UNIVERSITI SAINS MALAYSIA Secod Semester Examiatio 016/017 Academic Sessio Jue 017 MSG 466 - Multivariate Aalysis [Aalisis Multivariat] Duratio : 3 hours [Masa : 3 jam] Please check that this examiatio paper cosists of TWENTY-ONE pages of prited material before you begi the examiatio. [Sila pastika bahawa kertas peperiksaa ii megadugi DUA PULUH SATU muka surat yag bercetak sebelum ada memulaka peperiksaa ii.] Istructios: [Araha: Aswer all eight [8] questios. Jawab semua lapa [8] soala.] I the evet of ay discrepacies, the Eglish versio shall be used. [Sekiraya terdapat sebarag percaggaha pada soala peperiksaa, versi Bahasa Iggeris hedaklah digua pakai]. /-
- - Questio 1 (a) What are the three properties that must be satisfied by a distace measure, d (P, Q) betwee two poits P ( x1, x, K, x p ) ad Q (y 1, y, K, y p )? (b) The distace from the poit P = ( x1, x ) to the poit Q = ( y1, y ) is defied as d( P, Q) max x y, x y. 1 1 (i) Does this measure of distace satisfy the three coditios that have bee stated i (a) above? Justify your aswer. Soala 1 (a) (ii) Compute the distace from the poit P = (1, 3) to the poit Q = (, 1). (iii) Plot the locus of poits whose distace from P = (1, 3) is. [ 0 marks ] Apakah tiga sifat yag mesti dipeuhi bagi suatu ukura jarak, d (P, Q) atara dua titik P ( x1, x, K, x p ) da Q (y 1, y, K, y p )? (b) Jarak dari titik P = ( x1, x ) ke titik Q = ( y1, y ) ditakrifka sebagai d( P, Q) max x y, x y. 1 1 (i) Adakah ukura jarak ii memeuhi ketiga-tiga syarat yag diyataka dalam (a) di atas? Tetusahka jawapa ada. (ii) Kira jarak atara titik P = (1, 3) ke titik Q = (, 1). (iii) Lakar lokus titik-titik yag jarakya dari P = (1, 3) adalah. Questio [ 0 markah ] (a) Suppose X : N ( μσ, ). Based o a radom sample of size 30, the sample mea vector ad the sample variace-covariace matrix are obtaied as below: x 9 3 1 ad 4 S 1 (i) Is S a positive defiite matrix? 3/-
- 3 - (ii) Determie the geeralized sample variace ad total sample variace. (iii) Fid the maximum likelihood estimates of the 1 mea vector μ ad the variace-covariace matrix Σ. (b) Let X be N3 ( μσ, ) where X1 4 9 0 X X, μ 3 ad Σ 0 4 1. X 3 1 1 (i) Fid the distributios of X1 X. 3 (ii) Determie the coditioal distributio of X give that X1 x ad X3 x3. [ 30 marks ] Soala (a) Adaika X : N ( μσ, ). Berdasarka suatu sampel rawak bersaiz 30, vektor mi sampel da matriks varias-kovarias sampel diperoleh seperti di bawah: x 9 3 1 da 4 S 1 (i) Adakah S matriks tetu positif? (ii) Tetuka varias sampel teritlak da varias sampel keseluruha. (iii) Cari aggara kebolehjadia maksimum bagi vektor mi 1, μ da matriks varias-kovarias, Σ. (b) Biar X sebagai N3 ( μσ, ) yag maa X1 4 9 0 X X, μ 3 da Σ 0 4 1. X 3 1 1 (i) Cari tabura bagi X1 X. 3 (ii) Tetuka tabura bersyarat bagi X diberi bahawa X1 x da X3 x3. [ 30 markah ] 4/-
- 4 - Questio 3 Let X1, X ad X 3 be mutually idepedet ad idetically distributed 3 1 radom vectors from N3 ( μσ,, ) where μ (3,1, )' ad 1 0 Σ 4 3. 0 3 9 (a) Fid the margial distributio for each of the radom vectors ad V 1 X 1 X 1 X 3 3 3 1 1 3 V X X X. 1 3 (b) Are V 1 ad V idepedet? Justify your aswer. (c) Specify the joit distributio of V 1 ad V. [ 5 marks ] Soala 3 Biar X1, X da X 3 sebagai vektor rawak 3 1 secama daripada N3 ( μσ,, ) yag maa μ (3,1, )' da 1 0 Σ 4 3. 0 3 9 bertabura salig tak bersadar da (a) Cari tabura sut bagi setiap vektor rawak da V 1 X 1 X 1 X 3 3 3 1 1 3 V X X X. 1 3 (b) Adakah V 1 da V tak bersadar? Tetusahka jawapa ada. (c) Pericika tabura tercatum V 1 da V. [ 5 markah ]...5/-
- 5 - Questio 4 Madible measuremets were made o eight male moder dogs from Thailad. Two of the measuremets are legth of first molar (i mm) ad breadth of first molar (i mm), which are assumed to have a bivariate ormal distributio. The sample mea vector ad the sample variace-covariace matrix are as follows: x m 19.8 0.79 0.19 ad m 7.8 S 0.19 0.08 (a) Test Ho: μ m (18.5, 8.5)' at 0.05. State ay assumptios that you have made ad give your coclusio from this test. (b) Similar measuremets are also made for eight female dogs of the same species. The sample mea vector ad the sample variace-covariace matrix are give by x 19.0 0.57 0.6 f ad f 7.5 S 0.6 0.37. Is there evidece of differeces betwee the madible size of the male ad the female dogs? Perform a appropriate test ad state your assumptios. Soala 4 [ 0 marks ] Ukura-ukura rahag dibuat pada lapa ekor ajig jata mode daripada Thailad. Dua daripada ukura adalah pajag molar pertama (dalam mm) da lebar molar pertama (dalam mm), yag diaggap mempuyai tabura ormal bivariat. Vektor mi sampel da matriks varias-kovarias sampel adalah seperti berikut: x m 19.8 0.79 0.19 da m 7.8 S 0.19 0.08 (a) Uji Ho: μ m (18.5, 8.5)' pada 0.05. Nyataka sebarag adaia yag telah ada buat da berika kesimpula daripada ujia ii. (b) Ukura yag sama juga telah dibuat pada lapa ekor ajig betia daripada spesies yag sama. Vektor mi sampel da matriks variaskovarias sampelya diberi oleh x 19.0 0.57 0.6 f da f 7.5 S 0.6 0.37. Adakah terdapat bukti perbezaa saiz rahag atara ajig jata da ajig betia? Lakuka ujia yag bersesuaia da yataka adaia-adaia ada. [ 0 markah ] 6/-
- 6 - Questio 5 The table below displays the edited data of four measuremets made o 40 male skulls from the area of Thebes i Egypt. There are two samples of 0 skulls each from the early predyastic period ad the Roma period. For each skull, the measuremets (i mm) are: X1 = maximum breadth X = basibregmatic height X3 = basialveolar legth X4 = asal height Skull X1 X X3 X4 1 131 138 89 49 M M M M M 0 13 131 101 49 1 137 13 91 50 M M M M M 40 145 19 89 47 Skulls 1 to 0 are from the early predyastic period while skulls 1 to 40 are from the Roma period. Statistical aalyses usig MINITAB have bee performed for this dataset ad the output is displayed i Appedix A. (a) (b) (c) (d) Based o the output, i your opiio, what aalyses have bee performed? How are the four measuremets related? Do the skulls from the early predyastic ad Roma periods have statistically sigificat differece for their mea values of the variables? Discuss your aswer. Suppose, a skull is just foud from the area ad the four measuremets o the skull are (16, 133, 10, 51). Usig the iformatio i the output, determie whether the skull is most likely from the early predyastic or Roma period. Discuss your aswer. [ 5 marks ] 7/-
- 7 - Soala 5 Jadual di bawah mempamerka data yag telah disutig bagi empat ukura yag dibuat pada 40 tegkorak lelaki daripada kawasa Thebes di Mesir. Terdapat sampel dega 0 tegkorak setiap satuya daripada zama awal pradiastik da zama Roma. Bagi setiap tegkorak, ukura-ukuraya (dalam mm) adalah: X1 = lebar maksimum X = tiggi basibregmatik X3 = pajag basialveolar X4 = tiggi hidug Tegkorak X1 X X3 X4 1 131 138 89 49 M M M M M 0 13 131 101 49 1 137 13 91 50 M M M M M 40 145 19 89 47 Tegkorak 1 ke 0 adalah daripada zama awal pradiastik semetara tegkorak 1 ke 40 adalah daripada zama Roma. Aalisis statistik megguaka MINITAB telah dijalaka pada set data ii da outputya dipamerka dalam Lampira A. (a) (b) (c) (d) Berdasarka output, pada pedapat ada, aalisis apa yag telah dijalaka? Bagaimaa keempat-empat ukura ii berkait? Adakah tegkorak pada zama awal pradiastik da tegkorak pada zama Roma mempuyai perbezaa bererti secara statistik bagi ilai mi pembolehubah-pembolehubah mereka? Bicagka jawapa ada. Adaika suatu tegkorak baru sahaja dijumpai daripada kawasa itu da empat ukura pada tegkorak adalah (16, 133, 10, 51). Megguaka maklumat daripada output, tetuka sama ada tegkorak itu berkemugkia besar daripada zama awal pradiastik atau daripada zama Roma. Bicagka jawapa ada. [ 5 markah ] Questio 6 (a) Aswer the followig questios o Factor Aalysis: (i) (ii) (iii) (iv) What is the purpose of the aalysis? How does it differ from Pricipal Compoet Aalysis? How are the variables grouped? What is the purpose of factor rotatio? 8/-
- 8 - (b) A set of data cosists of 130 observatios geerated by scores o a psychological test admiistered to Peruvia teeagers (see table below). The scores were accumulated ito five subscale scores labeled idepedece (idep), support (supp), beevolece (beev), coformity (coform) ad leadership (leader). Idep Supp Beev Coform Leader 7 13 14 0 11 1 13 4 5 6 M M M M M 7 19 7 9 10 17 8 Soala 6 Factor aalysis was performed o this data set usig the pricipal compoet method ad the maximum likelihood method. The output of the aalysis is displayed i Appedix B. Iterpret ad discuss the results. [ 5 marks ] (a) Jawab soala-soala berkeaa Aalisis Faktor berikut: (i) (ii) (iii) (iv) Apakah tujua aalisis tersebut? Bagaimaa ia berbeza daripada Aalisis Kompoe Utama? Bagaimaa pembolehubah-pembolehubah dikelaska? Apakah tujua putara faktor? (b) Suatu set data megadugi 130 cerapa yag dijaa daripada skor suatu ujia psikologi terhadap remaja-remaja Peru (lihat jadual di bawah). Skor-skor dikumpul dalam lima skor subskala berlabel berdikari (idep), sokoga (supp), kebajika (beev), pematuha (coform) da kepimpia (leader). Idep Supp Beev Coform Leader 7 13 14 0 11 1 13 4 5 6 M M M M M 7 19 7 9 10 17 8 Aalisis faktor telah dijalaka pada set data ii megguaka kaedah kompoe utama da kaedah kebolehjadia maksimum. Output aalisis dipamerka dalam Lampira B. Tafsir da bicag keputusaya. [ 5 markah ] 9/-
- 9 - Questio 7 The percetages of the labour force i three differet types of idustry (agriculture, miig ad maufacturig) for five Europea coutries are show i the followig table Coutry AGR MIN MAN Belgium.6 0. 0.8 Romaia.0.6 37.9 Switzerlad 5.6 0.0 4.7 Frace 5.1 0.3 0. Bulgaria 19.0 0.0 35.0 (a) (b) Determie the distace matrix for the data usig the city-block metric. Cluster the five coutries usig the complete likage hierarchical procedure. Draw a dedogram ad discuss your results. [ 30 marks ] Soala 7 Peratusa teaga kerja dalam tiga jeis idustri berbeza (pertaia, perlomboga da pembuata) bagi lima egara Eropah ditujukka dalam jadual di bawah : Negara AGR MIN MAN Belgium.6 0. 0.8 Romaia.0.6 37.9 Switzerlad 5.6 0.0 4.7 Peracis 5.1 0.3 0. Bulgaria 19.0 0.0 35.0 (a) (b) Tetuka matriks jarak bagi data megguaka metrik blok-badar. Klusterka kelima-lima egara tersebut megguaka tatacara berhierarki pauta legkap. Lukis dedogram da bicag keputusa ada. [ 30 markah ] 10/-
- 10 - Questio 8 Observatios o two resposes are collected for four treatmets. The summary statistics of the two variables for Treatmet 1, Treatmet, Treatmet 3 ad Treatmet 4 are as follows: Treatmet 1: x 5 3 1, 1, 1 18 3 S 1 4 1 Treatmet : x,, 13 1 S 1 Treatmet 3: x 6 3 1 3, 3, 3 0 4 S 1 1 Treatmet 4: x 3 4, 4, 4 14 4 S 1 Costruct a oe-way MANOVA table ad test for treatmet effects usig 0.05. Give your coclusio ad state ay assumptios that you have made. [ 5 marks ] Soala 8 Cerapa daripada dua respo dikumpul bagi empat rawata. Statistik rigkas daripada dua pembolehubah bagi Rawata 1, Rawata, Rawata 3 da Rawata 4 adalah seperti berikut: Rawata 1: x 5 3 1, 1, 1 18 3 S 1 4 1 Rawata : x,, 13 1 S 1 Rawata 3: x 6 3 1 3, 3, 3 0 4 S 1 1 Rawata 4: x 3 4, 4, 4 14 4 S 1 Bia jadual MANOVA satu-hala da uji kesa rawata megguaka 0.05. Berika kesimpula ada da yataka sebarag adaia yag ada telah buat. [ 5 markah ] 11/-
- 11 - Appedix/Lampira A *Period: Early Predyastic (1), Roma () Descriptive Statistics: X1, X, X3, X4 Total Variable Period Cout Mea Variace Miimum Maximum X1 1 0 13.00 9.68 119.00 141.00 0 135.90 8.73 16.00 145.00 X 1 0 134.10 1.88 11.00 143.00 0 130.90 8.5 10.00 138.00 X3 1 0 97.80 9.43 89.00 109.00 0 94.350 15.9 86.000 101.000 X4 1 0 50.500 7.105 44.000 56.000 0 51.700 1.64 45.000 58.000 Correlatios: X1, X, X3, X4 X1 X X3 X -0.09 0.861 X3-0.068 0.07 0.678 0.01 X4 0.1 0.40 0.060 0.170 0.136 0.714 Cell Cotets: Pearso correlatio P-Value ANOVA: X1, X, X3, X4 versus Period Factor Type Levels Values Period fixed 1, Aalysis of Variace for X1 Source DF SS MS F P Period 1 15.10 15.10 5.1 0.08 Error 38 1109.80 9.1 Total 39 161.90 S = 5.40419 R-Sq = 1.05% R-Sq(adj) = 9.74% Aalysis of Variace for X Source DF SS MS F P Period 1 10.40 10.40 4.06 0.051 Error 38 957.60 5.0 Total 39 1060.00 S = 5.01996 R-Sq = 9.66% R-Sq(adj) = 7.8% 1/-
- 1 - Aalysis of Variace for X3 Source DF SS MS F P Period 1 119.03 119.03 5.3 0.07 Error 38 849.75.36 Total 39 968.78 S = 4.7883 R-Sq = 1.9% R-Sq(adj) = 9.98% Aalysis of Variace for X4 Source DF SS MS F P Period 1 14.400 14.400 1.46 0.35 Error 38 375.00 9.874 Total 39 389.600 S = 3.144 R-Sq = 3.70% R-Sq(adj) = 1.16% MANOVA for Period s = 1 m = 1.0 = 16.5 Test DF Criterio Statistic F Num Deom P Wilks' 0.67173 4.76 4 35 0.006 Lawley-Hotellig 0.48869 4.76 4 35 0.006 Pillai's 0.387 4.76 4 35 0.006 Roy's 0.48869 SSCP Matrix for Period X1 X X3 X4 X1 15.1-14.8-134.6 46.80 X -14.8 10.4 110.4-38.40 X3-134.6 110.4 119.0-41.40 X4 46.8-38.4-41.4 14.40 SSCP Matrix for Error X1 X X3 X4 X1 1109.80 91.80 59.70 108.40 X 91.80 957.60 99.10 19.40 X3 59.70 99.10 849.75 78.10 X4 108.40 19.40 78.10 375.0 Partial Correlatios for the Error SSCP Matrix X1 X X3 X4 X1 1.00000 0.08905 0.06148 0.16799 X 0.08905 1.00000 0.10986 0.3098 X3 0.06148 0.10986 1.00000 0.1383 X4 0.16799 0.3098 0.1383 1.00000 Discrimiat Aalysis: Period versus X1, X, X3, X4 Liear Method for Respose: Period Predictors: X1, X, X3, X4 13/-
- 13 - Group 1 Cout 0 0 Summary of classificatio True Group Put ito Group 1 1 17 3 3 17 Total N 0 0 N correct 17 17 Proportio 0.850 0.850 N = 40 N Correct = 34 Proportio Correct = 0.850 Squared Distace Betwee Groups 1 1 0.00000 1.85703 1.85703 0.00000 Liear Discrimiat Fuctio for Groups 1 Costat -746.05-737.1 X1 3.87 4.01 X 4.38 4. X3 3.50 3.33 X4 1.0 1. Summary of Misclassified Observatios True Pred Squared Observatio Group Group Group Distace Probability 6** 1 1 8.4 0.14 4.50 0.876 1** 1 1 10.17 0.080 5.91 0.90 18** 1 1.108 0.370 1.040 0.630 6** 1 1.564 0.931 7.775 0.069 8** 1 1 5.709 0.640 6.861 0.360 36** 1 1 3.866 0.550 4.67 0.450 14/-
- 14 - Appedix/Lampira B Descriptive Statistics: Idep, Supp, Beev, Coform, Leader Total Variable Cout Mea StDev Variace Idep 130 15.669 5.895 34.750 Supp 130 17.077 4.185 17.513 Beev 130 18.785 5.463 9.845 Coform 130 15.500 5.748 33.043 Leader 130 11.731 5.19 6.958 Correlatios: Idep, Supp, Beev, Coform, Leader Idep Supp Beev Coform Supp -0.173 0.049 Beev -0.561 0.018 0.000 0.836 Coform -0.471-0.37 0.98 0.000 0.000 0.001 Leader 0.187-0.401-0.49-0.333 0.033 0.000 0.000 0.000 Cell Cotets: Pearso correlatio P-Value.5 Scree Plot of Idep,..., Leader.0 Eigevalue 1.5 1.0 0.5 0.0 1 3 Factor Number 4 5 15/-
- 15 - Factor Aalysis: Idep, Supp, Beev, Coform, Leader Pricipal Compoet Factor Aalysis of the Correlatio Matrix Urotated Factor Loadigs ad Commualities Variable Factor1 Factor Commuality Idep -0.77 0.101 0.606 Supp 0.180 0.9 0.88 Beev 0.813-0.009 0.660 Coform 0.651-0.574 0.753 Leader -0.696-0.4 0.66 Variace.1966 1.368 3.5649 % Var 0.439 0.74 0.713 Rotated Factor Loadigs ad Commualities Varimax Rotatio Variable Factor1 Factor Commuality Idep 0.775 0.076 0.606 Supp 0.033-0.939 0.88 Beev -0.794-0.174 0.660 Coform -0.764 0.413 0.753 Leader 0.583 0.568 0.66 Variace.1544 1.4105 3.5649 % Var 0.431 0.8 0.713 Factor Score Coefficiets Variable Factor1 Factor Idep 0.359 0.007 Supp 0.07-0.675 Beev -0.36-0.077 Coform -0.383 0.34 Leader 0.39 0.37 Factor Aalysis: Idep, Supp, Beev, Coform, Leader Pricipal Compoet Factor Aalysis of the Correlatio Matrix Urotated Factor Loadigs ad Commualities Variable Factor1 Factor Factor3 Commuality Idep -0.77 0.101-0.580 0.943 Supp 0.180 0.9 0.163 0.909 Beev 0.813-0.009 0.100 0.670 Coform 0.651-0.574-0.56 0.819 Leader -0.696-0.4 0.563 0.979 Variace.1966 1.368 0.7559 4.307 % Var 0.439 0.74 0.151 0.864 Rotated Factor Loadigs ad Commualities Varimax Rotatio Variable Factor1 Factor Factor3 Commuality Idep -0.971 0.018-0.003 0.943 Supp 0.136-0.31 0.890 0.909 Beev 0.700-0.418-0.081 0.670 Coform 0.419-0.379-0.707 0.819 Leader -0.155 0.971-0.111 0.979 16/-
- 16 - Variace 1.6506 1.3587 1.3114 4.307 % Var 0.330 0.7 0.6 0.864 Factor Score Coefficiets Variable Factor1 Factor Factor3 Idep -0.75-0.36-0.147 Supp 0.119-0.19 0.690 Beev 0.37-0.17-0.010 Coform 0.073-0.77-0.545 Leader 0.40 0.83 0.008 Factor Aalysis: Idep, Supp, Beev, Coform, Leader Maximum Likelihood Factor Aalysis of the Correlatio Matrix * NOTE * Heywood case Urotated Factor Loadigs ad Commualities Variable Factor1 Factor Commuality Idep 0.679 0.173 0.49 Supp 0.000-1.000 1.000 Beev -0.689-0.018 0.475 Coform -0.668 0.37 0.553 Leader 0.59 0.401 0.441 Variace 1.660 1.985.9605 % Var 0.33 0.60 0.59 Rotated Factor Loadigs ad Commualities Varimax Rotatio Variable Factor1 Factor Commuality Idep 0.679 0.173 0.49 Supp 0.000-1.000 1.000 Beev -0.689-0.018 0.475 Coform -0.668 0.37 0.553 Leader 0.59 0.401 0.441 Variace 1.660 1.985.9605 % Var 0.33 0.60 0.59 Factor Score Coefficiets Variable Factor1 Factor Idep 0.310 0.000 Supp 0.034-1.000 Beev -0.305 0.000 Coform -0.347-0.000 Leader 0.19 0.000 * WARNING * Too may factors, solutio is ot uique 17/-
- 17 - Factor Aalysis: Idep, Supp, Beev, Coform, Leader Maximum Likelihood Factor Aalysis of the Correlatio Matrix * NOTE * Heywood case Urotated Factor Loadigs ad Commualities Variable Factor1 Factor Factor3 Commuality Idep -0.788 0.187 0.587 1.000 Supp -0.464-0.401-0.790 1.000 Beev 0.53-0.49-0.086 0.53 Coform 0.664-0.333 0.194 0.589 Leader 0.000 1.000 0.000 1.000 Variace 1.5591 1.5486 1.0133 4.111 % Var 0.31 0.310 0.03 0.84 Rotated Factor Loadigs ad Commualities Varimax Rotatio Variable Factor1 Factor Factor3 Commuality Idep -0.99 0.034 0.1 1.000 Supp 0.048-0.19-0.980 1.000 Beev 0.56-0.454 0.098 0.53 Coform 0.515-0.371 0.43 0.589 Leader -0.19 0.968 0.13 1.000 Variace 1.584 1.3199 1.170 4.111 % Var 0.317 0.64 0.43 0.84 Factor Score Coefficiets Variable Factor1 Factor Factor3 Idep -1.016-0.130-0.04 Supp -0.13 0.19-1.069 Beev -0.000 0.000 0.000 Coform -0.000 0.000-0.000 Leader 0.011 1.081-0.11 18/-
- 18 - Appedix/Lampira C FORMULAE SHEET 1. Suppose X has E( X) c' Σc. μ ad Cov( X) Σ. Thus cx ' has mea. Bivariate ormal p.d.f.: 1 1 f ( x, x ) exp 11 1 1 1 1 1 x1 1 x x1 1 x 1 11 11 c' μ ad variace 3. Multivariate ormal p.d.f.: 4. If X : N ( μσ, ), the p f( x) 1 p/ (a) a' X : N a' μ, a' Σa (b) AX : N q Aμ, AΣA ' (c) Xd : N p μ d, Σ (d) AX d : N q Aμ d, AΣA ' (e) 1 X μ' Σ X μ : p Σ 1/ e 1 ( xμ)' Σ ( xμ)/ 5. Let X j : N p( μj, Σ), j1,..., be mutually idepedet. The V1 c jxj : N p c j j, c j μ Σ. Moreover, V 1 ad V bjx j are j1 j1 j1 j1 joitly multivariate ormal with covariace matrix j1 c j Σ bc ' Σ b ' c Σ. b j Σ j1 X1 μ1 6. Let X be distributed as X : N p( μσ, ) with μ X μ, Σ11 Σ1 Σ ad Σ1 Σ Σ 0. The the coditioal distributio of X 1, give that X x, is ormal ad 1 has mea = μ Σ Σ x μ ad covariace = 1 1 1 11 1 1 Σ Σ Σ Σ. 19/-
- 19-7. Oe-sample results: 1 (a) T X μ' S X μ X 1 1, ' X S X X X X j j j j1 1 j1 p T : Fp, p 1 (b) 100(1 )% simultaeous cofidece itervals for a' μ : p p ( 1) a' X Fp, p( ) a' Sa ( p) (c) 100(1 )% Boferroi cofidece iterval for i : x i p t1 (d) 100(1 )% large sample cofidece iterval for i : sii x i sii p( ) 8. Two-sample results (Paired comparisos): 1 (a) T Dδ' δd Dδ D 1 1, ' D S D D D D j d j j j1 1 j1 p T : Fp, p 1 (b) 100(1 )% simultaeous cofidece itervals for i : p d i p ( 1) sd F, ( ) i p p ( p) 9. Two-sample results (Idepedet samples): 1 1 1 1 p 1 1 1 (a) T X X μ μ ' S X X μ μ 1 0/-
- 0-1 p 1 T : F 1 p 1 p, p1 S p S i 1S 1 i j1 1 1 1 S ij i ij i x x x x i 1 ' (b) 100(1 )% simultaeous cofidece iterval for ' a μ μ : 1 1 1 a' X1 X c a' S pa 1 c 1 p F 1 p 1 1 p, p1 ( ) (c) For large 1 p ad p, a approximate 100(1 )% simultaeous cofidece iterval for ' a μ μ : 1 1 1 a' X1 X c a' S1 S a 1 c ( ) p 10. Oe-way MANOVA g l1 B x x x x l l l ' g l 1 1 W x x x x ' ( 1) S ( 1) S... ( 1) S l1 j1 lj l lj l g g * W B + W Distributio of * : g1 * For p 1, g : : Fg 1, g 1 * g g11 * p, g : F g 1 * : For ( g1),( g1) 1/-
- 1 - p11 * For p 1, g : : Fp, p1 p * p 1 * p 1, g 3: F p * : For p, ( p) l 11. The Estimated Miimum ECM Rule for two ormal populatios: Allocate x 0 to populatio 1 if 1 1 1 c(1 ) p x1 x ' Sp x0 x1 x ' Sp x1 x l c( 1) p1 Allocate x 0 to populatio otherwise. - ooo O ooo -