2006 STAT3613 Past Paper 1
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The University of Hong Kong *
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Course
3613
Subject
Statistics
Date
Nov 24, 2024
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THE UNIVERSITY
OF
HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2313
MARKETING ENGINEERING
December 23, 2006
Time: 9:30a.m.
-11:30
a.m.
Candidates taking examinations that permit the use
of
calculators may use any calculator
which fulfils the following criteria:
(a)
it should be self-contained, silent, battery-operated
and
pocket-sized
and
(b)
it should have numeral-display facilities only
and
should be used
only
for
the purposes
of
calculation.
It is the candidate's responsibility to ensure that the calculator operates satisfactorily
and
the
candidate must record the name
and
type
of
the calculator on the front page
of
the
examination scripts. Lists
of
permitted/prohibited calculators will not be made available to
candidates
for
reference, and the onus will be on the candidate
to
ensure that the calculator
used will not be in violation
of
the criteria listed above.
Answer ALL FOUR questions.
The
questions are of equal value.
1.
The information
of
a promotion resource allocation exercise that Company A performed
is given in the table. The price
of
the product is $3.7 and the cost is $3.2.
Promotional spending
($million)
0
0.52
1
1.56
2.21
2.73
Sales
(million units)
5.6
6.75
8
10.83
12.4
13.23
The ADBUDG model is employed to study the data given as
xc
Y=a+b---
Xc+d
where
Y
is the market response,
X
is the marketing activity, and
a,
b,
c,
and
dare
the
parameters. Suggested initial estimates for the parameters are
a
= lower bound
of
the response curve
b
=(upper
bound)-
(lower bound
of
the response curve)
Suppose
X=
1 and
Y=
1
d=-b--1
1-a
Y'(d+IY
c
=
where
Y'
=
the
1st
derivative
of
Ywith respect to X (slope)
of
the
bxd
response curve at
X=
1
1
STA T2313 Marketing Engineering
(a)
(b)
(c)
Normalize the sales with reference level at promotional spending=
$1
million.
Calculate the initial estimates based
on
the provided formulae.
Substitute the initial estimates in (b) into the ADBUDG model,
(i)
Find the predicted values for the normalized sales for the various amounts
of
(ii)
(iii)
(iv)
promotional spending given
in
the table.
Find the error sum
of
squares.
FindR
2
•
Calculate the net profit for the various amounts
of
promotional spending given
in the table.
(v)
Based on (iv), plot the net profit against the various amounts
of
promotional
spending.
(vi)
Estimate from the plot the optimal promotional spending so as to maximize
the net profit.
2.
Six products (A, B, C, D, E, and F) from different companies are ranked by a number
of
customers in terms
of
5 attributes -
Product quality, Overall service, Delivery speed,
Salsesforce image, and Price flexibility. The overall ranking
of
the products are obtained
as a contingency table. Correspondence analysis is employed to study the data. Parts
of
the SPS S output are provided as follows.
Summary
Confidence Singular
Prooortion
of
Inertia
Value
Singular
Standard
Correlation
Dimension
Value
Inertia
Chi Square
Sig.
Accounted for
Cumulative
Deviation
2
1
.422
.178
.714
.714
.051
.007
2
.255
.065
.261
.975
.056
3
.066
.004
.018
.992
4
.043
.002
.008
1.000
Total
.249
72.191
.oooa
1.000
1.000
a.
20
degrees
of
freedom
Overview
Row
PointS'
Score
in
Dimension
Contribution
Of
Point to Inertia of
Dimension
Of Dimension to Inertia
of
Point
Row
Mass
1
2
Inertia
1
2
1
2
Total
Product quality
.138
1.380
-.280
.114
.623
.043
.970
.024
.995
Overall service
.200
-.055
.598
.020
.001
.281
.013
.911
.924
Delivery speed
.245
.251
.174
.011
.037
.029
.573
.167
.740
Salsesforce image
.228
-.664
.210
.046
.238
.039
.924
.056
.979
Price flexibility
.190
-.473
-.904
.058
.100
.608
.310
.686
.996
Active Total
1.000
.249
1.000
1.000
a.
Symmetrical normalization
2
·.
STA T2313 Marketing Engineering
Overview
Column
PointS'
Score
in
Dimension
Contribution
Of Point to Inertia of
Dimension
Of Dimension to Inertia of Point
Column
Mass
1
2
Inertia
1
2
1
2
Total
A
.186
-.358
.593
.028
.057
.257
.365
.603
.968
B
.145
-.689
-.488
.038
.163
.135
.766
.232
.998
c
.152
.733
-.641
.051
.194
.244
.668
.308
.976
D
.183
-.245
.501
.018
.026
.180
.259
.653
.911
E
.193
.971
.207
.080
.432
.032
.961
.026
.987
F
.141
-.619
-.523
.034
.129
.152
.669
.288
.958
Active Total
1.000
.249
1.000
1.000
a.
Symmetrical normalization
Confidence Column Points
Confidence Row Points
Standard Deviation in
Standard Deviation in
Dimension
Correlation
Dimension
Correlation
Column
1
2
1-2
Row
1
2
1-2
A
.218
.206
.-269
Product quality
.160
.291
.454
B
.161
.142
-.763
Overall service
.242
.201
.044
c
.277
.219
.464
Delivery speed
.192
.215
-.140
D
.208
.228
.193
Salsesforce image
.137
.223
.305
E
.155
.237
-.453
Price flexibility
.280
.158
-.494
F
.201
.231
-.154
(a)
What is proportion
of
the total variation explained by each
of
the 4 dimensions and
what is the proportion
of
the total variation explained by all 4 dimensions?
(b)
A researcher suggests that the 2-dimensional solution is adequate. Justify the
suggestion.
(c)
Based on the 2-dimensional solution,
(i)
Comment on the fitness
of
the model.
(ii)
Interpret the 2 dimensions in terms
of
the 5 attributes.
(iii)
Interpret the 2 dimensions in terms
of
the 6 products.
(iv)
Construct the perceptual map.
(v)
How many segments are suggested by the perceptual map?
(vi)
Describe the segments.
3
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ST A T23I3
Marketing Engineering
3.
A study is conducted for the types
of
emotional responses to consumption experience
based
on
a sample
of
owners
of
newly purchased cars. I25 questionnaires were completed
regarding respondents' emotional reactions to their automobiles. The questionnaire
1
consists
of
IO
variables:-
X1: Interest; X2: Joy; X3: Surprise; X4: Sadness; X5: Anger;
X6: Disgust; X7: Contempt; X8: Fear; X9: Shame;
XI
0: Guilt.
A Principal component factor analysis was applied to the standardized values. The
eigenvalues are given as
Eigenvalue
I
5.87I3
2
1.42I7
3
0.9538
4
0.4984
5
0.43I6
6
0.2642
7
O.I863
8
O.I800
9
O.I376
10
0.0550
(a)
Determine the number
of
factors based
on
(i)
the scree plot.
(ii)
latent root criterion.
(b)
A 2-factor model was employed and the initialloadings are given as
Factor I
Factor 2
XI
0.3I3I
0.4765
X2
-0.1950
0.8847
X3
0.5073
0.5358
X4
0.8577
-0.2249
X5
0.8720
-O.I429
X6
0.9395
-O.II77
X7
0.9460
-0.0438
X8
0.8862
O.I329
X9
0.8II4
0.1427 -
XIO
0.8723
-0.0144
(i)
Interpret the factors.
(ii)
Calculate and comment
on
the communalities.
(c)
V arimax rotation was applied and the rotated loadings for the first rotated factor is
given as
Factor I
XI
0.2II7
X2
-0.3676
X3
0.390I
X4
0.8853
X5
0.8830
X6
0.9441
X7
0.9357
X8
0.84I8
X9
0.7666
XlO
0.8576
4
ST AT2313 Marketing Engineering
(i)
Find the loadings
of
the second rotated factor.
(ii)
Interpret the rotated loadings.
(iii)
Calculate the variance explained
by
each rotated factor.
(d)
The background information
of
the respondents are also collected which are the age
(in years), educational level (as primary
or
below, secondary, or above secondary)
and income (in dollars). Suggest statistical method(s) to study the relationship
between the 2 factors and the background information.
4.
A researcher studies the switching behaviour
of
a number
of
consumers across 5 different
brands and types
of
instant coffee. The table below shows the transaction matrix that .
reports the switching behaviour for the consumers.
To
Brand A
Brand B
Brand B
Brand C
Brand D
decaffeinated
caffeinated
decaffeinated
caffeinated
caffeinated
regular
freeze dried
freeze dried
regular
regular
Brand A
decaffeinated
93
7
17
19
18
regular
Brand B
caffeinated
9
80
12
11
24
freeze dried
Brand B
From
decaffeinated
9
14
46
3
7
freeze dried
Brand C
caffeinated
19
18
4
82
29
regular
Brand D
caffeinated
26
11
6
35
184
regular
For example, that the number
of
times
we
observe a switch from Brand A decaffeinated
regular to Brand B caffeinated freeze dried is 7 and the number
of
times we observe a
switch from Brand A decaffeinated regular to Brand B decaffeinated freeze dried is 17.
The researcher claims that the transaction matrix could be used to reflect the level
of
competition between the instant coffee. Let
Pif
be the proportion
of
the consumers who
selected product
i
but switched to
productj.
The proposed measure
of
competition
between product
i
and
productj
is given as
(pu
+
pji)
/2.
(a)
Justify the researcher's claim.
(b)
Calculate the values
ofpij
for all products.
(c)
Calculate the proximity matrix based
on
the researcher's proposed formula. Is it a
similarity
or
dissimilarity matrix?
(d)
A single-linkage hierarchical cluster analysis
by
the build-up method was employed
to study the data based
on
the proposed proximity matrix in (c).
The
icicle graph is
provided.
5
STAT2313 Marketing Engineering
Vertical Icicle
Case
CD
N
CD
C)
CD
N
C)
C)
CD
CD
CD
CD
CD
....
.:::
~
....
....
11:::
11:::
11:::
11:::
C1S
C1S
11:::
C1S
C1S
0
0
C1S
0
0
CD
CD
0
Cl
u
"C
"C
cc
aO
~
~
cc
N
.....
Number
of
clusters
~
1
X
X
X
X
X
X
X
X
X
2
X
X
X
X
X
X
X
X
3
X
X
X
X
X
X
X
4
X
X
X
X
X
X
(i)
Report the cluster solution at each stage
of
the build-up method.
(ii)
Find the value
of
(dis)similarity
of
concern to
join
two clusters at each stage.
(iii)
Sketch a dendrogram for the cluster analysis. The distances are not required to
be in scale.
(iv)
Describe the competitive structure
of
the instant coffee market.
**************END OF PAPER**************
6
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