MST 564 Statistical Reliability [Kebolehpercayaan Statistik]

Transcription

1 UNIVERSITI SAINS MALAYSIA Firs Semeser Examinaion 2012/2013 Academic Session January 2013 MST 564 Saisical Reliabiliy [Kebolehpercayaan Saisik] Duraion : 3 hours [Masa : 3 jam] Please check ha his examinaion paper consiss of SIXTEEN pages of prined maerial before you begin he examinaion. [Sila pasikan bahawa keras peperiksaan ini mengandungi ENAM BELAS muka sura yang berceak sebelum anda memulakan peperiksaan ini.] Insrucions: [Arahan: Answer all four [4] quesions. Jawab semua empa [4] soalan.] In he even of any discrepancies, he English version shall be used. [Sekiranya erdapa sebarang percanggahan pada soalan peperiksaan, versi Bahasa Inggeris hendaklah diguna pakai]. 2/-

2 - 2 - [MST 564] 1. (a) The reliabiliy of an iem is he probabiliy ha i will adequaely perform is specified purpose for a specified period of ime under specified environmenal condiions. Give wo examples and reasoning for uilizing such a definiion. (b) (c) In reliabiliy daa analysis, once he definiion of a failure even is in place, you have o deermine wha daa is imporan o fully describe he failure even. Provide a lis and descripion of criical informaion ha is imporan o collec for any failure even. When we examine reliabiliy as a funcion of ime, his leads us o he definiion of failure rae. (i) How would you define failure rae? (ii) Wha sor of curve would represen a ime-dependen failure rae? (iii) How does he curve in (ii) reveal he behaviour of failure raes wih ime? (iv) Is he curve in (ii) sill valid in our digial age? If no, explain why no? Is here anoher curve which will beer represen he failure rae? [100 marks] 1. (a) 'Kebolehpercayaan sesuau benda adalah kebarangkalian bahawa ia akan melaksanakan ujuan erenu dengan sempurna bagi empoh masa yang dinyaakan di bawah syara-syara alam sekiar erenu.' Berikan dua conoh dan penjelasan unuk menggunakan definisi yang dinyaakan. (b) (c) Dalam analisis daa kebolehpercayaan, seelah definisi perisiwa kegagalan dienukan, anda perlu menenukan daa apa adalah pening unuk menggambarkan sepenuhnya perisiwa kegagalan. Sediakan suau senarai dan perihalan makluma kriikal yang pening unuk mengumpul sebarang perisiwa kegagalan. Apabila kia memeriksa kebolehpercayaan sebagai fungsi masa, ianya membawa kia kepada akrif kadar kegagalan. (i) Bagaimana anda akan mendefinisikan kadar kegagalan? (ii) Apakah jenis lengkungan yang akan mewakili kadar kegagalan yang berganung kepada masa? (iii) Bagaimanakah lengkungan dalam (ii) mendedahkan ingkah laku kadar kegagalan dengan masa? (iv) Adakah lengkungan dalam (ii) masih sah dalam zaman digial kia? Jika idak, jelaskan mengapa idak? Adakah erdapa sau lagi lengkungan yang akan mewakili kadar kegagalan dengan lebih baik? [100 markah]...3/-

3 - 3 - [MST 564] 2. (a) A componen has a consan failure rae of 0.02 per hour. (i) Wha is he probabiliy ha i will fail during he firs 10 hours of operaion? (ii) Suppose ha he componen has successfully operaed for 100 hours. Wha is he probabiliy ha i will fail during he nex 10 hours of operaion? (b) Given a Weibull failure disribuion wih a shape parameer of 1/3 and a scale parameer of 16,000, compleely characerize he failure process. (i) Wha is he reliabiliy funcion of his process? (ii) How does he shape parameer provide insigh ino he behaviour of his failure process? (iii) Compue he MTTF. If he disribuion is highly skewed, how would you represen a beer average ime o failure? Then compue he beer alernaive. (iv) If a 90% reliabiliy is desired, deermine he design life. [100 marks] 2. (a) Suau komponen mempunyai kadar kegagalan malar 0.02 seiap jam. (i) Apakah kebarangkalian bahawa ia akan gagal dalam operasi pada 10 jam perama? (ii) Kaakan bahawa komponen elah berjaya dikendalikan selama 100 jam. Apakah kebarangkalian bahawa ia akan gagal dalam operasi 10 jam seerusnya? (b) Diberi aburan kegagalan Weibull dengan parameer benuk 1/3 dan parameer skala 16,000, mencirikan proses kegagalan dengan sepenuhnya. (i) Apakah fungsi kebolehpercayaan proses ini? (ii) Bagaimanakah parameer benuk memberi penjelasan ke aas ingkah laku proses kegagalan ini? (iii) Kirakan MTTF. Jika aburan adalah sanga pencong, bagaimana anda akan mewakili suau puraa masa kegagalan yang lebih baik? Seerusnya, kira alernaif yang lebih baik. (iv) Jika kebolehpercayaan 90% diingini, enukan masa hidup reka benuk. [100 markah]...4/-

4 - 4 - [MST 564] 3. (a) Suppose ha you follow hireen paiens for survival and observe he following imes (in monhs) : 5, 12, 25, 26, 27, 28, 32+, 33+, 34+, 37, 39, 40+, 42+ (The noaion + denoes censoring.) (i) (ii) (iii) (iv) (v) (vi) Calculae he Kaplan-Meier survival esimaes for he hireen paiens. Inerpre he Kaplan-Meier survival funcion for he hireen paiens. Find S(36) where S is he survival funcion. Obain he acuarial life able aking inervals of widh six monhs. Find he esimaed hazard a he midpoin of each inerval. Discuss he srenghs and weaknesses of using he produc limi mehod and he acuarial mehod. (b) The daa in Table 1 gives he remission imes (in weeks) for paiens subjeced o wo differen reamens. Tweny paiens were assigned o each of he wo reamens; + denoes righ censoring. Table 1. Remission imes (in weeks) for weny paiens Treamen A Treamen B (i) (ii) (iii) Compare he survival prospecs of he wo groups by inerpreing he graphs of heir survival funcions. Obain esimaes of he probabiliy of survival beyond 12 weeks. Wha is your conclusion based on he insigh from pars (i) and (ii)? [100 marks] 3. (a) Andaikan bahawa anda mengikui haya hidup iga belas orang pesaki dan memerhaikan masa beriku (dalam bulan): 5, 12, 25, 26, 27, 28, 32 +, 33 +, 34 +, 37, 39, 40 +, 42 + (Caaan '+' menandakan 'censoring'.) (i) (ii) (iii) (iv) Kirakan anggaran haya Kaplan-Meier bagi 13 orang pesaki. Tafsirkan fungsi haya Kaplan-Meier bagi 13 orang pesaki. Cari S (36) di mana S adalah fungsi haya. Dapakan jadual jangka haya akuari dengan selang lebar enam bulan....5/-

5 - 5 - [MST 564] (v) (vi) Cari anggaran bahaya di iik engah seiap selang. Bincangkan kekuaan dan kelemahan menggunakan kaedah had produk dan kaedah akuari. (b) Daa dalam Jadual 1 memberikan masa remian (dalam minggu) unuk pesaki yang diberi dua rawaan yang berbeza. Dua puluh pesaki elah diugaskan unuk seiap dua rawaan; '+' menandakan righ censoring. Jadual 1. Masa remian (dalam minggu) unuk dua puluh pesaki Rawaan A Rawaan B (i) (ii) (iii) Bandingkan prospek survival kedua-dua kumpulan dengan menafsir graf fungsi haya mereka. Dapakan anggaran kebarangkalian haya melebihi 12 minggu. Apakah kesimpulan anda berdasarkan pandangan dari bahagian (i) dan (ii)? [100 markah] 4. (a) The daa se given in Table 2 was obained by Nelson & Hahn (1972). Hours o failure of componens are given as a funcion of operaing emperaures (classified as Low and High). There is severe censoring (0 = censored, 1 = uncensored) wih only 17 ou of 40 componens failing. The experimen is conduced a higher emperaures o speed up failure ime. (i) Find he disribuion which provides he bes fi o he failure daa in Table 2. (ii) Apply a suiable model which akes ino accoun he acceleraed process. Inerpre he applicaion of his model. (iii) Use your favourie saisical package o obain he acceleraion facor for he componens o fail. (iv) Wha is your overall conclusion?...6/-

6 - 6 - [MST 564] Table 2. Hours o failure of fory componens. Componen Time (hours) Saus Temperaure Low Low Low Low Low Low Low Low Low Low Low Low Low Low Low Low Low Low Low Low High High High High High High High High High High High High High High High High High High High High...7/-

7 - 7 - [MST 564] (b) One hundred and fory-nine diabeic paiens were followed for 17 years (a subse of daa from Lee e al., 1988). Table 3 (only 5 cases are displayed) gives he survival ime from baseline examinaion, survival saus, and several poenial prognosic facors a baseline: age (years), body mass index (BMI), age a diagnosis of diabees (years), smoking saus (0=No, 1=Ex-smoker, 2=Curren), sysolic blood pressure (SBP), diasolic blood pressure (DBP), elecrocardiogram reading (ECG) (1=Normal, 2=Borderline, 3=Abnormal), and wheher he paien had any coronary hear disease (CHD) (0=No, 1=Yes). (i) Deermine he quesions you ough o ask in he analysis of he daa in Table 3. (ii) Perform an appropriae analysis of he daa in Table 3 (use he daa file provided). (iii) Inerpre he resuls obained in (ii). Wha can you conclude? Paien Saus Survival Time (yrs) Table 3. Daa of 149 diabeic paiens (firs 5 cases displayed) Age (yrs) BMI Age a Diagnosis (yrs) Smoking Saus SBP (mmhg) DPB (mmhg) ECG Noe ha: Saus (0 = alive, 1 = dead) Smoking saus (0 = no, 1 = ex-smoker, 2 = curren) ECG (1 = normal, 2 = borderline, 3 = abnormal) CHD (0 = no, 1 = yes) [100 marks] CHD...8/-

8 - 8 - [MST 564] 4. (a) Se daa yang diberikan dalam Jadual 2 elah diperoleh oleh Nelson & Hahn (1972). Waku kegagalan komponen diberikan sebagai fungsi suhu operasi (diklasifikasikan sebagai Rendah dan Tinggi). Terdapa censoring eruk (0 = censored, 1 = uncensored ) dengan hanya 17 daripada 40 komponen gagal. Eksperimen dijalankan pada suhu yang lebih inggi unuk mempercepakan masa kegagalan. (i) Cari aburan yang menyediakan penyuaian erbaik unuk daa kegagalan dalam Jadual 2. (ii) Aplikasikan model yang sesuai yang mengambil kira proses dipercepakan. Tafsirkan aplikasi model ini. (iii) Gunakan pakej saisik kegemaran anda unuk mendapakan fakor pecuan bagi komponen gagal. (iv) Apakah kesimpulan anda secara keseluruhan? Jadual 2. Jam unuk kegagalan empa puluh komponen. Komponen Masa (jam) Saus Suhu Rendah Rendah Rendah Rendah Rendah Rendah Rendah Rendah Rendah Rendah Rendah Rendah Rendah Rendah Rendah Rendah Rendah Rendah Rendah Rendah Tinggi Tinggi Tinggi Tinggi Tinggi Tinggi Tinggi Tinggi Tinggi...9/-

9 - 9 - [MST 564] Tinggi Tinggi Tinggi Tinggi Tinggi Tinggi Tinggi Tinggi Tinggi Tinggi Tinggi (b) Seraus dan empa puluh sembilan pesaki kencing manis elah diikui selama 17 ahun (subse daa daripada Lee e al., 1988). Jadual 3 (hanya 5 kes dipaparkan) memberikan masa survival daripada pemeriksaan asas, saus hidup, dan beberapa fakor ramalan poensi pada asas: umur (ahun), indeks jisim badan (BMI), umur semasa diagnosis diabees (ahun), saus merokok (0 = Tidak, 1 = Bekas-perokok, 2 = Semasa), ekanan darah sisolik (SBP), ekanan darah diasolik (DBP), bacaan elekrokardiogram (ECG) (1 = Normal, 2 = Sempadan, 3 = Tidak Normal), dan sama ada pesaki mempunyai sebarang penyaki janung koronari (CHD) (0 = Tiada, 1 = Ya). (i) Tenukan soalan yang anda harus beranya dalam analisis daa dalam Jadual 3. (ii) Lakukan analisis yang sesuai pada daa dalam Jadual 3 (menggunakan fail daa yang disediakan). (iii) Tafsirkan kepuusan yang diperoleh dalam (ii). Apa yang anda boleh simpulkan?...10/-

10 Pesaki Saus Masa Haya (ahun) [MST 564] Jadual 3. Daa 149 pesaki kencing manis (5 kes perama dipaparkan) Umur (ahun) BMI Umur semasa Diagnosis (ahun) Saus Merokok SBP (mmhg) DPB (mm Hg) ECG CHD Perhaikan bahawa: Saus (0 = hidup 1, = mai) Saus merokok (0 = idak, 1 = bekas perokok, 2 = semasa) ECG (1 = normal, 2 = sempadan, 3 = idak normal) CHD (0 = idak, 1 = ya) [100 markah]...11/-

11 [MST 564] APPENDIX Summary of Reliabiliy Formulae F( ) f ( ) d 0 R( ) 1 F( ) df( ) dr( ) f() d d h () f() R () H ( ) h( ) d 0 R() e H() H ( ) ln R( ) MTTF f ( ) d R( ) d /-

12 [MST 564] APPENDIKS Ringkasan Rumus-Rumus Kebolehpercayaan F( ) f ( ) d 0 R( ) 1 F( ) df( ) dr( ) f() d d h () f() R () H ( ) h( ) d 0 R() e H() H ( ) ln R( ) MTTF f ( ) d R( ) d /-

13 [MST 564] APPENDIX (cond) Summary of Reliabiliy Formulae (cond) Lifeime following an Exponenial Disribuion: f () e F( ) 1e R() e h () H () 1 MTTF Lifeime following a Weibull Disribuion: f 1 ( ) exp F( ) 1e R() e h() 1 H () 1 MTTF 1 Design Life ( ln R) R 1/...14/-

14 [MST 564] APPENDIKS (sambung) Ringkasan Rumus-Rumus Kebolehpercayaan (sambung) Masahaya mengiku Taburan Eksponen: f () e F( ) 1e R() e h () H () 1 MTTF Masahaya mengiku Taburan Weibull: f 1 ( ) exp F( ) 1e R() e h() 1 H () 1 MTTF 1 Design Life ( ln R) R 1/...15/-

15 [MST 564] APPENDIX (cond) Summary of Reliabiliy Formulae (cond) Lifeime following a Normal Disribuion: 1 1 ( ) f( ) exp F () R ( ) 1 f() h () 1 Lifeime following a Lognormal Disribuion: f( ) exp ln 2 2 s 2s median 1 F ( ) ln s median 1 R ( ) 1 ln s median 2 s MTTF median exp 2 exp sz R median 1R...16/-

16 [MST 564] APPENDIKS (sambung) Ringkasan Rumus-Rumus Kebolehpercayaan (sambung) Masahaya mengiku Taburan Normal: 1 1 ( ) f( ) exp F () R ( ) 1 f() h () 1 Masahaya mengiku Taburan Lognormal: f( ) exp ln 2 2 s 2s median 1 F ( ) ln s median 1 R ( ) 1 ln s median 2 s MTTF median exp 2 exp sz R median 1R - ooo O ooo -