Introduction to Statistics inPsychologyPSY 201Professor Greg FrancisLecture 21Estimation of population meanThe great census debate.LAST TIMEwe know how to check if a samplemean, X, is statistically significantlydifferent from a hypothesizedpopulation mean, µ.but sometimes we have no idea what µis!we would like to be able to estimateµ using the sample data we havestatistical estimation2POINT ESTIMATIONsingle value that represents the bestestimate of a population valuewhen we want to estimate µ, the bestpoint estimate is the sample mean Xbut the estimate depends on whichsample we select!0 20 40 60 80 100 120Sample Mean 00.010.020.03Frequency3INTERVAL ESTIMATIONwe get a better idea of the value of µby considering a range of values thatare likely to contain µwe will show how to build upconfidence intervals using theproperties of the sampling distributionof the mean41 42 43 44 45Sample Mean00.20.40.60.84σ KNOWNto demonstrate our technique, supposewe have a population of scores withµ = 43, σ = 10from the population we get thesampling distribution for samples ofsize n = 400 withσX=σ√n=0.5041 42 43 44 45Sample Mean00.20.40.60.85INTERVAL ESTIMATIONwith the sampling distribution we cancalculate (using the standard normaltable) that 95% of all sample meanswill lie between 42.02 and 43.98but since we do not really know thevalue of µ, we must estimate it41 42 43 44 45Sample Mean00.20.40.60.86CONFIDENCE INTERVALSconstruct an interval around theobserved statistic, XCI = statistic±(critical value) (standard error of the statistic)CI = X ± (tcv)(sX)where• X is the sample mean• tcvis the critical value using the ap-propriate t distribution for the de-sired level of confidence• sXis the estimated standard errorof the meansX=s√n7LEVEL OF CONFIDENCEdegree of confidence that computedinterval contains µusually complement of level ofsignificance, αlevel of confidence is (1 − α)calculating the critical value tcvis thesame!e.g., for α =0.05, (1 − α)=0.95, andtcv=1.9659(using the inverse t-distributioncalculator with df = n − 1 = 399)8CONFIDENCE INTERVALsuppose we calcul ate X = 44.6the confidence interval is thenCI = X ± (tcv)(sX)CI95= 44.6 ± (1.9659)(0.50)CI95= (43.61705, 45.58295)9CONFIDENCE INTERVALthis means we are 95% confident thatthe interval (43.62, 45.58) contains theunknown value µnote: if µ = 43 like was said originally,we are wrong!CI does not contain µ (no way to avoiderror completely)!41 42 43 44 45Sample Mean00.20.40.60.810EXAMPLEGuess the height of this room in feet,and write down your guess on thepapers going around the room.Now go around the room and get 10guesses from other peopleCalculate the mean and standarddeviation for your sampleX =!Xins ="#######$!iX2i− [(!iXi)2/n]n − 1I’ll calculate the population mean for the classeach of you will calculate a confidence interval,for your sample, with α =0.0511CONFIDENCE INTERVALCI = X ± (tcv)(sX)Calculate standard error of the meansX=s√n=s√10=we haved.f. = n − 1 = 10 − 1=9so from the t-distribution table we findthattcv=2.26212CONFIDENCE INTERVALSthusCI95= X ± (tcv)(sX)CI95= X ± (2.262)(sX)CI95=( , )13WHAT DOES THIS MEAN?we conclude with 95% confidence thatyour interval contains µthis is a probabilistic statement aboutthe intervalµ is a parameter, a fixed numberµ =different samples produce differentconfidence intervals, but 95% of thetime the interval would contain µcheck14CONFIDENCEwe never say that the confidence intervalcontains µ with probability 0.95either the interval contains µ or it does notwe can say that the procedure of producingCI’s produce intervals that contain µ withprobability 0.95we do talk about the confidence that aninterval includes µwe would say that the confidence intervalcontains µ with confidence of 0.95the confidence is in the procedure ofcalculating CIs15CENSUS DEBATEthe US Constitution mandates thatthe government take a full censusevery 10 yearswith 250 million people this is difficultespecially for certain groups of people(Hispanics, working mothers,...)census information is used to guideFederal aid (billions of dollars!)census bureau wants to use statisticalestimation to complement the census,especially for groups that they knowget missed in the straight head countcongress will not allow it16CONCLUSIONSestimationconfidence intervalsnormal distribution, t distributioninterpretation17NEXT TIMEmore on estimationrelationship between confidenceintervals and hypothesis testingstatistical precisionLess than 5% of publishedpsychological data is