math306_mid2 - SULUWMS TD MID'TEIZM 1F [2—1- A popuhtiun...

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Unformatted text preview: SULUWMS TD MID'TEIZM 1F [2—1- A popuhtiun haS the probability denaity funflian 1913 E 92.1: 'f . -. {I F I -_ _ 1' 2" f? | J? .-‘ 'l: 1' { U eifiewhere 1'1 EamPlE {1'11121"'1-Enj i5 taken from this papillatiun 335196 on this 53mph: find the Maximum Likelihood Btimatur [MLE] of the parameter E. {M1165} ’3 f5 #5123 U”:- w‘fi J9) - fl LE9 if e 2. 11" 1 {in 2 ,5: E" '2. '2 P -- 121-1 6%] '2 n T: {Q (XI _ :(n )1 Eli E3 129% if“ .- " n n “LP AL (3’5, -_.g'x -L:;Nl-: (5n tat-3L— mini 4‘35wi/F'1XL'1PQLEZC h} . _ fl MHQL it _‘2E?.Z1’c :a M? 9‘ ' Fl Eh, #:257121- 6: 1/;— :‘FC 9 a J 9%:- {2—2. In. ho Their veriai 11] We 5ha|lfitirnate In. [I1 of the questions below we are considering two' normal populations with unknown. means In, and Iii_.. Ice: are known as of - Fifi and mi = 455. .t-Lu using. a 95% two-sided fifillfidonte Interval by taking earanes From the two populatimm II we Iarant that the maximum error of the interval estimation be £1 units, what should be the equal sample sures. H1 -- f|__I H '5' {minted h] SuppoSE Illat we have detrded to take Samples of different EIIEE Itl — [2 and H3 .— 2“. With what degree of confidence can we assert that the maxamum error of our estimation w1|| be lees than 4.3nni15? {Warning Hus queetion is independent oF the question in part a] The degree of confidencrr i5 r1o_.more 95%). {H.l‘pth'] '1...- ’L r“ _ Eu, Far ‘31 4 H ‘— L i. 11"] fiifiig.’ I fit+fl “Hoffii Kifli¢+E°5 EL 4.15:1 1..- a I I" fl ,1 .. f 1 (It: Grl—F-oi 12 6.11:1) firth} . -—~ #1 l'J‘f'laI l'JEI—Tail— FT‘J211E-f; 3w: Ep+flio 1 - 'L Fir n2 1. L , air-wth 1112‘: F: +5; :1.g5{36+4? :4 1 in r1 H‘— il 1 "‘ fi¢{1~%fl§j ,I my one may | d . _""’L r E- :- m W t tEiELr(E-56) £437“?! J‘th (€.o?}fifl.é£0? t—fituteeéme magma art-ecwwm sci-103$? (3—3. 15. population has the follow-mg probability density function . _;.i 1': _ .2 - . . _L. “law? _ magi-[‘3 J. :I If [i t. ..I . H [H p I” ll L'I‘iE‘WI'HEFE' A single meafiunnlwr'ul X i5 taken From this population (1.: _ 1] Consider the random variable 11' -.= g- all Find the cligtrioution of II'. (First obtain the cumulative distribution Function Firth-l and then the deneity fwtu'll- [50!!!3} b} Ufiing a] Show that 'H' can be a pivotal tendon: variable for the parameter {-3. [51:55] oi Exolain the way that you would obtain a won—om onesicleci confidence interval of the form {l}. é, -} where Hr; is tho upper confidence limit for Do not try to find the root of a diFficoEt onuation Call it. say r1}. [ 'Fpn'fi] an Emmott Pl'wem 2 PH: to): Pl’Xelezfi {9w} ’ Y e»! or: 1 . E; , El 1* r :i fee—t2 '1. 3 are) ngéfost :14: mg gll Eggox—g) :24ch {7:0 3 E 3; 1” ,‘LQW? ,3 3 . . W‘wi’zeoriéav if)“; W”? "fléw‘L‘I'Jg'iw‘o’i'ilel 4w {W}: % (lo—WE"): Oiwél ,3 flea? eioewiuxrf '9] w 56144: if“? 5. 5‘] K‘ri’wl‘ oii prerelw cf or pivotal fin“. (9 FM [ifi)c‘3£(fi“§)1 Di 61W.“ at we Cam fume (or «F-Pw-ximtfi'] (J, 1‘0 (W E‘Ji‘laf-Zéealc—n r51] 7: [*‘3‘1 E3. . 1 ' ' fl __'li}'l-b"-J' i—(fiifijll—cc" W43. MW 1X D w WQ,HLHWHT xii“? 7 Ftkfl (-Lfix "1—: its“ Iflfi 1:1", | ("1—4. .5- Study is conducted to estimate the dil'fererioe in the mean occupational exposure to radioactivity in utility I-Ieorhers in the years 19H altd 19W. Follmuir'lg data [in appropriate units] based on independent workers for the two veers are obtained : lUTLl lEJT'lJ' Jr: —]ii H3215 .i'] = [L5H .53 = on”; .~:'-f : l].lI-:|[| bf : Home 2 3.} Assuming that ol for In] - m. #31 —- 0“ find a 99% law-rel" sided [I 9. having only lower tonEidence limit} confidence interval Efflpn’h] h} Lookmg at the confidente interval in a} can we say that the mean expo-Sure In 19?}. was, in fact, higher than Cool : mm -1. 1. 1.31.11 in 19?? 7' Explain why [Ewe] E} Was Ifi'ur assumtiou of — r15 .5 reasonable one ? To anewer construct a suitable two-sided Qflz’u confidence interval. I: i'l'lplfa} ’3 e , .. :53} 5101:. Liam”? T Era 5”} .c: @0544 9“ ‘9? 10111856 | 3 +_ J?_.—2_ l 3L3? lngEr £11444 dEMC€ lfll'ewf'tl J Pol? —_ JC. 3 L . 1 l 1 To fil 1'- n1 LIILII_H.L 1-? al- 4-32 —.—I—'—' .f_._l— CL’).'3>?.G£3._-Ejl FREE: 1:}... fl-‘Ejzla—fi.5'1r-£_é¥3rflilféélJ'ILi I6 51 a 14171 z: 1.441 lg‘] V. ~lul >121 2. Hey; We: ("L ’7'— ‘E-q'“ ut- CF i I~_l_#d__‘ {L if r}; “L 1—,: £9.05.— ("1- {1 E’L '3 H—lI’lr—l .2 '2. l 1 9 r 1 lla'Emlaimri 1“ F . ,3 r1 la.05;iiii5 fluid” iu_ugjifi_)|¢ 2'6 o. 040 Fj_ 4‘17: a 9‘04? (em) —I—'_'_.—'—\—'_ ' 1 0.5 L3 Lee “‘1 Cam ’3’ [51' lane; 45,1 (33141:; '51? . «zit-n- ew 14m? Hi 5 in larval twe-l..'w.el_.&5- al.- ? W ME A F '1' ii beefimqble- 2w {emf— qfllel-JHM 15155;” Q-En The IECOVEW time [in hm} iron! a certain SUrgew is fiSStll'l'lnE'El to be approximately nnrntally disttihuted with [he Etflflflfitti deviatifln U — ti hrs With the present surgeryl technique the mean recovery time is £12 his. A new 5mg.er Iflfihllieut‘ claims In shorten the recovery time. To test this claim we IEHtSlOthl‘f seiect n patients In be. operated Is; the new technique and record their recovery trmes. We use the data for the Following statistical test: H” : gt T 4'2 HI :3: ~':' 41! [Here In refers to the recovery.» time uncler the netsI technique} it] Using the sample mean as the test statistic we want tn construct a Rejectinn Region {Ftth oi the {aim Hii' [.r' '. .t' «t it} Determine the 'u'alIJE Di .i' e I] such 1hai the significance level {size at the critical region} WI“ be u. =1].[}.‘3, {Hint-e} h} If n = 1} eaticntfi are tested . find .1‘ — PtType ll Ermr] ifthe true mean remvew time [under the new technique] is ,It : 313 his. [time] c] Find the power at the test when H. — Li patients are tested ancl the true recovery time under the new surgery teehnique is u — 4U hours. Unite} d] Usil'ig 33] and h} rlrew a rough S'ltEitcl'i Di J probabilities as a function Jillri] of the true mean u. [."iph'j a) 0:19.05 :Pf‘fes int-um salw flaw L: . k; {’L —r Edi l€«42_ fir )ffilfllfl _—e —ilé":l5 h E ifl J rm , t5 fizfl'fi'e l<'=§1-i.54’5' q .- "a 6 q-ii'le_ $25.5}; new: Hm $21.13;”wa 15 $1.11! I, a: (4'1, l,é4§'(-—-—--)B la) “at? Pésfizleflg‘firtflteeeél‘f‘flelePOKEW‘Hll‘flé) -: Pres M) = Prize-1355) mg I _ dilin “‘ng (DAMS; 441199-361”!le J le'viLeflesTl 5'0}in Prggresfisngmn Hem e"; “— We}??? N we} tree-i Ho lwvetfeée e ii:tgucet'zt?t":“—‘*”) L t :- Pf ages-4's?) 42:1th {amfienlee ft? : delta-.051 or mall-gag— ‘ifl'l’ii Ira-Ctfi'i'TflLIF‘ZJ. , time rinses-t ‘3' 0.11“: s ...
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This note was uploaded on 03/17/2012 for the course MATHEMATIC 306 taught by Professor U during the Spring '12 term at Sabancı University.

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math306_mid2 - SULUWMS TD MID'TEIZM 1F [2—1- A popuhtiun...

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