Semester 1 Assessment, 2022
School of Mathematics and Statistics
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
This assignment is worth 7% of your total mark.
You may choose to either typeset your assignment in LATEX, or handwrite and scan it to
produce an electronic version.
You may use R for this assignment, including the lm function unless otherwise specified.
If you do, include your R commands and output.
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
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.
Submit your scanned assignment as a single PDF file and carefully review the submission
in Gradescope. Scan again and resubmit if necessary.
University of Melbourne 2022 Page 1 of 4 pages Can be placed in Baillieu Library
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 (τ?A ? τ?C) + var (τ?B ? τ?C)? 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.
Question 4 (11 marks)
Maple trees have winged seeds called samara. An experiment is conducted to investigate the
effect of shape on the speed of descent. Samara were collected from three trees, and their “disk
loading” (a quantity based on size and weight, which was used to quantify shape) and descent
velocity are calculated. The data is given in the file heli.csv, available on the LMS.
(a) Plot the data, using different colours and/or symbols for each tree. What do you observe?
(b) Test for the presence of interaction between disk loading and tree.
(c) Use backward elimination from the model with interaction to select variables for the data.
(d) Add lines corresponding to model from part (c), and the full model with interaction, to
the plot from question (a).
(e) In the full model with interaction, test the hypothesis that a samara from tree 2 with a
disk loading of 0.2 has an average descent velocity of 1.
Page 3 of 4 pages
MAST30025 Linear Statistical Models Assignment 3 Semester 1, 2022
Question 5 (9 marks)
An experiment is to be set up to test the effectiveness of a new teat disinfectant in controlling
mastitis in dairy cows. The disinfectant is applied to the cow’s four teats immediately after
milking. There are only two treatments in the experiment: disinfectant, or no disinfectant. The
disinfectant is applied as a spray by the experimenter. There are 24 cows available, and there
are 4 sections of the milking shed where the treatment can be applied; 6 cows can fit into each
section. Each section is managed by a different farm worker. Following a week of milking, each
of the four teats on each cow is given an infection rating on a 7-point scale.
For each of the following six designs, state the following:
What the experimental unit is;
What type of design is used (completely randomised, randomised block, or neither). If it
is a randomised block design, state the blocking factor;
Any flaws in the experiment (statistically unsound aspects).
Here are six possible experimental designs:
(a) Twelve cows are randomly chosen to get the disinfectant.
(b) The two left or the two right teats on each cow are randomly chosen to get the disinfectant.
(Assume that the teats respond to any treatment independently of each other.)
(c) Three cows in each section are randomly chosen to get the disinfectant.
(d) The first three cows to be milked in each section get the disinfectant.
(e) Two sections are randomly chosen, and the cows in those sections get the disinfectant.
(f) All 24 cows get the disinfectant, and the results are compared with measurements taken
before the experiment.
End of Assignment — Total Available Marks = 47