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MAST30025 Linear Statistical Models
创建日期：2024-05-09 17:35:25
所属分类：数学mathematics
标签： MAST30025
Linear Statistical Models

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MAST30025 Linear Statistical Models Assignment 3

Submission deadline: Friday May 27, 5pm
This assignment consists of 4 pages (including this page) with 5 questions and 47 total marks
Instructions to Students
Writing
ˆ This assignment is worth 7% of your total mark.
ˆ You may choose to either typeset your assignment in LATEX, or handwrite and scan it to
produce an electronic version.
ˆ You may use R for this assignment, including the lm function unless otherwise specified.
If you do, include your R commands and output.
ˆ Write your answers on A4 paper. Page 1 should only have your student number, the
subject code and the subject name. Write on one side of each sheet only. Each question
should be on a new page. The question number must be written at the top of each page.
Scanning and Submitting
ˆ Put the pages in question order and all the same way up. Use a scanning app to scan all
pages to PDF. Scan directly from above. Crop pages to A4.
ˆ Submit your scanned assignment as a single PDF file and carefully review the submission
in Gradescope. Scan again and resubmit if necessary.

MAST30025 Linear Statistical Models Assignment 3 Semester 1, 2022
Question 1 (7 marks)
Let X =
[
1 2 1
2 1 1
]
and let A = XTX.
(a) Calculate A.
(b) Find a conditional inverse Ac such that r(Ac) = 1, or show that no such conditional inverse
exists.
(c) Find a conditional inverse Ac such that r(Ac) = 2, or show that no such conditional inverse
exists.
(d) Find a conditional inverse Ac such that r(Ac) = 3, or show that no such conditional inverse
exists.
Question 2 (11 marks)
Consider a one-way classification model
yij = µ+ τi + εij
for i = 1, 2, 3 and j = 1, 2, . . . , ni. The following data is collected:
Factor level: A B C
ni 12 8 16
Mean response: 11.3 8.4 10.2
We are also given s2 = 4.9.
For this question, you may not use the lm function in R.
(a) Calculate a 95% confidence interval for τA − τB.
(b) Calculate the F -test statistic for the hypothesis τA = τB = τC , and state the degrees of
freedom for the test.
(c) Test the hypothesis H0 : τC − τB ≥ 2 against H1 : τC − τB < 2 at the 5% significance level.
(d) Suppose the above data is collected through a completely randomised design with total
sample size n = 36. Does this design minimise 2var (τˆA − τˆC) + var (τˆB − τˆC)? If not,
what is the optimal allocation for nA, nB, and nC?
Page 2 of 4 pages
MAST30025 Linear Statistical Models Assignment 3 Semester 1, 2022
Question 3 (9 marks)
Consider the two-factor model with interaction
yij = µ+ τi + βj + ξij .
Suppose that there are a and b levels of the factors respectively. Now consider the set of
equations
ξij − ξ1j − ξi1 + ξ11 = 0, i = 2, . . . , a, j = 2, . . . , b.
(a) Show that the equations are not redundant.
(b) Show that these equations are equivalent to the hypothesis of no interaction.
(c) Thereby calculate the rank of the hypothesis of no interaction.
(d) Show that the hypothesis is testable, provided there exists at least one sample from each
combination of factor levels.

案例分类
标签云
关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 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