Question 1. Almost all U.S. light-rail systems use electric cars that run on tracks built at street level. According to the Federal Transit Administration, light-rail is one of the safest modes of travel, with an accident rate of .99 accidents per million passenger miles as compared to 2.29 for buses. The following data show the miles of track and the weekday ridership in thousands of passengers for selected light-rail systems (USA Today, January 7, 2003).
|
City
|
Miles
|
Riders
|
|
|
|
|
|
|
22
|
70
|
|
Los Angeles
|
|
San Diego
|
47
|
75
|
|
Portland
|
38
|
81
|
|
Sacramento
|
21
|
31
|
|
San Jose
|
31
|
30
|
|
San Francisco
|
73
|
164
|
|
Philadelphia
|
69
|
84
|
|
Boston
|
51
|
231
|
|
Denver
|
17
|
35
|
|
Salt Lake City
|
18
|
28
|
|
Dallas
|
44
|
39
|
|
New Orleans
|
16
|
14
|
|
St. Louis
|
34
|
42
|
|
Pittsburgh
|
18
|
25
|
|
Buffalo
|
6
|
23
|
|
Cleveland
|
15
|
15
|
|
Newark
|
9
|
8
|
a. Develop a scatter diagram for these data, treating the number of miles of track as the independent variable. Does a simple linear regression model appear to be appropriate?
b. Use a simple linear regression model to develop an estimated regression equation to predict the weekday ridership given the miles of track. Construct a standardized residual plot. Based upon the standardized residual plot, does a simple linear regres¬sion model appear to be appropriate?
c. Perform a logarithmic transformation on the dependent variable. Develop an estimated regression equation using the transformed dependent variable. Do the model assump¬tions appear to be satisfied by using the transformed dependent variable?
d. Perform a reciprocal transformation on the dependent variable. Develop an estimated regression equation using the transformed dependent variable.
e. What estimated regression equation would you recommend? Explain.
Question 2 :In a regression analysis involving 30 observations, the following estimated regression equation was obtained.
y' = 17.6 + 3.8x1 - 2.3x2 + 7.6x3 + 2.7x4
For this estimated regression equation SST = 1805 and SSR = 1760.
a. At α - .05, test the significance of the relationship among the variables. Suppose variables x, and x4 are dropped from the model and the following estimated regression equation is obtained.
y' = 11.1 - 3.6x2 + 8.1x3
For this model SST = 1805 and SSR = 1705.
b. Compute SSE(x1, x2, x3, x4 ).
c. Compute SSE(x2, x3 ).
d. Use an F test and a .05 level of significance to determine whether xl and x4 contribute significantly to the model.