EXERCISE 7.1
Use Theorem
7.1 to derive a test for testing
the hypothesis that a dice is balanced, based on

tosses of that dice.
(Hint: use the multinomial probability function.)
EXERCISE 7.2
Consider

.
Formulate the hypothesis

in terms
of

.
EXERCISE 7.3
Simulate a normal sample with

and

and test

first with

known and then
with

unknown. Compare the results.
EXERCISE 7.4
Derive expression (
7.3) for the likelihood ratio test
statistic in Test Problem
2.
EXERCISE 7.5
With the simulated data set of Example
7.14, test the hypothesis of equality of the
covariance matrices.
EXERCISE 7.6
In the U.S. companies data set, test the equality of means between the
energy and manufacturing sectors,
taking the full vector of observations

to

. Derive the simultaneous confidence
intervals for the differences.
EXERCISE 7.8
Repeat the preceeding exercise with

unknown and

.
Compare the results.
EXERCISE 7.10
Consider two independent i.i.d. samples, each of size 10, from two
bivariate normal populations. The results
are summarized below:
Provide a solution to the following tests:
Compare the solutions and comment.
EXERCISE 7.11
Prove expression (
7.4) in the Test Problem 2
with log-likelihoods

and

.
(Hint: use (
2.29).
EXERCISE 7.13
The yields of wheat have been measured in 30 parcels that
have been randomly attributed to 3 lots prepared by one of 3
different fertilizer A B and C.
The data are
Fertilizer Yield |
A |
B |
C |
1 |
4 |
6 |
2 |
2 |
3 |
7 |
1 |
3 |
2 |
7 |
1 |
4 |
5 |
5 |
1 |
5 |
4 |
5 |
3 |
6 |
4 |
5 |
4 |
7 |
3 |
8 |
3 |
8 |
3 |
9 |
3 |
9 |
3 |
9 |
2 |
10 |
1 |
6 |
2 |
Using Exercise 7.12,
- a)
- test the independence between the 3 variables.
- b)
- test whether
and compare this to
the 3 univariate
tests.
- c)
- test whether
using simple ANOVA and the
approximation.
EXERCISE 7.14
Consider an i.i.d. sample of size

from a bivariate normal
distribution
where

is a known parameter. Suppose

. For what value of

would the
hypothesis

be rejected in favor of

(at the 5% level)?
EXERCISE 7.15
Using Example
7.14, test the last two cases described there
and test the sample number one
(

), to see if they are from a normal population with

(the sample covariance matrix to be used is given by

).
EXERCISE 7.16
Consider the bank data set. For the counterfeit bank notes, we want to know
if the length of the diagonal
(

) can be predicted by a linear model in

to

. Estimate the linear model and test if
the coefficients are significantly different from zero.
EXERCISE 7.17
In Example
7.10, can you predict the vocabulary score
of the children in eleventh grade,
by knowing the results from grades 8-9 and 10? Estimate a linear model and test its significance.
EXERCISE 7.18
Test the equality of the covariance matrices from the two groups in the WAIS subtest
(Example
7.19).
EXERCISE 7.20
Using Theorem
6.3 and expression (7.16), construct an asymptotic rejection region
of size

for testing, in a general model

, with

against

.
EXERCISE 7.21
Exercise
6.5
considered the pdf

.
Solve the problem
of testing

from an iid sample of size

on

, where

is large.
EXERCISE 7.22
In
Olkin and Veath (1980), the evolution of citrate concentrations
in plasma is observed at 3 different times of day,

(8 am),

(11 am) and

(3 pm), for two groups of patients who
follow a different diets.
(The patients were randomly attributed to each group under a
balanced design

).
The data are:
Group |
(8 am) |
(11 am) |
(3 pm) |
|
125 |
137 |
121 |
|
144 |
173 |
147 |
I |
105 |
119 |
125 |
|
151 |
149 |
128 |
|
137 |
139 |
109 |
|
93 |
121 |
107 |
|
116 |
135 |
106 |
II |
109 |
83 |
100 |
|
89 |
95 |
83 |
|
116 |
128 |
100 |
Test if the profiles of the groups are parallel,
if they are at the same level and if they are horizontal.