g e in w e ig h t l i f t = r x c h a n g e in w e ig h t a t h l e t e x S D (w e ig h t l i f t ) / S D ( w e ig h t a t h l e t e ) = = r x c h a n g e in w e ig h t a t h l e t e ( c ) tr u e ; w e ig h t a t h l e t e a b o v e a v e ra g e a n d a p o s itiv e c o rre la tio n c o e ffic ie n t (h e re 0 .6 ) a re a lw a y s a s s o c ia te d w ith h ig h e r w e ig h t l i f t (a s lo n g a s th e in d iv id u a ls b e lo n g to th e g ro u p fo r w h ic h th e c o r re la tio n c o e ffic ie n t w a s c a lc u la te d ) (d ) tr u e ; r( x ,y ) = r( y ,x ) a n d , th e re fo re , th e in s ig h t fro m (c ) c a n b e a p p lie d a n a lo g o u s ly to th e r e la tio n s h ip b e t w e e n w e ig h t l i f t a n d w e ig h t a t h l e t e (e ) fa ls e ; th e s ta te m e n t im p lie s c a u s a lity , b u t c a u s a lity c a n n e v e r b e d e riv e d fr o m a s s o c ia tio n ; e v e n if w e ig h t a t h l e t e a n d w e ig h t l i f t w o u ld b e c a u s a lly c o n n e c te d a c a u s a l e ffe c t o f 6 0 % w o u ld d e p e n d o n th e re la tio n s h ip b e tw e e n S D (w e ig h t l i f t ) a n d S D (w e ig h t a t h l e t e ) (s e e p a rt (a )) R e v ie w E x e r c is e 1 0 .6 ( m e a n t to re m e m b e r th e e x is te n c e o f 2 re g r e s s io n lin e s p e r s c a tte r d ia g ra m a n d b e a b le to id e n tify t h e m ) d a s h e d lin e : re g r e s s io n lin e fo r y o n x ; n e c e s s a r ily h a s a s lo p e th a t is le s s th a n th e S D lin e d o tte d lin e : S D lin e ; lie s a lw a y s b e tw e e n th e 2 r e g re s s io n lin e s s o lid lin e : re g re s s io n lin e fo r x o n y ; n e c e s s a rily h a s a s lo p e th a t is b ig g e r th a n th e S D lin e If o n e re fle c ts th e p o in ts o f th e re g re s s io n lin e fo r y o n x o n th e S D lin e o n e o b ta in s th e re g re s s io n lin e fo r x o n y . R e v ie w E x e r c is e 1 0 .7 ( 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) T h e w h o le s c e n a rio ( a te s t ‐ re te s t s itu a tio n ; in d iv id u a ls h a v in g b lo o d p r e s s u re v a lu e s fa r a b o v e th e a v e ra g e d u rin g th e firs t te s t te n d to h a v e lo w e r b lo o d p re s s u re v a lu e s d u rin g th e s e c o n d te s t; in d iv id u a ls h a v in g b lo o d p r e s s u re v a lu e s fa r b e lo w th e a v e ra g e d u r in g th e firs t te s t te n d to h a v e h ig h e r b lo o d p re s s u r e v a lu e s d u rin g th e s e c o n d te s t) w o u ld b e in lin e w ith th e re g re s s io n e ffe c t. T h e re g re s s io n e ffe c t is a lw a y s p r e s e n t if c o rr e la tio n b e tw e e n 2 d a ta s e rie s is n o t p e rfe c t a n d m ig h t b e m is ta k e n fo r a re a l e ffe c t in a te s t ‐ re te s t s itu a tio n . T o fin d o u t if a re a l e ffe c t b e y o n d th e re g re s s io n e ffe c t is p r e s e n t o n e w o u ld h a v e to c a lc u la te s u m m a r y s ta tis tic s lik e a v e ra g e s a n d s ta n d a rd d e v ia tio n s o f th e 2 d iffe re n t d a ta s e rie s (s e e re v ie w e x e rc is e 1 0 .8 ). If, fo r e x a m p le , th e a v e r a g e o f b lo o d p re s s u r e v a lu e s is h ig h e r in th e s e c o n d te s t th a n in th e firs t te s t o n e h a s e v id e n c e fo r th e h y p o th e s is th a t p a tie n ts b e c o m e m o r e n e rv o u s d u rin g th e s e c o n d te s t. R e v ie w E x e r c is e 1 0 . 9 ( m e a n t to u n d e rs ta n d th e re g re s s io n c o n n e c tio n b e tw e e n s ta n d a r d u n its a n d p e rc e n tile s ) T h e s c a tte r d ia g r a m s a r e s a id to b e fo o tb a ll s h a p e d . T h is m e a n s th a t th e n o r m a l a p p ro x im a tio n a p p lie s . T h e re fo re , s ta n d a rd u n its c a n b e e a s ily c o n v e r te d in to p e rc e n tile s th ro u g h th e k n o w le d g e o f th e a r e a b e tw e e n th e in te rv a ls [ ‐ z ,z ]. (a ) 2 1 % ; 5 % o n th e m id te rm m e a n s th a t th e s tu d e n t w a s 1 .6 5 S D ’s (z = ‐ 1 .6 5 ) b e lo w m id te rm a v e ra g e : 1 .6 5 S D ’s b e lo w m id te rm a v e r a g e im p lie s b y r e g r e s s io n th a t h e is p re d ic te d t o b e r x 1 .6 5 S D ’s ≈ 0 .8 S D ’s b e lo w fin a l a v e ra g e . 0 .8 S D ’s (z = ‐ 0 .8 ) tr a n s la te s in to th e 2 1 % p e rc e n tile : (b ) 6 5 % ; a n a lo g o u s to (a ) ju s t th a t s tu d e n t is 0 .8 5 S D ’s a b o v e m id te r m a v e ra g e (c ) 5 0 % ; m id te rm a v e ra g e le a d s to th e p re d ic tio n o f fin a l a v e r a g e ( = 5 0 % p e rc e n tile ) (d ) 5 0 % ; a s n o th in g is k n o w n th e fin a l a v e ra g e is th e b e s t p r e d ic to r (= 5 0 % p e rc e n tile ) R e v ie w E x e r c is e 1 3 .1 ( m e a n t to s h o w a b a s ic u n d e rs ta n d in g o f p ro b a b ility ) (a ) fa ls e ; p ro b a b ilitie s h a v e to b e b e tw e e n 0 % a n d 1 0 0 % ; a p r o b a b ility o f 1 0 0 0 % d o e s h a v e n o m e a n in g in p ro b a b ility th e o r y (b ) tr u e ; if th e s e tu p c a n b e re p e a te d m a n y tim e s u n d e r id e n tic a l c o n d itio n s o n e w o u ld e x p e c t th e s o m e th in g to h a p p e n 9 tim e s o u t o f 1 0 a n d its o p p o s ite to h a p p e n 1 tim e s o u t o f 1 0 R e v ie w E x e r c is e 1 3 .2 ( m e a n t to s h o w a p o s s ib le p itfa ll w h e n c o m p a r in g p r o b a b ilitie s o f e v e n t s ) ( i) is b e tte r; in o p tio n ( ii) th e s a m e e v e n t lik e in (i) h a s to h a p p e n , p lu s a n a d d itio n a l s p e c ific e v e n t; th is re n d e rs th e p r o b a b ility o f ( ii) lo w e r th a n th e p r o b a b ility o f ( i) ; if o n e w o u ld re p e a t th e s e tu p u n d e r id e n tic a l c o n d itio n s o n e w o u ld s e e a k in g a s th e firs t c a rd m o re o fte n th a n th e c o m b in a tio n o f a k in g a s th e firs t c a rd a n d a q u e e n a s th e s e c o n d c a r d R e v ie w E x e r c is e 1 4 .1 ( m e a n t to illu s tra te th e c a lc u la tio n o f p ro b a b ili tie s o f e v e n ts th a t c o n s is t o f e q u a lly lik e ly o u tc o m e s ) (a ) 1 / 3 6 ; o n ly 1 o u tc o m e o u t o f 3 6 e q u a lly lik e ly o u tc o m e s re s u lts in th e e v e n t th a t b o th d ic e s h o w 3 s p o ts (b ) 6 / 3 6 ; 6 o u tc o m e s o u t o f 3 6 e q u a lly lik e ly o u tc o m e s re s u lt in th e e v e n t th a t b o th d ic e s h o w th e s a m e n u m b e r o f s p o ts R e v ie w E x e r c is e 1 4 .3 ( m e a n t to s h o w p o s s ib le w h e n c a lc u la tin g p r o b a b ilitie s o f e v e n ts th a t c o n s is t o f e q u a lly lik e ly o u tc o m e s ) (a ) fa ls e ; p ro b a b ilitie s o f e v e n ts th a t a re n o t m u tu a lly e x c lu s iv e c a n n o t b e a d d e d u p ; th e rig h t e v e n t p r o b a b ility is : 1 ‐ (5 / 6 )^ 3 (s e e s e c tio n 1 4 .4 ) (b ) fa ls e ; a n a lo g o u s to ( a ) ; th e rig h t e v e n t p ro b a b ility is : 1 – (1 / 2 )^ 2 ...
View Full Document

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.

Ask a homework question - tutors are online