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MTH5120: Statistical Modelling

创建日期：2024-05-06 15:31:18

所属分类：数学mathematics

标签： MTH5120

Statistical Modelling

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MTH5120: Statistical Modelling I
You should attempt ALL questions. Marks available are shown next to the ques-
tions.
In completing this assessment, you may use books, notes, and the internet. You
may use calculators and computers, but you should show your working for any
calculations you do. You must not seek or obtain help from anyone else.
At the start of your work, please copy out and sign the following declaration:
I declare that my submission is entirely my own, and I have not sought or obtained
help from anyone else.
All work should be handwritten, and should include your student number.
You have 24 hours in which to complete and submit this assessment. When you have
finished your work:
scan your work, convert it to a single PDF file and upload this using the upload tool
on the QMplus page for the module;
e-mail a copy to [email protected] with your student number and the module code in
the subject line;
with your e-mail, include a photograph of the first page of your work together with
either yourself or your student ID card.
You are not expected to spend a long time working on this assessment. We expect you to
spend about 2 hours to complete the assessment, plus the time taken to scan and upload
your work. Please try to upload your work well before the end of the assessment period, in
case you experience computer problems. Only one attempt is allowed – once you have
submitted your work, it is final.
IFoA exemptions
This module counts towards IFoA actuarial exemptions. For your submission to be eligible for
IFoA exemptions, you must submit within the first 3 hours of the assessment period. You
may then submit a second version later in the assessment period if you wish, which will count
only towards your degree. There are two separate upload tools on the QMplus page to enable
you to submit a second version of your work.
Examiners: L I Pettit, D S Coad
© Queen Mary University of London (2020) Continue to next page
MTH5120 (2020) Page 2
Question 1 [15 marks].
You should answer this question using a calculator. You should show all working.
(a) For the data given below find the values of x¯, y¯, Sxx, Syy and Sxy. [5]
x 7 6 5 1 5 4 7 3 4
y 97 86 78 10 75 62 101 39 y9
Note: The value of y9 is equal to 43 plus the sum of the seventh and ninth digits of
your student number.
(b) Hence find the least squares estimates of β0 and β1 in the simple linear regression model
of y on x. [3]
(c) Find the Analysis of Variance table to test the hypothesis that β1 = 0 against a two
sided alternative using a 5% significance level. [7]
You are given the following values from R:
qf(0.95,1,6)= 5.99, qf(0.95,1,7)= 5.59, qf(0.95,1,8)= 5.32,
qf(0.95,1,9)= 5.12, qf(0.95,2,6)= 5.14, qf(0.95,2,7)= 4.74,
qf(0.95,2,8)= 4.46.
Question 2 [34 marks]. Crickets are insects which make a characteristic chirping sound.
It has been observed that the frequency of chirps seems to be related to the temperature. A
biologist measured the number of chirps (y) per second a cricket made at various
temperatures (x) measured in degrees Fahrenheit.
(a) State the assumptions made about the errors in a simple linear regression model. [2]
(b) The biologist fitted a simple linear regression model to the data and obtained the
following plot. Comment on whether a simple linear regression model seems to fit the
data. [3]
© Queen Mary University of London (2020) Continue to next page
MTH5120 (2020) Page 3
(c) The following output was obtained from R. Two quantities in the Table of Coefficients
are recorded as A and B. Find their values. [4]
> crickets<-lm(y~x)
> summary(crickets)
Call:
lm(formula = y ~ x)
Residuals:
Min 1Q Median 3Q Max
-1.62746 -0.56464 0.08213 0.76762 1.54563
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.46951 2.96747 A 0.876716
x B 0.03727 5.447 0.000112 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.9799 on 13 degrees of freedom
Multiple R-squared: 0.6953,Adjusted R-squared: 0.6719
F-statistic: 29.67 on 1 and 13 DF, p-value: 0.0001119
(d) Two t tests can be carried out using this output. For each write down the null and
alternative hypotheses, the distribution of the test statistic if H0 is true and state your
conclusions. [8]
(e) The biologist looked at the plot of standardised residuals versus x.
(i) Comment on this plot. [3]
© Queen Mary University of London (2020) Continue to next page
MTH5120 (2020) Page 4
(ii) The biologist obtained the following output. Explain what assumption he is
checking and the conclusion. What graph could he have looked at to examine this
assumption? [4]
> shapiro.test(stdres)
Shapiro-Wilk normality test
data: stdres
W = 0.96628, p-value = 0.7996
(iii) The biologist is interested in predicting the mean number of chirps per second
when the temperature is 85 degrees Fahrenheit. He obtains the following output.
Explain what it shows. [5]
> pred1 <- predict(crickets, newdata=data.frame(x=85), interval=’confidence’)
> pred1
fit lwr upr
1 17.72361 17.01161 18.4356
(iv) The biologist wonders if it would be possible to predict the temperature based on
the number of chirps. Discuss how you could do that. Looking at the data
comment on how good the prediction would be. [5]
Question 3 [19 marks]. For the general linear model Y = Xβ+ ε, where ε is a vector of
errors assumed to be uncorrelated with zero mean and constant variance σ2, the formula for
the least squares estimator βˆ is
βˆ = (XTX)−1XTY .

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COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 CIVL2010 ECMT2150 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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 PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 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