Topic 2: Distributions, hypothesis testing, and sample size determination

Transcription

1 Topc : Drbuo, hypohe eg, ad ample ze deermao. The Sude - drbuo [ST&D pp. 56, 77] Coder a repeaed drawg of ample of ze from a ormal drbuo. For each ample, compue,,, ad aoher ac,, where: ( ) The ac he devao of a ormal varable from hypohezed mea meaured adard error u. Phraed aoher way, he umber of adard error ha eparae ad µ. The drbuo of h ac kow a he Sude' drbuo. There a uque drbuo for each uque value of. For ample ze, he correpodg drbuo ad o have ( ) degree of freedom (df). Now coder he hape of he frequecy drbuo of hee ampled value. Though ( ) ymmerc ad appearg que mlar o a ormal drbuo of ample mea ( Z µ ), he drbuo wll be more varable (.e. have a larger dpero, or broader peak) ha Z becaue vare from ample o ample. The larger he ample ze, he more precely wll emae, ad he cloer approache Z. value derved from ample of ze 60 are approxmaely ormally drbued. Ad a à, à Z. ( ) µ ( ) ( ) Fgure Drbuo of (df 5 4) compared o Z. The drbuo ymmerc ad omewha flaer ha he Z, lyg uder a he ceer ad above he al. The creae he value relave o Z he prce we pay for beg ucera abou he populao varace

2 . Cofdece lm [S&T p. 77] Suppoe we have a ample {,..., } wh mea draw from a populao wh ukow mea µ, ad we wa o emae µ. If we mapulae he defo for he ac, we ge: µ + ( ) Noe ha µ a fxed bu ukow parameer whle ad are kow bu radom ac. The ac drbued abou µ accordg o he drbuo; ha, afe: P( - µ +,, ) - Noe ha for a cofdece erval of ze, you mu ue a value correpodg o a upper percele of / ce boh he upper ad lower percele mu be cluded (ee Fg. ). Coder he erm bracke above: - µ +,, The wo erm o eher de repree he lower ad upper ( - ) cofdece lm of he mea. The erval bewee hee erm called he cofdece erval (CI). For example, he barley mal exrac daa e, ad.7 / Table A.3 gve a correpodg (0.05,3) value of.6, whch we mulply by o coclude a he 5% level ha µ ± 0.7. Tha, µ ± 0.05, ±.6(0.379) ± 0.7 Wha doe h mea exacly? I correc o ay ha here a probably of 95% ha he rue mea wh he erval ± 0.7. The correc way o hk abou h: If we repeaedly drew radom ample of ze 4 from he populao ad coruced a 95% CI for each, we would expec 95% of hoe erval (9 ou of 0) o coa he rue mea. True mea Fgure The vercal le repree 0 95% cofdece erval. Oe ou of 0 erval doe o clude he rue mea (horzoal le). The cofdece level repree he perceage of me he erval cover he rue (ukow) parameer value [ST&D p.6].

3 . 3. Hypohe eg [ST&D p. 94] Aoher ue of he drbuo, more he le of expermeal deg ad ANOVA, hypohe eg. Reul of experme are uually o clear-cu ad herefore eed acal e o uppor deco bewee alerave hypohee. Recall ha h cae a ull hypohe H 0 eablhed ad a alerave H eed aga h. For example, we ca ue he barley mal daa (ST&D p. 30) o e H 0 : µ 78 aga H : µ 78. The defo of, a before: ( ) ( ) µ, wh ( ) So, for our example: ( ) / We rejec H o f he probably of drawg h ample from a populao wh µ 78 (afyg H 0 ) le ha ome pre-aged umber. Th pre-aged umber kow a he gfcace level or Type I error rae (). For h example, le' e From Table A.3, he crcal value aocaed wh h gfcace level (for 4) : (, ) 0.05 (,4).60 Sce our calculaed (- 6.8) larger, abolue value, ha he crcal value (0.05,3) (.60), we coclude ha he probably of correcly rejecg H 0 le ha Th mehod equvale o calculag a 95% cofdece erval aroud he ample mea ( ± 0.05,3 * ). I h example, he lower ad upper 95% cofdece lm for µ are [75.3, 76.65]. Sce 78 (H 0 ) o wh h cofdece erval, we rejec H 0. I oher word, a value of o expeced for a ample of 4 value from a populao wh µ 78 ad.7. Aga, h example, he value 0.05 called he gfcace level of he e ad deoed. I repree he probably of correcly rejecg H 0 whe acually rue, a Type I error. The oher poble error, a Type II error, o correcly accep H o whe fale [S&T p. 8]. The probably of h eve deoed β. The relaohp bewee hypohee, deco, ad error ca be ummarzed a follow: H 0 rejeced o rejeced rue Type I error Correc deco fale Correc deco Type II error 3

4 .3. Power of a e for a gle ample. For h dcuo, he ull hypohe H 0 : µ µ (.e. he paramerc mea of he ampled populao equal o ome value µ.) The magude of β drecly deped o oly o he choe bu alo o he acual dace bewee he wo mea uder coderao he ull hypohe. A he value of µ approache ha of µ, β creae o a maxmum value of ( - ). A mpora cocep coeco wh hypohe eg he power of a e (ST&D p.9): Power β P(Z > Z µ µ ) OR P( >, µ µ ) The fr equao for a populao wh kow ad he ecod for a populao wh ukow (.e. emae ). The ubraced erm he dfferece bewee he wo mea expreed u of adard error. Power he probably of rejecg H 0 whe fale ad he alerave hypohe correc. I h cae, a meaure of he ably of he e o dffereae wo mea. Suppoe, for example, ha he rue mea of he populao he ame a our calculaed Wha he power of a e for H 0 : µ 74.88? I oher word, wha our ably o declare ha o he mea of h ampled populao? Refer o Fgure 3. Sce 0.05 ad r 4, 0.05,3.60; ad Ug he prevou formula for ukow : Power β P( > ) P( > -.07) The fal ep he proce above, agg a value of 0.85 o he probably aeme, doe by referecg Table A.3. The Type II error (β) he haded area o he lef of he lower curve Fgure 3. I he probably of falg o rejec a fale H 0. 4

5 Fal o rejec H 0 Rejec H 0 / 74.7 ( *0.38) ( *0.38) H 0 True H 0 Fale β Power Fal o accep H Accep H Fg. 3. Type I ad Type II error he Barley daa e..3. Power of he e for he dfferece bewee he mea of wo ample (-e). For h dcuo, he ull hypohe H 0 : µ - µ 0, veru ) H : µ - µ 0 (woaled e), ) H : µ - µ < 0 (oe-aled e), or 3) H : µ - µ > 0 (oe-aled e). [Noe: Oe-aled e are almo ever approprae bac reearch, o we wll focu h eco o he wo-aled e oly.] The power formula ued above for a gle ample wh ukow hould be modfed for he dfferece bewee wo mea. Specfcally, he geeral power formula for boh equal ad uequal ample ze : Power P( > µ µ ) P( > µ µ ), pooled pooled where a weghed varace: pooled pooled ( ) + ( ( ) + ( ) ) 5

6 ad pooled +. Noce he pecal cae of equal ample ze (where ), he formula mplfy: ( ) + ( ) ( )( + ) + pooled ( ) + ( ) ( ) pooled + Power P( > µ µ ) P( > µ µ ) Take oe of wha h expreo dcae: The varace of he dfferece bewee wo radom varable he um of her varace (.e. error alway compoud) (ST&D 3-5). If he varace are he ame, he he varace of he derved varable (.e. he dfferece of he wo radom varable) * average. Th he uve explaao for he mulplcao by he formula of he adard error of he dfferece bewee he wo mea: pooled pooled The degree of freedom for he crcal / ac are: Geeral cae: ( -) + ( -) For equal ample ze: *(-) Summary of hypohe eg [S&T p. 9] Formulae a meagful, falfable hypohe for whch a e ac ca be compued. Chooe a maxmum Type I error rae (), keepg md he power ( - β) of he e. Compue he ample value of he e ac ad fd he probably of obag, by chace, a value more exreme ha ha oberved. If he compued, e ac greaer ha he crcal, abular value, rejec H 0. Somehg o coder: H 0 almo alway rejeced f he ample ze oo large ad almo alway o rejeced f he ample ze oo mall. 6

7 .4 Sample ze deermao.4. Facor affecg replcao There are may facor ha affec he deco of how may replcao of each reame hould be appled a experme. Some of he facor ca be pu o acal erm ad be deermed acally. Oher are o-acal ad deped o experece ad kowledge of he ubjec reearch maer or avalable reource for reearch.. Noacal facor: co ad avalably of expermeal maeral, he rera ad co of meaureme, he dffculy ad applcably of he expermeal procedure, he kowledge or experece of he ubjec maer of reearch, he objecve of he udy or eded ferece populao. Example: Ue exg formao o he expermeal u for beer amplg eleco. Suppoe everal ew varee are o be eed a dffere locao. Iead of radomly elecg, ay, 0 locao from 00 poble eg e, oe hould udy hee locao fr. Maybe poble o clafy hee 00 locao o a few dc clae of e baed o varou phycal or evromeal arbue of hee locao. Thu, may be approprae o chooe oe locao radomly from each cla o repree cera evromeal characerc for he vareal eg experme. Th example llurae he po ha order o deerme a uable umber of replcao, o uffce oly o fd a umber bu alo he propere of replcaed maeral ough o be codered. Example: The dered bologcal repoe dcae he ample ze. I a experme of eg reame dfferece, he mpora bologcal dfferece ha eed o be deeced mu be clearly uderood ad defed. A releva ample ze ca oly be deermed acally o aure ucce wh hgh probably, o deec h gfca dfferece. Thu, eg a herbcde effec o weed corol, wha coue a meagful bologcal effec? I 60% weed corol eough, or do you eed 90% weed corol for ayoe o care? Bewee wo compeve herbcde, 5% dfferece corollg weed mpora eough, or 0% eeded order o clam uperory of oe over he oher? I hould be poed ou, he maller he dfferece o be deeced, he larger he requred ample ze. A pove dfferece ca alway be how gfca acally, o maer how mall, by mply creag ample ze (of coure, aga, f he dfferece oo mall magude, o oe wll care). Bu a lack of dfferece bewee reame (H 0 ) ca ever be prove wh ceray, o maer how large he ample ze he udy.. Sacal facor: Sacal procedure o deerme he umber of replcao are prmarly baed o coderao of he requred preco of a emaor or requred power of a e. Some commoly ued procedure are decrbed he ex eco. Noe ha for a gve µ ad, f of he 3 quae, β, or he umber of obervao are pecfed, he he hrd oe ca be deermed. I choog a ample ze o deec a parcular dfferece, oe mu adm he pobly of Type I ad Type II error ad chooe accordgly. 7

8 We wll pree hree bac example:. Sample ze emao for buldg cofdece erval of cera legh (.4. ad.4.3);. Sample ze emao for comparg wo mea, gve Type I ad II error cora (.4.4); ad 3. Sample ze emao for buldg cofdece erval for adard devao ug he Ch-quare drbuo (.4.5)..4. Sample ze for he emao of he mea wh kow, ug he Z ac Where aeo coceraed o parameer emao raher ha hypohe eg, a procedure for deermg he requred umber of obervao avalable for couou daa. The problem o deerme he eceary ample ze o emae a mea by a cofdece erval guaraeed o be o loger ha a precrbed legh. If he populao varace kow, or f dered o emae he cofdece erval erm of he rue populao varace, he Z ac for he adard ormal drbuo may be ued. No al ample requred o emae he ample ze. The ample ze ca be compued a oo a he requred cofdece erval legh decded upo. Recall ha: Z µ, ad he cofdece erval CI ± Z /. y Le d repree he half-legh of he cofdece erval: d Z Z Th ca be rearraged o gve a expreo for, gve ad a dered d: Z d.4.3 Sample ze for he emao of he mea. Se' Two-Sage Sample [S&T p. 4] Whe he varace ukow ad he adard error ued ead of he paramerc varace, he drbuo hould replace he ormal drbuo. The addoal complcao geeraed by he ue of ha, whle he Z drbuo depede of, he drbuo o. Therefore, a erave approach requred. 8

9 Coder a ( - )% cofdece erval abou ome mea µ: - µ +,, The half-legh (d) of h cofdece erval herefore: d,, A he prevou eco, h formula ca be rearraged o fd he ample ze requred o emae a mea by a cofdece erval o loger ha d: Z, d d Se' procedure volve ug a plo udy o emae. Noce ha appear o boh de of h equao. Before olvg h equao va a erave approach, oe ha we may alo expre erm of he coeffce of varao (CV /, ST&D p. 6):, CV d Here (d / ) e he legh of he cofdece erval a a fraco of he emaed mea. For example (d / ) 0. mea ha he legh of d (half-legh of he cofdece erval) hould be o larger ha oe eh of he populao mea. Example [ST&D, p. 5] The dered maxmum legh of he 95% cofdece erval for he emao of µ 0 mm. A prelmary ample of he populao yeld value of, 9, 3,, ad 3 mm. Wh ( 0.05, 4 ) , 6.7, ad d 5 we ge: Therefore, we eed lghly more ha fve obervao. Sce here o uch hg a a fraco of a obervao, we coclude ha we eed x obervao. Now, f he reul much greaer ha he of he plo udy, we have a problem becaue he ' o he wo de of he equao wll o agree. I oher word, wll be oo far away from he umber of degree of freedom ued o deerme he ac. I uch cae, we mu erae ul he equao afed wh he ame value o boh de of he equal g. 9

10 Example : A expermeer wa o emae he mea hegh of cera maure pla. From a plo udy of 5 pla, he fd ha 0 cm. Wha he requred ample ze, f he wa o have he oal legh of a 95% cofdece erval abou he mea be o loger ha 5 cm? Ug, he ample ze emaed eravely:, d Ial 0.05, Calculaed (.776) (0) / (.96) (0) / Thu, wh 64 obervao, oe could emae he rue mea wh a preco 5 cm, a he gve. Noe ha f we ared wh a Z approxmao, he: Z / d (.96) (0) /.5 6, whch o oo far from he more exac emae, 64. I fac, oe may ue he Z approxmao a a hor-cu o bypa he fr few roud of erao, producg a good emae of o he refe ug he more approprae drbuo..4.4 Sample ze emao for he comparo of wo mea Whe eg he hypohe H 0 : µ µ, we hould alo ake o accou he pobly of a Type II error ad, hereby, he power of he e ( - β). To calculae β, we eed o kow eher he alerave mea or a lea he mmum dfferece we wh o deec bewee he mea (δ µ - µ. The approprae formula for compug, he requred umber of obervao from each reame, : ( / δ) (Z / + Z β) For he ypcal value 0.05 ad β 0.0, you fd Z/.96, Zβ 0.846, ad (Z/ + Zβ) So, o dcrmae wo mea ha are apar, approxmaely 4 replcao per reame are requred ( * (½) * ). If he wo mea are or 0.5 adard devao apar, he requred creae o 6 ad 63, repecvely. The expreo above ha everal obvou dffcule. The fr ha we rarely kow ad we cao hoely ue he Z ac. A approxmae oluo o mulply he reula by a fal "correco" facor: * (error d.f. +3) / (error d.f. +) [ST&D p. 3]; 0

11 where (error d.f.) (umber of par - ) for meagfully pared ample [ST&D p. 06] ad ( - ) for depede ample. The ame equao ad correco ca be ued o emae he umber of block a radomzed complee block deg aaly of varace [ST&D p. 4]. Aoher opo, he cae of depede ample, o emae wh he pooled ample adard devao pooled (ST&D p00 Eq.5.8) ad replace Z by : pooled + β, + δ, + Here, emaed eravely, a.4.3. If kow, h equao ca be ued o emae he power of he e hrough he deermao of β, +-. Aga, f o emae of avalable, he equao may be expreed erm of he coeffce of varao ad he dfferece δ bewee mea a a proporo of he mea: [(/µ) / (δ/µ)] (Z/ + Zβ) (CV / δ%) (Z/ + Zβ) The problem may alo be obvaed by defg δ erm of. For example, we mgh wa o deec a dfferece bewee mea of oe adard devao ze. I h cae, (Z/ + Zβ). Example: Two varee wll be compared for yeld, wh a prevouly emaed ample varace of.5. How may replcao are eeded o deec a dfferece of.5 o/acre bewee varee? Aume 5% ad β 0%. Le' fr approxmae he requred ug he Z ac: Approxmae (/δ) (Z / + Z β) (.5/.5) ( ) 5.7 We'll ow ue h reul a he arg po a erave proce baed o he ac, where ( / δ) (/ + β) : Ial df , 0.0, Calculaed The awer ha here hould be 7 replcao of each varey. I geeral, 7 ample per reame are eceary o dcrmae wo mea ha are oe adard devao apar, wh 5% ad β 0%.

12 .4.5 Sample ze o emae populao adard devao The Sude- drbuo ued o eablh cofdece erval aroud he ample mea a a way of emag he populao mea of a ormally drbued radom varable. The chquared drbuo ued a mlar way o eablh cofdece erval aroud he ample varace a a way of emag he populao varace The Ch- quare drbuo [ST&D p. 55]. q re F df 4 df 6 df Ch-quare Fgure 4 Drbuo of χ, for, 4, ad 6 degree of freedom. Relao bewee he ormal ad ch-quare drbuo: The reao ha he ch quare drbuo provde a cofdece erval for ha Z ad χ are relaed o oe aoher a mple way. If Z, Z,..., Z are radom varable from a adard ormal drbuo, he he um Z + Z Z ha a χ drbuo wh degree of freedom. χ defed a he um of quare of depede, ormally drbued varable wh zero mea ad u varace. Therefore: 5 6 χ, df Z, df, df e.g. χ 3. 84, Z , ad , Noe ha Z value from boh al of he N drbuo go o he upper al of he χ for d.f. becaue of he dappearace of he mu g he quarg.

13 3 Le' rewre h um of quared Z varable: Z ) ( µ If we emae he paramerc mea µ wh a ample mea, h expreo become: Z ) ( Recall ow he defo of ample varace: ) ( Therefore, ) ( ) ( Ad we fd: ) ( ) ( Z Th expreo, whch ha a χ - drbuo, mpora becaue provde a relaohp bewee he ample varace ad he paramerc varace Cofdece erval formula Baed o he expreo gve above, follow ha a ( - )% cofdece erval for he varace ca be derved a follow: Suppoe,...,, are radom varable draw from a ormal drbuo wh mea µ ad varace. We ca make he followg probablc aeme abou he rao (-) / : P{ χ -/, - ( - ) / χ /, -} - Smple algebrac mapulao of he quae wh he bracke yeld P {χ -/, - / ( - ) / χ /, - / ( - )} - OR P {( - ) / χ /, - ( - ) / χ -/, -} -

14 The fr form parcularly ueful whe he dered preco of ca be expreed erm of a proporo of. The ecod form approprae whe you have a acual emae of. For example, he barley daa from before,.5057 ad 4. If we le 0.05, he from Table A.5 we fd χ 5. 0 ad χ Therefore, he 95% cofdece erval for [ ] ,3 0.05,3 Example: Wha ample ze requred f you wa o oba a emae of ha you are 90% cofde devae o more ha 0% from he rue value of? Tralag h queo o aeme of probably: P (0.8 < / <.) 0.90 OR P (0.64 < / <.44) 0.90 hu χ -/, - / ( - ) 0.64 AND χ /, - / ( - ).44 Sce χ o ymmerc, he value of ha be afy he above wo cora may o be exacly equal f ample ze mall. However, mall ample geerally do o provde good emae of ayway. I pracce, he, he requred ample ze wll be large eough o avod h problem. The acual compuao volve a erave proce. our arg po a gue. Ju pck a al o beg; you wll coverge o he ame oluo regardle of where you beg. For h excere, we've choe a al of. df - / 95% / 5% (-) χ (-) χ (-) /(-) χ (-) χ (-) /(-) Wh h, we fd < 0.64 ad.57 >.44. To coverge beer o he dered value, we eed o ry a larger. Le' ry 4. df - / 95% / 5% (-) χ (-) χ (-) /(-) χ (-) χ (-) /(-) Now we have 0.66 > 0.64 ad.40 <.57. We eed o go lower. A you coue h proce, you wll eveually coverge o a "be" oluo. See he complee able o he ex page: 4

15 df - / 95% / 5% (-) χ (-) χ (-) /(-) χ (-) χ (-) /(-) Thu a rough emae of he requred ample ze approxmaely 35. 5