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Prelim 1 Solutions
Problem 1 (Total 10 points)
Give the least square line that has the best ﬁt to the following data points:
(0
,
2), (1
,

2), (2
,

2) and (3
,
2).
Hint:
Recall that the slope and
y
intercept of the best ﬁt line are given by the
formulas
nb
+
±
n
X
i
=1
x
i
!
m
=
n
X
i
=1
y
i
and
±
n
X
i
=1
x
i
!
b
+
±
n
X
i
=1
x
2
i
!
m
=
n
X
i
=1
x
i
y
i
.
Solution:
We begin by computing the sums required for the equations given in the hint:
n
X
i
=1
x
i
= 6
n
X
i
=1
y
i
= 0
n
X
i
=1
x
2
i
= 14
n
X
i
=1
x
i
y
i
= 0
Plugging these values into the equations gives
(4)
b
+ (6)
m
= 0 and
(6)
b
+ (14)
m
= 0
Solving these equations by any method yields the unique solutions
m
= 0 and
b
= 0. Therefore the best ﬁt line is
y
= 0, a horizontal line through the origin.
Problem 2 (Total 16 points)
Draw Venn diagrams that satisfy the following requierments.
Let
A
⊆
U
,
B
⊆
U
, and
C
⊆
U
and
(a)
( 5 points)
A
∩
B
6
=
∅
,
C
⊆
B
and
C
∩
A
6
=
∅
.
(b)
( 5 points)
A
∩
B
6
=
∅
and
C
⊂
A
∩
B.
(c)
( 6 points)
A
⊆
B
,
C
∩
B
6
=
∅
, and
C
∩
A
0
6
=
∅
.
Solution:
(Other satisfactory solutions exist.
..)
1
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View Full Document Problem 3 (Total 16 points)
(1) Give the equations for the lines that go through the following pairs of
points. Write all equations in point slope form.
(a)
( 3 points)
(

1
,
3) and (2
,
4).
(b)
( 3 points)
(

6
,

2) and (9
,
3).
(c)
( 3 points)
(1
,
2) and (

4
,
2).
(2)
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This note was uploaded on 04/21/2008 for the course MATH 1105 taught by Professor Hurtado during the Fall '07 term at Cornell University (Engineering School).
 Fall '07
 HURTADO
 Math, Slope, YIntercept

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