Recall, general format for all sampling distributions in Ch. 9: The sampling distribution of the sample statistic is approximately normal, with:
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1 Today: Fiih Chaper 9 (Secio 9.6 o 9.8 ad 9.9 Leo 3 ANNOUNCEMENTS: Qui #7 begi afer cla oday, ed Friday a oo. Jao Kramer will give he lecure o Friday. Pleae check your grade o eee ad le me kow if here are problem. HOMEWORK: (Due Friday Chaper 9: #66, 90, 9 Updae o he five iuaio we will cover for he re of hi quarer: Parameer ame ad decripio Populaio parameer Sample aiic For Caegorical Variable: [Doe!] Oe populaio proporio (or probabiliy p Differece i wo populaio proporio p p p For Quaiaive Variable: [Today, Fr, M] Oe populaio mea µ x Populaio mea of paired differece (depede ample, paired µ d d Differece i wo populaio mea µ (idepede ample µ x x For each iuaio will we: Lear abou he amplig diribuio for he ample aiic Lear how o fid a cofidece ierval for he rue value of he parameer Te hypohee abou he rue value of he parameer Recall, geeral forma for all amplig diribuio i Ch. 9: The amplig diribuio of he ample aiic i approximaely ormal, wih: Mea = populaio parameer (p, p p, µ, ec. Sadard deviaio = adard deviaio of ; he blak i filled i wih he aiic ( p,, x ec. Ofe he adard deviaio mu be eimaed, ad he i i called he adard error of. See ummary able o page for all deail! Today: Samplig diribuio for: oe mea mea differece for paired daa differece bewee mea for idepede ample Remember, wo ample are called idepede ample whe he meaureme i oe ample are o relaed o he meaureme i he oher ample. Could come from: Separae ample Oe ample, divided io wo group by a caegorical variable (uch a male or female Radomiaio io wo group where each ui goe io oly oe group Paired daa occur whe wo meaureme are ake o he ame idividual, or idividual are paired i ome way.
2 Samplig Diribuio for a Sample Mea (Secio 9.6 Suppoe we ake a radom ample of ie from a populaio ad meaure a quaiaive variable. Noaio: µ = mea for he populaio of meaureme. = adard deviaio for he populaio of meaureme. x = ample mea for a radom ample of idividual. = ample adard deviaio for he radom ample The amplig diribuio of he ample mea x i approximaely ormal, wih: Mea = populaio parameer = µ Sadard deviaio = adard deviaio of x = d..( x = Ofe he adard deviaio mu be eimaed, ad he i i called he adard error of x. Replace wih he ample adard deviaio, o e..( x = Suppoe we wa o eimae he mea weigh lo for he populaio of people who aed weigh lo cliic for 0 week. Suppoe he diribuio of weigh loe i approximaely ormal, µ = 8 poud, = 5 poud. (Empirical rule: ee picure Populaio of idividual weigh loe Deiy Weigh loe for 0 week cliic Normal, Mea=8, SDev= Number of poud lo (or gaied, if egaive value 3 We pla o ake a radom ample of 5 people from hi populaio ad record weigh lo for each pero, he fid ample mea x. We kow he value of he ample mea will vary for differe ample of = 5. How much will hey vary? Where i he ceer of he diribuio of poibiliie? Reul for four poible radom ample of 5 people, wih he correpodig ample mea x ad ample adard deviaio : Sample : x = 8.3 poud, = 4.74 poud. Sample : x = 6.76 poud, = 4.73 poud. Sample 3: x = 8.48 poud, = 5.7 poud. Sample 4: x = 7.6 poud, = 5.93 poud.
3 Noe: Each ample had a differe ample mea, which did o alway mach he populaio mea of 8 poud. Alhough we cao deermie wheher oe ample mea will accuraely reflec he populaio mea, aiicia have deermied wha o expec for all poible ample mea. µ = mea for populaio of iere = 8 poud = adard deviaio for populaio of iere = 5 poud. x = ample mea for a radom ample of idividual. The he amplig diribuio of x i approximaely ormal, wih Mea = µ Sadard deviaio =.d.( x = Example: Mea of 5 weigh loe, he diribuio of poible value i approximaely ormal wih: mea = 8 poud 5 adard deviaio = = 5 = poud Compare: idividual weigh lo, x for = 5, x for = 00 Idividual weigh lo Mea of 5 Mea of 00 Mea 8 poud 8 poud 8 poud S. Dev. 5 poud poud ½ poud Codiio for amplig diribuio of x o be approximaely ormal: Populaio (idividual value are approx. bell-haped OR Sample ie i large (a lea 30, more if oulier Comparig origial populaio wih amplig diribuio of x : Noe ha larger ample ie will reul i maller.d.( x Weigh lo for idividual, ad for mea of 5 idividual Normal, Mea=8 Compare amplig diribuio for = 5 ad = 00: Deiy SDev Samplig diribuio of he ample mea, = 5 ad = 00 Normal, Mea=8 = 00 SDev Deiy Weigh lo or average weigh lo From he empirical rule: 68% 95% 99.7% Idividual 3 o 3 poud - o 8 poud -7 o 3 poud Mea of = 5 7 o 9 poud 6 o 0 poud 5 o poud Mea weigh lo I oher word, for larger ample, x will be cloer o µ i geeral, ad hu will be a beer eimae for µ. = 5 0
4 Example where he origial populaio i o bell-haped: A bu ru every 0 miue. Whe you how up a he bu op, i could come immediaely, or ayime up o 0 miue. So he ime you wai for i i uiform, from 0 o 0 miue, ad idepede from day o day. 0 Populaio mea = µ = 5 miue, populaio.d. = = =.9 Wha i he amplig diribuio for x for = 40 day? Eve hough he origial ime are uiform (fla hape, he poible value of he ample mea x are: Approximaely ormal Mea = 5 miue.9 Sadard deviaio = = 40 = 0.46 miue Deiy Origial value ad amplig diribuio of mea for = Diribuio Mea SDev Normal Diribuio Lower Upper Uiform Waiig ime or mea waiig ime for = 40 Example of poible ample: x = 5.9 x = 4.3 Secio 9.7 ad 9.8: Samplig diribuio for mea of paired differece, ad for differece i mea for idepede ample Need o lear o diiguih bewee hee wo iuaio. Noaio for paired differece: d i = differece i he wo meaureme for idividual i =,,..., µ d = mea for he populaio of differece, if all poible pair were o be meaured d = he adard deviaio for he populaio of differece d = he mea for he ample of differece d = he adard deviaio for he ample of differece Example: IQ meaured afer lieig o Moar ad o ilece d i = differece i IQ for ude i for he wo codiio µ d = populaio mea differece, if all ude meaured (ukow d = he mea for he ample of differece = 9 IQ poi Baed o ample, we wa o eimae mea populaio differece Noaio for differece i mea for idepede ample: µ = populaio mea for he fir populaio µ = populaio mea for he ecod populaio Parameer of iere i µ µ = he differece i populaio mea x = ample mea for he ample from he fir populaio x = ample mea for he ample from he ecod populaio The ample aiic i x x = he differece i ample mea = populaio adard deviaio for he fir populaio = populaio adard deviaio for he ecod populaio = ample adard deviaio for he ample from he populaio = ample adard deviaio for he ample from he d populaio = ie of he ample from he populaio = ie of he ample from he d populaio
5 Example where idepede ample migh be ued: Compare weigh lo for me ad wome a he weigh lo cliic. Compare UCI ude wih ude from aoher campu o quaiaive meaure like hour pe udyig per week, icome, ec Compare umber of ick day off from work for people who had a flu ho ad people who did Compare chage i blood preure for people radomly aiged o a mediaio program or a exercie program for 3 moh. Example where paired daa migh be ued: Eimae average differece i icome for hubad ad wive Compare SAT core before ad afer a raiig program Weigh lo example ca be hough of a paired differece, wih weigh before ad weigh afer he program Noe ha paired differece are imilar o he oe mea iuaio, excep pecial oaio ell u ha he mea are for differece. Codiio for he amplig diribuio for hee wo iuaio are he ame a for a igle mea, wih a ligh wi: For paired differece, populaio of differece mu be bellhaped OR ample mu be large. For differece i mea for idepede ample, boh populaio mu be bell-haped OR boh ample ie mu be large. I boh cae, he amplig diribuio for he ample aiic i approximaely ormal, wih mea = populaio parameer of iere. d For paired differece:.d.( d = (ame a oe mea, bu wih d For differece i wo mea:.d.( x x = Sadardied Saiic: For all 5 cae i Chaper 9, a log a he codiio are aified for he amplig diribuio o be approximaely ormal, he adardied aiic for a ample aiic i: = ample aiic - populaio parameer.d.(ample aiic Example of weigh lo cliic. Suppoe i a ample of 5 clie he average weigh lo i 0. If populaio mea weigh lo i really 8 poud wih = 5 poud, how ulikely i a ample mea of 0 poud for = 5? Samplig diribuio for x-bar whe mu=8, igma=5, =5 Normal, Mea=8, SDev= Noe ha he deomiaor ha.d., o.e. For oe mea: x µ x µ ( x µ = = =. d.( x Probabiliy of x-bar beig 0 or le i 6.0E Poible value of x-bar
6 How o compue hi awer: Sude diribuio Sample mea for = 5 are approximaely ormal wih mea of 8 poud ad.d. of poud. So, he adardied core for 0 i: = 5(0 8 = 8 5 A -core of 8 i o very likely! So if we aw a 0 mea weigh lo, we would o believe ha he populaio mea i 8 poud! Noe: Whe i o kow, we mu ue he ample adard. e.( x = deviaio iead. Sadard error of x i I ha cae, adardied aiic ha a -diribuio, alo called Sude diribuio. I 908 William Sealy Goe figured ou he formula for he diribuio. Called Sude becaue explaied i cla! Sadardied Saiic Uig Sadard Error Uually we do kow (populaio adard deviaio, o we eed o ue (ample adard deviaio. I ha cae, he adardied aiic for x i x µ x µ ( x µ = = = e..( x / Thi ha a Sude diribuio wih degree of freedom = I look almo exacly like he ormal diribuio I i compleely pecified by kowig he df I ge cloer ad cloer o he ormal diribuio, ad whe degree of freedom = ifiiy, i i exacly he ormal diribuio. Compario of diribuio wih df = 5 ad adard ormal diribuio Sadardied aiic Sadard ormal diribuio diribuio wih df = 5 For example, middle 95% for wih df = 5 i.57 o.57 For adard ormal, i i abou o I Chaper (Friday we will lear how o fid probabiliie. 3 4
7 Summary of amplig diribuio for he 5 parameer (p. 38: The aiic ha a amplig diribuio. I i approximaely ormal if he ample( i (are large eough. The mea of he amplig diribuio = he parameer. The adard deviaio of he amplig diribuio i i he able below, i he colum adard deviaio of he aiic. Someime i eed o be eimaed, he adard error i ued. Parameer Saiic Sadard Deviaio of he Saiic Oe p ( p proporio p Differece p ( p p ( p Bewee p p p Proporio Oe Mea Mea Differece, Paired Daa Differece Bewee Mea µ x µd d µ µ x x d Sadard Error of he Saiic p ( ( p ( p d Sadardied Saiic wih.e. Parameer Saiic Sadard Deviaio of he Saiic Oe p ( p proporio p Differece Bewee Proporio Oe Mea Mea Differece, Paired Daa Differece Bewee Mea p p p ( p p ( p µ x d µ d d µ µ x x Sadard Error of he Saiic p ( ( p p( p d or? (wih.e.
Recall, general format for all sampling distributions in Ch. 9:
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