Introduction to Hypothesis Testing
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1 Noe for Seember, 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?
2 Noe for Seember, (. 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 amlig 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 : µ 98.6, i rue, how far i he aiic 98.3 from 98.6 i erm of adard core (or z-core)? Te Saiic : µ z µ 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 , he we rejec he ull hyohei. ( z.5.64 i alo called criical value) Rejecio Regio α Oe-ided Te: The deciio rule i baed oe ide of he amle diribuio Z
3 Noe for Seember, Level of igificace for he e (α) 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 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. 3
4 Noe for Seember, 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: Te Saiic : µ 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 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. 4
5 Noe for Seember, 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: µ 5 Ha: µ > 5. Which e ca be ued for eig he hyohei above? (Check aumio.) Oe Samle Z e, ice he amle i from a ormal oulaio wih kow adard deviaio. 3. Comue Te Saiic: Wrie dow he e aiic formula µ µ 6 5 Te Saiic : z The value of he e aiic i.78 wih -value Deciio Rule:.78 Secify a level of igificace, α, for he e. α.5. Criical value aroach: Rejec Ho if z >.64. P-value aroach: Rejec Ho if -value < Cocluio: Sice -value.7 <.5, (or z.78 >.64) we rejec he ull hyohei. The daa rovide ufficie evidece o uor he aleraive hyohei ha he average coamia level i hi locaio eceed 5 mg/kg.?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? wo-ided e 5
6 Noe for Seember, Wha would be he -value baed o he e aiic calculaed above for eig wheher he mea i differe from 5 mg/kg?.54 Wha would be he criical value baed o he e aiic calculaed above for eig wheher he mea i differe from 5 mg/kg? -.96 ad.96 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: µ 5 Ha: µ > 5. Which e ca be ued for eig he hyohei above? (Check aumio.) Oe amle -e. The radom amle wa from a ormal oulaio ad ukow variace.. Comue Te Saiic: µ Wrie dow he e aiic formula Te Saiic : The value of he e aiic i 4.3 wih a -value Deciio Rule: Secify a level of igificace, α, for he e. α.5. Criical value aroach: Rejec Ho if >.6. P-value aroach: Rejec Ho if -value < Cocluio: Sice -value.96 <.5, (or 4.3 >.6) we rejec he ull hyohei. The daa rovide ufficie evidece o uor he aleraive hyohei ha he average coamia level i hi locaio eceed 5 mg/kg. 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. 6
7 Noe for Seember, Cofidece Ierval Eimae Work Shee. Ue he cofidece level o fid he cofidece coefficie: The give cofidece level i.95_ α, he α.5, α/ _.5_. Cofidece coefficie.96. Fid he amle mea 6. If oulaio adard deviaio i o give he fid adard deviaio The cofidece ierval for mea i 6 ± 7.6 (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.95_ α, he α.5, α/ _.5. Cofidece coefficie.57. Fid he amle mea If oulaio adard deviaio i o give he fid adard deviaio.7 3. The cofidece ierval for mea i 63.9 ± 7.8 (Ue he cofidece ierval eimae formula.) 7
8 Noe for Seember, 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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