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A p=6 dimensional random vector 'I'r = [ Y1, Y2, Ya, Ya, Y5, Ya] I from a sample of n=60 motor cyclists was studied. Data consists of the following 6 measures of headfface dimensions on each cyclist: Y1: HWWDE = head width at widest dimension, Y2: HDCIRCUNI = head otrcmference, Y3: EYEIEVELFB = nentto-back measurement: at eye level, Y4: EYETOPHD = eyetotopofhead measurement, Y5: EARTOPHD = eaItotopofhead measurement, Ya: WIDTHJAW = jaw Widl Referring to the SAS output in Appendix 1, answer the following questions. NOTE: You can round up 5A5 numbers in Appendix 1 to two decimal places, for ease of writing. a} bi a} hi it Give the formulation of the first 4 principal components Prinj,j=1, ..., 4. {2 marks} Find the variance and the cumulative proportion explained by each of the 5 principal components. {3 marks] Interpret the first 4 Principal Components. Justify your answers carefully according to the Principal Component Pattern Profile plot displayed in Appendix 1. [4 marks] How many PC's would you retain based on the scree plot and your answer in b]? Justify your answer. [4 marks} Perform formal statistical tests to ascertain the optimal number of principal components to retain. HINT: Test the signicance of the ''larger" components, that is, the components corraponding to the larger eigenvalues. {5 marks] Construct the 95 96 CI for in. Show your formula and working along with the result. {2 marks] Construct the 95 946 CI for 153. Show your formula and working along with the result. {2 marks] Which variables corrtribute the most to PC1? {1 mark] From your answer in c} what motor cyclist head attribute profile would you give as a name and dacription in your own words to PC: and PC3? {4 marks} Find the correlation coefcient between the variable Y3 and PCT [3 marks} APPENDIX 1: Motor Cyclists PCA Y1= HWWIDE = head width at widest dimension, Y= HDCIRCUM = head circumference, Y;= EYELEVELFB = front-to-back measurement at eye level, Y4= EYETOPHD = eye-to-top-of-head measurement, Y;= EARTOPHD = ear-to-top-of-head measurement, Y= WIDTHJAW = jaw width Observations 60 Variables Simple Statistics y1 v2 v6 Mean 15.50000000 57.57483333 19.80666667 10.51343433 13.57500000 11.87333333 SID 0.60841556 1.62141442 0.67694895 1.21396767 0.69857603 0.56893200 Covariance Matrix yl y3 y4 y5 y6 0.370169492 0.602033898 0. 148813559 0.044406780 0. 107118644 0.209322034 0.602033898 2.62898-47 18 0.801475706 0.665629379 0. 102783898 0.376859587 0.1486 13559 0.801475706 0. 458259887 0.011265537 -0.013220339 0.119841808 y4 0.044406780 0.665629379 0.011265537 1.473717514 0.252203390 -0.054384181 Q. 1071 18644 0. 102783898 -0.013220339 0.252203390 0. 488008475 -0.035593220 0.209322034 0.376859867 0. 119841808 -0.054384181 -0.035593220 D.323683616 Total Variance 5.7428237006 Eigenvalues of the Covariance Matrix Eigenvalue Difference Proportion Cumulative 3.32341443 1.94910637 0.5787 1.5787 2 1.37430806 0.89823926 0.2393 0.8180 3 0.47606880 0.15138456 0.0829 0.9009 0.32468424 0.16818700 0.0565 0.9575 5 0. 15649723 0.06864629 0.0273 0.9847 6 0.08785095 0.0153 1.0000 Eigenvectors Print Prin2 Prin3 Prin4 Prins PrinG 0.207444 . 141526 0.421553 0.442546 -.168262 0.731486 V2 0.872845 -.219128 .094338 .130981 -330410 -. 238085 y3 0.261265 -.231401 -. 120880 -.381913 0.767628 0.358-430 0.325862 0.891178 .173086 0.173308 0.164103 0.112657 0.065639 0.222030 0.867467 -.354519 0.114432 -.234771 0. 127883 -.186846 0. 13-4576 0.695721 0.482950 746039413 Motor Cyclists PCA 1.00 0.75 0.50 r Correlation 0.25 0.00 - 0.25 HWWIDE HDCIRCUM EYELEVELFB EYETOPHD EARTOPHD WIDTHJAW Variable Component -0-1 -0-2 -0 -0 :4 -0 5-0 -6 The PRINCOMP Procedure Scree Plot Variance Explained 1.0 0.8 N Eigenvalue Proportion 0.4 1 07 0.0 2 3 5 6 3 5 6 Principal Component Principal Component 1 9 : Cumulative Proportion