Statistics 404
Fall 2009
FINAL EXAM
Solutions
Name _______________________________
Google has hired you to evaluate a variety of formats
for its “Google Shopping” (GS) website. When
potential shoppers access the site, Google wants to
be sure that they are not “turned off” by what they
see and that they are able to quickly find the
products they seek. That is, Google wishes to
ensure that potential customers have positive web experiences in terms of both
the ascetics and the efficiency of its web site. In close cooperation with the GS
web-development team, you modify the GS website as follows:
First, you develop two color schemes for the site—one with “cool” colors (blue,
green) and one with “warm” colors (red, orange). Next, you offer two types of
music—“background” music (classical) and “foreground” music (contemporary).
People who access the site are randomly assigned to one of the following four
versions of the GS website: cool colors with background music, cool colors with
foreground music, warm colors with background music, warm colors with
foreground music. Your sample consists of the first 100 people to access each of
these four website versions (for a total sample size of 400 subjects) who were
paid $20 each by Google to have filled in a short online survey before leaving the
website. Accordingly, in addition each subject’s website version and the amount
of money they spent prior to leaving the site, you also have information on each
from this survey on each subject’s emotional state, satisfaction with the webshopping experience, and intention to revisit the website.
In your preliminary report to Google executives, you state the following: “The
400 subjects in the study spent an average of $30 during their visit to the GS
website. When the website’s colors were warm during their visit, they spent an
average of $5 more than this; when their website’s music was background, they
spent an average of $2 more. Independent of these effects, there is a bonus of $1
additional spending when the website had both warm colors and background
music.” The four dollar-values in this statement correspond to unstandardized
regression coefficients from the following equation in which S measures dollars
spent, C measures color scheme, M measures music type, and B measures the
“bonus” referred to in the statement:
Sˆ  30  5C  2 M  1B
a. Assuming that both this equation and your above-quoted statement are true,
indicate whether each of the following 8 sentences is true or false.
[weight 8]
C=1 means that the subject was assigned to a website
with a warm color scheme.
true
false
M=1 means that the subject was assigned to a website
with foreground (i.e., contemporary) music.
true
false
1
Statistics 404
Fall 2009
FINAL EXAM
Solutions
Name _______________________________
Variable B is a contrast.
true
false
Variable B is a polynomial.
true
false
Variables C and M are contrasts.
true
false
There is a bonus of $1 additional spending when the
website had both cool colors and foreground music.
true
false
Subjects who visited the website with warm colors
spent $35 on average.
true
false
Subjects who visited the website with warm colors and
foreground music spent $32 on average.
true
false
The above-mentioned bonus effect turns out not to be significantly different from
zero, so you proceed without it and estimate the following path model (Model A)
using data from your online survey:
Color
(Z1)
.104
BBB
AAA
.200
Emotions
(Z3)
.468
Satisfaction
(Z4)
.584
Intention
to revisit
(Z5)
.046
Music
(Z2)
.733
.913
.799
The correlation matrix among these 5 standardized variables is as follows:
Z1
Z2
Z3
Z4
Z5
Z1
Z2
Z3
Z4
Z5
1.00
CCC 1.00
–.65 .20 1.00
–.20 .35 .40 1.00
.30 .25 .55 .60 1.00
2
STAT 404 / FINAL EXAM
b. What is the value of Model A’s unanalyzed path, AAA? (Hints: Note the
description of the sampling design described in this exam’s second
paragraph, and be sure to explain your answer. If you are stumped, just pick a
number and proceed with the exam.)
[weight 2]
AAA  rCM  0
The correlation (and thus the unanalyzed path) equals zero as a direct
result of the sampling design in which equal numbers of subjects had each
combination of color scheme and music type.
c. Find values for the path, BBB, and for the correlation, CCC? (Hint: Again be
sure to explain how you obtained these values.)
[weight 2]
As explained in the answer to part b, the correlation, CCC equals zero as a
result of how subjects were sampled. The partial slope, BBB, equals the
correlation between Color and Emotions because there is no collinearity
between Color and Music. This can be demonstrated as follows:
ˆ1 
r13  r23r12  .65  .2  .0  .65


 .65
1  r122
1  .0 2
1
3
STAT 404 / FINAL EXAM
d. According to Model A, what are the total effects of the GS website’s color
scheme on subjects’ intention to revisit the website? Likewise, what are the
total effects of the GS website’s music type on subjects’ intention to revisit the
website? Based on this model, what color scheme and music type would you
recommend to Google as the optimal combination for stimulating subjects to
revisit its GS website? (Show your work!)
[weight 4]
Effect of color scheme on intention to revisit:
Direct:
p51  0
p41 p54  p31 p43 p54  .104  .584    .65  .468  .584 
 .061  .178  .117
Total effect = Direct + Indirect  0  .117  .117
Indirect:
Given the negative total effect of Z1 (Color, where high scores indicate a warm
color scheme) on Z5 (Intention to revisit), I would recommend a cool color
scheme to Google for its GS website.
Effect of music type on intention to revisit:
Direct:
p52  .046
p32 p43 p54  .20  .468  .584  .055
Total effect = Direct + Indirect  .046  .055  .101
Indirect:
Given the positive total effect of Z2 (Music, where high scores indicate a
background music) on Z5 (Intention to revisit), I would recommend classical,
or background music to Google for its GS website.
4
STAT 404 / FINAL EXAM
e. In the space provided below, draw a revised path model (Model B) to which
two overidentifying restrictions have been added, namely that p41 = p52 = 0. Be
sure to draw all paths, and to provide values for each (including unanalyzed
and residual paths).
[weight 5]
Color
(Z1)
–.650
.000
.200
.400
Emotions
(Z3)
Satisfaction
(Z4)
.600
Intention
to revisit
(Z5)
Music
(Z2)
.733
.917
.800
Work space:
The two revised paths between endogenous variables equal the respective
variables’ correlations, because each path equals the standardized slope from
a bivariate regression. The two revised residual slopes are obtained as
follows:
e2  1  R42.3  1  .4 2  .917
e3  1  R52.4  1  .6 2  .8
5
STAT 404 / FINAL EXAM
f. Which path model (Model A or Model B) provides the better fit to your data?
(Use the .05 significance level, and show your work!)
[weight 4]
Table of
e 2 s:
Less Restricted
More Restricted
Q
Z3
.7332  .537
.7332  .537
Z4
.9132  .834
.917 2  .840
Z5
.799 2  .638
.8 2  .64
.537  .834  .638 .532

 .990
.537  .840  .640 .538
W  n  d 2  d1  lnQ   400  5  3 ln.990   398   .010  4.099
 22,.05  5.991  4.099  W
Thus the less restricted model (Model A) does not provide a significantly better fit
to the data than the more restricted model (Model B). The two paths (i.e., p41 and
p52) may be assumed to equal zero.
6