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 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

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
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 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

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

=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

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

620-371: Linear Models
Practice Class 2
10th March, 2009
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
ad bc
A1 =
d b
c
a
.
Solut

620-371: Linear Models
Assignment 2
Due: Monday, 4th May, 2009
This assignment is worth 5% of your total mark.
You may use R (and only R) for this assignment, but for matrix calculations
and statistical distributions only. You may not use the lm function.

620-371: Linear Models
Assignment 3
Due: Monday, 25th May, 2009
This assignment is worth 5% of your total mark.
You may use R (and only R) for this assignment, but only for questions 1
and 2e. You may not use the lm function for question 1, but you may us

620-371: Linear Models
Practice Class 5
31st March, 2009
In this practice class, we will replicate and extend the results of the practice
class from last week, using R. From last week: we model an individuals income
at age 30 against the number of years o

620-371: Linear Models
Practice Class 10
12th May, 2009
In a manufacturing plant, lters are used to remove pollutants. We are
interested in comparing the lifespan of 5 dierent types of lters. Six lters of
each type are tested, and the time to failure in h

620-371: Linear Models
Practice Class 9
5th April, 2009
1. Show that in a mutually orthogonal full rank model, X T X is diagonal.
Calculate (X T X)1 (expressed in terms of the elements of the X matrix).
Solution: The (i, j)th element of X T X is
T
Xki Xkj

620-371: Linear Models
Practice Class 8
28th April, 2009
1. It is known that R( 1 | 2 ) has a noncentral 2 distribution with r degrees
of freedom and noncentrality parameter
=
1 T T
T
T
X [X(X T X)1 X T X2 (X2 X2 )1 X2 ]X.
2 2
Show that if H0 : 1 = 0 is

620-371: Linear Models
Practice Class 6
7th April, 2009
1. Derive the formula for the condence interval of the individual parameter
0 from the formula for the condence interval of a linear combination of
parameters:
tT b t/2 s tT (X T X)1 t.
Solution: We