Julie Novak, Tung Phan, and Wei Han
Professor Rakhlin
STAT 101 HW3 Solutions
September 29th, 2011
1. Question 1
Question.
#26:
A fitness center weighed 53 male athletes and then measured the
amount that they were able to lift.
The correlation was found to be
r
=0.75.
Can
you interpret this correlation without knowing whether the weights were in pounds or
kilograms or a mixture of the two?
Yes, because the correlation does not depend on the scales of the two variables, but we
would like to see a graph.
Question.
#28: After calculating the correlation, the anlayst at the fitness center in
Exercise 26 discovered that the scale used to weigh the athletes was off by 5 pounds;
each athlete’s weight was measured as 5 pound more than it actually was. How does
this error affect the reported correlation?
Mathematically, the covariance between
x
and
y
is
Cov
(
x, y
) =
∑
n
i
=1
(
x
i

¯
x
)(
y
i

¯
y
)
n

1
. Be
cause the original weight, (let’s say
x
), was five pounds greader than the true weight,
(let’s say
x
0
), we have to subtract 5 pounds from the original weight,
x
, to get the true
weight (i.e.
x
0
=
x

5). Then, the new mean of
x
0
is
¯
x
0
=
n
X
i
=1
x
0
i
n
=
n
X
i
=1
x
i

5
n
=
n
X
i
=1
x
i
n

5
n
=
n
X
i
=1
x
i
n

n
X
i
=1
5
n
= ¯
x

n
*
5
n
= ¯
x

5
Also, the new standard deviation of
x
0
is
sd
x
0
=
v
u
u
t
1
n

1
n
X
i
=1
(
x
0
i

¯
x
0
)
2
=
v
u
u
t
1
n

1
n
X
i
=1
(
x
i

5

(¯
x

5))
2
=
v
u
u
t
1
n

1
n
X
i
=1
(
x
i

5

¯
x
+ 5)
2
=
v
u
u
t
1
n

1
n
X
i
=1
(
x
i

¯
x
)
2
=
sd
x
1
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Basically, the standard deviation remains the same if we subtracted five from
x
. Now,
the new covariance,
Cov
(
x
0
, y
), between
x
0
and
y
is
Cov
(
x
0
, y
) =
∑
n
i
=1
(
x
0
i

¯
x
0
)(
y
i

¯
y
)
n

1
=
∑
n
i
=1
((
x
i

5)

(¯
x

5))(
y
i

¯
y
)
n

1
=
∑
n
i
=1
((
x
i

5

¯
x
+ 5))(
y
i

¯
y
)
n

1
=
∑
n
i
=1
(
x
i

¯
x
)(
y
i

¯
y
)
n

1
=
Cov
(
x, y
)
Therefore, the new covariance
Cov
(
x
0
, y
) equals the original covariance
Cov
(
x, y
) (i.e.
Cov
(
x
0
, y
) =
Cov
(
x, y
)). In addition, using the facts about the new standard deviation,
sd
x
‘
, we see that the new correlation is going to be the same as the original correlation.
Mathematically,
Corr
(
x
0
, y
) =
Cov
(
x
0
, y
)
sd
x
0
sd
y
=
Cov
(
x, y
)
sd
x
sd
y
=
Corr
(
x, y
)
You can also arrive at the answer without the math. In class and in the book, corre
lation is a direct measure of the slope of the line that we fit in a scatterplot of
x
and
y
. Suppose we subtract five from
x
because we measured it incorrect. By doing so,
we shift all the values of
x
to the left on the scatter plot while
y
still has the same
values. But, when we fit the line on the scatterplot of the shifted values of
x
and
y
,
the slope remains the same as the line we fit on the scatterplot with the original
x
and
y
; only the intercept of the line changes. A visualization of this is posted below: the
red points represent the weights shifted 5 pounds while the blue points are the original
data. Notice how the slopes do not chang, but the intercept does.
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 Fall '08
 Heller
 Statistics, Correlation, Correlation does not imply causation, sdx sdy

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