Question

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	13	14
Denver	15	34
Portland	36	80
Sacramento	19	30
San Diego	45	74
San Jose	29	29
St. Louis	32	41

1. Use these data to develop an estimated regression equation that could be used to predict the ridership given the miles of track.
   
   Compute b₀ and b₁ (to 2 decimals).
   b₁   
   b₀   
   
   Complete the estimated regression equation (to 2 decimals).
   =   +  x
   
2. Compute the following (to 1 decimal):

   SSE
   SST
   SSR
   MSE

   
   
3. 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?
   SelectYes, it even provides an excellent fitYes, it provides a good fitNo, it does not provide a good fitItem 10  
   
4. Develop a 95% confidence interval for the mean weekday ridership for all light-rail systems with 30 miles of track (to 1 decimal).
   (  ,   )
   
5. 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).
   (  ,   )

Answer 1

1.

x(miles)	y(ridership)
13	14
15	34
36	80
19	30
45	74
29	29
32	41

x(miles)	y(ridership)	mean x	mean y	x - mean x	(x - mean x)^2	y - mean y	(y - mean y)^2	(x - mean x)(y-mean y)
13	14	27	43.14	-14	196	-29.14	849.1396	407.96
15	34	27	43.14	-12	144	-9.14	83.5396	109.68
36	80	27	43.14	9	81	36.86	1358.6596	331.74
19	30	27	43.14	-8	64	-13.14	172.6596	105.12
45	74	27	43.14	18	324	30.86	952.3396	555.48
29	29	27	43.14	2	4	-14.14	199.9396	-28.28
32	41	27	43.14	5	25	-2.14	4.5796	-10.7

a. The formula for b1 is SS_xy/SS_xx

from the table we calculate SS_xy= (x-mean x)(y- mean y) summing over for all cities = 1471 and SS_xx = (x-mean x)^2 summing over for all cities = 838. So b₁ = 1471/838 = 1.76

b₀ = mean y - b₁mean x

b₀ = 43.14 - (1.76 * 27)

b₀ = -4.38

So the estimated regression equation can be written as :y = -4.38 + 1.76 x

b. The sum of squares total (SST), is the squared differences between the observed dependent variable and its mean

So here SST = summation of (y - mean y)^2 for all cities.

SST = 3620.86

The sum of squares due to regression (SSR) is the sum of the differences between the predicted value and the mean of the dependent variable. The predicted value is calculated using the estimated regression equation:

predicted y	predicted y - mean y	(predicted y - mean y)^2
18.5	-24.64	607.1296
22.02	-21.12	446.0544
58.98	15.84	250.9056
29.06	-14.08	198.2464
74.82	31.68	1003.6224
46.66	3.52	12.3904
51.94	8.8	77.44

The summation of square of predicted y minus mean y over all cities gives the SSR.

SSR = 2595.79

SSE is the sum of square of error which is the residual after SSR. So SSE = SST- SSR

SSE = 1025.07

MSE is the mean squared error which takes the summation of the square of the difference between actual y and predicted y.

error	error^2
-4.5	20.25
11.98	143.5204
21.02	441.8404
0.94	0.8836
-0.82	0.6724
-17.66	311.8756
-10.94	119.6836

MSE = 1038.73

c. Coefficient of determination is SSR/SST = 2595.79/3620.86 = .72

The value is closer to 1 so a greater part of the variation in dependent variable can be explained by the independent variable. So, the model is a good fit.