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1.
A highway data set contains the number of deaths per million vehicle miles
from the year 1945 to 1984 for the entire United States and the state of New Mexico.
A
Linear regression analysis is conducted. The R output is given below
where predictor Year is coded as 1, 2, … 40 corresponding to year 1945, year 1946,
…year 1984.
(a)
What is the
X
matrix
in the linear regression analysis reported in R output.
X=
1
1
0
0
1
2
0
0
┊┊
┊┊
┊┊ ┊ ┊
1
40
0
0
1
1
1
1
1
2
1
2
┊┊
┊┊
┊┊ ┊ ┊
1
40
1
40

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*Sign up*(b) What are the estimated trends? In which year do we expect the death rate for
New Mexico to be equal to the death rate for the United States?
The estimated trend for New Mexico is
Y
NM
= 13.15385-0.23616 Year
The estimated trend for United States is
Y
US
= 13.15385-4.41692+(0.07778-0.23616)Year
= 8.73693-0.15838 Year
When the two death rate are equal, we need to slove the equation Y
NM
= Y
US
13.15385-0.23616 Year = 8.73693-0.15838 Year
Year=56.79
≈
57
1945+57-1=2001
Therefore, in the year 2001, the death rate are supposed to be equal.
(c) State the null and alternative hypotheses for the test based on the reported F-
statistic, 310
:
4 on 3 and 76 degrees of freedom .
For the linear regression model, Y=X
β
+e, E(e)=0,Var(e)=
2
80
I
σ
The null and alternative hypothesis are:
Ho:
1
2
3
0
=
=
=
Ha: at least one of them is not equal to zero
(d) Compute the predicted death rate for both New Mexico and the United States in 1985.
Year=1985+1-1945=41

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