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    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 Preview

    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 ...

    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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