Linköpings Universitet
IDA/Statistik
BW
732A37 Multivariate Statistical Methods, 6 hp
Exam in Multivariate Statistical Methods, 2012-03-16
Time allowed:
Allowed aids:
Examinator:
Grades:
kl: 14-18
Calculator, The book: Johnson, Wichern: Applied Multivariate
Statistical Analysis. Notes in the book and Copy of the book are
allowed.
Bertil Wegmann
A=22-24 points, B=19-21p, C=16-18p, D=13-15p, E=10-12p
Provide a detailed report that shows motivation of the results.
GOOD LUCK!
_________________________________________________________________________________________
1
 4
3
.
Let the random vector X  ( X1, X 2 )' have covariance matrix   

3
2


Determine the principal components and find the proportion of the total
variance of X explained by the first component.
3p
2
Observations on two responses (X1 = maximum jump height, X2 = maximum
jump length (in metres)), were collected for top performance athletes that
either participated in fitness treatment 1 or 2 during a three-month period.
The observation vectors x1 , x2  are:
Treatment 1:
Treatment 2:
(2.18, 6.8), (2.25, 7.2), (2.16, 6.9), (2.20, 7.0), (2.20, 7.1)
(2.15, 6.5), (2.18, 6.7), (2.14, 6.4), (2.16, 6.6)
Assume for each treatment that the data on  X1, X 2  is a random sample
from a bivariate normal population.
a) Evaluate T 2 , for testing, H 0 : µ’ = (2.2, 7.1), for treatment 1. 3p
b) Specify the distribution of T 2 for the situation in a) and test H 0 at
significance level   0.05. State your conclusions clearly.
2p
c) Construct the one-way MANOVA table for comparing population mean
vectors between treatments.
4p
d) Evaluate Wilks’ lambda,  * , and use Table 6.3 to test for differences
in the population mean vectors between the treatments. Use
significance level   0.05 for the test.
2p
3
Let X be distributed as
where
and
(
)
a) Are  X 2 , X 3  and X 1 independent? Explain.
1p
b) Are X 2 and 3 X1  X 2  2 X 3 independent? Explain.
1p
c) Find the distribution of 3 X1  X 2  2 X 3 .
2p
4
Consider a bivariate normal distribution with 1  1, 2  4,11  2, 22  5, and
12  2 / 5. Determine the constant-density contour that contains 95 % of the
probability. Hence, specify the midpoint and the axes of the constant-density
ellipse. You do not need to sketch the resulting ellipse.
3p
5
Let the random vector X  ( X1(1) , X 2(1)  X1( 2) , X 2( 2) )' have covariance matrix








(

)
Verify that the first pair of canonical variates are U1  X 2(1) , V1  X1( 2) with
canonical correlation 1*  0.95 .
3p