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STAT2008 FINANCE, ACTUARIAL STUDIES
创建日期：2024-04-12 15:59:20
标签： STAT2008
FINANCE, ACTUARIAL STUDIES

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RESEARCH SCHOOL OF FINANCE, ACTUARIAL STUDIES AND STATISTICS REGRESSION MODELLING STAT2008 / STAT4038 / STAT6038 Final Examination for Semester 1, 2020 

Reading Time: 0 minutes Writing Time: 240 minutes Exam Conditions: Centrally Scheduled Examination Permitted Materials: Any Instructions: • This examination paper comprises a total of 21 pages and there is a separate file of R Output which also has a total of 15 pages. • You can write your answer on this question sheet or on blank sheets of paper. Please include the cover sheet as the first page when you submit. • Please write your student number and course code in the space provided at the top of this page. • There is one part (Q2c) which is only to be attempted by students enrolled in STAT4038 and STAT6038. • There are 5 questions worth a total of 103 marks for STAT2008 students, excluding Q2c. • There are 5 questions worth a total of 113 marks for students enrolled in STAT4038 and STAT6038. • Statistical tables (generated using R) are provided on Wattle site. • Unless otherwise indicated, use a significance level (α) of 5% and 4 decimal places. Question 1 2 3 4 5 Total STAT2008/4038/6038 8 15 16 36 28 103 STAT4038/6038 only 0 10 0 0 0 10 Score: Final Examination - Semester 1, 2020 Page 1 of 21 Question 1 [8 Marks] You want to conduct some simulation studies. You have written out an algorithm as follows: 1. Generate Xi = i, i = 1, . . . , 10 2. Generate Yi = 4−2X+εi, i = 1, . . . , 10, where εi are i.i.d. and follow normal distribution with mean 0 and variance 5. 3. Fit Xi and Yi in a simple linear regression model using least squares and record the parameter estimates b0 and b1. Ŷ = b0 + b1X. 4. Construct a 99% confidence interval for the intercept parameter. 5. Repeat 1000 times steps 2, 3 and 4, so that you have 1000 b0 values and confidence intervals at hand. (a) 6 marks What distribution do you expect best describes these 1000 b0 values? You should state the shape, mean, and variance of this distribution. (b) 2 marks How many of these 1000 confidence interval do you expect to include the number 4? Final Examination - Semester 1, 2020 Page 2 of 21 Question 2 [15 Marks] For STAT4038 and STAT6038 students: [25 Marks] The Fnord Motor Company is about to embark on a major marketing campaign for their newly released vehicle, the Imposta II, under the slogan “Have more left in the tank after a ride in an Imposta II”. Before they can make such a statement, Fnord’s crack legal department suggested that the company might conduct a scientific study to support their claim. Fnord’s slick marketing team decided to put 30 Fnord Imposta IIs through their paces at Fnord’s top-secret test facilities in Oodnadatta. There, Fnord’s expert drivers were each given an Imposta II full of petrol (40 litres) and each car was driven for a different number of kilometres, denoted X, and at their return, the amount of petrol remaining in the tank, Y , was measured. In this question, we will attempt to model the amount of petrol left in terms of the distance driven. The attached R printout gives details of the modelling exercise attempted. From the attached printout, answer the following questions: (a) 3 marks State the true model as in model1 and then write down the fitted model (b) 4 marks Find a 95% confidence interval for β0. Is the value 40 contained in your confidence interval? Explain why you might expect it to be. Final Examination - Semester 1, 2020 Page 3 of 21 (c) 10 marks [For 4038 and 6038 students only] The manager argued that all Imposta IIs are given full tank of petrol so the intercept is known to be 40, thus the model should be Yi = 40 + β1Xi + i. Using least squares, you find that the estimator for β1 in this model is b1 = ∑ XiYi−40 ∑ Xi∑ X2i . Find the variance of this estimator and then develop a formula for a 95% confidence interval for the mean response given X = x0 under this model. You do not need to use the actual numbers but make sure you specify the appropriate degrees of freedom used. Final Examination - Semester 1, 2020 Page 4 of 21 (d) 8 marks The manager claims that the new Imposta II is more energy efficient than their old model Imposta I. He asked the same drivers to also drive 30 old Imposta I’s for a different number of distances W , and measure the amount of petrol remaining in the tank denoted as Z. He fitted a second model as model2. Assume the true error variance of model1 and model2 are the same. Conduct a test testing the claim of the manager. Final Examination - Semester 1, 2020 Page 5 of 21 Question 3 [16 Marks] You are given a response variable Y and 4 covariates X1, X2, X3 and X4. A number of different models are fitted in R. Use the printout to answer the following questions. (a) 2 marks Write down the model fit as model1 making sure you write down the values estimated for the parameters in the model. (b) 2 marks Write down the parameter estimates for the model fit as model2. (c) 2 marks What proportion of the variability in Y is explained by the linear model in X1, X2, X3 and X4 fit as model2? Final Examination - Semester 1, 2020 Page 6 of 21 (d) 4 marks Test whether the linear model fit in model2 plausibly passes through the origin. Use a 5% test. Final Examination - Semester 1, 2020 Page 7 of 21 (e) 6 marks You want to see whether X1 and X4 are given equal weighting in the formula for Y and whether X2 and X3 are given equal weighting in the formula for Y . Test the hypothesis H0 : β1 = β4, β2 = β3. Use information contained in the printout to test this hypothesis at the 5% level. Final Examination - Semester 1, 2020 Page 8 of 21 Question 4 [36 Marks] Drought, flood, super-cell storms. . . climate change is upon us! One of the indicators of the progress of climate change is the onset and nature of the so-called El Nin˜o and La Nin˜a effects used to describe ocean temperatures. Scientists think that these effects are instrumental parts of our weather patterns, and that these effects dictate the prevalence of floods and drought in south-eastern Australia and other parts of the world. The data for this question concerns the number of tropical storms and hurricanes in the Atlantic Basin between 1950 and 1997. Several variables were recorded: the year (Year); a record of whether the year was a cold, warm or neutral El Nin˜o year (elnino); a record of whether West Africa was wet, dry or normal that year (wa); the number of tropical storms each year; the number of hurricanes each year; and a storm index (called NTC) which is a composite variable measuring the overall intensity of the hurricane season (an average of the number of tropical storms, the number of hurricanes, the number of days of tropical storms, the number of days of hurricanes, the total number of intense hurricanes, and the number of days they last). In this question, we will focus on the relationship between the storm index NTC and its relationship with time and the variables elnino and wa. (a) 3 marks The data set contains 48 years worth of data. How many of the 48 years were cold El Nin˜o years? Warm El Nin˜o years? Neutral El Nin˜o years? How many of the 48 years were wet years in West Africa? 

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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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