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midterm2_2019_04_02_solutions.pdf-Second Midterm Examination Econ 103, Statistics
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midterm2_2019_04_02_solutions.pdf-Second Midterm Examination Econ 103, Statistics
##### Page 6
Econ 103
Midterm II, Page 6 of 8
April 2nd, 2019
Solution:
We have
E
[
X
2
1
]=0
.
5
×
5
2
+0
.
3
×
10
2
+0
.
2
×
20
2
=0
.
5
×
25 + 0
.
3
×
100 + 0
.
2
×
400
= 122
.
5
Hence, by the shortcut rule
V ar
(
X
1
) = 122
.
5
-
(9
.
5)
2
= 32
.
25
grams
2
.
(c)
8
Is
W
an unbiased estimator of the weight of the bag? Explain brieﬂy.
Solution:
The true weight of the bag is
50
×
5+30
×
10 + 20
×
20 = 950
grams.
By the linearity of expectation,
E
[
W
]=
E
[20
×
(
X
1
+
···
+
X
5
)]
= 20
×
(
E
[
X
1
]+
···
+
E
[
X
5
])
= 20
×
5
×
9
.
5
= 950
grams
Hence,
W
is an unbiased estimator of the weight of the bag.
(d)
8
Calculate the standard error of
W
.
Solution:
The RVs
X
1
,X
2
,...,X
5
are iid. Hence,
V ar
(
W
)=
V ar
[20
×
(
X
1
+
···
+
X
5
)]
= 400
×
[
V ar
(
X
1
)+
···
+
V ar
(
X
5
)]
= 400
×
5
×
32
.
5
= 64500
grams
2
Therefore,
SE
(
W
)=
p
V ar
(
W
)
254
grams.
6. This question is based on a recent paper examining how “organic” labeling changes
people’s perceptions of diﬀerent food products.
Researchers recruited volunteers at a
local mall in Ithaca, New York and gave each two samples of yogurt to taste. Although
both yogurts were in fact identical, the volunteers were
told
that one of them was organic
while the other was not. After tasting both, each volunteer was asked to estimate how
many calories each of the samples of yogurt contained. (Since, unknown to the volunteer,
both samples contained exactly the same kind of yogurt, each in fact contained the same
number of calories.) To prevent confounding from anchoring or other behavioral eﬀects,
Name:
Student ID #:

##### Page 7
Econ 103
Midterm II, Page 7 of 8
April 2nd, 2019
the order in which a given volunteer tasted the two yogurts, i.e. “organic” ﬁrst or “organic”
second, was chosen at random. The results are stored in the dataframe
yogurt
:
regular organic
1
60
40
2
5
0
3
200
100
Each row in this dataframe corresponds to a single individual’s guess of the number of
calories contained in each of the two yogurts. For example, the values 60 and 40 in row
1 mean that volunteer number one guessed that the regular yogurt sample contained 60
calories and the organic sample contained 40. Summary statistics are as follows:
regular
organic
Sample Mean
113
90
Sample Var
3600
2916
Sample SD
60
54
Sample Corr.
0.8
Sample Size
115
(a)
10
Sara thinks that this experiment should be analyzed as independent samples data.
Assume that she is correct and construct an approximate 95% CI for the diﬀerence
of means (
regular
-
organic
) based on the CLT.
Solution:
The diﬀerence of means (regular minus organic) is 23 calories. Sara
calculates her standard error assuming independent samples:
q
σ
2
X
/n
+
σ
2
Y
/m
=
p
3600
/
115 + 2916
/
115 =
p
6516
/
115
7
.
5
so her conﬁdence interval is approximately
23
±
15
, in other words
(8
,
38)
.
(b)
10
Kevin thinks that this experiment should be analyzed as matched pairs data. As-
sume that he is correct and construct an approximate 95% CI for the diﬀerence of
means (
regular
-
organic
) based on the CLT.
Solution:
Kevin takes into account the sample correlation between columns
when calculating his standard error. He does this by using the sample statistics
from the table to calculate the sample variance of the
diﬀerences
: regular minus
organic. In particular, he calculates:
s
2
D
= 3600 + 2916
-
2
·
0
.
8
·
60
·
54 = 1332
Name:
Student ID #:

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