Introduction to Statistics inPsychologyPSY 201Professor Greg FrancisLecture 18Hypothesis testing of the meanWhy I don’t use herbal medicinesSUPPOSEwe think the mean value of apopulation of SAT scores is µ = 455we can take a sample of the populationand calculate the sample mean of SATscores X = 535we can make some statement abouthow rare it is to get X = 535(what we did two lectures ago)orwe can make a statement about howunreasonable it is that our originalthought is true!2HYPOTHESIS TESTINGin hypothesis testing we consider howreasonable a hyp othesis is, given thedata that we haveif the hyp othesis is reasonable(consistent with the data), we assumeit is trueif the hyp othesis is unreasonable(inconsistent with the data), weassume it is falsedeciding on what hypotheses to test iscritically important!3HYPOTHESIS TESTINGfour steps:1. State the hyp othesis.2. Set the criterion for rejecting the hy-pothesis.3. Compute the test statistic.4. Decide about the hypothesis.4HYPOTHESISconjecture about one or morepopulation parameterse.g.µ = 455σ =3.5r =0.76...in inferential statistics we always testthe null hypothesis: H05NULL HYPOTHESISH0is the assumption of norelationship, or no differencee.g.H0: no relationship between variablesH0: no difference between treatmentgroupsthe alternative hypothesis, Hais theother possibilitye.g.H0: µ = 455Ha: µ != 455(does not say what µ is, but says whatit is not!)6NULL HYPOTHESISwhat’s wrong with herbal medicines?nothing necessarily, but I don’t knowthat they are any good (and they maybe bad)lots of reports that they help people(but how can they be sure?)need to start by assuming that amedicine does nothing, and provethat the assumption is false!anecdotal reports are just aboutworthless7NULL HYPOTHESISoften times (almost always) the goal ofstatistical research is to reject the nullhypothesis, so that the only alternativeis to accept the Hasimilar to an indirect proofe.g.show that the angles of a triangle sumto 180oby assuming that they do notand then finding a contradictionwhy this approach?it is much easier to show thatsomething is false (H0) than to showthat something is true (Ha)understanding a relationship betweenvariables or differences between groupsoften requires many experiments!8STATE THE HYPOTHESISbefore doing anything else, we need tomake certain that we understand thetested hypothesisfor the SAT exampleH0: µ = 455Ha: µ != 455sometimes this is the most difficultstep in designing an experimentto start, we will worry only abouthypotheses about the populationmean, µ9CRITERIONwe will examine the data to see if weshould reject or accept H0we will do that by comparing thesample mean, X, to the hypothesizedvalue of the population mean, µthe bottom-line is whether X issufficiently different from µ to rejectH0but we have to consider three things toquantify the term sufficientlydifferent• errors in hyp othesis testing• level of significance• region of rejection10DECISIONSafter deciding to reject or not rejectH0there are four possible situations• A true hyp othesis is rejected.• ** A true hypothesi s is not rejected.• A false hyp othesis is not rejected.• ** A false hypothesis is rejected.errors are unavoidablewe want to minimize the probability ofmaking errors, given the particulardata set we have11ERRORStwo types of errors:• Type I error: when we reject atrue null hypothesis.• Type II error: when we do notreject a false null hypothesis.State of natureDecision made H0true H0falseReject H0Type I error Correct decisionDo not reject H0Correct decision Type II errorgenerally, decreasing the probability ofmaking one typ e of error increases theprobability of making the other type oferror12USEsuppose you have a new, untested, andexpensive treatment for canceryou run a test to judge whether thedrug is better than existing drugsif you reject H0, indicating that thedrug is more effective, when in fact itis not, people will spend a lot of moneyfor no reason (Type I error)if you fail to reject H0, indicating thatthe drug is not effective, when in factit is, people will not use the drug(Type II error)scientific research tends to focus onavoiding Type I errors13SIGNIFICANCE LEVELalpha (α) levelindicates probability of Type I errorfrequently we choose α =0.05 orα =0.01that is, the corresponding decision toreject H0may produce a Type I error5% or 1% of the timea statement ab out how much error wewill acceptusually chosen before the data isgathereddepends upon use of the analysis14REGION OF REJECTIONα is a probabilityit identifies how much risk of Type Ierror we are willing to take (rejectingH0when it is true)consider our example of SAT scoresH0: µ = 455suppose we have also been told thefollowingσ = 100and our sample size is n = 14415REGION OF REJECTIONwe know that the samplingdistribution is:1. Normal.2. Has a mean of µ = 455, if H0is true3. Has a standard error of the meanσX=σ√n=100√144=8.33400 420 440 460 480 500Sample Mean 00.010.020.030.0416REGION OF REJECTIONarea under the curve represents theprobability of getting thecorresponding sample means, giventhat µ is where specified in H0the extreme tails of the samplingdistribution correspond to what shouldbe very rare sample means, if the H0istrue400 420 440 460 480 500Sample Mean 00.010.020.030.0417REGION OF REJECTIONwe shade in the extreme α percentageof the sampling distributioncalled the region of rejectionif our sample mean, X, is in the regionof rejection, we reject H0because it isunlikely that we would get such asample mean for the corresponding µ.400 420 440 460 480 500Sample Mean 00.010.020.030.0418CRITICAL VALUESvalues of sample means at thebeginning of the region of rejectionNOTE: α is split up in each tailcalled a two-tailed or non-directionaltest400 420 440 460 480 500Sample Mean 00.010.020.030.0419CRITICAL VALUEScritical values represent the beginningof the region of rejectionfor α =0.05, we find from thestandard normal table that the criticalvalues (in z-scores) are ±1.96-4 -2 0 2 4X00.10.20.30.420TEST STATISTICif the z-score for X is beyond ±1.96z-scores from µ = 455, it is very, veryunlikely to have occurred if the H0istrue.we have the following data:• µ = 455, H0• n = 144, sample size• X, observed value for sample statis-tic• σ = 100, value of the standard devi-ation of the population• σX=8.33, standard error (calcu-lated earlier)from this we can calculate the z-scorecorresponding to X21TEST STATISTICwe want to know how different X