Practice Problems # 11 - Solutions
1. This is a continuation of exercise # 4 in Practice Problem set # 9. Use the same data.
a. What proportion of observed variation in mileage can be attributed to the simple linear
regression relationship between weight and mileage?
28.859128 0.8923
9.928171
R==
b. Does the fitted model provide enough evidence for us to say that there is a significant
relationship between weight and fuel consumption? That is, test the hypothesis that the slope is
zero using the F test.
Sum of Mean Prob Power
Source DF Squares Square F-Ratio Level (5%)
Slope 1 8.859128 8.859128 41.434844 0.001345 0.998745
Error 5 1.069043 0.2138087
Adj. Total 6 9.928171 1.654695
To test the hypothesis that β1 = 0 we use the F test from the above ANOVA table.
H
0: β1 = 0
H
a: β1 ≠ 0
fobs = 41.434844 with an associated p-value of 0.001345.
Since this is a very small p-value, we would reject the null hypothesis.
c. One way to monitor fuel consumption and environmental emissions for an entire group of vehicle
types is to pay attention to the slope of the fitted regression line relating weight and fuel
consumption (the less steep the slope of the line, the less differential there is between weight
increase and fuel consumption). Does the data provide evidence that the slope is greater than
-0.16? Test a hypothesis about
β
1.
To test this hypothesis we must use the t test.
H
0: β1 = -0.16
H
a: β1 > -0.16
()
()
110
1
ˆ0.155131 0.16 0.20203
ˆ0.024100
ts
ββ
β
−−−
−
== =
p-value =
()
(
)
(
)
5
2 [ 0.20203] 2 0.4239 0.8478PT⋅> = = . If our significance level α is
smaller than this value then we would fail to reject the null hypothesis that β1 = -0.16.
If we set α = 0.05, then since this is a two tailed test, the rejection regions and critical
values are given by:
