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Answer :
Sure, let's go through the questions step-by-step:
(a) Is a linear model appropriate? Clearly explain your reasoning.
To determine if a linear model is appropriate, we need to examine whether the relationship between the variables is linear. This usually involves looking at a scatterplot to see if the data points form a line or if the residual plot indicates randomness (i.e., no patterns). If the residuals show a random scatter without a discernible pattern, a linear model may be appropriate. However, without specific data or a plot, this question can't be definitively answered without more information about the residuals and the overall fit of the data.
(b) Suppose the fuel consumption cost is [tex]$25 per unit. Give a point estimate (single value) for the change in the average cost of fuel per mile for each additional rail car attached to a train. Show your work.
You're asked to find how much the cost changes for each additional rail car. Given:
- Fuel consumption cost per unit = $[/tex]25
- Change in fuel consumption per additional rail car = 2.15 units
To find the change in the average cost of fuel per mile for each additional rail car, multiply these two values:
[tex]\[ \text{Change in cost per additional rail car} = 25 \times 2.15 = 53.75 \][/tex]
So, the point estimate for the change in the average cost of fuel per mile for each additional rail car is $53.75.
(c) Interpret the value of [tex]\( r^2 \)[/tex] in the context of the problem.
The value of [tex]\( r^2 \)[/tex], the coefficient of determination, indicates how well the data fits a statistical model – in this case, how well the number of rail cars predicts fuel consumption. In context, an [tex]\( r^2 \)[/tex] value closer to 1 means that a larger proportion of the variability in fuel consumption can be explained by the number of rail cars. Conversely, an [tex]\( r^2 \)[/tex] value closer to 0 suggests that the number of rail cars does not explain much of the variability in fuel consumption.
(d) Would it be reasonable to use the fitted regression equation to predict the fuel consumption for a train on this route if the train had 65 railcars? Explain.
To decide if it's reasonable to use the regression equation for prediction, consider:
- Extrapolation: If the number of rail cars for the prediction (65) is within the range of the data used to create the model, prediction is safer. If 65 rail cars is outside this range, the prediction may be unreliable.
- Model Fit: The appropriateness of the model (from part a), as shown by [tex]\( r^2 \)[/tex], also matters. If the model fits well, predictions are more likely to be accurate.
Without knowing the range of the data or the model fit, it’s difficult to definitively conclude. However, if 65 is within the configuration the model is validated on, and the model has a good fit, it could be considered reasonable.
(a) Is a linear model appropriate? Clearly explain your reasoning.
To determine if a linear model is appropriate, we need to examine whether the relationship between the variables is linear. This usually involves looking at a scatterplot to see if the data points form a line or if the residual plot indicates randomness (i.e., no patterns). If the residuals show a random scatter without a discernible pattern, a linear model may be appropriate. However, without specific data or a plot, this question can't be definitively answered without more information about the residuals and the overall fit of the data.
(b) Suppose the fuel consumption cost is [tex]$25 per unit. Give a point estimate (single value) for the change in the average cost of fuel per mile for each additional rail car attached to a train. Show your work.
You're asked to find how much the cost changes for each additional rail car. Given:
- Fuel consumption cost per unit = $[/tex]25
- Change in fuel consumption per additional rail car = 2.15 units
To find the change in the average cost of fuel per mile for each additional rail car, multiply these two values:
[tex]\[ \text{Change in cost per additional rail car} = 25 \times 2.15 = 53.75 \][/tex]
So, the point estimate for the change in the average cost of fuel per mile for each additional rail car is $53.75.
(c) Interpret the value of [tex]\( r^2 \)[/tex] in the context of the problem.
The value of [tex]\( r^2 \)[/tex], the coefficient of determination, indicates how well the data fits a statistical model – in this case, how well the number of rail cars predicts fuel consumption. In context, an [tex]\( r^2 \)[/tex] value closer to 1 means that a larger proportion of the variability in fuel consumption can be explained by the number of rail cars. Conversely, an [tex]\( r^2 \)[/tex] value closer to 0 suggests that the number of rail cars does not explain much of the variability in fuel consumption.
(d) Would it be reasonable to use the fitted regression equation to predict the fuel consumption for a train on this route if the train had 65 railcars? Explain.
To decide if it's reasonable to use the regression equation for prediction, consider:
- Extrapolation: If the number of rail cars for the prediction (65) is within the range of the data used to create the model, prediction is safer. If 65 rail cars is outside this range, the prediction may be unreliable.
- Model Fit: The appropriateness of the model (from part a), as shown by [tex]\( r^2 \)[/tex], also matters. If the model fits well, predictions are more likely to be accurate.
Without knowing the range of the data or the model fit, it’s difficult to definitively conclude. However, if 65 is within the configuration the model is validated on, and the model has a good fit, it could be considered reasonable.
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