Statistics 401E
Fall 2010
SECOND EXAM
Solutions
Name _______________________________
The Gay Liberation Movement (GLM) is generally thought
of as having begun on June 17, 1969, at the Stonewall
Inn—a gay bar in New York City. Although police raids
of such bars were commonplace at the time, this date
marks the first instance when police officers met violent
resistance from homosexual men while attempting to
raid a gay bar. Along with the Women’s Movement, GLM
has sought to undermine gender-related discrimination
in the US. One strategy endorsed by both movements
has been toward the reduction of stereotypical depictions of male and female
demeanor in the mass media.
In Gender Advertisements (1976) Erving Goffman provided numerous illustrations
of such gender-stereotypical depictions within advertisement photography. For
example, in magazine advertisements women’s hands are typically depicted as
“barely touching, even caressing” objects, whereas men’s hands are more
typically “grasping or manipulating” them. Men instruct women, but not the
reverse. Women may sit or lie on a bed, but not men. Many studies have
subsequently examined US magazine advertisements, and have found in them
pervasive evidence of such gender-cues. However, all of the magazines studied
until now have been “straight” ones (i.e., magazines with advertisements that
target traditional heterosexual consumers).
You decide to fill this gap in the research by studying advertisements in lesbiangay-bisexual-transgender (LGBT) as well as straight magazines. From among all
issues published between 1970 and 2010 of the straight magazine, Glamour, and
the LGBT magazine, Advocate, you randomly sample 121 issues from each
magazine. You then select the first instance of advertisement photography in
each of these issues, yielding a sample of 242 photographs—121 from the
straight magazine, Glamour, and 121 from the LGBT magazine, Advocate. Your
first hypothesis is that advertisement photography in straight magazines (e.g.,
Glamour) is more likely to have gender-cues than photography in LGBT
magazines (e.g., Advocate). Your selection of the 1970 starting date is due to its
proximity to the beginning of GLM. Since you believe that GLM (along with the
Women’s Movement) has been effective, your second hypothesis is that the
number of gender-cues in the two magazines has declined linearly over the past
four decades. You have data on the following three variables:
YEAR (Y)
= Year of the issue in which the photograph appeared
NUMCUE (N)
= Number of gender-cues depicted in the photograph
TYPEMAG (T) = 1 if photograph is from the straight magazine, Glamour
0 if photograph is from the LGBT magazine, Advocate
1
Statistics 401E
Fall 2010
SECOND EXAM
Solutions
Name _______________________________
Correlations, means, and standard deviations among these three variables are as
follows:
——Correlation Coefficients——
Standard
YEAR
NUMCUE TYPEMAG
Mean
Deviation
YEAR
1.00
–.05
.00
1990.0
11.8
NUMCUE
–.05
1.00
.24
6.0
1.5
TYPEMAG
.00
.24
1.00
0.5
0.5
(Notes: Please be sure that you understand these data. For example, the
correlation between YEAR and NUMCUE is –.05 for your sample of 242
photographs. The mean value of YEAR is 1990 years; the standard deviation of
YEAR is 11.8 years.)
More data:
 On average there are 6.24 gender-cues (with a variance of 1.52) among the 121
photographs from Glamour.
 On average there are 5.55 gender-cues (with a variance of 2.49) among the 121
photographs from Advocate.
a. Using the .05 significance level, test the hypothesis that the two variances
listed in the just-mentioned bulleted sentences (namely, 1.52 and 2.49) are
significantly different. Show your work and state your conclusion in words!
[weight 4]
120
120
F
ˆ A2 2.49
120
 2 
 1.64  1.35  F120
,.05
ˆG 1.52
Thus the variances are significantly different.
b. What null and alternative hypotheses would you use to test your first
hypothesis as stated in the third paragraph of this problem? (Hint: Remember
to take into consideration the levels of measurement of the variables you will
be using from the data provided above.)
[weight 2]
H o : N G  N L
H A : N G  N L
Here N G is the population mean value on the NUMCUE variable
among photos from Glamour, and
N
L
is the population mean
NUMCUE value among photos from Advocate.
2
STAT 401E / SECOND EXAM
c. Using the .05 significance level throughout, test the null hypothesis stated
in part b. (Hint: Be sure to show your work and to state your conclusion in
clear English!)
[weight 5]
Given the finding in part a separate variance estimate should be used.
2
ˆ SEPARATE

And thus
1.52 2.49 4.01


 .033
121 121 121
ˆ SEPARATE  .033  .182 .
Regarding to the hypotheses in part b,   z ˆ SEPARATE  1.645  .182  .299 .
The rejection rule is thus, reject
Ho
if X N G  X N L  .299 .
Since X NG  X N L  6.24  5.55  .690  .299 , we reject
H o and conclude that
during 1970-2010 gender-cues in first instances of advertisement photography
were significantly more likely in Glamour than in Advocate.
d. After regressing NUMCUE on YEAR, you obtain the following slope:
ˆ
1 .5
bˆY  rNY N  .05
 .0064 State in plain English (i.e., as discussed in
ˆY
11.8
class) what this slope means.
[weight 3]
One would estimate for each subsequent year between 1970 and 2010 a
decrease of .0064 in the number of gender-cues in Glamour and Advocate
magazines’ first instances of advertisement photography.
3
STAT 401E / SECOND EXAM
e. What null and alternative hypotheses would you use to test your second
hypothesis as stated in the third paragraph of this problem?
[weight 2]
Hypotheses about a slope or a correlation may be used here.
H o : bY  0
H A : bY  0
, where b̂Y is from
Nˆ  aˆ  bˆY Y
OR
H o :  NY  0
H A :  NY  0
f. What rejection rule would you use in testing the hypotheses stated in part e
of this problem at the .05 significance level? (Hint: Be sure to show your
work, and to remember that a rejection rule is a sentence that always begins
with the phrase, “Reject the null hypothesis if…”.)
[weight 5]
If part e states hypotheses about a slope:
SS N  n  1  ˆ N2  241  1.52  542.25  SS TOTAL
SSY  n  1  ˆY2  241  11.82  33,556.84


2
  SSTOTAL  1   .05  542.25  540.894
SS ERROR  1  rNY
ˆ 2  MSE 
ˆ b2ˆ 
Y
2
SS ERROR 540.894

 2.254
n  k 1
240
2.254
MSE

 .000067 thus ˆ bˆ 
Y
33,556.84
SSY
.000067  .0082
  z.05ˆ bˆ  1.645  .0082  .0135
Y
Reject Ho if
bˆY  .0135 .
OR
If part e states hypotheses about a correlation:
ˆ
2
1  rNY
1   .05


 .0042 thus ˆ rNY  .0042  .0645
n2
240
2
2
rNY
  z.05ˆ rNY  1.645  .0645  .1061
Reject Ho if rNY
 .1061 .
4
STAT 401E / SECOND EXAM
g. Apply the rejection rule obtained in part f, and using clear English explain
your conclusion in terms of the second hypothesis.
[weight 1]
I do not have statistically significant evidence (at α=.05) of a linear decline in
gender-cues in the 2 magazines’ photographs over the past 4 decades, because
bˆY  .0064  .0135
(or, equivalently, because rNY
 .05  .1061 ).
h. If between 1970 and 2010 there were actually a decline of .01 gender-cue per
year in the two magazines, what is the probability that you committed a Type 2
[weight 3]
error in part g? As always, show your work!
z
 .0135   .01
 .425
.0082
Pr(z<-.425)=Pr(z>.425)=.3354
β=Pr(Type II error)=1-.3354=.6646
Hypothetical truth
  6646
-.01
-.0135
Rejection region
Acceptance region
5