| Course | HS345 Biostatistics |
|---|---|
| Unit | Unit 9 |
| Paper type | Regression exercise |
| Length | About 600 words, 5 pages double-spaced |
| Format | APA 7 student paper |
| School | Purdue University Global |
| Program | BS in Health Science |
| Updated | October 2026 |
Free sample paper for HS345 Unit 9
A Line That Slopes Down and What It Does Not Show: A Unit 9 Regression Exercise on Physician Supply and Preventable Hospital Stays
[Student Name]
Purdue University Global
HS345: Biostatistics
Unit 9 Regression Exercise
[Instructor Name]
[Date]
The forty-eight counties and their values are composite; the county rankings measure, the review of ecologic studies and the statistics text cited are real.
The Data and the Plot
Each point is a county. The horizontal axis is primary care physicians per 100,000 residents, ranging from about 25 to 120. The vertical axis counts preventable stays per 100,000 enrollees in Medicare, a county rankings measure built from admissions for conditions that timely outpatient treatment can usually head off (University of Wisconsin Population Health Institute, 2025). Rural counties are drawn as open circles; most sit at the left of the plot, with fewer physicians and more preventable stays.
The Fitted Line
Predicted preventable stays equal 4,946 minus 16.5 times physicians per 100,000.
Slope: minus 16.5, with a standard error of 2.45. 95 percent interval for the slope: minus 21.4 to minus 11.6, using t on 46 degrees of freedom. Correlation: minus 0.70. R squared: 0.50.
Reading the slope: counties with ten more primary care physicians per 100,000 residents had, on average, about 165 fewer preventable stays per 100,000 enrollees. The interval excludes zero by a wide margin.
R squared of 0.50 means physician supply accounts for about half the variation between counties in preventable stays; the other half reflects everything else.
Two Predictions
At 40 physicians per 100,000: 4,946 minus 660, or about 4,290 stays. At 100 physicians per 100,000: 4,946 minus 1,650, or about 3,300 stays.
Both are rounded to the nearest ten, because the residual standard deviation is close to 500 stays, so any single county could easily sit several hundred stays from the line. Predicting outside the range of the data, for example at 200 physicians, would be extrapolation the data cannot support.
The slope describes how counties differ from each other, not what would happen inside one county if it hired more doctors, and certainly not what happens to any individual patient.
What the Slope Does Not Show
It does not show cause. Counties with more physicians also tend to have higher incomes, more insurance coverage, more hospitals and fewer older residents. Any of these could lower preventable stays independently of physician supply.
It does not show anything about individuals. Inferring from county averages to people risks the ecological fallacy, since the relationship between groups need not hold within them (Morgenstern, 1995).
It does not separate rural from urban. The open circles cluster at the left and high, so part of the slope may reflect rurality itself. A model that included a rural indicator would show whether the slope holds among rural counties alone.
Checking the Model
A residual plot shows no curve and roughly even spread, so a straight line is reasonable. Two counties have large positive residuals; both host regional hospitals that admit patients from neighboring counties, which may inflate their rates. Removing them changes the slope only to about minus 15.8 (Pagano et al., 2022).
What a County Health Officer Could Take From This
The analysis supports the idea that primary care access is worth examining in counties with high preventable stays. It does not support a promise that recruiting a set number of physicians would reduce stays by a set amount. A more useful next step would be a model with income, insurance and rurality added, or a look at counties that increased physician supply over time.
References
Morgenstern, H. (1995). Ecologic studies in epidemiology: Concepts, principles, and methods. Annual Review of Public Health, 16, 61-81. https://doi.org/10.1146/annurev.pu.16.050195.000425
Pagano, M., Gauvreau, K., & Mattie, H. (2022). Principles of biostatistics (3rd ed.). CRC Press.
University of Wisconsin Population Health Institute. (2025). County health rankings and roadmaps: Preventable hospital stays. https://www.countyhealthrankings.org
What the HS345 Unit 9 assignment asks for
The ninth unit typically supplies paired data and asks for a scatter plot, a fitted least squares line, interpretation of the slope and R squared, a prediction and a discussion of limitations. Exercises usually fill two to three pages. Points go to a correctly fitted and interpreted line, a slope interval or test, predictions kept inside the data's range and an honest account of what regression on aggregate data cannot show. Many sections ask for a residual plot. Explain the slope in the units of both variables. Support the method with a statistics text, and when each point is a place rather than a person, add a source on ecological inference.
How the HS 345 Unit 9 example is put together
The exercise describes the scatter plot and both axes, citing the county rankings source for the outcome. The fitted line, slope with standard error and interval, correlation and R squared follow, and the slope is read in plain words for ten more physicians. Two predictions are rounded with the residual spread in mind, and extrapolation is ruled out. The highlighted sentence states the ecological limit. A section lists what the slope does not show: cause, anything about individuals, and the separate effect of rurality, citing a review of ecologic studies. Residual and influence checks follow, and a closing section tells a county health officer what the analysis supports and what model to fit next.
Reading the HS345 Unit 9 grading rubric
Regression exercises earn credit chiefly for correct interpretation of the slope and for recognizing the model's limits. Interpretation points require the slope in the units of both variables, R squared explained as shared variation and predictions with sensible precision. Limits credit goes to naming confounding, extrapolation and, for aggregate data, the ecological fallacy. A slope interval or test adds statistical support. Residual checks show the line's fit was examined. Clear labeling of the plot and equation, and a conclusion aimed at a real decision maker, complete the work that graders typically expect at the top level. Stating the next model you would fit, with the extra variables named, shows the limits were taken seriously.
Common HS345 Unit 9 mistakes, and how to avoid them
The commonest error is reading the slope as a cause, as if adding physicians would lower stays by a fixed amount. Another is applying a county-level slope to individual patients. Some students report R squared as a percentage of counties explained rather than variation explained. Others predict far outside the data's range. Skipping the residual plot can hide a curve or an influential point. If your data pair other measures, perhaps smoking rates and lung cancer or income and life expectancy, share the variables and the unit of analysis, and the fit and its limits can be worked through together. Reporting the equation without saying which variable is the predictor leaves the slope's meaning ambiguous.
Get an HS345 Unit 9 example written to your instructions
Regression prompts in HS345 Unit 9 vary in their data, from county measures to patient-level readings. Attach yours, or the output it produced, along with the instructions and rubric. Delivered within 24-48 hours, the first custom sample is free and interprets the slope in the data's own units, plots the residuals, and says plainly what the line cannot establish. The paper above is an original model document written by our desk, not a submitted student paper and not an official Purdue University Global document.
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HS345 Unit 9 questions, answered
What does it mean that the data are ecological?
Each point is a county, not a person. A pattern across counties does not show that any individual patient with better physician access avoided a hospital stay. Individual-level data would be needed for that claim. The example flags this in its conclusion, because ecological associations can be stronger, weaker or even reversed at the individual level.
Can I report the correlation instead of the slope?
Report both if the prompt allows. The correlation of minus 0.70 describes strength and direction without units, while the slope says how much stays change per physician, which is what a planner needs. Most regression rubrics expect the slope interpreted in context, so leaving it out usually costs more than leaving out the correlation.
Where can I find a free HS345 Unit 9 sample paper?
The whole Unit 9 exercise is here, fitting a line to physician supply and preventable hospital stays across forty-eight counties.
What is the ecological fallacy?
Drawing conclusions about individuals from data on groups. A pattern across counties may not hold for the people within them.
What does R squared of 0.50 mean?
Half of the variation between counties in preventable stays is accounted for by the straight-line relationship with physician supply; the other half reflects other factors.