# 辅导MAST30025、R程序设计辅导

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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

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