Companies in the U.S. car rental market vary greatly in terms of the size of the
ID: 3227491 • Letter: C
Question
Companies in the U.S. car rental market vary greatly in terms of the size of the fleet, the number of locations and annual revenue. The following data show the number of cars in service (measured in thousands) and the annual revenue (in $ millions) for 6 small car rental companies.
Company Cars(1000s) Revenue($Millions)
U-Save Auto Rental System, Inc 11.5 118
Payless Car Rental System, Inc 10.0 135
ACE Rent a Car 9.0 100
Rent-A-Wreck of America 5.5 37
Triangle Rent-A-CAR 4.2 40
Affordable/Sensible 3.3 32
Suppose that you are interested in the relationship between cars in service and annual revenue. Let the number of cars in service be the independent variable (x) and annual revenue be the dependent variable (y). (a) Use the least squares method to derive the regression equation. Show your work. [10 Points] (b) Fox Rent-A-Car has 11,000 cars in service. Use the estimated regression equation developed in Part (a) to predict annual revenue for Fox Rent-A-Car. [2 Points] (c) Calculate the coefficient of determination (r2 ) [10 Points] (d) Calculate the correlation coefficient (rxy) [3 Points]
Explanation / Answer
a)The regression equation is given by:
Y = b0 + b1X + e
Where b0 is the intercept
b1 is the slope
e is the residual
E = e2 = (Y- b0 –b1X)2
dE/db0 = -2( Y- b0 –b1X)=0
y – nb0 – b1 X=0
dE/db1 = -2X( Y- b0 –b1X)=0
XY – b0 X – b1X2 = 0
obs
cars(x)in1000s
revenue(y)in$millions
xy
x*x
1
11.5
118
1357
132.25
2
10
135
1350
100
3
9
100
900
81
4
5.5
37
203.5
30.25
5
4.2
40
168
17.64
6
3.3
32
105.6
10.89
total
43.5
462
4084.1
372.03
462 – 6b0 –43.5 b1 =0
4084.1 – 43.5b0 – 372.03b1 =0
By solving both the equations, we get :
b0 = -17.0049 and b1 = 12.9662
The regression equation is given by:
Y = -17.0049 + 12.9662X
b) The number of cars in service = 11000 which means X =11
The predicted annual revenue = -17.0049 + 12.9662*11 = 125.6233.
c) Coefficient of determination =R2 = SSR/SST
where SSR = (y - y)2
SST = (y - y)2
obs
y
y - y
(y - y)2
(y - y)2
1
132.1063
55.10635
3036.709
1681
2
112.657
35.65705
1271.425
3364
3
99.69085
22.69085
514.8746
529
4
54.30915
-22.6908
514.8746
1600
5
37.45309
-39.5469
1563.958
1369
6
25.78351
-51.2165
2623.128
2025
total
9524.97
10568
y = -17.0049 + 12.9662*X
y = (118+135+100+37+40+32)/6 = 77
R2 = 9524.97/10568 = 0.901303
d) Correlation coefficient = r = b1 * s.d(x) / s.d(y)
Where s.d(x) is the standard deviation of X = [(X - X)2/(n-1)] = 3.366155
s.d(y) is the standard deviation of Y = [(Y - Y)2/(n-1)] = 45.97391
r = 12.9662*3.66155/45.973916 = 0.94937
obs
cars(x)in1000s
revenue(y)in$millions
xy
x*x
1
11.5
118
1357
132.25
2
10
135
1350
100
3
9
100
900
81
4
5.5
37
203.5
30.25
5
4.2
40
168
17.64
6
3.3
32
105.6
10.89
total
43.5
462
4084.1
372.03
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