Exa-tye practce: A Leve a Let X dete the dtrbut ae ad X dete the dtrbut eae The dee the rad varabe Y X X j j The expected vaue Y : E( Y) EX X j j EX EX j j EX E X 7 The varace : Var( Y) VarX VarX j j Var( X ) Var(X ) = Var( X ) Var( X ) Y ~ N (7,) I rder t cacuate P( Y ) we tadarde t 7 PZ PZ 4 Y 7 where Z I rder t ue the tattc tabe we rerat ur cacuat t: P Z 4 P( Z 4) P( Z 4) Thu ug the tabe we get: P( Z 4) P( Z 4) j b Ug the ae tat a the prevu quet, dee a ew rad varabe a W 4X X E( W) E(4 X X ) 4 E( X ) E( X ) 4 Var( W) 4 4 we have W ~ N,4 We wat t d P( W ) ad we tadarde the ue the tabe rder t get: PV PV 4 4 a Let dete the varace the apg wth erter ad dete the varace the apg wthut erter Our hypthee are: H : ad : H The gcace eve % (% at each ta) wth degree reed: v, v 47, 74 Fr the tabe, we d the crtca vaue F, 7 The tet tattc 7 7 7 there ucet evdece t reject H ad we ay aue that the tw ppuat have equa varace Pear Educat Ltd Cpyg pertted r purchag ttut y Th atera t cpyrght ree
b Let dete the ea heght apg wth erter ad dete the ea heght apg wthut erter The u hypthe that the derece betwee the ea The ateratve hypthe that the derece -zer: H : H : We have tadard devat ad ape ze :,, 4, We cacuate a ubaed etate the ppuat varace ug bth ape: p 4 4 7 : x x t p 7 ( dp) (Nte that we ued the u hypthe H : th cacuat) The % (tw-taed) crtca vaue r t wth degree reed t ur tet tattc vaue t gcat eugh t reject H Thu we aue that the ea heght pat bth ppuat are the ae c The tet part b requre that bth the varace are equa The tet part a etabhed that th wa reaabe a A % cdece terva wth degree reed ha a t vaue t 7 the cdece terva the r: ˆ x 7 I we take the hgher vaue the cdece terva,, we ay ve r ˆ ˆ x 7 ˆ 7 44 ˆ ˆ 4 ˆ 7 ( ) b The percetage pt are 7 ad We have cacuated ˆ 7 ad ca cacuate that the crtca pt are ˆ 7 4 7 ad ˆ 7 the % cdece terva r varace (, 4) The % cdece terva r the tadard devat ha the quare rt the t th terva a t t e (4, ) Pear Educat Ltd Cpyg pertted r purchag ttut y Th atera t cpyrght ree
c Let dete the pa a adut ae had We wat t d P( ) whch 4 a ca be tadarded t PZ I rder t axe the prbabty, we eed t be a a a pbe we che the bgget vaue r ad that are ur cdece terva PZ PZ P Z 74 Thu, the hghet etate the prprt adut ae wth had pa greater tha 7 ( dp) hh h h 74 4 c b te u the rate at whch the cdece creae wth creaed heght I th cae r every cetetre heght, cdece creae by 4 ut d It wud t ake ee t have a terpretat a ce t wud py that t pbe t have c heght e h c c 7 4h e 7 77 7 7 7 7 44 4 4 4 4 4 47 77 7 7 74 7 The crrect vaue 7 a the agtude t redua gcaty greater tha the ther c cc c 7 ad h c hc hc 444 4 b We cacuate hc 4 b 4 hh a c bh 4 c 74 h ( ) Pear Educat Ltd Cpyg pertted r purchag ttut y Th atera t cpyrght ree
4 hh h h 74 cc c 7 77 47 ad h c hc hc 4447 7 4 We cacuate hc 4 b 4 hh a c b h c 7 4 744 c 7444h g c 7444 7 a Let x rak dete the pace whch a per cae the actua race ad y rak dete the rak whch a per cae the quayg ap-te Nae x rak y rak d d Car 4 Paua arah 4 Davd 4 Dhruv Ay 7 Jake 7 A Nw we have the data, we u a ad get d r d ( ) b H : d vaue H : We have a ape ze ad the gcace eve the ta Fr the tabe, the crtca vaue r r a gcace eve wth a ape ze r 4, the crtca reg r 4 The berved vaue r de the crtca reg, we reject H There ucet evdece at the % eve gcace that there a ptve acat betwee quayg ap-te ad actua race reut c Race rak are t eaurabe a ctuu cae Pear Educat Ltd Cpyg pertted r purchag ttut y Th atera t cpyrght ree 4
d Data w have 4 vaue wth ted rak Ag a rak equa t the ea the ted rak Cacuate the PMCC drecty r the raked data rather tha ug the rua a Frt we d the prbabty dety uct by deretatg the cuuatve dtrbut uct We d that: x x x ( x) therwe I rder t d E( X), we cacuate: E( X) x ( x) dx x x x dx x x 4 x d 4 x x 4 ( ) ( ) E X x x x dx x 4 x x dx x dx x x, Var( X) E( X ) E( X) 4 7 4 b We ee that the axa pt ccur whe x (ce we have a creag uct) ad we have that the de c The de ea ad we ay that t egatvey kewed d Pk X k P( X k) P( X k) 7k k 4 4k k k k X( X) 7 Let A dete the area eced by the raewrk ad the grud X( X) E( A) E EX( X) EX X xx dx x x Pear Educat Ltd Cpyg pertted r purchag ttut y Th atera t cpyrght ree