Statistics 20: Quiz 2 Solutions
September 27, 2006
1. The
variance
of data
X
is defined as
V ar
(
X
) = [
SD
(
X
)]
2
.
Starting from this definition,
prove that
V ar
(
X
) =
1
n
∑
n
i
=1
X
2
i

(
¯
X
)
2
. (
Hint:
Start by writing
V ar
(
X
) in terms of the
standard deviation, then expand the sums to get the result. Going in the other direction is
more difficult.)
V ar
(
X
) = [
SD
(
X
)]
2
=
1
n
∑
n
i
=1
(
X
i

¯
X
)
2
2
=
1
n
∑
n
i
=1
(
X
i

¯
X
)
2
=
1
n
(∑
n
i
=1
X
2
i
)

2
(∑
n
i
=1
X
i
¯
X
)
+
(∑
n
i
=1
(
¯
X
)
2
)
=
1
n
∑
n
i
=1
X
2
i

2
¯
X
1
n
∑
n
i
=1
X
i
+
n
n
(
¯
X
)
2
=
1
n
∑
n
i
=1
X
2
i

2
¯
X
(
¯
X
) + (
¯
X
)
2
=
1
n
∑
n
i
=1
X
2
i

(
¯
X
)
2
.
2. Suppose the university conducted a study attempting to predict
Y
, a student’s first year GPA,
based upon
X
, the student’s SAT math score. Students at the university had an average SAT
math score of 500 with a standard deviation of 100, and the average first year GPA is 2.5 with
a standard deviation of 0.5. A scatter plot of the data is football shaped. The correlation of
X
and
Y
is
r
= 0
.
8. A particular student chosen at random has an SAT math score of 800.
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 Spring '08
 Staff
 Statistics, Standard Deviation, Variance, Yi, Deviation

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