More Statistics tutorial at 1. Introduction to mathematical Statistics

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1 Mor Sttstcs tutorl t wwwdumblttldoctorcom Itroducto to mthmtcl Sttstcs Fl Soluto A Gllup survy portrys US trprurs s " th mvrcks, drmrs, d lors whos rough dgs d ucompromsg d to do t thr ow wy st thm shrp cotrst to sor cutvs mjor Amrc corportos" (Wll Strt Jourl, My 985 O of th my qustos put to smpl of trprurs bout thr work hbts, socl ctvts, tc, cocrd th org of th cr thy prsolly drv most frqutly Th rsposs r gv th followg tbl US Europ Jp Do ths dt provd vdc of dffrc th prfrc of trprurs for domstc crs vrsus forg crs? Tst t α5 Soluto: Ths problm c b do two wys usg thr ( th tst o o populto proporto or ( th Ch-squr goodss-of-ft tst wth two ctgors Ths two pprochs r quvlt ( For th frst pproch, frc o o proporto, lrg smpl, w hv, 45 Lt p b th proporto of 45 trprurs wth domstc crs, w hv p ˆ, d w r tstg: : p 5 vrsus : 5 p pˆ 5 Th tst sttstcs s: Z 5 5 / Sc Z < 96 Z5, w c ot rjct th ull hypothss t th sgfcc lvl of 5 ( Altrtvly, d quvltly, you c us th Ch-squr goodss-of-ft tst Th bov tbl s rdly rducd to th followg two-ctgory tbl: Domstc crs Forg crs Lt p, p b th proporto of trprurs wth domstc or forg crs rspctvly, w r tstg: : p 5, p 5 vrsus : s ot tru c w hv 5, 5 ( Th tst sttstc s: W < χ,5, uppr ( Z5 (96 84 Thrfor w c ot rjct th ull hypothss t th sgfcc lvl of 5 Of cours you oly d to show o of th two pprochs bov to gt full crdt ~ N,,,, d thy r dpdt to ch othr, th ( Wht s th dstrbuto of (b Wht s th dstrbuto of? Prov your clm ( /? Prov your clm Soluto: ( M ( t E p( [ ] t E p( t E p( t M ( p p t t t ( t ( + + t Thrfor w hv show tht ~ (,( N (b Lt Z, th w hv

2 Mor Sttstcs tutorl t wwwdumblttldoctorcom MZ ( t E p t E p t p t p tm t p t p p t+ t t Z ~ N, Thrfor w hv show tht d Z Furthr, sc Z ~ N,,,, ; By th dfto of th ch-squr dstrbuto, w kow tht χ / ~ (Tht s, t follows th ch-squr dstrbuto wth dgrs of frdom Lt,, K, b rdom smpl from th tructd potl dstrbuto wth pdf f( p θ, f θ; d f(, f < θ Pls ( Drv th mthod of momt stmtor of θ; (b Drv th MLE of θ Soluto: ( Frst w drv th populto m s follows ( θ θ θ θ θ { θ θ } θ { } { θ } θ θ θ E f ( d d d d θ θ θ + + Sttg th populto m qul to th smpl m, w hv + θ ; thus th MOME for θ s ˆ θ L (b Th Lklhood s L p( + θ p ( + θ ; for θ,,, for θ m ( ˆ m Thus th lklhood s mmzd wh θ chvs ts lrgst vlu; thus th MLE for θ s θ ( 4 Jrry s plg to purchs sportg goods stor clcultd tht ordr to covr bsc pss vrg dly sls must b t lst $55 chckd th dly sls of 6 rdomly slctd busss dys Ad h foud tht th vrg dly sl for ths dys s $565 wth stdrd dvto of $5 ( At sgfcc lvl α5, c Jrry coclud tht th vrg dly sl s hghr th $55? Wht s th p-vlu? (b I ordr to stmt th vrg dly sl of th stor to wth $ wth 95% rlblty, how my dys should Jrry smpl? Pls drv th grl formul for smpl sz clculto bsd o lrg smpl of sz, mmum rror of E, d rlblty of (-α% frst (c If Jrry could oly chck th dly sls of 9 rdomly slctd busss dys (std of 6 rdomly slctd dys Suppos th dly sl for ths 9 dys r 5, 57, 548, 59, 5, 49, 6, 499 d 64 rspctvly At th sgfcc lvl α5, c Jrry coclud tht th vrg dly sl s hghr th $55? Wht s th p-vlu of th tst? (d For th sttg (c bov, tht s, w hv smll smpl of sz from orml populto Pls drv th twosdd tst o th populto m, t th sgfcc lvl of α, usg ( th pvotl qutty mthod (*pls clud th complt drvto of th pvotl qutty, th proof for th dstrbuto of th pvotl qutty, d th drvto of th rjcto rgo for full crdt, ( th lklhood rto tst (*pls clud th complt drvto of th MLE s, th lklhood rtos, d th rjcto rgo for full crdt Furthrmor, pls show tht ( th pvotl qutty d th lklhood rto tst pprochs r quvlt Soluto: Ifrc o o populto m 6 Populto vrc s ukow If you kow th dt, th you do ormlty tst (g, Shrpo-Wlk tst to s f th smpl s from orml dstrbuto If th populto s orml, th w oly us t-dstrbuto

3 Mor Sttstcs tutorl t wwwdumblttldoctorcom If t s ot orml but th smpl sz s lrg (>, th pvotl qutty Z ~ N(, (by Ctrl Lmt Thorm d Slusky Thorm ( 565, S5 : 55 : > Not: If 55(<55, th you should otc tht s ot sutbl Tst sttstc: Z 6 5 / 6 At th sgfcc lvl α, w rjct f Z cot rjct P-vlu PZ ( z PZ ( > α W c ot rjct Not: P-vlu P ( 565 PZ ( 6 (b Frst w drv th grl formul P( E α z α r z z α Sc z 6 < 645 z α, w P( E E α E E P( α / / / E zα / zα / / E Nt w plug th vlus to obt th swr for th gv problm zα / 96* E (c Suppos from th Shrpo-Wlk tst, w kow th dt/smpl s from orml populto : 55 : > 54667, S Tst sttstc: T 59 / At th sgfcc lvl α, w rjct f T t,α Sc t < t 8,5 86, w cot rjct P-vlu PT ( t PT ( From th t-tbl, w foud < p vlu < 5 W cot rjct (d ( [] Frst w drv th pvotl qutty d ts dstrbuto Pot Estmtor for : ~ N(, ; s NOT pvotl qutty sc s ukow Th w cosdr Z ~ N(, ; Ths s lso NOT pvotl qutty sc s ukow By th: Thorm Smpl from orml populto S Z ~ N (,, w kow W ~ χ

4 Mor Sttstcs tutorl t wwwdumblttldoctorcom 4 Z Ad by th: Dfto T ~ t T ( Z d W r dpdt W S ( T s pvotl qutty for [] Nt w drv th o-smpl t-tst d ts rjcto rgo For -sdd tst of : vrsus :, th tst sttstc s th pvotl qutty t, tht s, T Itutvly, w would rjct fvor of f T c Th problm s how to dtrm c By th dfto of th sgfcc lvl, w hv α P rjct_ P T c P T c Thus ( ( ( / PT c d subsqutly w hv α ( c t, α / Tht s, t th sgfcc lvl α, w rjct fvor of f T t, α / ( & For -sdd tst of : vrsus :, wh th populto s orml d populto vrc s ukow, w ow drv th lklhood rto tst [] Wrt dow your prmtr spc udr {(, :, } f [] Wrt dow th urstrctd/orgl prmtr spc {(, : R, } f [] Wrt dow th lklhood (of th dt L f(,, L, ; f( ; [4] Wrt dow your log-lklhood ( l l L l( [5] Fd MLEs udr d plug to gt m L ( dl + 4 d ˆ ( ˆ m L L(,, L, ;, ( ( ( 4

5 Mor Sttstcs tutorl t wwwdumblttldoctorcom 5 5 [6] Fd MLEs udr d plug to gt m L 4 dl d dl d + ˆ ˆ ˆ ˆ m (,,, ;, L L L [7] Gt th lklhood rto m m L LR L [8] Drv th dcso rul bsd o sgfcc lvl α

6 α P(Rjct s tru Rcll t-tst sttstc : T Mor Sttstcs tutorl t wwwdumblttldoctorcom 6? ( PLR ( c : P( c : (, t sgfcc lvl α, w rjct fvor of f S ~ t ( ( : c ( P ( + * ( c ( P : ( ( : c ( P ( + ( ( + ( * ( c ( P ( P + c * ( : ( At α, w rjct f T t, α : PT ( c : ** PT ( c : ** Th LR tst s quvlt to th t-tst T t, α 5 (tr crdt Suppos w hv two dpdt rdom smpls from two orml popultos,,,, ~ N, Y, Y, K, Y ~ N, K, d ( ( At th sgfcc lvl α, pls costruct tst of th hypothss o: r kow costts (b Suppos w hv cofrmd tht c d b vs : b r, b b At th sgfcc lvl α, pls costruct tst to tst whthr + or ot usg th pvotl qutty mthod r c, d, r kow costts Pls clud th drvto of th pvotl qutty, th proof of ts dstrbuto, d th drvto of th rjcto rgo for full crdt Soluto: Ths s frc o two orml populto ms, dpdt smpls ( Ths s th usul F-tst o two orml populto vrcs: : / b/ vrsus : / b/ Th tst sttstc s: F S / S S / S ~ F,, /, b/ At th sgfcc lvl α, w wll rjct f F s smllr th F α or F s grtr th,, /, L F,, α /, U 6

7 Mor Sttstcs tutorl t wwwdumblttldoctorcom 7 b (b Gv tht b, w st d thus r s smpl outl of th drvto of th tst: : c+ d vrsus : c+ d, whch r quvlt to: : c d vrsus : c d [] W strt wth th pot stmtor for th prmtr of trst( c N c, c b/ + / usg th mgf for ( ( : ( c Y Its dstrbuto s N, whch s M ( t p( t + t /, d th dpdc proprts of th rdom smpls From ths w hv ( c Y ( c Z ~ N (, Ufortutly, Z c ot srv s th pvotl qutty bcus s cb/ + / ukow [] W t look for wy to gt rd of th ukow followg smlr pproch th costructo of th pooldvrc t-sttstc W foud tht W ( S + ( S / ~ χ + b usg th mgf for χ k k / whch s M ( t, d th dpdc proprts of th rdom smpls t [] Th w foud, from th thorm of smplg from th orml populto, d th dpdc proprts of th rdom smpls, tht Z d W r dpdt, d thrfor, by th dfto of th t-dstrbuto, w hv c Y c obtd our pvotl qutty: T ( ( S + ( S b * cb/ ( + / + [4] Th rjcto rgo s drvd from P ( T c α, whr T c Y + d sgfcc lvl of α, w rjct fvor of ff T ~ t + ( S + ( S b * cb/ ( + / + t +, α / ~ t + Thus c t +, / Thrfor t th α 7

ERDOS-SMARANDACHE NUMBERS. Sabin Tabirca* Tatiana Tabirca**

ERDOS-SMARANDACHE NUMBERS. Sabin Tabirca* Tatiana Tabirca** ERDO-MARANDACHE NUMBER b Tbrc* Tt Tbrc** *Trslv Uvrsty of Brsov, Computr cc Dprtmt **Uvrsty of Mchstr, Computr cc Dprtmt Th strtg pot of ths rtcl s rprstd by rct work of Fch []. Bsd o two symptotc rsults