# Correlation coefficiient and square regression

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The Rio-River Railroad, headquartered in Santa Fe, New Mexico, is trying to devise a method for allocating fuel costs to individual railroad cars on a particular route between Denver and Santa Fe. The railroad thinks that fuel consumption will increase as more cars are added to the train, but it is uncertain how much cost should be assigned to each additional car. In an effort to deal with this problem, the cost accounting department has randomly sampled 10 trips between the two cities and recorded the data below.

Rail Cars Fuel used (units/mile)

18 55

18 50

35 76

35 80

45 117

40 90

37 80

50 125

40 100

27 75

Draw a scatter plot for these two variables and comment on the relationship between fuel consumption and the number of rail cars on the train.

(a.) Compute the correlation coefficient between fuel consumption and the number of train cars. (b.) Test Rio-River's preconception of the relation between fuel consumption and the number of train cars using a significance level of 0.025. (c.) Comment on the results of this test. Do these results necessarily indicate that adding more cars will increase fuel usage?

Develop the least squares regression model to help explain the variation in fuel consumption.

Interpret the regression results by addressing the issue of, on average, how much the addition of another train car will increase fuel consumption. Also, calculate the average fuel consumption, average number of cars per train, and average fuel consumption per car for the data given. Does this average equal the average increase in fuel consumption from adding an additional car using the regression model? Explain any difference.

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##### Solution Summary

The Rio-River Railroad, headquartered in Santa Fe, New Mexico, is trying to devise a method for allocating fuel costs to individual railroad cars on a particular route between Denver and Santa Fe. The railroad thinks that fuel consumption will increase as more cars are added to the train, but it is uncertain how much cost should be assigned to each additional car. In an effort to deal with this problem, the cost accounting department has randomly sampled 10 trips between the two cities and recorded the data below.

Rail Cars Fuel used (units/mile)

18 55

18 50

35 76

35 80

45 117

40 90

37 80

50 125

40 100

27 75

Draw a scatter plot for these two variables and comment on the relationship between fuel consumption and the number of rail cars on the train.

(a.) Compute the correlation coefficient between fuel consumption and the number of train cars. (b.) Test Rio-River's preconception of the relation between fuel consumption and the number of train cars using a significance level of 0.025. (c.) Comment on the results of this test. Do these results necessarily indicate that adding more cars will increase fuel usage?

Develop the least squares regression model to help explain the variation in fuel consumption.

Interpret the regression results by addressing the issue of, on average, how much the addition of another train car will increase fuel consumption. Also, calculate the average fuel consumption, average number of cars per train, and average fuel consumption per car for the data given. Does this average equal the average increase in fuel consumption from adding an additional car using the regression model? Explain any difference.

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we find a positive correlation between fuel consumption and the number of rail cars on the train: the more rail cars, the more fuel used.

2)(a.) Compute the correlation coefficient between fuel consumption and the number of train cars.

First, compute the covariance between the two variables:

COV(X, Y) = [(X-Xm)(Y-Ym)]/(N-1) = 243.78

And calculate Standard Deviation of both variables by

SDx = SQRT[(X-Xm)^2/(N-1)] = 10.64

SDy = SQRT[(Y-Ym)^2/(N-1)] = 24.11

Then correlation coefficient is

R = COV(X, Y) / (SDx*SDy) =0.9507

Please refer to EXCEL for calculation.

(b.) Test Rio-River's preconception of the relation between fuel ...

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