DS 533
Fall 2004
Exam # 3
Name: ___________________
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An automobile rental company wants to predict the yearly maintenance expense (Y) for
an automobile using the number of miles driven during the year ( X1 ) and the age of the
car ( X 2 , in years) at the beginning of the year. The company has gathered the data on 10
automobiles and the regression information from Excel is presented below. Use this
information to answer the following questions.
Summary measures
Multiple R
R-Square
Adj R-Square
Standard Error
0.9689
0.9387
0.9212
72.218
Regression coefficients
Constant
Miles Driven
Age of car
a.
Coefficient
33.796
0.0549
21.467
Std Err
48.181
0.0191
20.573
t-value
0.7014
2.8666
1.0434
p-value
0.5057
0.0241
0.3314
Use the information above to estimate the linear regression model.
yˆ  33.796  .0549 x1  21.467 x 2
x1  Miles driven
x 2  Age of Car
b.
Interpret each of the estimated regression coefficients of the regression model
in Question a.
For every extra 100 miles driven, the maintenance cost goes up by $5.49, given
the age of the car is fixed.
As the age of the car goes up by one year the maintenance cost goes up by
$21.467, give the miles driven is fixed.
c.
Identify and interpret the coefficient of determination ( R2 ), and the standard
error of the estimate (Sy.x) for the model in Question 3.
R2 = .9387. 93.87% of the variability in maintenance cost can be explained by
the age of the car and the miles driven.
1
S = 72.218. This measures the variability around the fitted model.
d.
Does the given set of explanatory variables do a good job of explaining
changes in the maintenance costs? Explain why or why not.
The R2 is high, indicating a good model, but the variable age of the car is not a
significant predictor of the maintenance car given the first variable (Miles driven)
in the model. The variable age of the car may not be needed in the model.
d.
Would you recommend that this company examine any other factors to predict
maintenance expense? If yes, what other factors would you want to consider?
Explain your answer.
This is a good model with R2 = 94%. Other variable that may be considered is
the make and model of the car.
f. Give a 95% confidence interval for the average yearly maintenance cost for an
automobile for every extra mile driven during the year ( X1 ).
b1  t * SE (b1 )
.0549  2.365(.0191)
.0549  .045  (.010, .10)
g.
What is the average yearly maintenance cost for a 10-year-old automobile that
drives 12000 miles per year?
yˆ  33.796  .0549 x1  21.467 x 2
yˆ  33.796  .0549(12000)  21.467(10)  873.47
2
Mid-Valley Travel Agency (MVTA) has offices in 12 cities. The company believes that
its monthly airline bookings are related to the mean income in those cities and has
collected the following data:
Location Bookings
1
1098
2
1131
3
1120
4
1142
5
971
6
1403
7
855
8
1054
9
1081
10
982
11
1098
12
1387
Income
43299
45021
40290
41893
30620
48105
27482
33025
34687
28725
37892
46198
The data are analyzed using regression analysis. The partial computer output is given
below:
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.879189
R Square
0.772974
Adjusted R Square 0.750271
Standard Error
78.16735
Observations
12
ANOVA
df
Regression
Residual
Total
1
10
11
Intercept
X Variable 1
a)
SS
MS
F
208036.3 208036.3 34.04775
61101.35 6110.135
269137.7
Coefficients Standard Error t Stat
P-value
371.6758
128.5571 2.891133 0.016076
0.019381
0.003322
What is the estimated least square regression line?
yˆ  371.6758  .019 x
3
b)
What is the standard error of the estimate?
S =78.167
c)
Forecast the number of bookings when the mean income is $51385.
yˆ  371.6758  (.019)(51385)  1347.99
d)
Test the significance of the regression coefficient at the 5% level (state the
null and alternative hypothesis, the value of your test statistic, the p-value or
the decision rule, and your conclusion).
H0 : 1 = 0
Ha ; 1 ≠ 0
T
.019
 5.72
.003322
P-value < 2(.005) = .01
Reject H0.
Mean income is a significant predictor of the air line bookings.
e)
Give an interval estimate of 1 with a 95% confidence coefficient.
b1  t * SE (b1 )
.019  (2.23)(. 003322)
.019  .00741
(.0116, .026)
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Multiple Choice Questions
Select the best answer
1. In choosing the “best-fitting” line through a set of points in linear regression, we
choose the one with the:
a.
b.
c.
d.
e.
smallest sum of squared residuals **
largest sum of squared residuals
smallest number of outliers
largest number of points on the line
none of the above
2. In a multiple regression analysis, there are 25 data points and 5 independent
variables, and the sum of the squared differences between observed and predicted
values of y is 160. The regression standard error will be:
a.
b.
c.
d.
e.
2.530
3.464
2.902**
5.657
none of the above
3. In a simple linear regression analysis, the following sum of squares are produced:
( y  y)
i
2
 400,
 (y  yˆ )
i
2
 80,
(yˆ  y )
i
2
 320
The proportion of the variation in y that is explained by the variation in x is:
a.
b.
c.
d.
e.
20%
80%**
25%
50%
none of the above
4. Given the least squares regression line ŷ  8 – 3x,
a.
b.
c.
d.
e.
the relationship between x and y is positive
the relationship between x and y is negative**
as x increases, so does y
as x decreases, so does y
there is no relationship between x and y
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5. A multiple regression equation includes 6 independent variables, and the
coefficient of multiple determination is 0.91. The percentage of the variation
in y that is explained by the regression equation is:
a.
b.
c.
d.
e.
A “fan” shape in a scatterplot indicates:
6.
a.
b.
c.
d.
7.
91%**
95%
83%
about 15%
none of the above
unequal variance**
a nonlinear relationship
he absence of outliers
sampling error
The values of the regression parameters i are not known. We estimate
them from the data.
a) True **
b) false
c) Not enough information
8. Residual plots can be used to check the aptness of the model for the data.
a) True**
b) False
c) Not enough information
9. We need to estimate the variance of the error terms because:
I)
It gives an indication of the variability of the distribution of y.
II)
It is needed for making inference concerning regression function
and the prediction of y.
a) Only (I) is true.
b) Only (II) is true.
c) Both (I) and (II) are true.**
d) Neither (I) nor (II) is true.
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