Introduction to Hypothesis Testing
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- Laurence Cook
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1 Iroducio o Hyohei Teig Iroducio o Hyohei Teig Scieific Mehod. Sae a reearch hyohei or oe a queio.. Gaher daa or evidece (obervaioal or eerimeal) o awer he queio. 3. Summarize daa ad e he hyohei. 4. Draw a cocluio. Saiical Hyohei Null hyohei (H ): Hyohei of o differece or o relaio (or o guily) ad ofe ha =,, or oaio i he mahemaical aeme of he hyohei. A heory abou he value of oe (or more) oulaio arameer(). The heory geerally reree he au quo, which we acce uil i i rove fale. Eamle: H : µ = 98.6F (average body emeraure i 98.6) Aleraive hyohei (H a ): Uually correod o reearch hyohei ad ooie o ull hyohei, (or guily) ad ha >, < or oaio i he mahemaical aeme of he hyohei. We acce hi hyohei oly whe ufficie evidece ei o uor i. Eamle: H a : µ < 98.6F (average body emeraure i le ha 98.6) Logic behid he hyohei eig: The jury rial of a accued murderer i aalogou o he aiical hyohei roce. The ull hyohei i a jury rial i ha he accued i ioce. The au quo hyohei i he US yem of juice i ha he accued i iocece, which i aumed o be rue uil rove beyod reaoable doub. I eig aiical hyohei, he ull hyohei i fir aumed o be rue. We collec evidece o ee if i i rog eough o rejec he ull hyohei ad, herefore, uor he aleraive hyohei. I ca be doe by uig our kowledge of he amlig diribuio of he e aiic baed he aumio ha he ull hyohei i rue, ad he ue he oberved value of he e aiic o ee wheher i i ereme eough (oo far away from mea uder ull hyohei) o rejec ull hyohei. Te Saiic: A amle aiic ued o decide wheher o rejec he ull hyohei. Se i hyohei eig. Sae hyohee. H ad H a.. Chooe a roer e aiic, collec daa ad comue he value of he aiic. Thi iclude checkig he aumio abou he amled oulaio ad he amlig rocedure. 3. Make deciio rule baed o level of igificace. Do we rejec or fail o rejec ull hyohei? 4. Draw cocluio. Oe-amle e for oulaio mea wih kow variace. Oe wihe o e wheher he average body emeraure for healhy adul a a regular evirome i le ha 98.6 F or o. Aume body emeraure for healhy adul uder regular evirome ha a ormal diribuio wih a adard deviaio of.4 F. A radom amle of 6 i choe wih a mea 98.3F. Wha doe hi ay abou he hyohei ha he average body emeraure for healhy adul a ormal evirome i le ha 98.6 F? Te he hyohei a a level of igificace.5. Oe-ided Te (. Sae hyohei) H : µ = 98.6 (or µ 98.6) H a : µ < 98.6 Wha will be he key aiic ha you would ue for hi iuaio? How hould we decide wheher he evidece i covicig eough? If ull hyohei i rue, wha i he amlig diribuio of he mea? A. Chag
2 Iroducio o Hyohei Teig (. Chooe a e, collec daa ad comue aiic) From reviou chaer, we kow ha he amlig diribuio of he mea of a radom amle of ize 6 amled from a ormal oulaio i ormal. If he ull hyohei i rue, he mea of he amlig diribuio i 98.6 ad he adard error i.4/4. 4 i he deomiaor i quare roo of amle ize, 6. If he ull hyohei, H : m = 98.6, i rue, how far i he aiic 98.3 from 98.6 i erm of adard core (or z-core)? - µ - µ Te Saiic : z = = = = = Thi imlie ha he aiic i.8 adard deviaio away from he mea 98.6 i H. I i ereme eough o covice u ha he average body emeraure i le ha 98.6? I i likely o occur if he ull hyohei i rue? Wha i he robabiliy ha he amle mea i le ha or equal o 98.3 uder ull hyohei? -value = P ( Z -.8) =. 3 (area o he lef of.8) Samlig diribuio Uder H Sadardized diribuio Z -.8 (3. Defie deciio rule) -value aroach: Comare -value wih he redeermied igificace level α. If hi robabiliy (-value) i le ha α=.5, he we rejec he ull hyohei. (The maller he - value he roger he evidece i o rejec ull hyohei.) Criical value aroach: Comare he e aiic wih he criical value defied by igificace level α. If he e aiic i le ha -z α = -z.5 = -.64, he we rejec he ull hyohei. ( z.5 =.64 i alo called criical value) = Rejecio Regio α=.5 Oe-ided Te: The deciio rule i baed oe ide of he amle diribuio Z A. Chag
3 Iroducio o Hyohei Teig Level of igificace for he e (a ) A robabiliy level eleced by he reearcher a he begiig of he aalyi ha defie ulikely value of amle aiic if ull hyohei i rue. ' -value ' The robabiliy of obaiig a e aiic more ereme ha acual amle aiic value give ull hyohei i rue. I i a robabiliy ha idicae he eremee of evidece agai H. (4. Draw cocluio) Coclude wheher he evidece uor he aleraive (reearch) hyohei or o. Sice from eiher criical value or -value aroach, we rejec ull hyohei. Therefore, here i ufficie evidece o uor he aleraive hyohei ha he average body emeraure i le ha 98.6 F. Poible aiical error i hyohei eig Tye I error: The ull hyohei i rue, bu we rejec i. Tye II error: The ull hyohei i fale, bu we do rejec i. Two-ided Te Ue he ame daa above o e wheher he average body emeraure i differe from 98.6 F. (. Sae hyohei) H : µ = 98.6 H a : µ 98.6 (. Chooe a e, collec daa ad comue aiic) Check if daa came from a ormal diribuio Te Saiic : z = = = value = P ( Z -.8 or Z.8) =.3 =.6 (area o he righ of.8 ad o he lef of.8) Samlig diribuio of Z.3.3 (3. Defie deciio rule) Z -value aroach: Comare -value wih he redeermied igificace level α. If hi robabiliy (-value) i le ha α=.5, he we rejec he ull hyohei. (The maller he - value he roger he evidece i o rejec ull hyohei.) Criical value aroach: Comare he e aiic wih he criical value defied by igificace level α. If he e aiic i le ha -z α/ = -z.5 = -.96 or greaer ha z.5 =.96, we rejec he ull hyohei. ( z.5 =.96 ad z.5 =.96 are boh criical value.) Rejecio Regio Rejecio Regio Z Two-ided Te: The deciio rule i baed boh ide of he amle diribuio. (4. Draw cocluio) We rejec ull hyohei. Why? Therefore, here i ufficie evidece o uor he aleraive hyohei ha he average body emeraure i differe from 98.6 F. A. Chag 3
4 Iroducio o Hyohei Teig Oe-amle e for oulaio mea wih ukow variace. I racice, oulaio variace i ukow mo of he ime. The amle adard deviaio i ued iead for. If he radom amle of ize i from a ormal diribued oulaio ad he ull hyohei i rue, he e aiic (adardized amle mea) will have a -diribuio wih degree of freedom. Te Saiic: = - µ Oe-ided Te Eamle If we have a radom amle ha ha a mea of 98. ad.4 i a amle adard deviaio. The e aiic will be a e aiic ad he value will be: - µ Te Saiic : = = = = Uder ull hyohei, hi -aiic ha a -diribuio wih degree of freedom, ha i, 5 = 6. Rejecio regio To e he hyohei a α level.5, he criical value i α =.5 =.753. For e, -value ca oly be aroimaed wih a rage becaue of he limiaio of -able. -value <. = P(T<.6) Sice he area o he lef of.6 i., he area o he lef of.8 i defiiely le ha Deciio Rule: If < -.753, we rejec he ull hyohei, or if -value <.5, we rejec ull hyohei. Cocluio: Sice = -.8 < -.753, or ay -value <. <.5, we rejec he ull hyohei. There i ufficie evidece o uor he reearch hyohei ha he average body emeraure i higher ha 98.6 F. If he amle ize i relaively large (>3) boh z ad e ca be ued for eig hyohei. The umber 3 i ju a referece for racicig roblem. I racice, if he amle i from a very kewed diribuio, we eed o icreae he amle ize or ue oarameric aleraive. May commercial ackage oly rovide e ice adard deviaio of he oulaio i ofe ukow. A. Chag 4
5 Iroducio o Hyohei Teig A radom amle of oe hudred 4-gram oil ecime were amled i locaio A ad aalyzed for cerai coamia. The amle reul i a mea coamia level of 6 mg/kg ad a adard deviaio of 36 mg/kg. Te he hyohei, a he level of igificace.5, ha he rue mea coamia level i hi locaio eceed 5 mg/kg. Hyohei Teig Work Shee. Wha i he hyohei o be eed? Ho: Ha:. Which e ca be ued for eig he hyohei above? (Check aumio.) 3. Comue Te Saiic: Wrie dow he e aiic formula. The value of he e aiic i ad -value i. 4. Deciio Rule: Secify a level of igificace, α, for he e. α =. Criical value aroach: Rejec Ho if. P-value aroach: Rejec Ho if. 5. Cocluio:?Wha if we wih o e wheher he mea i differe from 5 mg/kg? I i goig o be a oe-ided e or wo-ided e? Wha would be he -value baed o he e aiic calculaed above for eig wheher he mea i differe from 5 mg/kg? Wha would be he criical value baed o he e aiic calculaed above for eig wheher he mea i differe from 5 mg/kg? A. Chag 5
6 Iroducio o Hyohei Teig A radom amle of e 4-gram oil ecime were amled i locaio A ad aalyzed for cerai coamia. The amle daa are he followig: 65, 54, 66, 7, 7, 68, 64, 5, 8, 49 The coamia level are ormally diribued. Te he hyohei, a he level of igificace.5, ha he rue mea coamia level i hi locaio eceed 5 mg/kg. Hyohei Teig Work Shee. Wha i he hyohei o be eed? Ho: Ha: 5. Which e ca be ued for eig he hyohei above? (Check aumio.) 6. Comue Te Saiic: Wrie dow he e aiic formula. The value of he e aiic i ad -value i.. 7. Deciio Rule: Secify a level of igificace, α, for he e. α =. Criical value aroach: Rejec Ho if. P-value aroach: Rejec Ho if. 5. Cocluio: A. Chag 6
7 Iroducio o Hyohei Teig A radom amle of oe hudred 4-gram oil ecime were amled i locaio A ad aalyzed for cerai coamia. The amle reul i a mea coamia level of 6 mg/kg ad a adard deviaio of 36 mg/kg. Fid he 95% cofidece ierval eimae for he mea coamia level i locaio A. Cofidece Ierval Eimae Work Shee. Ue he cofidece level o fid he cofidece coefficie: The give cofidece level i = α, he α =, α/ =. Cofidece coefficie =. Fid he amle mea. If oulaio adard deviaio i o give he fid adard deviaio 3. The cofidece ierval for mea i ± (Ue he cofidece ierval eimae formula.) A radom amle of e 4-gram oil ecime were amled i locaio A ad aalyzed for cerai coamia. The amle daa are he followig: 65, 54, 66, 7, 7, 68, 64, 5, 8, 49 The coamia level are ormally diribued. Fid he 95% cofidece ierval eimae for he mea coamia level i locaio A. Cofidece Ierval Eimae Work Shee. Ue he cofidece level o fid he cofidece coefficie: The give cofidece level i = α, he α =, α/ =. Cofidece coefficie =. Fid he amle mea. If oulaio adard deviaio i o give he fid adard deviaio 3. The cofidece ierval for mea i ± (Ue he cofidece ierval eimae formula.) A. Chag 7
8 Iroducio o Hyohei Teig A. Chag 8 Saiic Formula Shee Cofidece Ierval Eimae: z-cofidece ierval: ± α z -cofidece ierval: ± α (d.f. = ) Cofidece ierval for roorio: ˆ ± α z ) ˆ ( ˆ Cofidece ierval for differece of wo mea: z α ± Cofidece ierval for differece of wo mea: If =, ± ) ( ) ( α, d.f. = If, ± α, d.f. = mi(, ) Hyohei Teig: z-e aiic: z µ = -e aiic: µ = (d.f. = ) z-e for roorio: z = ) ( ˆ z-e aiic for wo mea: D z = -e aiic for wo mea: If =, = ) ( ) ( D, d.f. = If, D =, d.f. = mi(, ) [rough aroimae]
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