Review - Week 10. There are two types of errors one can make when performing significance tests:
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1 Review - Week Read: Chaper -3 Review: There are wo ype of error oe ca make whe performig igificace e: Type I error The ull hypohei i rue, bu we miakely rejec i (Fale poiive) Type II error The ull hypohei i fale, bu we fail o rejec i (Fale egaive) The probabiliy ha a hypohei e will correcly rejec a fale ull hypohei i he power of he e To fid power, we mu pecify a paricular aleraive parameer value a he rue value The differece bewee he ull hypohei value ad he rue value of a model parameer i he effec ize Typically reducig he Type I error lead o a icreae i he Type II error The oly way o reduce boh ype of error i by icreaig he ample ize Exercie : The ower of a ore i cocered ice oly 3% of people who eer hi ore make a purchae Hi aia hik hi price are oo high ad ugge ha he give a 5% rebae o all purchae for a wo-week rial period The ower cou he umber of hopper ha eer he ore ad how may make a purchae, o deermie if here ha bee a igifica icreae i he proporio of people who make purchae, i which cae he will make he rebae permae (a) Help he ower formulae he appropriae ull ad aleraive hypohee (b) I hi coex, wha i a Type I error ad wha impac would uch a error have o he ore? (c) I hi coex, wha i a Type II error ad wha impac would uch a error have o he ore? (d) I hi coex, wha i a mea by he power of he e? (e) The ower decide o exed he rial period o la for oe moh o ha he ca gaher more daa Will he power of he e icreae, decreae or ay he ame?
2 Review: We ofe wa o compare he proporio of people, i wo populaio or group, which have ome characeriic Suppoe we ake ample of ize ad from each of he wo populaio ad obai he ample proporio p ad p, repecively To compare he wo populaio proporio we ue he differece bewee he ample proporio: D = p p Whe ad are large, D i approximaely N p p, A level C cofidece ierval for p p i p ( p ) p ( p ) ( p p ) ± z p ( p) p ( p) where z i he value for he adard ormal deiy curve wih area C bewee z ad z A igificace e for H : p p = ue he wo-proporio z-aiic z = ( p p ) p( p) wih P-value from he N(,) diribuio Here p i he pooled eimae of he commo value of p ad p, which i give by p X X = Exercie : Radom ample of ize = 55 ad = 63 were draw from Populaio ad, repecively The ample yielded X = 5 ad X = 5 (a) Coruc a 9% cofidece ierval for p p (b) Coruc a 99% cofidece ierval for p p (c) Te H : p p = agai H a : p p > Ca you rejec H a he 5% level of igificace?
3 Exercie : A SRS of adul America were aked if hey uppored a cerai propoal up for debae i Cogre Of he 8 people ha were aked heir opiio, 36 uppored he propoal while he re oppoed i I a follow up urvey performed a moh laer, 9 people were aked heir opiio ad 46 people replied ha hey uppored he propoal (a) I he differece bewee he wo poll igifica a he % level? Coruc a appropriae e of hypohei (b) Coruc a 95% cofidece ierval for he differece bewee he wo poll (c) Coruc a 9% cofidece ierval for he differece bewee he wo poll (d) Judgig by he cofidece ierval i par (b) ad (c), i he differece bewee he poll igifica a he 5% ad % level?
4 Review : σ The ample mea, y, follow a N µ, model We ca eimae σ uig he ample adard deviaio, = i= ( y i y) The adard error of he ample mea i SE ( y) = Whe σ i ukow, we eed o ue he adard error i place of he adard deviaio ad he reulig adardized ample mea = y µ follow a -model wih - degree of freedom The -model are ymmeric, igle-peaked, ad bell haped They are imilar i hape o he adard ormal model, hough heir pread i lighly larger A he degree of freedom icreae, he -model become icreaigly imilar o a ormal model A level C cofidece ierval for µ i y ± where i he value uder he model wih area C bewee ad A igificace e for he aeme H : µ = µ i baed o he aiic: y µ = where i he ize of he imple radom ample If <5 ue -procedure oly if he daa are cloe o ormal If 5 <4, ue -procedure a log a here are o oulier or rog kewe If 4 he -procedure ca be ued eve for clearly kewed diribuio
5 Exercie : A radom ample of obervaio i draw from a ormal populaio wih mea µ Specify he criical value eeded o coruc a cofidece ierval wih cofidece level C for each of he followig choice of C ad (a) C = 99%, =5 (b) C = 99%, =3 (c) C = 9%, =9 (d) C = 95%, = Exercie : Suppoe a SRS of ize i made from a ormally diribued populaio wih mea µ ad adard deviaio σ, boh ukow Uig he obaied daa we were able o calculae y = 987 ad = (a) Fid a 9% Cofidece Ierval for µ (b) Fid a 95% Cofidece Ierval for µ (c) Fid a 99% Cofidece Ierval for µ Exercie 3: Suppoe we e he ull hypohei H : µ agai he aleraive H : µ >, = for a ormal populaio wih mea µ ad adard deviaio σ, boh ukow A SRS of 5 obervaio i draw from he populaio Uig he obaied daa we were able o calculae y = 6 ad = 5 8 Would you rejec H a he 5% level of igificace? a
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