MAST90044 Thinking and Reasoning with Data: Lab 8b
Solutions for non-lab 8b Exercises
5. Vitamin C in tomato juice:
(a) Observational study.
(b) Continuous.
(c) Examples: mean vitamin C concentration in all cans of that brand of tomato juice stocked
at th

MAST30025: Linear Statistical Models
Week 9 Lab
1. Recall Question 5 from the Week 8 lab. In a manufacturing plant, filters are used to remove
pollutants. We are interested in comparing the lifespan of 5 different types of filters. Six filters
of each typ

MAST30025: Linear Statistical Models
Solutions to Week 6 Lab
1. In this question we consider the hypothesis H0 : = . Recall that the test statistic for this
hypothesis is
(b )T X T X(b )/p
.
SSRes /(n p)
(a) Show that
(b )T X T X(b ) = (y X )T (y X ) (y X

MAST30025: Linear Statistical Models
Week 8 Lab
1. Consider a less than full rank model with two factors. Factor 1 has two levels, and factor 2 has 3
levels. We take 2 samples from each possible combination of factor levels. We denote the response
variabl

MAST30025: Linear Statistical Models
Week 5 Lab
1. Derive a formula for the confidence interval of the individual parameter 0 from the formula for
the confidence interval of a linear combination of parameters, which is
q
tT b t/2 s tT (X T X)1 t.
Solution

MAST30025 (620-328) Linear Statistical Models
Semester 1 Exam, 2011
Department of Mathematics and Statistics
The University of Melbourne
Exam duration: 3 hours
Reading time: 15 minutes
This exam has 7 pages, including this page.
Authorised materials:
Scie

Student Number:
The University of Melbourne
Semester 1 Exam June, 2014
Department of Mathematics and Statistics
MAST30025 Linear Statistical Models
Exam Duration: 3 Hours
Reading Time: 15 Minutes
This paper has 8 pages
Authorised materials:
Scientific cal

MAST30025: Linear Statistical Models
Week 4 Lab
This practice class will use a single example. We model an individuals income at age 30 against the
number of years of formal education (with a linear model). The following data is collected:
Years of formal

MAST30025: Linear Statistical Models
Week 3 Lab
1. Let X be a 10 5 matrix of full rank and put H = X(X T X)1 X T .
Find tr(H) and r(H).
Solution: tr(H) = tr(X(X T X)1 X T ) = tr(X T X(X T X)1 ) = tr(I5 ) = 5. Since H is symmetric
and idempotent, r(H) = tr

MAST30025: Linear Statistical Models
Week 2 Lab
1. Show that X T X is a symmetric matrix.
Solution:
(X T X)T = X T (X T )T = X T X.
2. (a) Let
a b
c d
A=
be a nonsingular 2 2 matrix. Show by direct multiplication that
1
d b
.
A1 =
a
ad bc c
Solution:
1
ad

MAST30025 Linear Statistical Models
Semester 1 Exam, 2013
Department of Mathematics and Statistics
The University of Melbourne
Exam duration: 3 hours
Reading time: 15 minutes
This exam has 6 pages, including this page.
Authorised materials:
Scientific cal

MAST30025: Linear Statistical Models
Solutions to Week 7 Lab
1. Recall the joint confidence region for the parameters of a full rank linear model:
(b )T X T X(b ) ps2 f .
Use this to derive a test for the hypothesis H0 : = . Show that this test is equival

MAST30025: Linear Statistical Models
Week 10 Lab Solutions
1. An industrial psychologist is investigating absenteeism among production-line workers, based on
different types of work hours: (1) 4-day week with a 10-hour day, (2) 5-day week with a flexible

MAST30025: Linear Statistical Models
Week 11 Lab Solution
1. Suppose that y N (1, ) where
=
1
1
1
. . .
. . .
.
.
.
.
1
For what values of are the sample mean and sample variance independent?
Solution: We can write
y =
s2
=
1 T
1 y
n
1
yT (I n1 J)y

=0
General linear hypothesis
Splitting
Corrected SS
Sequential testing
Linear Statistical Models: Inference for the full
rank model
Notes by Yao-ban Chan and Owen Jones
Linear Statistical Models: Inference for the full rank model
1/123
=0
General linear

Params
Var
Diagnostics
MLE
CIs
PIs
Joint CIs
Generalised LS
Nonlinearities
Linear Statistical Models: Estimation for the Full
Rank Model
Notes by Yao-ban Chan and Owen Jones
Linear Statistical Models: Estimation for the Full Rank Model
1/163
Params
Var
Di

Classification
Conditional inverses
Normal equations
Estimability
2
Interval estimation
Linear Statistical Models: the less than full rank
model
Notes by Yao-ban Chan and Owen Jones
Linear Statistical Models: the less than full rank model
1/123
Classifica

Inverses
R
Orthogonality
Eigenthings
Rank
Idempotence
Trace
Theorems
Quadratic forms
Linear Statistical Models: Linear Algebra
Notes by Yao-ban Chan and Owen Jones
Linear Statistical Models: Linear Algebra
1/65
Inverses
R
Orthogonality
Eigenthings
Rank
Id

Random vectors
Random quadratic forms
Independence
Linear Statistical Models: Random vectors
Notes by Yao-ban Chan and Owen Jones
Linear Statistical Models: Random vectors
1/53
Random vectors
Random quadratic forms
Independence
Random vectors
The theory o

General linear model
Examples
Introduction
Notes by Yao-ban Chan and Owen Jones
I keep six honest serving-men
(They taught me all I knew);
Their names are What and Why and When
And How and Where and Who.
Rudyard Kipling
Linear Statistical Models: Introduc

MAST30025 Linear Statistical Models
Semester 1 Exam, 2012
Department of Mathematics and Statistics
The University of Melbourne
Exam duration: 3 hours
Reading time: 15 minutes
This exam has 6 pages, including this page.
Authorised materials:
Scientific cal

620-328 Linear Statistical Models
Semester 1 Exam June 18, 2010
Department of Mathematics and Statistics
The University of Melbourne
Exam duration: 3 hours
Reading time: 15 minutes
This exam has 12 pages, including this page.
Authorised materials:
Calcula

MAST30025: Linear Statistical Models
Assignment 1, 2015
Due: 3:15pm Thursday 2nd April (week 5)
This assignment is worth 6% of your total mark. Fill in a plagiarism declaration form and hand it
in together with this assignment.
1. Show that if A is symmet

620-371: Linear Models
Practice Class 7
21st April, 2009
In this practice class, we shall analyse a large(ish) dataset. Go to the
Datasets section of the 620-371 website and download the sleep dataset. This
dataset contains (among other things) data on th

MAST30025: Linear Statistical Models
Week 3 Lab
1. Let X be a 10 5 matrix of full rank and let H = X(X T X)1 X T .
Find tr(H) and r(H).
2. Let
3
A= 1
2
8
y1
1 , y = y2 .
4
y3
1
0
1
Let z = yT Ay. Write out z in full, then find
z
y
directly and using the m

MAST30025: Linear Statistical Models
Week 4 Lab
We model an individuals income at age 30 against the number of years of formal education with a
linear model. The following data is collected:
Years of formal education (x) Income ($k) (y)
8
8
12
15
14
16
16