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	16	16
Denver	18	36
Portland	39	82
Sacramento	22	32
San Diego	48	76
San Jose	32	31
St. Louis	35	43

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).
   (  ,  )
   
   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 confidence

Answer 1

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
16	16	196.00	849.306	408.000
18	36	144.00	83.592	109.714
39	82	81.00	1358.449	331.714
22	32	64.00	172.735	105.143
48	76	324.00	952.163	555.429
32	31	4.00	200.020	-28.286
35	43	25.00	4.592	-10.714

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	210	316	838.000	3620.9	1471
mean	30.00	45.14	SSxx	SSyy	SSxy

sample size ,   n =   7      
here, x̅ = Σx / n=   30.00   ,     ȳ = Σy/n =   45.14
              
SSxx =    Σ(x-x̅)² =    838.0000      
SSxy=   Σ(x-x̅)(y-ȳ) =   1471.0      

a)

estimated slope , ß1 = SSxy/SSxx =   1471.0   /   838.000   =   1.8
                  
intercept,   ß0 = y̅-ß1* x̄ =   -7.5   
                  
so, regression line is   Ŷ =   -7.5 +   1.8 *x

b)

SSE=   (SSxx * SSyy - SS²xy)/SSxx =    1038.708  

SSt=SSyy = 3620.9

SSR=SST-SSE

MSE=SSE/(n-2)=207.7

SSE	1038.7
SST	3620.9
SSR	2582.1
MSE=207.7

c)

R² =    (Sxy)²/(Sx.Sy) =    0.7131

Yes, it provides a good fit

d)

X Value=   30                      
Confidence Level=   95%                      
                          
                          
Sample Size , n=   7                      
Degrees of Freedom,df=n-2 =   5                      
critical t Value=tα/2 =   2.571   [excel function: =t.inv.2t(α/2,df) ]                  
                          
X̅ =    30.00                      
Σ(x-x̅)² =Sxx   838.0                      
Standard Error of the Estimate,Se=   14.41                      
                          
Predicted Y at X=   30   is                  
Ŷ =   -7.518   +   1.755   *   30   =   45.143
                          
standard error, S(ŷ)=Se*√(1/n+(X-X̅)²/Sxx) =    5.448                      
margin of error,E=t*Std error=t* S(ŷ) =   2.5706   *   5.4477   =   14.0037      
                          
Confidence Lower Limit=Ŷ +E =    45.143   -   14.0037   =   31.1   
Confidence Upper Limit=Ŷ +E =   45.143   +   14.0037   =   59.1   

e)

standard error, S(ŷ)=Se*√(1+1/n+(X-X̅)²/Sxx) =   15.4084              
margin of error,E=t*std error=t*S(ŷ)=    2.5706   *   15.41   =   39.6086
                  
Prediction Interval Lower Limit=Ŷ -E =   45.143   -   39.6086   =   5.5
Prediction Interval Upper Limit=Ŷ +E =   45.143   +   39.6086   =   84.8