231
Chapter 11: Linear Models and Estimation by Least Squares
11.1
Using the hint,
.
ˆ
)
ˆ
(
ˆ
ˆ
)
(
ˆ
1
1
1
0
y
x
x
y
x
x
y
=
β
+
β
−
=
β
+
β
=
K
11.2
a.
slope = 0, intercept = 1.
SSE = 6.
b.
The line with a negative slope should exhibit a better fit.
c.
SSE decreases when the slope changes from .8 to .7.
The line is pivoting around the
point (0, 1), and this is consistent with (
y
x
,
) from part Ex. 11.1.
d.
The best fit is:
y
= 1.000 + 0.700
x
.
11.3
The summary statistics are:
x
= 0,
y
= 1.5,
S
xy
= –6,
S
xx
= 10.
Thus,
y
ˆ = 1.5 – .6
x
.
The graph is above.
-2
-1
0
1
2
0.5
1.0
1.5
2.0
2.5
3.0
p11.3x
p11.3y
11.4
The summary statistics are:
x
= 72,
y
= 72.1,
S
xy
= 54,243,
S
xx
= 54,714.
Thus,
y
ˆ =
0.72 + 0.99
x
.
When
x
= 100, the best estimate of
y
is
y
ˆ = 0.72 + 0.99(100) = 99.72.
11.5
The summary statistics are:
x
= 4.5,
y
= 43.3625,
S
xy
= 203.35,
S
xx
= 42.
Thus,
y
21.575 + 4.842
x
.
Since the slope is positive, this suggests an increase in median prices
over time.
Also, the expected annual increase is $4,842.
11.6
a.
intercept = 43.362, SSE = 1002.839.
b.
the data show an increasing trend, so a line with a negative slope would not fit well.
c.
Answers vary.
d.
Answers vary.
e.
(4.5, 43.3625)
f.
The sum of the areas is the SSE.
11.7
a.
The relationship appears to be proportional to
x
2
.
b.
No.
c.
No, it is the best
linear
model.