HYPOTHESIS TESTING. four steps

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1 Irodcio o Saisics i Psychology PSY 20 Professor Greg Fracis Lecre 24 Correlaios ad proporios Ca yo read my mid? Par II HYPOTHESIS TESTING for seps. Sae he hypohesis. 2. Se he crierio for rejecig H Compe he es saisic. 4. Ierpre he resls. someimes hae fie seps if wa o iclde CIs. Sae he hypohesis. 2. Se he crierio for rejecig H Compe he es saisic. 4. Cofidece ieral. 5. Ierpre he resls. 2 3 SAMPLING DISTRIBUTION freqecy of di ere r ales, gie a poplaio parameer o sally a ormal disribio! ofe skewed o he lef or he righ cao fid area der cre! FISHER z TRANSFORM formla for creaig ew saisic z r 0 2 log +r B C A r where log e is he aral logarihm fcio also someimes desigaed as l We ge hese rasformed ales from he r o z 0 Calclaor FISHER z TRANSFORM for large samples, he samplig disribio of z r is ormally disribed (regardless of he ale of ) wih a mea z 2 log e + ad wih sadard error (sadard deiaio of he samplig disribio) s zr 3 where is he sample size 0 C A 4 5 6

2 CONFIDENCE INTERVAL Sppose we fid r 0.6 from a sample of size 30. BildCI 90. (e.g. family icome ad aides abo democraic childrearig) calclae sadard error s zr fid he criical ale from Ierse Normal Disribio Calclaor we fid ha he criical ale is.645 cosrc ieral i z r ales ad he coer back o r ales CI saisic ± (criical ale) (sadard error) CI 90 z r ± (.645)(s zr ) CI ± (.645)(0.92) CI 90 (0.393,.025) coer z r CI 90 io r, correlaio coe cies se he r o z 0 Calclaor i reerse z r 0.393! r z r.025! r so, i erms of r ales CI 90 (0.374, 0.772) A SPECIAL CASE Las ime we oed ha while we eeded Fisher s z rasformaio o coer he samplig disribio io a ormal disribio i is o ecessary for esig 0 0 EXAMPLE 32scorescalclaedoge r Sae he hypohesis. H 0 : 0,H a : Se he crierio for rejecig H Compe he es saisic r r2 ( 0.375) Use he -disribio Calclaor wih df 30ocompehep-ale p rejec H 0 4. Cosrc he cofidece ieral. 5. Ierpre he resls. BE CAREFUL!!! we js coclded ha we ca rejec H 0,whichmeasweaccephe saeme H a : 6 0 whe we cosrc a cofidece ieral we shold se he Fisher z rasform (we js coclded ha 6 0,sowe wold o wa o se he disribio o make he cofidece ieral) eeryhig is js like las ime. coer r! z r 2. Calclae CI i z rasformed scores. 3. Coer z r! r o ge CI i r scores 2

3 Cosrc he cofidece ieral Sice 0.05, we cosrc a 95% cofidece ieral cosrc ieral i z r ales ad he coer back o r ales CI saisic ± (criical ale) (sadard error) CI 95 z r ± (c)(s zr ) We fid he c sig he Ierse Normal Disribio Calclaor (se Area.05, ad choose Oside) c adr 0.375, so c.960 ad z r s zr plg i he iformaio CI ± (.960)(0.85) CI 95 ( 0.034, ) (i Fisher z scores) coer o r ales wih r o z 0 Calclaor CI 95 ( 0.30, 0.640) oe: 0 is o i he ieral, cosise wih hypohesis es! 4 PROPORTIONS may imes we wa o kow wha proporio (P )ofapoplaiohasa cerai rai Ow a phoe. Are a democra. Are a repblica. Ow a comper.... dichoomos poplaio (hae rai or do o) perceages 5 PROPORTIONS we ca ake a radom sample ad calclae a sample proporio p we ca es hypoheses abo he poplaio parameer P e.g. H 0 : P 0.5 H a : P he samplig disribio of p is he biomial disribio for large samples i is ery close o he ormal disribio STANDARD ERROR a esimae of he sadard error of he samplig disribio is: sadard error of he sample proporio s p P poplaioproporiopossessig characerisic Q P poplaioproporio o possessig characerisic samplesize ow we ca apply he echiqes of hypohesis esig! MIND READING Iamgoigopickoeofhefollowig words as a special word Yo ry o read my mid as o which oe is special wrie i dow o a shee of paper. I ll wrie dow my chose word o a shee of paper COMPUTER STEREO BICYCLE STAPLER BOOKCASE DESK 6 7 8

4 MIND READING Now, I ell yo my special word, ad we fid o how may of yo were correc. We are measrig p, he sample proporio we ca es wheher yo ca read my mid () Sae he hypohesis he ll hypohesis is ha yo cao read my mid, so we say ha H 0 : P H a : P where 0.67 is wha yo wold ge js by gessig s p MIND READING (2) Fid he criical ale we ll se 0.0 (3) Compe he es saisic (0.67)(0.833) z p P s p Which we plg i o he Normal Disribio Calclaor o fid he p-ale (4) Make a decisio 0.39 PEPSI CHALLENGE seeral years ago Pepsi sposored he Pepsi Challege where yo sampled Coke ad Pepsi ad decided which ased beer afer esig hdreds of people, hey fod ha more ha half he Coke drikers preferred Pepsi (63%) how wold we es o see if he proporio of people who preferred Pepsi oer Coke was a sigifica proporio (di ere from chace)? HYPOTHESIS Sep. Sae he hypohesis. by chace we wold expec he proporio of people ha preferred Pepsi wold be 50% H 0 : P 0.5 H a : P Le s sppose 300peoplewere esed CRITERION Sep 2. Se he crierio for rejecio. Le s se or leel of sigificace a 0.05, wo-ailed es Sep 3. Compe he es saisic. Sppose he sample proporio is p ad he sadard error of he sample proporio is: s p (0.5)(0.5) TEST STATISTIC he es saisic is: z p P s p We se he Normal Disribio Calclaor o compe p 0 we ca rejec H 0!

5 Sep 4. Cosrc cofidece ierals Sice we chose 0.05, we cosrc cofidece ierals wih leel of cofidece 0.95 The criical ale z c is fod from he Ierse Normal Disribio Calclaor z c.96 so CI 95 p ± (.96)(s p) Sice we rejeced H 0,wemsrecompehesadard error by sig he esimae from he sample s p pq (0.63)(0.37) CI ± (.96)(0.048) CI 95 (0.54, 0.72) which does o iclde he chace leel P 0.5 INTERPRETATION wha if we had failed o rejec H 0? (se P ad Q P isead of p ad q, respeciely) Sep 5. Ierpre he resls. H 0 is rejeced a he 0.05 sigificace leel The probabiliy of geig p 0.63 from a radom sample of 300 people, if P 0.5, is less ha The obsered di erece is a sigifica di erece. [Noe: I js made p he mber 300. p 0.63 is he mber gie by Pepsi.] CONCLUSIONS cofidece ierals for correlaios (carefl!) esig sigificace of proporios cofidece ierals for proporios NEXT TIME more hypohesis esig comparig meas from wo samples Why do we le people die? 28