Introduction to Statistics inPsychologyPSY 201Professor Greg FrancisLecture 29Hypothesis testing for two samplecaseother statisticsWhat do people think aboutdeath?DEPENDENT SAMPLESwhen the samples are notindependent, hypothesis testing ofproportions becomes a bit morecomplicatedsamples are dependent when eachscore in one sample is paired witha score in the other samplejust like dep endent samples forthe mean, the problem is that thesamples are not independent (nottruly random) and we need totake that into account2EXAMPLE• Testing the difference of propor-tions of individuals who pass eachof two similar items on a test.(e.g. comparing pass/fail for twosets of students who get betterthan 600 SAT)• Test the difference in proportionsof individuals who support some-thing before and after discussion.• Comparing proportions of hus-bands and wives on an issue.3HYPOTHESESfor dependent samples we letδp= P1− P2so our hyp othesesbecomeH0: δp= P1− P2=0Ha: δp= P1− P2"=04SAMPLINGDISTRIBUTIONwe need to know the samplingdistribution and the standard errorbut first we need to design a contingencytable which shows disagreement ordissimilarity in responsesGroup 1NO YESGroup YES AB A + B2 NO CD C + DA + CB+ D A + B + C + D = n• A is the number of scores that are “no” ingroup 1 and “yes” in group 2• B is the number of scores that are “yes” ingroup 1 and “yes” in group 2• C is the number of scores that are “‘no” ingroup 1 and “no” in group 2• D is the number of scores that are “yes” ingroup 1 and “no” in group 25CONTINGENCY TABLEwe then convert these toproportionsGroup 1NO YESGroup YES ab a + b2 NO cd c + da + cb+ d a + b + c + d =1.0• a = A/n is the proportion of scores that are“no” in group 1 and “yes” in group 2• b = B/n is the proportion of scores that are“yes” in group 1 and “yes” in group 2• c = C/n is the proportion of scores that are“‘no” in group 1 and “no” in group 2• d = D/n is the proportion of scores that are“yes” in group 1 and “no” in group 26PROPORTIONSfrom the contingency table we canget the proportions of scores withthe trait we are interested inthis is what we need for ourstatisticp1= b + d =B + Dnp2= a + b =A + Bnbut we also need the contingencytable for other reasons7CONTINGENCY TABLESthe sampling distribution isapproximately normal with amean of P1− P2ifA + D>10orB + C>10if not, do not use this testmoreover, our estimate ofstandard error of the differencebetween dependent proportions issp1−p2=!""""""#a + dnwhich we get from thecontingency table8HYPOTHESIS TESTINGso to actually carry out the test,we get the critical value zcvforthe corresponding α (bottom rowof t distribution table)and compare it to the test statisticz =(p1− p2) − δpsp1−p2or, since our null hypothesis isthat δp=0z =(p1− p2)sp1−p29SHORT-CUTwith some algebra you can showthat the formula for the teststatistic is alsoz =A − D√A + Dwhere A and D are thefrequencies from thecontingency tablethis saves the trouble ofcalculating sp1−p2(but if you want to calculateconfidence intervals you will needthe standard error)10EXAMPLEI want to know if there is adifference in the proportion ofstudents that support support guncontrol (Group 1) and and theproportion of students thatsupport the death penalty (Group2).One a sheet of paper answer thesequestions (you do not have to behonest if you do not want otherpeople to know your views):• Do you support legislation to re-strict the ownership of guns?• Do you support legislation to pro-hibit the death penalty for con-victed felons?11EXAMPLERaise your hand if• A: You do not support gun control, butsupport prohibiting the death penalty.(no, yes)• B: You support gun control and pro-hibiting the death penalty. (yes, yes)• C: You do not support gun control anddo not supp ort prohibiting the deathpenalty. (no, no)• D: You support gun control but do notsupport prohibiting the death penalty.(yes, no)12CONTINGENCY TABLEPROHIBIT DEATH PENALTYNO YESGUN YESCONTROL NOI want to testH0: P1− P2=0Ha: P1− P2"=0I will use α =0.05 and atwo-tailed test.Note:Group 1 is the set of responses to the question aboutrestricting gun ownershipGroup 2 is the set of responses to the question aboutprohibiting the death penalty13CRITERIONI need to check if I can use thenormal approximation to thesampling distributioncheck ifA + D>10orB + C>10if not, do not use this testif successful, then for α =0.05 Ifind from the bottom row of the tdistribution table thatzcv= ±1.9614PROPORTIONSI can calculate proportionsp1= b + d =B + Dn=p2= a + b =A + Bn=I could calculate the standarderror as:sp1−p2=!""""""#a + bnand then calculate the teststatisticz =p1− p2sp1−p2but I prefer to use the short-cut15SHORT-CUTz =A − D√A + D=√=compare to the critical valuez = <> zcvmake decision!16INTERPRETATIONIf we reject H0, that indicates theprobability of getting the difference ofproportions when the populationparameters were equal is less than 0.05.We interpret that as meaning thepopulation parameters are different.If we fail to reject H0, that indicates theprobability of getting the difference ofproportions when the populationparameters were equal is greater than0.05. We do not have strong enoughevidence to conclude that the populationparameters are different.17CONCLUSIONStwo-sample casedependent proportions18NEXT TIMEpowerdesigning experiments“Seeing”