Recall, general format for all sampling distributions in Ch. 9:

Transcription

1 Today: Fiih Chaper 9 (Secio 9.6 o 9.8 ad 9.9 Leo 3) ANNOUNCEMENTS: Quiz #7 begi afer cla oday, ed Moday a 3pm. Quiz #8 will begi ex Friday ad ed a 0am Moday (day of fial). There will be clicker queio i all lecure ex week. The la homework aigme (#8) will be from he lecure o Mo ad Wed ad will be due ex Friday. Problem o be aiged ex Friday are already o he web, wih oluio, ad are o help you review ha maerial (o o had i). Review for he fial exam i poed ad will be covered i dicuio ecio ex Friday (o i cla). Two file: o Maerial ice d miderm o Cocep from he quarer ha eed exra review HOMEWORK: (Due Moday, March ) Chaper 9: #68, 7, 46 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 pˆ Differece i wo populaio proporio p p pˆ ˆ p For Quaiaive Variable: [Today, M, W] 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: Aumig ample ize codiio are me, he amplig diribuio of he ample aiic: I approximaely ormal Mea = populaio parameer (p, p p, μ, ec.) Sadard deviaio = adard deviaio of ; he blak i filled i wih he aiic ( p ˆ, pˆ pˆ, x ec.) Ofe he adard deviaio mu be eimaed, ad he i i called he adard error of. See ummary able o page 353 for all deail! Today: Samplig diribuio for mea of quaiaive daa: 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) Radomizaio 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 ize from a populaio ad meaure a quaiaive variable. Noaio for Populaio (ue Greek leer): μ = mea of he populaio of meaureme. σ = adard deviaio of he populaio of meaureme. Noaio for Sample: x = ample mea of a radom ample of idividual. = ample adard deviaio of he radom ample The amplig diribuio of he ample mea x i: approximaely ormal Mea = populaio parameer = μ Sadard deviaio = adard deviaio of x = d..( x) Ofe he adard deviaio of x mu be eimaed, ad he i i called he adard error of x. Replace populaio σ wih ample adard deviaio, o e..( x) Coider he mea weigh lo for he populaio of people who aed weigh lo cliic for 0 week. Suppoe he populaio of idividual weigh loe i approximaely ormal, μ = 8 poud, σ= 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 of populaio of iere = 8 poud σ = adard deviaio of populaio of iere = 5 poud. x = ample mea of 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 Wha if = 00 iead of 5? Compare: idividual weigh lo, x for = 5, x for = 00 Idividual (w 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 ize i large (a lea 30, more if oulier) Comparig origial populaio wih amplig diribuio of x : Noe ha larger ample ize will reul i maller.d.( x ) Weigh lo for idividual, ad for mea of 5 idividual Normal, Mea=8 Compare amplig diribuio of x 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 lb o 8 lb 7 o 3 lb Mea of = 5 7 o 9 lb 6 o 0 lb 5 o lb Mea weigh lo I oher word, for larger ample, he ample mea 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 ay ime 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. = σ = =.9 Wha i he amplig diribuio of 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 of x = 5 miue.9 Sadard deviaio of x = 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 of he populaio of differece, if all poible pair were o be meaured σ d = he adard deviaio of he populaio of differece d = he mea of he ample of differece d = he adard deviaio of he ample of differece Example: IQ meaured afer lieig o Mozar 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 of 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 of he fir populaio µ = populaio mea of he ecod populaio Parameer of iere i µ µ = he differece i populaio mea x = ample mea of he ample from he fir populaio x = ample mea of he ample from he ecod populaio The ample aiic i x x = he differece i ample mea σ = populaio adard deviaio of he fir populaio σ = populaio adard deviaio of he ecod populaio = ample adard deviaio of he ample from he populaio = ample adard deviaio of he ample from he d populaio = ize of he ample from he populaio = ize of he ample from he d populaio

5 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 Differece bewee wha you eared i 0 ad wha you hope o have a your arig alary whe you graduae. Noe ha paired differece are imilar o he oe mea iuaio, excep pecial oaio ell u ha he mea are of he differece. Example where idepede ample migh be ued: Compare hour of udy for me ad wome i our cla. 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 o a exercie program for 3 moh. 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 ize mu be large. I boh cae, he amplig diribuio of he ample aiic i approximaely ormal, wih mea = populaio parameer of iere. For paired differece: d mea = μ d,.d.( d ) = For differece i wo mea: mea = μ μ,.d.( x x ) = (ame a oe mea, bu wih d ) Sadardized Saiic: For all 5 cae i Chaper 9, a log a he codiio are aified for he amplig diribuio o be approximaely ormal, he adardized aiic for a ample aiic i: z ample aiic - populaio parameer.d.(ample aiic) Noe ha he deomiaor ha.d., o.e. For oe mea: x x ( x ) z. d.( x) Ex: Sa 7, Wier 0, Hour of udy per week for he cla Speculaio over he log ru i ha ude udy a average of abou 5 hour per week for Saiic 7. Have daa from 64 ude i Wier 0, wih ample mea of 5.36 hour. Suppoe for he populaio of all poible Saiic 7 ude (o ju i Wier 0), populaio mea = μ = 5 hour a week, ad populaio adard deviaio σ = 4 hour a week (hey are o bell-haped defiiely kewed o he righ). For urvey, = 64. Wha are poible value of x (amplig diribuio of x )? Approximaely ormal Mea = 5 hour 4 d..( x)

6 I our ample, = 64 ad x Sadardized aiic for 5.36 hour i z = If populaio mea i really 5 hour, wih σ = 4 hour, how ulikely i a ample mea of 5.36 hour or more for = 64? Poible ample mea for 64 ude Normal, Mea=5, SDev=0.5 How o compue hi awer if give μ, σ,, ad x : Sample mea for = 64 are: approximaely ormal wih mea of 5 hour ad.d. of 4 hour. So, he adardized core for 5.36 i: x ( ) 64(5.36 5) z Poible value of x-bar Area above z-core of.44 i abou.075. So i i feaible ha he rue populaio mea for all Sa 7 ude (o ju Wier 0) i ideed 5 hour. Ukow Populaio Sadard Deviaio Sude diribuio Whe σ i o kow, we mu ue he ample adard deviaio iead. Sadard deviaio of x : Sadard error of x : d..( x). e.( x ) Major coequece: Whe uig adard error i iuaio ivolvig mea, adardized aiic ha a -diribuio iead of a z- diribuio; alo called Sude diribuio. I 908 William Sealy Goe figured ou he formula for he diribuio. Called Sude becaue explaied i cla!

7 Sadardized Saiic Uig Sadard Error Uually we do kow σ (populaio adard deviaio), o we eed o ue (ample adard deviaio). I ha cae, he adardized aiic for x i x x ( x ) 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 Sadardized 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 (Moday) we will lear how o fid probabiliie. 3 4 Summary of amplig diribuio for he 5 parameer (p. 353): 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 pˆ Differece p ( p) p ( p Bewee p p pˆ ˆ p Proporio Oe Mea Mea Differece, Paired Daa Differece Bewee Mea d x d x x ) d Sadard Error of he Saiic p ˆ( pˆ ) pˆ ( pˆ ˆ ˆ ) p ( p ) d Sadardized Saiic wih.e. z z The hree iuaio ivolvig mea: Oe Mea Mea Differece, Paired Daa Differece Bewee Mea Parameer Saiic Sadard Deviaio of he Saiic x d d d x x Sadard Error of he Saiic d z or? (wih.e.)