EXERCISE 3.1
The covariance

between

and

for the
entire bank data set is positive. Given the definitions of

and

, we would expect a negative covariance.
Using Figure
3.1 can you explain why

is positive?
EXERCISE 3.2
Consider the two sub-clouds of counterfeit and genuine bank notes in
Figure
3.1 separately. Do you still expect

(now calculated separately for each cloud) to be positive?
EXERCISE 3.3
We remarked that for two normal random variables, zero
covariance implies independence. Why does this remark not apply to
Example
3.4?
EXERCISE 3.4
Compute the covariance between the variables
from the car data set (Table
B.3).
What sign do you expect the covariance to have?
EXERCISE 3.5
Compute the correlation matrix of the variables in Example
3.2. Comment on the sign of the correlations and test
the hypothesis
EXERCISE 3.6
Suppose you have observed a set of observations

with

,

and

. Define the variable

. Can you
immediately tell whether

?
EXERCISE 3.7
Find formulas (
3.29) and (
3.30) for

and

by differentiating the objective function in
(
3.28) w.r.t.

and

.
EXERCISE 3.8
How many sales does the textile manager expect with a ``classic blue''
pullover price of

?
EXERCISE 3.9
What does a scatterplot of two random variables look like for

and

?
EXERCISE 3.10
Prove the variance decomposition (
3.38) and show that the coefficient of determination
is the square of the simple correlation between

and

.
EXERCISE 3.11
Make a boxplot for the residuals

for the ``classic blue'' pullovers data. If there
are outliers, identify them and run the linear regression again without
them. Do you obtain a stronger influence of price on sales?
EXERCISE 3.12
Under what circumstances would you obtain the same coefficients from
the linear regression lines of

on

and of

on

?
EXERCISE 3.13
Treat the design of Example
3.14 as if there were thirty shops
and not ten.
Define

as the index of the shop, i.e.,

. The null hypothesis is a constant regression
line,

. What does the alternative regression
curve look like?
EXERCISE 3.14
Perform the test in Exercise
3.13
for the shop example with a

significance level. Do you still
reject the hypothesis of equal marketing strategies?
EXERCISE 3.15
Compute an approximate confidence interval for

in Example (
3.2). Hint: start from a confidence interval for

and then apply the inverse transformation.
EXERCISE 3.16
In Example
3.2, using the exchange rate of 1 EUR = 106 JPY, compute the same empirical covariance using prices in Japanese Yen rather than in Euros. Is there a significant difference? Why?
EXERCISE 3.17
Why does the correlation have the same sign as the covariance?
EXERCISE 3.18
Show that

.
EXERCISE 3.19
Show that

is the
standardized data matrix, i.e.,

and

.
EXERCISE 3.20
Compute for the pullovers data the regression of

on

and of

on

. Which one has the better
coefficient of determination?
EXERCISE 3.21
Compare for the pullovers data the coefficient of determination for
the regression of

on

(Example
3.11), of

on

(Exercise
3.20) and of

on

(Example
3.15). Observe that this coefficient is
increasing with the number of predictor variables. Is this always the
case?
EXERCISE 3.22
Consider the ANOVA problem (Section
3.5) again. Establish the constraint
Matrix

for testing

. Test this hypothesis
via an analog of (
3.55) and (
3.56).
EXERCISE 3.24
Consider the linear model

where

is subject to
the linear constraints

where

is of rank

and

is of dimension

.
Show that

.
(Hint, let

where

and solve

and

).
EXERCISE 3.25
Compute the covariance matrix

where

denotes the matrix of observations on the counterfeit bank notes.
Make a Jordan decomposition of

. Why are all of the eigenvalues
positive?
EXERCISE 3.26
Compute the covariance of the counterfeit notes after they are linearly
transformed by the vector

.