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	14	17
Denver	16	37
Portland	37	83
Sacramento	20	33
San Diego	46	77
San Jose	30	32
St. Louis	33	44

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).
   
   
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).
   (  ,  )
   
   Do you think that the prediction interval you developed would be of value to Charlotte planners in anticipating the number of weekday riders for their new light-rail system?
   SelectYes, because this interval has high accuracyYes, because this interval has high confidenceYes, because this interval has both high accuracy and high confidenceNo, because this interval is too wideNo, because this interval has low confidenceItem 15

Answer 1

X	Y	XY	X²	Y²
14	17	238	196	289
16	37	592	256	1369
37	83	3071	1369	6889
20	33	660	400	1089
46	77	3542	2116	5929
30	32	960	900	1024
33	44	1452	1089	1936

Sample size, n =	7
Ʃ x =	196
Ʃ y =	323
Ʃ xy =	10515
Ʃ x² =	6326
Ʃ y² =	18525
x̅ =	28
y̅ =	46.14285714
SSxx = Ʃx² - (Ʃx)²/n =	838
SSyy = Ʃy² - (Ʃy)²/n =	3620.857143
SSxy = Ʃxy - (Ʃx)(Ʃy)/n =	1471

a) b1 = SSxy/SSxx = 1.75537 =  1.76
b0 = y̅ -b1* x̅ = -3.0075 = -3.01

Regression equation:

ŷ = -3.01 + 1.76 x

b)

SSE = SSyy -b*SSxy =	1039.708 = 1038.7
SST = SSyy = Ʃy² - (Ʃy)²/n =	3620.8571 = 3620.9
SSR = b*SSxy =	2582.1492 = 2582.1
MSE = SSE/(n-2) =	207.7

c) Coefficient of determination, R² = SSR/SST = 0.713

Yes, it provides a good fit.

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d) Predicted value at X = 30          
ŷ = -3.0075 + 1.7554 * 30 = 49.65  

Standard error, se = √(SSE/(n-2)) = 14.4132

At α = 0.05 and df = n-2 = 5, critical value, t_c = T.INV.2T(0.05, 5 ) = 2.5706

95%Confidence interval for the mean weekday ridership for all light-rail systems with 30 miles of track :
Lower limit = ŷ - tc*se*√((1/n) + ((x-x̅)²/(SSxx))) = 35.4
Upper limit = ŷ + tc*se*√( (1/n) + ((x-x̅)²/(SSxx))) = 63.9

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e) 95% prediction interval for the weekday ridership for the Charlotte system:

Lower limit = ŷ - tc*se*√(1 + (1/n) + ((x-x̅)²/(SSxx))) = 10.0
Upper limit = ŷ + tc*se*√(1 + (1/n) + ((x-x̅)²/(SSxx))) = 89.3

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