GB513 Unit 8 Simple Regression Analysis Example

Miles Explain Most of the Maintenance Bill: A Unit 8 Simple Regression for a Delivery Fleet Budget

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Purdue University Global

GB513: Business Analytics

Unit 8 Simple Regression Analysis

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[Date]

The fleet, its trucks and every figure are composite and illustrative; the methods cited are real.

The Business Question

Ridgeway Foodservice runs 45 refrigerated delivery trucks. The fleet manager, Tomas Reyes, must budget maintenance for next year, when planned mileage per truck ranges from about 25,000 to 75,000 miles depending on route. He wants to know how much maintenance cost rises with miles and what to budget for a truck planned at 60,000 miles.

Scatter Plot

A scatter plot of last year's data, miles in thousands on the horizontal axis and maintenance dollars on the vertical, shows a clear upward pattern. The fitted line has the equation predicted cost equals 1,896.9 plus 112.66 times miles in thousands, and the chart shows R squared of 0.761.

Regression Statistics

R Square: 0.7612. Standard error of the regression: $913.8. Observations: 45.

Reading this table: about 76 percent of the variation in maintenance cost across trucks is associated with differences in miles driven. The remaining 24 percent reflects other factors, such as truck age, route terrain and driver habits. The standard error means a typical truck's actual cost differs from the line's prediction by about $914 (Levine et al., 2021).

ANOVA Table

Regression sum of squares: about 114.5 million. Residual sum of squares: about 35.9 million. F statistic: about 137.1, with a significance F far below 0.001.

Reading this table: the F test asks whether the model explains more variation than chance would. An F of 137 with a p-value near zero means miles are clearly related to cost.

Coefficients

Intercept: 1,896.9, standard error 472.0.

Miles (thousands): 112.66, standard error 9.62, t statistic 11.71, p-value about 0.000000000000006. At 95 percent confidence, the slope lies between 93.25 and 132.07.

Reading this table: each additional thousand miles is associated with about $113 more maintenance cost, and we can be fairly sure the true figure lies between about $93 and $132. The intercept, about $1,900, can be read as the fixed annual cost of keeping a truck in service, such as inspections and refrigeration unit checks, although no truck drove zero miles, so it is an extrapolation and should not be pushed hard (Anderson et al., 2020).

The slope is the budget figure that matters: each thousand miles adds about $113, and the interval of $93 to $132 tells the fleet manager how far to trust it.

Residual Plot

A plot of residuals against miles shows points scattered evenly above and below zero across the range, with no curve and no widening funnel. A straight line therefore looks adequate for this range and that the spread of errors is similar for low- and high-mileage trucks. Three trucks sit about $1,800 to $2,000 above the line; all three are among the oldest in the fleet, which suggests truck age could improve the model.

Checking the Other Assumptions

Two more checks support the model. A histogram of the residuals is roughly bell shaped, with no long tail, so the t-based intervals are reasonable. And the trucks' costs were recorded independently, one invoice history per truck, so there is no reason to expect one truck's error to affect another's. The data cover about 21,600 to 76,400 miles, which includes the 60,000-mile prediction, so the forecast does not extrapolate beyond what the fleet has experienced.

What the Model Leaves Out

About a quarter of the variation in cost is unexplained. Truck age is the most likely cause, as the residual plot suggests, and Ridgeway's records include each truck's model year. Route type also matters: trucks on stop-and-go city routes wear brakes faster than highway trucks at the same mileage. A multiple regression adding age and a city route indicator would likely raise R squared and narrow the prediction interval, which would let the budget carry less contingency.

Prediction and Budget

For a truck planned at 60,000 miles, the predicted maintenance cost is 1,896.9 plus 112.66 times 60, or about $8,657. A single truck's actual cost is uncertain, so the 95 percent prediction interval for one truck is about $6,780 to $10,540. James Evans distinguishes this prediction interval for a single observation from the narrower confidence interval for the average of many trucks at the same mileage (Evans, 2020); the budget for one truck should respect the wider one.

For the budget, the fleet manager should use the predicted cost for each truck's planned miles and add a contingency based on the standard error. Across 45 trucks, individual errors partly cancel, so the fleet total is more certain than any one truck. Budgeting each truck at its prediction and adding about 5 percent fleet-wide contingency would cover most outcomes.

Recommendation

Budget each truck at 1,897 dollars plus 113 dollars per thousand planned miles, plus a 5 percent fleet contingency. Track actual costs quarterly and add truck age to next year's model, since the oldest trucks sit consistently above the line.

References

Anderson, D. R., Sweeney, D. J., Williams, T. A., Camm, J. D., & Cochran, J. J. (2020). Statistics for business and economics (14th ed.). Cengage.

Evans, J. R. (2020). Business analytics: Methods, models, and decisions (3rd ed.). Pearson.

Levine, D. M., Stephan, D. F., & Szabat, K. A. (2021). Statistics for managers using Microsoft Excel (9th ed.). Pearson.