In: Math
eBook Almost all U.S. light-rail systems use electric cars that run on tracks built at street level. The Federal Transit Administration claims 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 six light-rail systems.
City | Miles of Track | Ridership (1000s) | ||||||||
Cleveland | 17 | 17 | ||||||||
Denver | 19 | 37 | ||||||||
Portland | 40 | 83 | ||||||||
Sacramento | 23 | 33 | ||||||||
San Diego | 49 | 77 | ||||||||
San Jose | 33 | 32 | ||||||||
St. Louis | 36 |
44 a) Use these data to develop an estimated regression equation
that could be used to predict the ridership given the miles of
track. Complete the estimated regression equation (to 2
decimals). b) Compute the following (to 1 decimal):
c) What is the coefficient of determination (to 3 decimals)?
Note: report r2 between 0 and 1. Does the estimated regression equation provide a good fit? d) Develop a 95% confidence interval for the mean weekday ridership for all light-rail systems with 30 miles of track (to 1 decimal). e) Suppose that Charlotte is considering construction of a light-rail system with 30 miles of track. Develop a 95% prediction interval for the weekday ridership for the Charlotte system (to 1 decimal).
|
Sol:
Miles of Track(X) | Ridership (1000s)(Y) | XY | X2^ | Y^2 | |
17 | 17 | 289 | 289 | 289 | |
19 | 37 | 703 | 361 | 1369 | |
40 | 83 | 3320 | 1600 | 6889 | |
23 | 33 | 759 | 529 | 1089 | |
49 | 77 | 3773 | 2401 | 5929 | |
33 | 32 | 1056 | 1089 | 1024 | |
36 | 44 | 1584 | 1296 | 1936 | |
TOTAL | 217 | 323 | 11484 | 7565 | 18525 |
mean=xbar=31 | |||||
mean=ybar=46.143 | |||||
b=slope=7*11484-217*323/7*7565-(217)^2=1.7554
a=intercept=323-1.7554(217)/7=-8.275
Regression equaton is
y=-8.28+1.76*x
b0=y intercept=-8.28
b1=slope=1.76
Solutionb:
get the same outpur in excl by going to
data >data amalysis>Regression
SUMMARY OUTPUT | ||||||
Regression Statistics | ||||||
Multiple R | 0.844471 | |||||
R Square | 0.713132 | |||||
Adjusted R Square | 0.655758 | |||||
Standard Error | 14.41324 | |||||
Observations | 7 | |||||
ANOVA | ||||||
df | SS | MS | F | Significance F | ||
Regression | 1 | 2582.149 | 2582.149 | 12.42962 | 0.016819 | |
Residual | 5 | 1038.708 | 207.7416 | |||
Total | 6 | 3620.857 | ||||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
Intercept | -8.27361 | 16.36798 | -0.50548 | 0.634714 | -50.3489 | 33.80163 |
Miles of Track(X) | 1.75537 | 0.497897 | 3.525567 | 0.016819 | 0.475484 | 3.035256 |
SSE | 1038.7 | |
SST | 3620.9 | |
SSR | 2582.1 | |
MSE | 207.7 | |
Solutionc:
rsq=0.713
Does the estimated regression equation provide a good fit?
YES