Download Exercise Answers Chapter 04

Chapter 4 Answers to Problems (quantitative only)
4. In Figure 4-4 we showed the relationship between levels of GDP and GDP growth for
different countries, some members of the OECD and some not. These data are on the
website for the book (convergence.html).
b. Calculate the correlation coefficients for the data in these two scatterplots (GDP vs.
GDP growth for OECD and non-OECD countries).
Solution:
Use the binary (dummy) variable OECD to split the convergence data into
two subsets, one for OECD nations and one for non-OECD nations. Then
find the Pearson product moment correlation coefficient between GDP and
GDP growth in each dataset:
For OECD nations, r = -0.045.
For non-OECD nations, r = -0.912.
6. Measurements of sound levels and distance from the centerline of an urban expressway
are as follows
Observation Distance (ft.) Sound Level (dB)
1
45
83
2
63
81
3
160
66
4
225
68
5
305
69
6
390
66
7
461
58
8
515
57
9
605
55
10
625
61
c. Using the format of Table 4-8, estimate the slope and intercept of the regression
equation.
Solution:
Treating distance as the independent variable (X) and sound level as the
dependent variable (Y), we set up the table below and find the following:
n
X
i 1
n
Y
i 1
i
i
 3394
 664
n
X Y
i 1
i i
 208976
X  339.4
Y  66.4
n
X
i 1
n
Y
i 1
i
 1561740
2
i
2
 44906
Obs.
1
2
3
4
5
6
7
8
9
10
b
X
45
63
160
225
305
390
461
515
605
625
Y
83
81
66
68
69
66
58
57
55
61
XY
3735
5103
10560
15300
21045
25740
26738
29355
33275
38125
X2
2025
3969
25600
50625
93025
152100
212521
265225
366025
390625
Y2
6889
6561
4356
4624
4761
4356
3364
3249
3025
3721
n( XY )  ( X )(  Y ) 10(208976)  (3394)(664)

 0.03998
n(  X 2 )  (  X ) 2
10(1561740)  (3394) 2
a  Y  bX  66.4  (0.03998)339.4  79.97
d. Now fit the real regression line to the scatterplot.
Solution:
From a statistical software package: Y  79.97  0.03998 X
S Y  X  4.490
r 2  0.802
h. Calculate the coefficient of determination for this problem.
Solution:
Use the formula to compute the correlation coefficient and then square the
result.
 XY   X  Y / n
r
2
 X  ( X ) 2 / n  Y 2  ( Y ) 2 / n
r
208976  225361.6
409816.4 816.4

 16385.6
 0.896
18291.358
Therefore, r 2  0.802 , that is confirmed by the output above.
i. Calculate the standard deviation of the dependent variable and the standard deviation of
the independent variable. Use these values, together with the regression slope coefficient,
to estimate the correlation coefficient between your variables. Check this value by
finding the square root of the coefficient of determination.
Solution:
We know that b  r
sY
.
sX
sX
213.4
 0.03998
 0.896.
sY
9.52
Check with the values above.
Thus, r  b
7. Get the carbon dioxide and atmospheric temperature data (warming.html) from the
website for this book.
b. Calculate the regression equation for these variables.
Solution:
From a statistical software package: Y  2.6468  0.0082 X
assuming that temperature is the dependent variable.
d. Calculate the standard error of estimate and the coefficient of determination.
Solution:
sY  X
(Yi  Yˆi ) 2
0.41288


 0.1136 .
n2
32
 s
r   b X
 sY
2
2

12.6 

   0.0082
  0.458.
0.1526 


2
9. Obtain the Dow Jones stock market data from the book website (dowjones.html).
Calculate the first-order autocorrelation for these data.
Solution:
rYtYt 1  0.999 .