HYPOTHESIS TESTING. four steps. 1. State the hypothesis and the criterion. 2. Compute the test statistic. 3. Compute the p-value. 4.
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1 Inrodcion o Saisics in Psychology PSY Professor Greg Francis Lecre 24 Hypohesis esing for correlaions Is here a correlaion beween homework and exam grades? for seps. Sae he hypohesis and he crierion 2. Compe he es saisic. 3. Compe he p-ale. 4. Make a decision we need o know he properies of he sampling disribion for he mean, he cenral limi heorem ells s ha he sampling disribion is normal, and specifies he mean and sandard deiaion (sandard error) area nder he cre of he sampling disribion gies probabiliy of geing ha sampled ale, or ales more exreme (p-ale) for oher ypes of saisics, he sampling disribion is di eren area nder he cre of sampling disribion sill gies probabiliy of geing ha sampled ale, or ales more exreme correlaion coe cien 2 3 he approach is sill basically he same Tes saisic = we compe saisic - parameer sandard error of he saisic and se i o compe a p-ale, which we compare o CORRELATION COEFFICIENT from a poplaion wih scores X and Y,wecancalclaeacorrelaion coe cien le be he correlaion coe cien parameer of he poplaion le r be he correlaion coe cien saisic from a random sample of he poplaion Measre of Inelligence 0 SAMPLING = Nmber of cigarees smokes depending on which poins we sample, he comped r will ake di eren ales 4 6
2 RANDOM SAMPLING r =0.24 Measre of Inelligence Nmber of cigarees smokes r =0.67 SAMPLING DISTRIBUTION freqency of di eren r ales, gien a poplaion parameer no sally a normal disribion! ofen skewed o he lef or he righ canno find area nder cre! FISHER z TRANSFORM formla for creaing new saisic z r = 0 2 log +r B A e r where log e is he naral logarihm fncion also someimes designaed as 2 ln Measre of Inelligence 0 z r Nmber of cigarees smokes r exbook proides a r o z 0 calclaor FISHER z TRANSFORM for large samples, he sampling disribion of z r is normally disribed (regardless of he ale of ) wih a mean z = 2 log e + and wih sandard error (sandard deiaion of he sampling disribion) s zr = n 3 where n is he sample size 0 C A FISHER z TRANSFORM means we can se all or knowledge abo hypohesis esing wih normal disribions for he ransformed scores! online calclaor coners r o z r (i calls i z 0 ) e.g. r = 0.90! z r =.472 r =0! z r =0 r =0.4! z r =0.48 we can coner back he oher way from z r! r oo! Sppose we sdy a poplaion of daa ha we hink has a correlaion of 0.6. We wan o es he hypohesis wih a sample size of n =30. (e.g. family income and aides abo democraic childrearing) Sep. Sae he hypohesis and crierion H 0 : =0.6 H a : 6= 0.6 wo-ailed es =
3 Sep 2. Compe he es saisics sppose from or sampled daa we ge r =0.6 we need o coner i o a z r score r =0.6! z r =0.709 and calclae sandard error s zr = n 3 = 27 =0.92 now we calclae he es saisic saisic - parameer Tes saisic = sandard error of he saisic z = z r z = s zr = Sep 3. Compe he p-ale From he Normal Disribion calclaor, we compe p = > 0.0 = Sep 4. Make a decision. p = > 0.0 = H 0 is no rejeced a he 0.0 significance leel The probabiliy of geing r =0.6 (or aalefrherawayfrom0)wiha random sample, if =0.6, is greaer han 0.0. The obsered di erence is no a significan di erence. 3 4 A SPECIAL CASE hypohesis esing of correlaion coe ciens and r Fisher s z ransform H 0 : = a H a : 6= a special case a =0: H 0 : =0 H a : 6= 0 Is here a significan correlaion coe cien? 6 SAMPLING DISTRIBUTION while we needed Fisher s z ransformaion o coner he sampling disribion ino a normal disribion i is no necessary for esing =0 7 SAMPLING DISTRIBUTION for =0 he sampling disribion of he es saisic is a disribion wih df = (wo ses of scores, mins from each se) no need o coner wih Fisher z ransform we follow he same procedres as before. Sae he hypohesis. H 0 : =0 and se he crierion 2. Compe he es saisic. 3. Compe he p-ale 4. Make a decision. 8
4 eeryhing is he same, excep he es saisic calclaion is a bi di eren i rns o ha an esimae of he sandard error is: s r = r 2 so ha he es saisic is: = r = r s r r 2 we se his wih a disribion o compe a p-ale EXAMPLE n =32scorescalclaedoge r = Sae he hypohesis. H 0 : =0,H a : 6= 0 2. Se he crierion for rejecing H 0. = Compe he es saisics. 30 = r r2 =( 0.37) 0.89 = 2.26 se he Disribion calclaor wih df==30 p = Inerpre he resls: p = < 0.0 = rejec H 0 EXAMPLE Iookhepercenageofhefirssix homework grades and correlaed i wih he firs exam scores = Is his a significan correlaion? Homework Exam 9 2 CAREFULL! If I rea he class as a poplaion, he correlaion is wha i is. Significance is no an isse! If I rea he class as a sample of sdens who do homework and ake exams in saisics, hen I can ask abo saisical signfiicance CAREFULL! is r = significanly di eren from 0? I hae n =28scores Compe he es saisics. = r r 2 = se he Disribion calclaor wih df==26 p =0.769 Inerpre he resls: p =0.769 > 0.0 =, donorejec H 0 READING? For Homework and Reading, r = I hae n =28scores Compe he es saisics. = r r 2 =7.77 se he Disribion calclaor wih df==26 p 0 Inerpre he resls: p 0 < 0.0 =, rejech
5 CAREFUL! When we conclde a es is saisically significan, we base ha on he obseraion ha obsered daa (or more exreme) wold be rare if he H 0 were re B if we make mliple ess from a single sample, or calclaions of probabiliy may be inalid. We performed wo hypohesis ess from one sample of sdens. Each es has a chance of prodcing a significan resls, een if H 0 is re I is no appropriae o js rn arios ess wih one daa se, if all yo are doing is looking for significan resls (fishing) Yo hae o do a di eren ype of saisical analysis CONFIDENCE INTERVAL Always se he Fisher z ransform Bild ineral as a Fisher z score and hen coner o correlaion (r ale) CI = z r ± z c s zr For he correlaion beween homework and reading scores: CI 9 =.84±(.96)(0.2) = (0.86,.600) when we coner o r ales: (0.673, 0.922) POWER How wold we design a good experimen o es a correlaion? How big a sample do we need o hae a90%chanceofrejecingheheh 0? Concepally, his is he same isse as esimaing power or sample size for a hypohesis es of means We js need o se he sampling disribion for he Fisher z ransform of he sample correlaion insead of he sampling disribion for a sample mean POWER We js hae o specify he specific correlaion for he alernaie hypohesis Sppose we plan o es H 0 : =0,H a : 6= 0 and we se he specific alernaie as H a : a =0.8 Wha is he probabiliy ha a random sample of n =2willrejecheH 0? The on-line calclaor does all he work! POWER Higher han 99.9% chance of rejecing he nll hypohesis Wha sample size do we need o hae 90% power? CONCLUSIONS correlaion coe cien Fisher z ransform esing significance of correlaion confidence ineral power Howeer, wheher hese calclaions make sense depends on wheher =0.8 inrealiy
6 NEXT TIME hypohesis esing of wo means Why do we le people die? 3
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