Lecure 3 Topc : Drbuo, hypohe eg, ad ample ze deermao The Sude - drbuo Coder a repeaed drawg of ample of ze from a ormal drbuo of mea. For each ample, compue,,, ad aoher ac,, where: The ac he devao of a ormal varable from hypohezed mea meaured adard error u. For ay gve value of, cr alway larger ha Z cr. Th he prce we pay for beg ucera abou he populao varace
Cofdece lm baed o ample ac Takg o accou he mperfec formao provded by amplg, he emaed value of ay populao parameer λ ake he geeral form: λ Emaed λ ± Crcal Value * Sadard error of he emaed λ So, for a populao mea emaed va a ample mea: ±, The ac drbued abou accordg o he drbuo, afyg: P -, +, - 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. - CI for [ -, +, ], Example: Daa e of 4 barley mal exrac value 75.94,.7 / 4 0.379. By -able or R: 0.05,3. 6, 95% CI for 75.94 ±.60.379 75.94 ± 0.7 [75.3, 76.65] 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
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Hypohe eg ad power Example: Daa e of 4 barley mal exrac value 75.94,.7 / 4 0.379.. Chooe a ull hypohe: Te H 0 : 78 veru H : 78.. Chooe a gfcace level: Ag 0.05. 3. Calculae he e ac: 75.94 78.00 6.8 0.379 4. Compare he abolue value of he e ac o he crcal ac: - 6.8 >.6 5. Sce he abolue value of he e ac larger, we rejec H 0. Th equvale o calculag a 95% cofdece erval aroud. Sce 78 H 0 o wh he 95% CI [75.3, 76.65], we rejec H 0. H 0 rejeced o rejeced rue Type I error Correc deco fale Correc deco Type II error gfcace level Type I error rae he probably of correcly rejecg a rue H 0 β Type II error rae he probably of falg o rejec a fale H 0 Power β he probably of correcly rejecg a fale H 0 4
Power of a e for a gle ample Power β PZ > Z 0 OR P >, 0 Example: Ug he ame barley daa, wha he power of a e for H 0 : 74.88? Aga, 0.05, r 4, 0.05,3.60, ad 0.3795. Power β P > 75.94 74.88.60 P > -.07 0.85 0.3795 The magude of β deped upo:. The Type I error rae. The acual dace bewee he wo mea uder coderao 3. The umber of obervao à For a gve ad deeco dace, f ay wo of he quae, β, or are pecfed, he hrd deermed. Ue uffce replcao o keep Type I ad Type II error uder her dered lm. 5
Fal o rejec H 0 Rejec H 0 / 74.7 74.88 -.60*0.38 74.88 75.588 74.88 +.60*0.38 H 0 True H 0 Fale β Power 75.94 Fal o accep H Accep H Fg. 3. Type I ad Type II error he Barley daa e. H 0 almo alway rejeced f he ample ze oo large ad almo alway o rejeced f he ample ze oo mall. 6
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8 Power of he e for he dfferece bewee he mea of wo ample H 0 : - 0, veru: H : - 0 wo-aled e H : - < 0 or H : - > 0 oe-aled e The geeral power formula for boh equal ad uequal ample ze : pooled pooled P P Power > > where pooled a weghed varace: + + pooled ad pooled + I he pecal cae of equal ample ze where, he formula mplfy: pooled + pooled + + + + P P Power pooled > > The varace of he dfferece bewee wo radom varable he um of her varace.e. error alway compoud. The degree of freedom for he crcal / ac are: Geeral cae: - + - For equal ample ze: *-
Sample ze for emag, whe kow ug he Z ac If he populao varace kow, or f dered o emae he cofdece erval erm of he rue populao varace, he Z ac may be ued. Z, ad 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 : Z d For adard 0.05, Z Z 0.05.96 3.84 d d d d If d, 4 If d 0.5, 6 If d 0.5, 64 Th equao ca be re-expreed erm of he coeffce of varao: Z d Z CV d Example: The CV of yeld ral a our expermeal ao are ever greaer ha 5%. How may replcao are eeded o coruc a 95% CI for he rue mea wh a oal legh of o more ha 0% of he rue mea? d 0.0, o d 0.05.96 0.5 / 0.05 34.6 35 9
Sample ze for emag, whe ukow Coder a - % cofdece erval abou ome mea : + -, The half-legh d of h cofdece erval herefore:, d,, Z, d d Se' Two-Sage procedure volve ug a plo udy o emae. Example: A breeder 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 5.776.776 0 /.5 3.3 4.96.96 0 /.5 6.5 6.00 64 64.00 64 Thu, wh 64 obervao, oe could emae he rue mea wh a preco of 5 cm, a he gve. Noe ha f we ared wh a Z approxmao, he: Z / d.96 0 /.5 6 0
Sample ze emao for he comparo of wo mea Whe eg he hypohe H 0 :, we ca ake o accou he poble of Type I ad Type II error mulaeouly. To calculae, we eed o kow eher he alerave mea or a lea he mmum dfferece we wh o deec bewee he mea δ -. The approprae formula for compug, he requred umber of obervao from each reame, : / δ Z / + Z β For 0.05 ad β 0.0: Z /.96, Z β 0.846, ad Z / + Z β 7.849 8 If δ, 4 If δ, 6 If δ 0.5, 64 We rarely kow ad mu emae va ample varace: pooled + β, + δ, +, where + pooled Here, emaed eravely. If o emae of avalable, he equao may be expreed erm of he CV ad he dfferece δ a a proporo of he mea: We ca alo defe δ erm of. [/ / δ/] Z / + Z β CV / δ% Z / + Z β Example: Two varee are 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%. Approxmae /δ Z / + Z β.5/.5.96+0.846 5.7 Ial df - 0.05, 0.0, Calculaed 6 30.04 0.854 6.8 7 3.037 0.853 6.7 The awer ha here hould be 7 replcao of each varey.
For he ereed: Sample ze o emae populao adard devao The ch-quared drbuo ued o eablh cofdece erval aroud he ample varace a a way of emag he rue, ukow populao varace. The Ch- quare drbuo [ST&D p. 55]. q re F 0.5 0.4 0.3 0. df 4 df 6 df 0. 0.0 3 4 Ch-quare 5 6 The drbuo wh df defed a he um of quare of depede, ormally drbued varable wh zero mea ad u varace. df Z, df Z.96 3., df, df e.g. 3. 84, Z 84, ad.96 3. 84 0.05,
3 Reumg Z If we emae he paramerc mea wh a ample mea, we oba: Z due o: à Th expreo, whch ha a - drbuo, provde a relaohp bewee he ample varace ad he paramerc varace. Cofdece erval for We ca make he followg probablc aeme abou he rao - / :,, P Smple algebrac mapulao of he quae wh he bracke yeld,, P OR,, P
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 -/, - / - 0.64 AND /, - / -.44 df - / 95% / 5% - - - /- - - /- 0 0.90 0.545 3.4.57 4 40 6.50 0.66 55.8.40 3 30 8.50 0.66 43.8.46 36 35.46 0.64 49.8.4 35 34.66 0.637 48.6.43 Thu a rough emae of he requred ample ze approxmaely 35. 4