Inrodcion o Saisics in Psychology PSY Professor Greg Francis Lecre 23 Hypohesis esing for correlaions Is here a correlaion beween homework and exam grades? for seps. Sae he hypohesis. 2. Se he crierion for rejecing. 3. Compe he es saisics. 4. Inerpre he resls. 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 (aoid confsion wih Spearman s ) le r be he correlaion coe cien saisic from a random sample of he poplaion Measre of Inelligence SAMPLING =.22 3 4 Nmber of cigarees smokes depending on which poins we sample, he comped r will ake di eren ales 4 6
RANDOM SAMPLING r =.24 Measre of Inelligence 3 4 Nmber of cigarees smokes r =.67 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! formla for creaing new saisic z r = 2 log +r B C @ A e r where log e is he naral logarihm fncion also someimes designaed as 2 ln Measre of Inelligence z r - -2 3 4 Nmber of cigarees smokes - -.. r exbook proides a r o z calclaor 7 8 9 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 B @ C A 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 ) e.g. r =.9! z r =.472 r =! z r = r =.4! z r =.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.6. We wan o es he hypohesis wih a sample size of n =3. (e.g. family income and aides abo democraic childrearing) Sep. Sae he hypohesis. H : =.6 H a : 6=.6 wo-ailed es 2
Sep 2. Se he crierion for rejecing H =. Sep 3. Compe he es saisics sppose from or sampled daa we ge r =.6 we need o coner i o a z r score r =.6! z r =.79 and calclae sandard error s zr = n 3 = 27 =.92 now we calclae he es saisic saisic - parameer Tes saisic = sandard error of he saisic z = z r z = s zr.79.77.92 =.344 From he Normal Disribion calclaor, we compe p =.7346 >. = Sep 4. Inerpre he resls. H is no rejeced a he. significance leel The probabiliy of geing r =.6 (or aalefrherawayfrom)wiha random sample, if =.6, is greaer han.. 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 : = a H a : 6= a special case a =: H : = H a : 6= Is here a significan correlaion coe cien? while we needed Fisher s z ransformaion o coner he sampling disribion ino a normal disribion i is no necessary for esing = for = he sampling disribion is a disribion wih df = (wo ses of scores, mins from each se) no need o coner wih z ransform we follow he same procedres as before. Sae he hypohesis. H : = 2. Se he crierion for rejecing H. 3. Compe he es saisics. 4. Inerpre he resls. 6 7 8
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 =.37. Sae he hypohesis. H : =,H a : 6= 2. Se he crierion for rejecing H. =. 3. Compe he es saisics. 3 = r r2 =(.37).89 = 2.26 se he Disribion calclaor wih df==3 p =.344 4. Inerpre he resls: p =.344 <. = rejec H EXAMPLE Iookhepercenageofhefirssix homework grades and correlaed i wih he firs exam scores =.288 Is his a significan correlaion? Homework % 8 6 4 4 6 8 Exam score 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 =.288 significanly di eren from? I hae n =32scores Compe he es saisics. = r r 2 =(.288) 3.97 =.647 se he Disribion calclaor wih df==3 p =. Inerpre he resls: p =. >. =, donorejech STATLAB? For Exam and STATLAB, r =.46. I hae n =32scores Compe he es saisics. = r r 2 =(.46) 3.827 =2. se he Disribion calclaor wih df==3 p =.8 Inerpre he resls: p =.8 <. =, rejech 22 23 24
CAREFUL! When we conclde a es is saisically significan, we based ha on he obseraion ha obsered daa (or more exreme) wold be rare if he H 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 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 CONCLUSIONS hypohesis esing correlaion coe cien Fisher z ransform esing significance of correlaion NEXT TIME Confidence inerals wih correlaions hypohesis esing of proporions Can yo read my mind? Par II. 2 26 27