MST 565 Linear Models [Model Linear]

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1 UNIVERSITI SAINS MALAYSIA Second Semester Eamination 05/06 Academic Session June 06 MST 565 Linear Models [Model Linear] Duration : hours [Masa : jam] Please check that this eamination paper consists of TEN pages of printed material before you begin the eamination. [Sila pastikan bahawa kertas peperiksaan ini mengandungi SEPULUH muka surat yang bercetak sebelum anda memulakan peperiksaan ini.] Instructions: Answer FOUR (4) questions. [Arahan: Jawab EMPAT (4) soalan]. In the event of any discrepancies the English version shall be used. [Sekiranya terdapat sebarang percanggahan pada soalan peperiksaan versi Bahasa Inggeris hendaklah diguna pakai.] /-

2 - -. Eplain the following terms: (a) (c) (d) (e) Symmetric matri Idempotent matri Less than full rank model Estimable functions Testable hypothesis Give eamples to illustrate your answers. [ 5 marks ]. Terangkan sebutan-sebutan berikut: (a) (c) (d) (e) Matrik simetri Matrik Idempoten Model pangkat kurang penuh Fungsi teranggar Hipotesis teruji Beri contoh untuk menggambarkan jawapan anda. [ 5 markah ]. (a) Let y y matri y be a random vector with mean vector and covariance y 0 μ Σ 0 0 (i) Which variables are independent? (ii) Let z y y y and z y y y covz where z. z z E z and. Find.../-

3 - - (iii) Define w as follows: w y y y w w w w y y y w y y y Then by using z as defined in (ii) find cov zw. Let random vector v be N 4 μσ where μ and Σ are given as μ Σ. 6 7 If v is partitioned as v y y (i) What are the partitioned form of μ and Σ? (ii) Find covariance matri of partial correlations Σ y and matri of partial correlation P y. (c) If y is N μσ where 0 0 μ Σ 0 0 Find a symmetric matri A such that value of? yay is. Then find the [ 0 marks ]...4/-

4 (a) Biar y y y kovarian y sebagai vektor rawak dengan vektor min and matrik 0 μ Σ 0 0 (i) Pemboleh ubah yang mana adalah tidak bersandar? (ii) Biar z y y y dan z y y y covz yang mana z. (iii) Tafsirkan w w y y y w y y y w y y y w w w z z E z dan. Cari seperti berikut: Kemudian menggunakan z seperti di dalam (ii) cari cov zw. Biar vektor rawak v ialah N sebagai μ Σ. 6 7 Jika v terpetak sebagai μσ yang mana μ dan Σ diberikan v y y (i) Apakah bentuk terpetak bagi μ dan Σ? (ii) Cari matrik kovarian bagi korelasi separa Σ y dan matrik korelasi separa P y....5/-

5 (c) Jika y adalah N μσ yang mana 0 0 μ Σ 0 0 Cari matrik A yang simetri supaya cari nilai? yay adalah. Kemudian [ 0 markah ]. (a) The data in Table. relate grams plant dry weight y to percent soil organic matter and kilograms of supplemental soil nitrogen added per 000 square metres. y Table.. Plant Dry Weight Data (i) Propose a suitable model for the data in Table.. (ii) What is the X matri for such a model? (iii) Find XX XX and Xy. (iv) Find the least squares estimator for β. (v) Write the regression equation. [ 0 marks ]...6/-

6 - 6 - In an effort to obtain maimum yield in a chemical reaction the value of the following variables were choosen by the eperimenter: = temperature ( 0 C ) = concentration of a reagent (%) = time of reaction (hours) One response variable was observed: y = percent of unchanged starting material The data are in the following Table.: (i) Table.. Chemical Reaction Data y Find and estimate cov ˆβ. (ii) Find R Ra R and adjusted. (iii) Consider model y 0 test the hypothesis H0 : by epressing the H 0 in the form H : Cβ 0 0 versus H : Cβ 0. [ 0 marks ]...7/-

7 (a) Data dalam Jadual. menghuraikan y gram berat kering tanaman terhadap peratus bahan organik tanah dan kilogram tanah tambahan bernitrogen per seribu m. Jadual.. Data Berat Kering Tumbuhan y (i) Cadangkan suatu model yang sesuai untuk data dalam Jadual.. (ii) Apakah matrik X bagi model tersebut? (iii) Cari X X X X dan Xy. (iv) Cari penganggar kuasa dua terkecil bagi β. (v) Tulis persamaan regressi tersebut. [ 0 markah ] Dalam usaha untuk mendapatkan hasil yang maksimum dalam suatu tindakbalas kimia nilai pemboleh ubah berikut telah dipilih oleh seorang pengkaji: = suhu ( 0 C ) = kepekatan reagen (%) = masa tindakbalas (jam) Satu pemboleh ubah respon yang diperhatikan ialah: y = peratusan bahan asal yang tidak berubah...8/-

8 - 8 - Data adalah dalam Jadual.: Jadual.. Data Tindakbalas Kimia y (i) Cari dan anggar cov ˆβ. (ii) Cari R Ra R terlaras dan. (iii) Pertimbangkan model y 0 uji hipotesis H0 : dalam bentuk H : Cβ 0 0lawan H Cβ 0 : [ 0 markah ]...9/-

9 Table 4. shows plasma inorganic phosphate levels (mg/dl) one hour after a standard gluscose tolerance test for obese subjects with or without hyperinsulinemia and controls (data from Jones 987). Table 4.. Plasma Phosphate Levels in Obese and Control Subjects. Hyperinsulinemic Obese Non- hyperinsulinemic Obese Controls (a) Assuming that the one-way classification model with fied effects is appropriate use these data to test hypothesis that there are no mean differences among the three groups. What is your conclusion? Obtain a 95% confidence interval for the difference in means between the two obese groups and comment on the interval. [ 5 marks ]...0/-

10 Jadual 4. menunjukkan tahap fosfat tak organik plasma (mg/dl) sejam selepas suatu ujian piawai bagi toleransi glukosa terhadap subjek-subjek yang berat badan berlebihan dengan atau tanpa hiperinsulinemik dan subjek kawalan (data daripada Jones 987). Jadual 4.. Tahap Fosfat dalam Plasma Subjek yang Berat Badan Berlebihan dan Subjek Kawalan. Hyperinsulinemic Obese Non- hyperinsulinemic Obese Kawalan (a) Andaikan bahawa pengkelasan sehala dengan kesan tetap adalah sesuai gunakan data tersebut untuk menguji hipotesis bahawa tidak wujud perbezaan min antara ketiga-tiga kumpulan. Apakah kesimpulan anda? Dapatkan selang keyakinan 95% bagi perbezaan min antara dua kumpulan subjek yang berat berlebihan dan berikan komen tentang selang keyakinan tersebut. [ 5 markah ] - ooo 0 ooo -

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