Introduction to Statistics inPsychologyPSY 201Professor Greg FrancisLecture 19Hypothesis testing of the meanWhy clinical studies use thousandsof subjects.SUPPOSEthe we think the mean value of apopulation of SAT scores is µ = 455with σ = 100we take a sample of 144 from thepopulation and calculate the samplemean of SAT scores X = 5352HYPOTHESIS TESTINGfour steps1. State the hyp othesis.2. Set the criterion for rejecting H0.3. Compute the test statistic.4. Decide whether to reject H0.3RECAP OF LAST TIME(1) State the hypothesesH0: µ = 455Ha: µ != 455α =0.05(2) Set the criterion:Find the z-scores that identify the 5%of scores farthest from µzcv= ±1.964RECAP OF LAST TIME(3) Compute the test statisticz =X − µσXz =535 −4558.33=9.60(4) Decide:z > zcv, so reject H0the found sample mean would be veryrare if H0were true5DIFFERENT MEANsuppose we had the same situation asbefore, but we had instead foundX = 465(1) State the hypothesesH0: µ = 455Ha: µ != 455α =0.05(2) Set the criterion:Find the z-scores that identify the 5%of scores farthest from µzcv= ±1.966DIFFERENT MEAN(3) Compute the test statisticz =X − µσXz =465 − 4558.33=1.20(4) Decide:z < zcv, so do not reject H0the found sample mean would not bevery rare if H0were truethe probability that X = 465 would befound by random sampling is greaterthan .057SAMPLE SIZEsuppose we had the same situation asbefore, but we had instead foundX = 465with a sample size of n = 500(1) State the hypothesesH0: µ = 455Ha: µ != 455α =0.05(2) Set the criterion:Find the z-scores that identify the 5%of scores farthest from µzcv= ±1.968SAMPLE SIZE(3) Compute the test statisticz =X − µσXwe need to recompute σXσX=σ√n=100√500=4.47z =465 − 4554.47=2.24α =0.05(4) Decide:z > zcv, so do reject H0the found sample mean would be very rare if H0were truethe probability that X = 465 would be found by randomsampling is less than .059CLINICAL TRIALSoften hear about medical studies thattrack thousands of patientswhy do they need so many people?a larger sample makes for less variationin the sampling distributionthus, if the null hypothesis really isfalse, you are more likely to reject itwith a larger sampleif the null hypothesis is really true, youare not more likely to reject it (noextra mistakes with a larger sampl esize!)10COMMENTSseveral things are worth noting1. Even when we reject H0, there is always achance that it is true.2. Even when we do not reject H0, there isalways a chance that it is false.3. The statement p<0.05 is about the statis-tic given the hypothesis, not about the hy-pothesis. We never conclude that H0is falsewith probability 0.95.4. Technically, we have done all of this before.5. No inclusion of knowledge about the direc-tion of difference.6. We have to know σ (population standarddeviation).7. These techniques are quantifiable.11DIRECTIONALHYPOTHESISwe choose a significance level, αindicates probability of Type I errorearlier, we split this error across thetwo tails of the sampling distribution400 420 440 460 480 500Sample Mean 00.010.020.030.0412DIRECTIONALHYPOTHESISsuppose we know (or strongly suspect)that if the sample mean X is differentfrom the population mean µ, it will begreaterthen we don’t need to worry ab out theleft-side tailH0: µ = 455Ha: µ>45513REGION OF REJECTIONif we only have to worry about onetail, the region of rejection (in thattail) is larger!last 5% starts with a z-score of 1.645we can reject H0when the differencebetween X and µ is smaller!-4 -2 0 2 4X00.10.20.30.414EXAMPLEwe know that the samplingdistribution is:1. Normal.2. Has a mean of µ = 455.3. Has a standard error of the meanσX=σ√n=100√144=8.33400 420 440 460 480 500Sample Mean 00.010.020.030.0415REGION OF REJECTIONarea under the curve represents theprobability of getting thecorresponding sample means, giventhat µ is where specifiedthe extreme right tail of the samplingdistribution corresponds to whatshould be very rare sample meanscritical z-score value is 1.645-4 -2 0 2 4X00.10.20.30.416REGION OF REJECTIONwe convert sample mean to z-scorez =X − µσXz =535 − 4558.33=9.60greater than critical value9.60 > 1.645reject H0-4 -2 0 2 4X00.10.20.30.417EXAMPLEsuppose everything was the same,except we had hypotheses:H0: µ = 455Ha: µ<455then we would shift the region ofrejection to the left tail400 420 440 460 480 500Sample Mean 00.010.020.030.0418EXAMPLEthe critical z-score value becomes-1.645with our sample mean of X = 535,and z =9.60, we cannot reject H0-4 -2 0 2 4X00.10.20.30.419TABLENOTE: you only need to know a fewcritical value z-scoresLevel of Level of CriticalSignificance, Significance, Value ofTwo-tailed test One-tailed test Test Statistic0.20 0.10 1.2820.10 0.05 1.6450.05 0.025 1.9600.02 0.01 2.3260.01 0.005 3.2910.001 0.0005 3.29120SIGNIFICANCE vs.IMPORTANCEsomething may make a big differencein the value of X but not bestatistically significantsomething can be statisticallysignificant without b eing practicallyimportantwill you change your diet to avoidcancer?depends on the diet and how much ofa difference it makes21CONCLUSIONShypothesis testingsample sizedirectional testsignificance22NEXT TIMEwhen σ is unknownt-distributiont-testDo nuclear plants make you