1.
A marketing firm wishes to determine whether or not there is a relationship between
the number of television commercials broadcast and the sales of its product. The data,
obtained from 5 different cities, are shown in the following table.
Number of TV
Commercials
x
Sales Units
y
3
7
7
19
5
13
9
15
6
11
Σ
x
= 30,
Σ
y
= 65,
Σ
x
2
= 200,
Σ
y
2
= 925,
Σ
x
y
= 420,
Σ
(
x
–
x
)
2
= 20,
Σ
(
y
–
y
)
2
= 80,
Σ
(
x
–
x
)
(
y
–
y
)
=
Σ
(
x
–
x
)
y
= 30.
Consider the model
Y
i
=
β
0
+
β
1
x
i
+
ε
i
., where
ε
i
’s are i.i.d.
N
(
0,
σ
2
)
.
a)
Find the equation of the leastsquares regression line. Add the leastsquares regression
line to the scatter plot.
b)
In Anytown, 20 commercials aired. What is your prediction of the sales? Why is it
dangerous to predict sales for this particular value of
x
.
c)
Find an estimate for
σ
, the standard deviation of the observations about the true
regression line?
d)
What proportion of the observed variation in the sales is explained by a straightline
relationship with the number of television commercials for the product?
e)
Construct a 90% confidence interval for
β
1
.
f)
Test for the significance of the regression at a 5% level of significance. That is, test
H
0
:
β
1
= 0 vs.
H
1
:
β
1
≠
0 at a 5% level of significance.
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g)
Construct a 95% prediction interval for the sales corresponding to
x
= 8 TV
commercials.
h)
Test
H
0
:
μ
(
x
= 8
)
= 20 vs.
H
1
:
μ
(
x
= 8
)
< 20 at a 10% level of significance.
i)
Test
H
0
:
β
0
= 0 vs.
H
1
:
β
0
≠
0 at a 10% level of significance.
2.
An agronomist experimented with different amounts of liquid fertilizer on a
sample of equalsize plots. The amount of fertilizer and the yields are:
Plot
Amount of
Fertilizer
(tons)
Yield
(hundreds
of bushels)
A
0
4
B
1
4.5
C
1
5.5
D
1.5
5.5
E
2
4.5
F
2.5
6.5
G
3
5.5
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 Spring '07
 STEPANOV
 Statistics, Least Squares, Linear Regression, Regression Analysis, Yi

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