-distributed random variables consisting of n samples each. Determine the asymptotic confidence intervals for

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1 Assgme Sepha Brumme Ocober 8h, h semeser, PREFACE I 004, I ed o sped wo semesers o a sudy abroad as a posgraduae exchage sude a he Uversy of Techology Sydey, Ausrala. Each opporuy o ehace my Eglsh (boh wre ad spoke) wll ake me a sep closer o a successful ad uforgeable year Dow Uder. As a resul, all assgmes wll be wre Eglsh. I suppose ha hs decso of me should o cause ay problems sce he laguage used owadays mahemacal papers s Eglsh. I apprecae all hs cocerg a mproper or wrog use of Eglsh erms. Thaks for your kd udersadg. 003 Sepha Brumme PROBLEM? Geerae 00 B(, ) -dsrbued radom varables cossg of samples each. Deerme he asympoc cofdece ervals for. = 0. 8 ad = 50. = 0. 5 ad = = 0. 8 ad = 80 Compare he wdhs of hese ervals. The huge amou of daa o be geeraed forced me o use dedcaed sofware esvely. I my prevous lecures I go o coac wh he excelle program Maple 8 (ral verso, avalable for free). Therefore, wll accompay me aga o my kowledge-seekg way hrough hs (ad maybe he followg) problem assgme. Bascally, each X ~ B(, ) dsrbuo where he acual ( 0,) s ukow has he characerscs EX = ad ( ) VarX = EX ad ( ) If we furher defe µ = = σ = VarX = ad ake o cosderao ha all our radom varables are..d. (depedely ad decally dsrbued, because hey are he oucome of a close-o-perfec radom umber geeraor) he he Law of Large Numbers helps fdg he cofdece ervals. I allows us o approxmae = X by he ormal dsrbuo µ ~ N σ The cossecy gves ( 0,) S σ where S = ( ) Sassche Daeaalyse - - wer erm 003/004

2 Assgme Sepha Brumme Ocober 8h, h semeser, Sassche Daeaalyse - - wer erm 003/ Sepha Brumme Therefore; we ge ( ) 0, ~ N S µ A ormal dsrbuo s cofdece erval for µ s defed by + = S S I ; ;, α α whch leads us o ( ) ( ) + = I α α, ; ; If we se 05 = 0. α he ;0.975 ; = α. O he ex page I wll expla he Maple 8 (ral) program used o geerae he accordg radom umbers, her cofdece ervals ad s wdhs. I was my eo o do que more ha requesed order o ge back he feelg for sascal problems ad Maple s programmg laguage.

3 Assgme Sepha Brumme Ocober 8h, h semeser, Frs, some packages are requred order o defe all used fucos: > wh(sas): wh(radom): wh(plos): wh(saplos): A real programmer (lke me ) frequely srves o move he ma roues o separae, dedcaed fucos. Oe of he mos-called fucos s observao. Is purpose s o geerae all eves for a sgle radom varable X,.e. a seres of 0 s ad s: > observao:=(hea, sze) -> [bomald[, hea](sze)]: Uforuaely, here s o deeper eres ha seres. Wha we acually eed s he frequecy of s fally gvg he esmaed. > esmao:=(hea, sze) -> sum(observao(hea, sze)[], =..sze)/sze; 003 Sepha Brumme Oe hudred esmaos of form a que sold experme. Thus, we ca deerme he cofdece ervals. > expermesze:=00: > experme:=(hea, sze) -> sor([seq(esmao(hea, sze), j=..00)]): The sude- dsrbuo plays such a mpora role ha deserves s ow fuco: > sude:=(alpha, sze) -> saevalf[cdf, sudes[sze]](-alpha/): Throughou he whole assgme, he level of cofdece ad he umber of esmaos remas uchaged so I creased Maple s performace by magudes by defg a cosa : > :=sude(0.05, expermesze); Nex comes cofdeceiervals whch s resposble for geerag he requesed cofdece ervals: > cofdeceiervals:=(samples, sze) -> seq([samples[]-*sqr(samples[]*(-samples[])/sze), samples[]+*sqr(samples[]*(-samples[])/sze)], =..expermesze); All he meoed fucos perform he core work. They are saaed by jus hese sx les: > samples:=experme(0.8, 50): > samples:=experme(0.5, 50): > samples3:=experme(0.8, 80): > ervals:=cofdeceiervals(samples, 50): > ervals:=cofdeceiervals(samples, 50): > ervals3:=cofdeceiervals(samples3, 80): Numbers ca be que absrac, hece, a plo ofe helps o ga furher sghs. Here s he code used for experme, does o dffer much from experme or 3: > ervalplo:=seq(plo([[(ervals[][]),], [ervals[][],]], color=coversthea(ervals[][], ervals[][],0.8, red,blue)), =..00) : > dsplay([ervalplo, plo([[0.8,0],[0.8,00]], color=black)]); Sassche Daeaalyse wer erm 003/004

4 Assgme Sepha Brumme Ocober 8h, h semeser, Experme For experme ( = 0. 8 ad = 50 ) we observe ha 96 ou of 00 ervals,.e. 96%, acually cover whch comes prey close o our cofdece level of 95%. I he followg plo all ervals ha overlap are colored red, self s represeed by a black vercal le: 003 Sepha Brumme Fgure Cofdece ervals of experme Sassche Daeaalyse wer erm 003/004

5 Assgme Sepha Brumme Ocober 8h, h semeser, Experme Experme faled some way. I mssed he proposed cofdece level of 95% by far ad reached oly 9%. O he oher had, fve of he msses erval borders are prey close o. Therefore, he geeraed umbers are o compleely uusable. If you ake a look a he shape of he ervals you may oce ha here are a b more symmercally dsrbued ha experme. 003 Sepha Brumme Fgure Cofdece ervals of experme Sassche Daeaalyse wer erm 003/004

6 Assgme Sepha Brumme Ocober 8h, h semeser, Experme 3 The las experme exacly maches our cofdece level of 95%. The overall mpresso fs my expecaos so here s ohg more lef o say: 003 Sepha Brumme Fgure 3 Cofdece ervals of experme 3 Sassche Daeaalyse wer erm 003/004

7 Assgme Sepha Brumme Ocober 8h, h semeser, Sepha Brumme Comparg he wdhs Fally, I compared he wdhs of all hree expermes. The covcg sghs gaed from he plos empowered me o use a plo for he wdhs, oo. Mos of he Maple code s resposble for a properly sored order: > wdh:=sor([seq(ervals[][] - ervals[][], =..00)]): > wdh:=sor([seq(ervals[][] - ervals[][], =..00)]): > wdh3:=sor([seq(ervals3[][] - ervals3[][], =..00)]): > wdhplo:=plo([seq([, wdh[]], =..00)], =..00, syle=po, color=black): > wdhplo:=plo([seq([, wdh[]], =..00)], =..00, syle=po, color=blue): > wdhplo3:=plo([seq([, wdh3[]], =..00)], =..00, syle=po, color=red): > dsplay([wdhplo, explo([90, 0.3, 'experme']), wdhplo, explo([90, 0.75, 'experme']), wdhplo3, explo([90, 0.85, 'experme3'])]); Fgure 4 Wdhs of he cofdece ervals The plo cofrms my hypohess: experme produces he larges spread sce for = 0. 5 he varace ( ) s maxmzed. Small varaos of f s close o 0.5 do o subsaally affec he wdh whch heavly depeds o he varace as see Fgure 5 below. Sassche Daeaalyse wer erm 003/004

8 Assgme Sepha Brumme Ocober 8h, h semeser, > plo(hea*(-hea), hea=0..); 003 Sepha Brumme Fgure 5 Varace depedg o Experme cosss of 00 radom varables each geeraed by 50 B (,0.8) observaos. I coras, experme 3 was bul of 80 B (,0.8) observao, abou 50% more. Ths dfferece leads o a sgfcaly smaller average wdh of experme 3 cofdece ervals. I coclude from he expermes ha he accuracy of a B(, )-based experme ca be serously creased by eher shfg away from 0. 5 or by rsg he umber of repeos. I he mos cases, he laer s he oly possble soluo whe amg for a more precse esmao. Sassche Daeaalyse wer erm 003/004