Introduction to Statistics inPsychologyPSY 201Professor Greg FrancisLecture 20Hypothesis testing of the meanDo nuclear plants make you sick?UP TO NOWif we know• µ. From H0• X. From the sample.• n. The sample size.• σ. Standard deviation of the popu-lation.then we can calculate standard errorσX=σ√nand apply the techniques of last time2UNKNOWN σof all the information we need, only σis hard to getfor a populationσ2=Σ(X − µ)2Nwhich implies that we already know µ!!so why would we want to test H0about µ?3UNKNOWN σwhen σ is unknown we use our bestguess, the standard deviation of thesamples =!"""""""#Σ(X − X)2n −1then the estimated standard error ofthe mean (for the samplingdistribution) issX=s√n4UNDERLYINGDISTRIBUTIONusing s in place of σ affects thesampling distribution of the meanchanges its shap e away from a normaldistribution (especially for small n)for large sample sizes, the shape is veryclose to a normal distributioncalled the Student’s t distributionssymmetrical, bell-shaped, centered onthe mean, changes shape as samplesize changes5t SCORESwe can transform the sample meansinto t scores(just like z scores)t =X − µsXdistribution centered on zero, standarddeviation of one, shape depends onsample size (degrees of freedom)6DEGREES OF FREEDOMmathematical conceptnumber of observations less thenumber of restrictions placed on theobservationse.g.if I knowX1+ X2= 10then knowing one of the X1,X2values means that I can calculate theother oneone degree of freedom among thescores7DEGREES OF FREEDOMor suppose I want to know the degreesof freedom of deviation scores fromthe sample data set{21, 23, 24, 27, 30}I can calculate X = 25and can calculate deviation scores of{−4, −2, −1, 2, 5}but if I only knew four of the deviationscores, I could get the other onebecause I knowΣxi=0these deviation scores have fourdegrees of freedom!8DEGREES OF FREEDOMfor the t distribution, calculatingdegrees of freedom (d.f.) is easywith a sample of size nd.f. = n − 1there is a unique t distribution for eachd.f.9HYPOTHESIS TESTINGwe use the t distribution just like weused the normal distribution forhypothesis testingarea under the curve indicatesprobability of those scoresfor hypothesis testing we just want toknow the critical values of regions ofrejectionone-tailed, two-tailed10TABLEfind the critical values with a table, which lists the criticalvalues for common α values, for different d.f. t distributionsTables•T-11Table entry for p and C isthe critical value t∗withprobability p lying to itsright and probability C lyingbetween −t∗and t∗.Probability pt*TABL E Dt distribution critical valuesUpper-tail probability pdf .25 .20 .15 .10 .05 .025 .02 .01 .005 .0025 .001 .00051 1.000 1.376 1.963 3.078 6.314 12.71 15.89 31.82 63.66 127.3 318.3 636.620.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9. 9 2 5 14.09 22.33 31.6030.765 0.978 1.250 1.638 2.353 3.182 3.482 4.541 5. 8 4 1 7.453 10.21 12.9240.741 0.941 1.190 1.533 2.132 2.776 2.999 3.747 4. 6 0 4 5.598 7.173 8.61050.727 0.920 1.156 1.476 2.015 2.571 2.757 3.365 4. 0 3 2 4.773 5.893 6.8696 0.718 0.906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 4.317 5.208 5.9597 0.711 0.896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 4.029 4.785 5.4088 0.706 0.889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 3.833 4.501 5.0419 0.703 0.883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 3.690 4.297 4.78110 0.700 0.879 1.093 1.372 1.812 2.228 2.359 2.764 3.169 3.581 4.144 4.587110.697 0.876 1.088 1.363 1.796 2.201 2.328 2.718 3. 1 0 6 3.497 4.025 4.437120.695 0.873 1.083 1.356 1.782 2.179 2.303 2.681 3. 0 5 5 3.428 3.930 4.318130.694 0.870 1.079 1.350 1.771 2.160 2.282 2.650 3. 0 1 2 3.372 3.852 4.221140.692 0.868 1.076 1.345 1.761 2.145 2.264 2.624 2. 9 7 7 3.326 3.787 4.140150.691 0.866 1.074 1.341 1.753 2.131 2.249 2.602 2. 9 4 7 3.286 3.733 4.07316 0.690 0.865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 3.252 3.686 4.01517 0.689 0.863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 3.222 3.646 3.96518 0.688 0.862 1.067 1.330 1.734 2.101 2.214 2.552 2.878 3.197 3.611 3.92219 0.688 0.861 1.066 1.328 1.729 2.093 2.205 2.539 2.861 3.174 3.579 3.88320 0.687 0.860 1.064 1.325 1.725 2.086 2.197 2.528 2.845 3.153 3.552 3.850210.686 0.859 1.063 1.323 1.721 2.080 2.189 2.518 2. 8 3 1 3.135 3.527 3.819220.686 0.858 1.061 1.321 1.717 2.074 2.183 2.508 2. 8 1 9 3.119 3.505 3.792230.685 0.858 1.060 1.319 1.714 2.069 2.177 2.500 2. 8 0 7 3.104 3.485 3.768240.685 0.857 1.059 1.318 1.711 2.064 2.172 2.492 2. 7 9 7 3.091 3.467 3.745250.684 0.856 1.058 1.316 1.708 2.060 2.167 2.485 2. 7 8 7 3.078 3.450 3.72526 0.684 0.856 1.058 1.315 1.706 2.056 2.162 2.479 2.779 3.067 3.435 3.70727 0.684 0.855 1.057 1.314 1.703 2.052 2.158 2.473 2.771 3.057 3.421 3.69028 0.683 0.855 1.056 1.313 1.701 2.048 2.154 2.467 2.763 3.047 3.408 3.67429 0.683 0.854 1.055 1.311 1.699 2.045 2.150 2.462 2.756 3.038 3.396 3.65930 0.683 0.854 1.055 1.310 1.697 2.042 2.147 2.457 2.750 3.030 3.385 3.646400.681 0.851 1.050 1.303 1.684 2.021 2.123 2.423 2. 7 0 4 2.971 3.307 3.551500.679 0.849 1.047 1.299 1.676 2.009 2.109 2.403 2. 6 7 8 2.937 3.261 3.496600.679 0.848 1.045 1.296 1.671 2.000 2.099 2.390 2. 6 6 0 2.915 3.232 3.460800.678 0.846 1.043 1.292 1.664 1.990 2.088 2.374 2. 6 3 9 2.887 3.195 3.4161000.677 0.845 1.042 1.290 1.660 1.984 2.081 2.364 2. 6 2 6 2.871 3.174 3.39010000.675 0.842 1.037 1.282 1.646 1.962 2.056 2.330 2. 5 8 1 2.813 3.098 3.300z∗0.674 0.841 1.036 1.282 1.645 1.960 2.054 2.326 2. 5 7 6 2.807 3.091 3.29150% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9%Confidence level CIntegre Technical Publishing Co., Inc. Moore/McCabe November 16, 2007 1:29 p.m. moore page T-11all you do is look up the critical value for the correct d.f., α(significance), and type of tail-test. Why only positive values?11TABLEIn this table, you look for the probability α/2 (in the uppertail only)Tables•T-11Table entry for p and C isthe critical value t∗withprobability p lying to itsright and probability C lyingbetween −t∗and t∗.Probability pt*TABL E Dt distribution critical valuesUpper-tail probability pdf .25 .20 .15 .10 .05 .025 .02 .01 .005 .0025 .001 .00051 1. 0 0 0 1.376 1.963 3.078 6.314 12.71 15.89 31.82 63.66 127.3 318.3 636.620.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.09 22.33 31.6030.765 0.978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 7.453 10.21 12.9240.741 0.941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173 8.61050.727 0.920 1.156 1.476 2.015 2.571