| Course | BU 321 Business Statistics |
|---|---|
| Unit | Unit 7 |
| Paper type | Regression output interpretation |
| Length | About 1,020 words, 6 pages |
| Format | APA 7 student paper |
| School | Herzing University |
| Program | BS in Business Management |
| Updated | September 2026 |
Free sample paper for BU 321 Unit 7
About Three Minutes a Mile: Interpreting a Regression of Delivery Time on Distance for a Composite Pizza Chain, and the Causal Sentence It Does Not Support
Student Name
Business, Herzing University
BU 321: Business Statistics
Instructor Name
April 15, 2026
About Three Minutes a Mile: Interpreting a Regression of Delivery Time on Distance for a Composite Pizza Chain, and the Causal Sentence It Does Not Support
The Variables and the Direction
This interpretation works from a simple linear regression fitted to the October 2025 sample of 240 orders from Brickoven Express, the invented pizza chain behind every unit of this course. The outcome being predicted is delivery time in minutes, from order to doorstep. The predictor is delivery distance in miles from the store to the customer. The direction matters: distance is known when an order is placed, and the chain would like to predict how long a delivery will take from it. Predicting distance from delivery time would answer no question anyone at the chain is asking.
The Output
The output below was produced in Python using ordinary least squares; the same figures appear in the Excel regression tool. It stays visible because every statement that follows can be checked against it.
| Term | Coefficient | Standard error | t | p |
|---|---|---|---|---|
| Intercept | 14.806 | 0.740 | 20.00 | below 0.001 |
| Distance (miles) | 3.139 | 0.204 | 15.41 | below 0.001 |
| n = 240; R-squared = 0.499; adjusted R-squared = 0.497 | ||||
| Residual standard error = 5.85 minutes; F = 237.4 on 1 and 238 df |
The Equation and the Slope
The estimated equation is: predicted delivery time = 14.81 + 3.14 times distance in miles. The slope says that, across these orders, each additional mile of distance is associated with about 3.1 more minutes of delivery time on average. A 95 percent confidence interval for the slope, using the standard error of 0.204 and a critical value of about 1.97, runs from 2.74 to 3.54 minutes per mile. In practical terms, a delivery 5 miles away is predicted to take about 30.5 minutes and one 2 miles away about 21.1 minutes, a difference of roughly 9 minutes for 3 extra miles.
The slope's t statistic of 15.41 and p-value below 0.001 show that a slope this large would be extremely unlikely if distance had no linear association with delivery time. That establishes that the relationship is not a sampling accident; it does not establish anything more.
The Intercept
The intercept of 14.8 minutes is the predicted delivery time at a distance of zero miles. A zero-mile delivery is not a real order, but the shortest trips in the sample were 0.4 miles, so zero is close to the observed range rather than far outside it. The intercept can therefore be read, cautiously, as the time an order spends on things other than driving: preparing and baking the order, waiting for a driver, loading and walking the order to the door. For a manager, that is useful. It suggests that roughly 15 minutes of every delivery happens before the wheels turn or after they stop, and that time does not shrink with a shorter route.
How Well the Line Fits
The R-squared of 0.499 means that knowing an order's distance removes roughly half of the order-to-order variation in delivery time. That figure should be put in proportion. It is high enough to make distance the single most useful piece of information for predicting a delivery's length, and low enough to show that the other half of the variation comes from something else. The residual standard error of 5.85 minutes gives the typical size of a prediction miss: for a given distance, actual delivery times commonly fall about 6 minutes above or below the line. A 5-mile delivery predicted at 30.5 minutes could easily take 25 or 36.
The adjusted R-squared of 0.497 is almost identical to the unadjusted figure because the model has only one predictor and 240 observations; the adjustment matters mainly when many predictors are added to a small sample. The F statistic of 237.4 tests the same question as the slope's t statistic in a one-predictor model, and it is simply that t value squared.
What the Residuals Show
Residuals are the differences between actual and predicted delivery times. Their spread is about the same for short and long deliveries, with a standard deviation of 5.75 minutes for orders in the lower half of predicted times and 5.95 minutes for the upper half, so the line's errors do not grow with distance. The 23 orders of 6 miles or more have an average residual of minus 0.44 minutes, meaning the line fits the longest trips about as well as the rest, with no sign of a curve at the far end.
A pattern does appear when residuals are grouped by time of day. Dinner orders have an average residual of plus 2.43 minutes, while lunch and late-night orders average minus 2.80 and minus 2.59 minutes. In other words, at any given distance, dinner deliveries take about five minutes longer than lunch deliveries. Time of day is a second influence that the one-variable model leaves out, and it accounts for part of the unexplained half of the variation. Anscombe (1973) showed with four datasets that share the same regression line but look entirely different when plotted, that output alone can hide patterns like this, which is why residuals should always be examined, not only the fit statistic (National Institute of Standards and Technology, 2012).
The Causal Sentence the Numbers Seem to Invite
The tempting conclusion is: "Each mile causes 3.1 minutes of delay, so cutting the delivery zone by one mile would cut delivery times by 3.1 minutes a mile for the affected customers." The physics of driving make some causal link obvious, but the regression does not establish the size of that effect. The data are observational. Distance may travel together with other factors, such as the roads serving outlying neighborhoods, apartment complexes where drivers must find a building, or the stores whose zones are largest, and the slope absorbs all of them. The residual pattern by time of day shows that at least one other factor is at work. What the output supports is a predictive statement: for planning, a delivery that is one mile farther can be expected to take about three minutes longer, give or take the uncertainty shown. A causal estimate for a policy change, such as shrinking a zone, would need either a controlled test or a model that accounts for time of day, store and neighborhood (Anderson et al., 2020). The dataset behind this output is a teaching composite, not a real chain's records.
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.
Anscombe, F. J. (1973). Graphs in statistical analysis. The American Statistician, 27(1), 17-21. https://doi.org/10.1080/00031305.1973.10478966
National Institute of Standards and Technology. (2012). NIST/SEMATECH e-handbook of statistical methods. https://www.itl.nist.gov/div898/handbook/
Reading the BU 321 Unit 7 instructions
The Unit 7 assignment asks you to interpret regression output, usually a simple linear regression produced from a business dataset. Most versions want the variables identified with the direction of prediction justified, the estimated equation written out, the slope and intercept interpreted in context, the fit statistics explained and the residuals examined. Many rubrics ask whether the relationship is causal and expect you to explain why regression alone cannot establish causation. Some ask for a prediction at a given value of the predictor. Keep the output visible in the paper and cite the textbook and any sources on regression diagnostics in APA 7. If the prompt supplies output from software you have not used, focus on the coefficients, their standard errors and the fit statistics, which every package reports in some form. Practicing on a small dataset of your own first makes the output far easier to read.
Inside the BU 321 Unit 7 example
The sample names the outcome and predictor and explains why distance predicts time rather than the reverse. The output appears as a table with coefficients, standard errors, t values, R-squared and the residual standard error. The slope is interpreted in minutes per mile with an interval and two example predictions. The intercept is interpreted cautiously after checking the observed range. The fit section explains what half the variation means and what a 6-minute typical miss implies. The residual section examines spread, the longest trips and time of day, finding a dinner effect. The final section quotes the tempting causal claim and replaces it with a predictive one.
Where the points sit in the BU 321 Unit 7 rubric
Herzing's criteria for the regression unit usually include identification of variables, interpretation of coefficients, evaluation of fit, residual analysis and the handling of causation. Variable points depend on naming the predictor and outcome and justifying the direction. Coefficient interpretation earns full credit when the slope is read in the business's units and the intercept is interpreted or declared meaningless for a stated reason. Fit is judged by explaining R-squared in proportion rather than as a grade. Residual analysis rewards specific patterns found and their meaning, such as the dinner effect. The causation criterion rewards papers that decline an unsupported causal claim and explain why.
Common BU 321 Unit 7 mistakes, and how to avoid them
A frequent Unit 7 mistake is reading the slope in the software's terms, the coefficient is 3.139, without saying 3.1 minutes per mile. Another is interpreting an intercept far outside the data, such as the value of a house with zero square feet. Students often treat R-squared as a grade, calling 0.50 poor or 0.90 proof, rather than explaining what it measures. Residuals are frequently ignored, which hides patterns like the dinner effect. The biggest error is writing that the predictor causes the outcome based only on a significant slope. Finally, give a prediction with its uncertainty; a single predicted number without the typical miss overstates what the model knows.
Get a BU 321 Unit 7 example written to your instructions
Send the Unit 7 instructions and the rubric from your BU 321 classroom, with the dataset or the output the assignment supplies. We write a custom example that defends the direction, writes the equation out, reads the slope in real units, tests the model on its residuals and keeps the prediction inside the data. First custom sample free, back in 24 to 48 hours.
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BU 321 Unit 7 questions, answered
Can I say the predictor causes the outcome?
Not from a regression on observational data, and the sentence that does is the most expensive one in this unit. Report the association, state its direction and size, and name at least one other explanation the data cannot rule out. If your instructions ask for a causal discussion, write about what design would be needed rather than claiming this one delivered it.
What does the fit statistic actually tell me?
How much of the variation in the outcome the model accounts for within this sample, and nothing about whether the relationship is useful, correct or causal. A modest figure can sit alongside a slope that matters commercially, and a high one can come from a model that will predict nothing new. Report it and interpret the slope separately.
How do I read the residual plot?
Look for pattern rather than for beauty. Scatter with no shape supports the assumptions your reading names, while a funnel, a curve or a run of points on one side each say something specific about where the model is failing. Describe what you see, say which assumption it bears on, and note what you would do about it.
Where can I find a free BU 321 Unit 7 sample paper?
The complete BU 321 Unit 7 interpretation, output block and residual discussion included, is posted above free of charge. For output from your own dataset, paste it into your request with the prompt and a free interpretation will follow, typically within 48 hours.
What does R-squared tell me in BU 321 Unit 7?
The share of variation in the outcome that the predictor accounts for in your data. The sample's 0.499 means distance explains about half the variation in delivery times, which makes it useful but leaves room for other factors. It is not a grade and does not prove causation.