A Conclusion Should Say What the Evidence Supports
A regression conclusion is not stronger because it sounds certain. It is stronger when its wording matches the evidence, identifies the situation the data describe, and makes clear what the analysis cannot establish. In “Selecting Evidence for a Written Conclusion,” you matched claims to evidence such as \(r\), \(r^2\), the slope, and residual plots. Now the focus is how to express those claims without stretching them.
Three questions help calibrate a conclusion: What kind of relationship does the analysis show? Which cases and setting does it describe? How much variation remains around the fitted line? Answering these questions helps avoid claiming that an association proves cause, that a sample represents everyone, or that a fitted value is exact.
Caution does not mean avoiding a clear statement. If \(r\) and a scatterplot support a strong positive linear association, say so. But do not change “associated with” into “causes,” or write “for everyone” when the data describe a narrow group. The aim is to be direct about what the analysis shows and precise about its boundaries.
Three Boundaries to Keep in View
Association is not causation. A regression line describes how the observed response tends to vary with the explanatory variable. In observational data, other variables or features of the data collection may help explain the pattern. A slope does not, by itself, show that changing \(x\) would cause \(y\) to change. Use language such as “is associated with,” “tends to be higher when,” or “the fitted model predicts,” rather than “leads to” or “results in.”
The cases and setting matter. As discussed in “Scope of Inference for a Regression Model,” a model’s scope depends on the population and setting represented by its cases, as well as the observed range of \(x\). A conclusion about students in one school, for example, does not automatically describe students everywhere. State the group actually studied when that information is available.
Predictions are not exact. The fitted line gives predicted responses, not guaranteed outcomes. The residual standard deviation \(s\), explained in “Using Residual Standard Deviation to Gauge Prediction Error,” describes the typical size of residuals in response units. It does not mean every prediction misses by exactly \(s\), or that all predictions are equally accurate. Use the residual plot to check whether the scatter around the line changes across the range.
“Uncertainty” also needs careful wording here. A noticeable \(s\) describes variation in observed responses around the fitted line. It is not, by itself, a confidence interval or a measure of uncertainty in the slope. Do not imply that the analysis provides a formal margin of error unless such an interval has actually been calculated.
A Practical Revision Routine
Before submitting a conclusion, check each phrase against the evidence. This is especially useful when a draft uses words such as “proves,” “always,” “exactly,” or “everyone.” Those words may promise more than a descriptive regression analysis can deliver.
Use the scatterplot and \(r\) to describe direction and strength, and the residual plot to discuss patterns left by the line.
Name the variables, units when they matter, and the cases or setting represented by the data.
Compare a requested \(x\)-value with the observed range and avoid generalizing beyond the represented cases without supporting evidence.
Describe predictions as model estimates and use \(s\) and residual patterns to communicate typical error or changing spread.
This routine does not require adding every caveat to every sentence. Include qualifications that matter to the claim being made. A conclusion about association should not become a paragraph of unrelated warnings, but a causal claim needs to be corrected directly. A prediction beyond the observed range needs an extrapolation qualification, as in “Writing an Extrapolation Critique.”
Worked Example: Association Does Not Establish Cause
Worked Example: Screen Time and Sleep
Original AP-style question. In an invented observational data set, researchers record daily recreational screen time \(x\), in hours, and sleep duration \(y\), in hours, for 30 students at one high school. The scatterplot is roughly linear and decreasing. The regression report gives \(\hat{y}=8.6-0.35x\), \(r=-0.72\), \(r^2=0.5184\), and \(s=0.72\) hours. The observed screen times range from 1 to 6 hours. A draft says, “Each extra hour of screen time causes students to lose 0.35 hours of sleep, and this is true for teenagers.” Revise the conclusion.
State. Describe the observed association and explain why the draft’s causal and population-wide claims are not supported by this analysis.
Plan. Use the negative \(r\) and decreasing scatterplot to describe direction and strength. Interpret the slope as a feature of the fitted line, not a causal effect. Limit the conclusion to the students and setting represented, and use \(s\) to describe typical residual size.
Do. The correlation is negative, and its magnitude is \(0.72\), so the reported value and scatterplot support a moderately strong negative linear association between recreational screen time and sleep duration among these observed students. The slope, \(-0.35\) hours of sleep per hour of screen time, means that for each additional hour of screen time, the fitted model predicts 0.35 fewer hours of sleep on average across the line. It does not show that screen time caused that decrease.
The value \(r^2=0.5184\) means that 51.84% of the variation in observed sleep durations is accounted for by the fitted linear model. The residual standard deviation \(s=0.72\) hours indicates a typical residual size of about 0.72 hours. Individual students’ sleep durations will not necessarily match their fitted values, and this overall residual size does not guarantee equal prediction accuracy at every screen-time value.
Conclude. Among the 30 students observed at this high school, recreational screen time and sleep duration have a moderately strong negative linear association. The fitted model predicts 0.35 fewer hours of sleep for each additional hour of screen time, on average, but these observational data do not establish that screen time causes less sleep or that the relationship applies to teenagers generally.
Worked Example: Put a Prediction in Its Proper Scope
Worked Example: Commute Distance and Time
Original AP-style question. In an invented data set, 16 volunteer students report their bicycle commute distance \(x\), in kilometers, and commute time \(y\), in minutes. Distances range from 1 to 8 kilometers. A roughly linear scatterplot and residual plot with no clear pattern support a line \(\hat{y}=4+5.2x\), with \(r=0.91\), \(r^2=0.8281\), and \(s=3.6\) minutes. Write a conclusion that includes the model’s prediction for a 6-kilometer commute without overclaiming.
State. Describe the association, calculate the fitted prediction at 6 kilometers, and state the scope and uncertainty appropriately.
Plan. The 6-kilometer value is inside the observed distance range of 1 to 8 kilometers, so this is interpolation rather than extrapolation. Use the slope and line for the model prediction, and \(s\) to describe typical residual size. Because the students volunteered from a limited setting, do not claim that the line applies to all cyclists.
Do. The correlation \(r=0.91\) and the upward scatterplot support a strong positive linear association between commute distance and commute time for these students. At \(x=6\) kilometers, the fitted value is
Thus, the model predicts a commute time of 35.2 minutes for a 6-kilometer commute. The slope means that the fitted time increases by 5.2 minutes for each additional kilometer of distance, on average across the line. Since \(s=3.6\) minutes, observed commute times typically differ from their fitted values by about 3.6 minutes. That is a typical residual size, not a guarantee that a particular student’s commute will be within 3.6 minutes of 35.2.
Conclude. For the 16 volunteer students, commute distance and time have a strong positive linear association. The fitted model predicts 35.2 minutes at 6 kilometers, a distance within the observed range. Actual times vary around the line, and these data do not establish the time for every cyclist or for students in other settings.
Worked Example: Keep a Specific Claim, Qualify Its Reach
Worked Example: Watering Time and Garden Yield
Original AP-style question. An invented data set records weekly watering time \(x\), in hours, and tomato yield \(y\), in kilograms, for 18 plots in one community garden during one season. Watering times range from 1 to 6 hours. The scatterplot is roughly linear and increasing, and the residual plot shows no clear systematic pattern. The fitted line is \(\hat{y}=2.5+1.8x\), with \(r=0.86\), \(r^2=0.7396\), and \(s=1.4\) kilograms. A student writes, “Watering more always increases tomato yield, and the line can predict yield at any watering time.” Identify what can be retained and revise the rest.
Solution. The positive correlation \(r=0.86\), the increasing scatterplot, and the residual plot without a clear pattern support describing a strong positive linear association between watering time and tomato yield for these plots. The slope of 1.8 kilograms per hour means that the fitted model predicts 1.8 kilograms more yield for each additional hour of weekly watering, on average across the line. It does not establish that additional watering causes more yield, nor that yield always increases for every plot.
The observed watering times range from 1 to 6 hours. Applying the line beyond that range would be extrapolation, and the observed pattern alone does not justify predictions at any watering time. The reported \(s=1.4\) kilograms gives the typical size of residuals in yield units; it does not guarantee accuracy for an individual plot or outside the observed range.
Conclude. In the 18 plots observed in this community garden during one season, watering time and tomato yield have a strong positive linear association. The fitted model predicts an increase of 1.8 kilograms in yield per additional hour of weekly watering, on average. These data do not show that watering causes the increase, and predictions are best limited to the observed range of 1 to 6 hours and this garden setting.
Common Mistakes and AP Exam Tips
- Turning a slope into a cause. “The model predicts” or “is associated with” is appropriate for a descriptive regression relationship. “Causes,” “makes,” or “leads to” needs support beyond a fitted line.
- Generalizing from a narrow group. Name the cases or setting described by the data. Do not write “all students,” “all gardens,” or “everyone” when the analysis concerns a particular sample or location.
- Presenting a fitted value as a guaranteed outcome. Use “predicts” or “estimated by the model.” If helpful, describe \(s\) as the typical residual size, not as an exact bound for every case.
- Ignoring the observed range. A prediction outside the observed \(x\)-range is extrapolation. A plausible numerical answer is not automatically a well-supported prediction.
- Using vague disclaimers instead of relevant qualifications. “There may be limitations” is less useful than naming the actual boundary: the sample is from one setting, the relationship is observational, the prediction is outside the range, or residual spread changes.
- Calling residual variation a confidence interval. \(s\) describes the typical size of residuals in response units. Unless an interval is calculated, do not claim a formal range of likely values or uncertainty in the slope.
A full-credit written conclusion typically identifies the variables and relevant group, states the direction and strength when supported, interprets the slope or prediction in context when asked, and includes a specific qualification when the claim otherwise reaches too far. It does not need to sound hesitant: it needs to be accurate.
Check Your Understanding
For each situation, write or revise a conclusion so its wording matches the regression evidence.
- An observational study finds a negative linear association between hours of gaming and hours of sleep. Why is “gaming causes less sleep” too strong? Write a more defensible sentence.
- A line is fitted to data from volunteer runners at one local club. What group should a conclusion explicitly name, and why should it not claim to describe all runners?
- A fitted model predicts a response of 42 units, and \(s=5\) units. What does \(s\) say about prediction errors, and what does it not guarantee for one case?
- A model uses observed \(x\)-values from 2 to 10. A student applies it at \(x=14\). What qualification belongs in the conclusion?
- A draft says, “The fitted line proves that every additional unit of \(x\) increases \(y\) by 3 units.” Rewrite it to describe a positive slope without claiming causation or an identical change for every case.