hw_answers9 - E c o n 3 2 1 A n s w e rs H o m e w o rk 0 9...

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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 9 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 2 0 .3 ( m e a n t to illu s tra te th e m e a n in g o f b o x m o d e ls in in fe re n tia l s ta tis tic s ) (a ) T h e re s h o u ld b e 5 0 ,0 0 0 tic k e ts in th e b o x . T h e b o x is th e p o p u la tio n — a ll 5 0 ,0 0 0 fo r m s . (b ) E a c h tic k e t s h o w s a 0 ( g ro s s in c o m e u n d e r $ 5 0 ,0 0 0 ) o r a 1 (g ro s s in c o m e o v e r $ 5 0 ,0 0 0 ) . O n e is o n ly in te re s te d in th e 2 c a te g o rie s o f in c o m e b e lo w $ 5 0 ,0 0 0 a n d in c o m e a b o v e $ 5 0 ,0 0 0 . If o n e w e r e in te re s te d in a ll d iffe r e n c e o f in c o m e th e tic k e ts w o u ld h a v e g ro s s in c o m e a s n u m b e r s . (c ) F a ls e : th e S D o f th e b o x is √ ( 0 .2 × 0 .8 ) = 0 .4 (a s th e fo rm u la fo r th e s ta n d a rd d e v ia tio n re d u c e s to √ (p × (1 ‐ p )) (d ) T r u e . In b o x m o d e ls fo r in fe re n tia l s ta tis tic s t h e n u m b e r o f d ra w s e q u a ls th e s a m p le s iz e . (e ) If o n e a s s u m e s th a t th e n o rm a l a p p ro x im a tio n o f p r o b a b ility h is to g r a m s h o ld s o n e c a n d e riv e th e p ro b a b ility th a t b e tw e e n 1 9 % a n d 2 1 % o f th e a u d ite d fo rm s h a v e in c o m e s o v e r $ 5 0 ,0 0 0 . T h e a p p ro x im a te n o r m a l c u rv e o f th e s u m w ill b e c e n te r e d a ro u n d th e e x p e c ta tio n a n d its s p re a d w ill b e m e a s u re d b y th e s ta n d a r d e rro r. € T h e in te rv a l [E ( X n ) ‐ 1 % , E ( X n ) + 1 % ] tra n s la te s in to s ta n d a rd u n its a s [0 ‐ 0 .7 5 ,0 + 0 .7 5 ] = [0 .7 5 ,0 .7 5 ]. O n e c a n re a d o ff th e p r o b a b ility th a t th e p e rc e n ta g e f a lls in to th e in te rv a l fro m th e ta b le o f th e n o r m a l c u rv e a n d fin d s th a t th e p ro b a b ility a m o u n ts to 5 5 % . X 1+.. + X n n n E ( X ) = X i = 0.2 = 20% SDX i 0.4 SE ( X n ) = = ≈ 1.3% n 900 Xn = F re e d m a n th in k s a b o u t th e p r o b le m in a d iffe r e n t w a y : T h e n u m b e r o f s a m p le fo rm s w ith g ro s s in c o m e s o v e r $ 5 0 ,0 0 0 is lik e th e s u m o f 9 0 0 d r a w s f r o m th e b o x . T h e e x p e c te d v a lu e fo r th e s u m is 1 8 0 . T h e S E fo r th e s u m is √ 9 0 0 × 0 .4 = 1 2 . T h e n u m b e r o f s a m p le fo r m s w ith g ro s s in c o m e s o v e r $ 5 0 ,0 0 0 w ill b e a ro u n d 1 8 0 , g iv e o r ta k e 1 2 o r s o . N o w 1 2 o u t o f 9 0 0 is a b o u t 1 .3 3 % . T h e p e rc e n ta g e o f fo rm s in th e s a m p le w ith g ro s s in c o m e s o v e r $ 5 0 ,0 0 0 w ill b e 2 0 .0 0 % , g iv e o r ta k e 1 .3 3 % o r s o . T h e c h a n c e is a b o u t 55%. (f) C a n ’t b e d o n e w ith th e in fo rm a tio n g iv e n . Y o u n e e d to k n o w th e p e rc e n ta g e o f fo rm s w ith g r o s s in c o m e s o v e r $ 7 5 ,0 0 0 . A s th e in c o m e d is trib u tio n fo llo w s d o e s n o t fo llo w a n o rm a l d is trib u tio n , b u t is h ig h ly s k e w e d o n e c a n n o t a p p ro x im a te th e p e rc e n ta g e w ith th e h e lp o f th e a v e r a g e a n d th e s ta n d a rd d e v ia tio n g iv e n in th e p ro b le m . R e v ie w E x e r c is e 2 0 .6 ( m e a n t to s h o w th a t a c c u ra c y d e p e n d s m a in ly o n th e a b s o lu te s a m p le s iz e a n d n o t o n th e s a m p le s iz e r e la tiv e to th e p o p u l a tio n ) O p tio n ( ii) is rig h t. A s th e s a m p le s iz e o f a s a m p le w ith 1 % o f th e C a lifo rn ia p o p u la tio n w ill b e m u c h la rg e r in a b s o lu te te r m s th a n a s a m p le w ith 1 % o f th e N e v a d a , th e a c c u ra c y o f th e C a lifo r n ia s a m p le w ill b e m u c h . T h e c h a n c e e r ro r o f th e s a m p le is c a p tu re d b y th e s ta n d a rd e rr o r o f p e rc e n ta g e s a n d th is s ta n d a r d e rro r d e c re a s e s w ith 1 / √ n (n = s a m p le s iz e ). T h e s iz e o f th e s a m p le re la tiv e to th e s iz e o f th e s ta te is b a s ic a lly ir re le v a n t if o n e is in te re s te d in th e a c c u ra c y o f p e rc e n ta g e ( s e e c h a p te r 2 0 s e c tio n 4 ). R e v ie w E x e r c is e 2 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 e x p e c ta tio n a n d s ta n d a rd e r ro r o f p e rc e n ta g e s in i.i.d . p ro c e s s e s ) (a ) is tru e . E (X n ) = ( X 1 + ..+ X n ) / n = A v e (X i ) = (6 0 ,0 0 0 x 0 + 2 0 ,0 0 0 x 1 ) / 8 0 ,0 0 0 = 0 .2 5 = 2 5 % (b ) is fa ls e . T h e e x p e c te d v a lu e is c o m p u te d fro m th e p ro b a b ility d e n s ity fu n c tio n ( i.e . th e b o x ) a n d h a s n o c h a n c e e r ro r. T h e e x p e c te d v a lu e is 2 5 % S e e (a ) . (c ) is tru e . T h e p e rc e n ta g e o f 1 ’s a m o n g th e d ra w s w ill b e o ff its e x p e c te d v a lu e , d u e to c h a n c e e r ro r. T h e s ta n d a rd e rro r m e a s u re s th e s p r e a d o f th e c h a n c e flu c tu a tio n s . T h e S E fo r X n = (X 1 + ..+ X n )/ n , i. e . fo r th e p e rc e n ta g e o f 1 ’s a m o n g th e d ra w s , is √ ( 0 .2 5 x (1 ‐ 0 .2 5 )) / √ 5 0 0 ) ≈ 0 .0 2 = 2 % . (d ) is fa ls e . C h a n c e v a ria b ility w i ll re s u lt in d iffe re n t v a lu e s fo r th e p e rc e n ta g e o f 1 ’s . S e e ( c ) . (e ) is tru e . O n e k n o w s th e p ro b a b ility d e n s ity fu n c tio n fo r X i , i.e . o n e k n o w s w h a t i s in th e b o x . T h is is fo r w a rd re a s o n in g . (f) is fa ls e . A s p ro b a b ility d e n s ity fu n c tio n fo r X i is k n o w n th e re is n o u n c e rta in ty . S e e ( e ) . R e v ie w E x e r c is e 2 0 .9 ( m e a n t to re m in d y o u o f th e e x p e c ta tio n a n d s ta n d a rd e r ro r o f s u m s in i.i .d . p ro c e s s e s ) T h e n u m b e r o f 1 ’s is th e s u m o f th e d ra w s . T h e e x p e c te d v a lu e is 2 0 0 , a n d th e S E is a b o u t 1 2 . T h is fo llo w s d ire c tly fro m F re e d m a n ’s fo rm u la s fo r th e e x p e c ta tio n a n d s ta n d a r d e rro r o f th e s u m . A lte r n a tiv e ly , o n e c a n s a y th a t th e s p e c ific b o x m o d e l h e re c o rr e s p o n d s to a n i.i.d . p ro c e s s w h e re th e r a n d o m v a ria b le s X i a r e c h a ra c te riz e d b y a p r o b a b ility fu n c tio n f X i ( x i ) a n d th e fin a l ra n d o m v a ria b le o f in te re s t X n is th e s u m o v e r th e in d iv id u a l ra n d o m v a ria b le s . T h e i.i.d . p r o c e s s h a s 6 0 0 re p e titi o n s . T h is a llo w s im m e d ia te c a lc u la tio n o f th e e x p e c ta tio n a n d th e s ta n d a rd e r ro r. € 1 1 SE ( X n ) = n SDX i = 600 (1 − ) ≈ 11.6 3 3 R e v ie w E x e r c is e 2 0 .1 2 ( m e a n t to s h o w a c o m b in a tio n o f b o x m o d e l w ith e x p e c ta tio n a n d s ta n d a r d e r ro r o f s u m s in i.i.d . p ro c e s s e s ) T h is r e v ie w e x e rc is e is v e ry s im ila r to re v ie w e x e rc is e 2 0 .9 . T h e to ta l n u m b e r o f in te rv ie w s is lik e th e s u m o f 4 0 0 d r a w s fro m a b o x . T h e a v e ra g e o f th e b o x a n d th e S D o f th e b o x is 1 .8 7 . T h e to ta l n u m b e r o f in te rv ie w s w ill b e a r o u n d 4 0 0 × 2 .3 8 = 9 5 2 , g iv e o r ta k e √ 4 0 0 × 1 .8 7 ≈ 3 7 o r so. R e v ie w E x e r c is e 2 3 .1 ( m e a n t to illu s tra te th e a n a lo g y b e tw e e n b o x m o d e ls a n d r e a l w o rld e n titie s ) T h is r e v ie w e x e rc is e is a n a lo g o u s to m a n y r e v ie w e x e r c is e s in c h a p te r 2 0 . H e re a v e ra g e a n d s ta n d a r d d e v ia tio n o f th e b o x a r e k n o w n . F o rw a r d re a s o n in g , n o t b a c k w a rd re a s o n in g / in fe re n tia l s ta tis tic s is a p p lie d . T h e s u m o f 4 0 0 d r a w s w ill b e a ro u n d 4 0 0 × 1 0 0 = 4 0 ,0 0 0 (= e x p e c ta tio n o f s u m ) , g iv e o r ta k e √ ( 4 0 0 × 2 0 ) = 4 0 0 (= s ta n d a rd e rro r o f s u m ) o r s o . T h e a v e ra g e o f 4 0 0 d r a w s w ill b e a ro u n d 1 0 0 (= e x p e c ta tio n o f a v e ra g e ) , g iv e o r ta k e 1 ( = s ta n d a rd e rro r o f a v e ra g e ) o r s o . (a ) T h e in te rv a l [8 0 , 1 2 0 ] c o r re s p o n d s to th e in te r v a l [1 0 0 – 2 0 x 1 , 1 0 0 + 2 0 x 1 ], i.e . th e in te r v a l c o v e rs 2 0 s ta n d a rd d e v ia tio n s a b o v e a n d 2 0 s ta n d a rd d e v ia tio n s b e lo w th e e x p e c ta tio n . A s th e n o rm a l a p p ro x im a tio n o f p r o b a b ility h is to g r a m s h o ld s fo r (s u m s a n d ) a v e ra g e s o n e k n o w s th a t th e p r o b a b ility o f a s a m p le a v e r a g e fa llin g in th e in te r v a l is v e ry , v e r y c lo s e to 100%. (b ) T h e in te rv a l [9 9 , 1 0 1 ] c o r re s p o n d s to th e in te r v a l [1 0 0 – 1 x 1 , 1 0 0 + 1 x 1 ], i.e . th e in te rv a l c o v e rs 1 s ta n d a rd d e v ia tio n a b o v e a n d 1 s ta n d a r d d e v ia tio n b e lo w th e e x p e c ta tio n . A s th e n o r m a l a p p ro x im a tio n o f p r o b a b ility h is to g r a m s h o ld s fo r (s u m s a n d ) a v e ra g e s o n e k n o w s th a t th e p r o b a b ility o f a s a m p le a v e ra g e fa llin g in th e in te rv a l is a b o u t 6 8 % . N e v e r c o n fu s e s ta n d a rd e r ro r o f a ra n d o m v a ria b le X n a n d th e s ta n d a rd d e v ia tio n o f th e in d iv id u a l ra n d o m v a ria b le s X i , i.e . th e s ta n d a rd d e v ia tio n o f th e b o x . 1 if x i = 1 fX i ( xi ) = 3 2 if x = 0 i 3 X n = X 1+.. + X n 1 E ( X n ) = nX i = 600 ⋅ = 200 3 € R e v ie w E x e r c is e 2 6 .1 ( m e a n t to c h e c k y o u r b a s ic u n d e rs ta n d in g o f h y p o th e s is te s tin g ) (a ) T r u e . B y d e fin itio n th e p v a lu e o f a te s t is a ls o c a lle d th e o b s e r v e d s ig n ific a n c e le v e l o f a te s t ( s e e F re e d m a n p .4 7 9 ). (b ) F a ls e . T h e n u ll h y p o th e s is s a y s th a t a n o b s e r v e d d iffe re n c e b e tw e e n s a m p le e s tim a te a n d a s s u m e d p o p u la tio n p a ra m e te r is d u e to c h a n c e . T h e a lte rn a tiv e h y p o th e s is s a y s th a t th e o b s e rv e d d iffe re n c e re s u lts fro m a d iffe r e n t p o p u la tio n p a ra m e te r ( s e e F re e d m a n p p .4 7 7 – 7 8 ). R e v ie w E x e r c is e 2 6 .2 ( m e a n t to c h e c k y o u r b a s ic u n d e rs ta n d in g o f h y p o th e s is te s tin g ) (a ) T h e d a ta c a n b e s e e n a s b e in g g e n e ra te d fro m a b o x m o d e l w h e re th e b o x c o n ta in s a n u n k n o w n n u m b e r o f 0 ’s a n d 1 ’s ( w ith 1 = re d ) a n d o n e d ra w s 3 8 0 0 tim e s a t r a n d o m w ith re p la c e m e n t . T h is is a n a lo g o u s to s a y in g th a t th e d a ta w e re g e n e ra te d b y 3 8 0 0 re p e titio n s o f a n i.i.d . p ro c e s s w h e re th e p ro b a b ility d e n s ity fu n c tio n f X i (x i ) o f th e ra n d o m v a ria b le s X i is : θ if x i = 1 fX i ( xi ) = 1 ‐ θ if x i = 0 O n e th e n s tu d ie s th e ra n d o m v a ria b le X n = (X 1 + ..+ X n )/ n a n d m a k e s s ta te m e n ts a b o u t th e v a lu e o f θ . T h e n u ll h y p o th e s is w o u ld c la im th a t th e ro u le tte w h e e l is fa ir a n d th e o b s e rv e d d iffe r e n c e s a re d u e to c h a n c e v a r ia b ility . T h e a lte rn a tiv e h y p o th e s is w o u ld c la im th a t th e ro u le tte w h e e l is u n fa ir a n d fa v o r s re d . (b ) N u ll: th e fra c tio n o f 1 ’s in th e b o x is 1 8 / 3 8 , o r 4 7 .4 % . A lt: th e fra c tio n o f 1 ’s in th e b o x is m o re th a n 1 8 / 3 8 . S e e (a ). (c ) T h e e x p e c te d n u m b e r o f re d s (c o m p u te d u s in g th e n u ll) is 1 8 0 0 . T h e S D o f th e b o x ( a ls o c o m p u te d u s in g th e n u ll) is n e a rly 0 .5 , s o th e S E fo r th e n u m b e r o f re d s is √ 3 8 0 0 × 0 .5 ≈ 3 1 . S o z = (o b s − e x p )/ S E = (1 8 9 0 − 1 8 0 0 )/ 3 1 ≈ 2 .9 , a n d P ≈ 2 / 1 0 0 0 . (d ) Y e s . If th e n u ll h y p o th e s is is tru e th a n th e o b s e rv a tio n o f 1 8 9 0 tim e s re d in 3 8 0 0 g a m e s is u n lik e ly . It w o u ld h a p p e n o n ly in 2 o u t 1 0 0 0 c a s e s . The null hypothesis removes the lack of knowledge over the box/probability density function, because it assumes a specific composition and then sees how well the sample is in line with the assumed box. ...
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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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