20. Scatterplots, Linear Models, Slope, and Residuals

Session 20: Scatterplots, Linear Models, Slope, and Residuals

Learning goals

By the end of this session, you will be able to:

• Interpret slope and intercept with units.
• Calculate predictions and residuals.
• Evaluate fit, extrapolation, and causal limits.

Big idea

A linear model is useful when its slope, intercept, predictions, and residuals have sensible contextual meaning within the range supported by the data.

Rich concept explanation

A scatterplot shows paired numerical data. Describe its direction, form, strength, and unusual points before using a model.

A linear model has the form:
\[ \hat y=mx+b, \]
where (m) is the predicted change in (y) for each one-unit increase in (x), and (b) is the predicted value when (x=0). Interpret both values with units and context.

If a study model is (hat y=4x+62), where (x) is hours studied and (y) is predicted score, the slope means each additional study hour is associated with about 4 more score points. For (x=5), the predicted score is (4(5)+62=82).

A residual measures prediction error:
\[ \text{residual}=\text{actual}-\text{predicted}. \]
If the actual score is 78 and the predicted score is 82, the residual is (78-82=-4). The model overpredicted by 4 points.

Interpolation predicts within the observed range and is usually safer. Extrapolation predicts outside the data range, where the trend may change. An influential outlier can noticeably change the slope or intercept.

Common misconception: a line of best fit must pass through every point. It summarizes the overall trend, so points are expected on both sides of the line.

GED® strategy: distinguish three answers—what the graph shows, what the model predicts, and what the context justifies.

A reliable data-reasoning routine

Use this five-part routine whenever a problem combines words, data, and a calculation:

1. Understand: restate the target in plain language; identify the population or event, variables, units, and any words that restrict the group.
2. Represent: organize the given information in a table, ordered list, graph, equation, sample space, or labeled probability statement.
3. Solve: choose a rule that matches the data and carry out one labeled calculation at a time without rounding too early.
4. Check: verify the denominator, units, scale, and reasonableness; then interpret the result in the original context.
5. Communicate: state what the evidence supports and avoid stronger claims, especially causal claims, that the design cannot justify.

The routine is deliberately slower than guessing at the first arithmetic operation. On the GED®, a correct setup usually saves more time than it costs because it prevents denominator, scale, and interpretation errors.

Ten real-life worked examples

The examples below move from direct application to deeper interpretation. Read the complete reasoning, cover the solution, and then solve the problem again from the prompt alone.

Worked example 1

Real-life problem. A model for weekly sales is y = 120 + 35x, where x is advertising hours and y is sales. Interpret the slope.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Attach change-in-output units to a one-unit increase in the input.

Step 3 - Solve step by step.

1. The slope is the coefficient 35.
2. The input unit is one hour of advertising.
3. The output unit is sales per week.
4. The model predicts about 35 additional weekly sales for each additional advertising hour.

Step 4 - Check and interpret. A slope interpretation describes predicted change and does not claim advertising alone causes the change.

Why this matters. Managers use model slopes to translate input changes into estimated business outcomes.

Common trap. Reporting a bare slope or residual without units, reversing actual minus predicted, or extrapolating beyond the model's evidence.

Worked example 2

Real-life problem. A taxi-fare model is y = 4.50 + 2.20x, where x is miles. Interpret the intercept.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Set the input to zero and explain the modeled output in context.

Step 3 - Solve step by step.

1. At x = 0, y = 4.50 + 2.20(0).
2. Therefore, y = 4.50.
3. The model predicts a $4.50 fare before any miles are traveled.
4. This likely represents the base or starting charge.

Step 4 - Check and interpret. The interpretation is meaningful because zero miles is plausible for the fare model.

Why this matters. Intercepts often represent fixed costs in transportation and service pricing.

Common trap. Reporting a bare slope or residual without units, reversing actual minus predicted, or extrapolating beyond the model's evidence.

Worked example 3

Real-life problem. Use y = 62 + 3.5x to predict a trainee's score after 6 hours of practice.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Substitute x = 6 and evaluate in order.

Step 3 - Solve step by step.

1. Start with y = 62 + 3.5x.
2. Substitute 6: y = 62 + 3.5(6).
3. Compute 3.5 x 6 = 21.
4. Predicted score: y = 62 + 21 = 83 points.

Step 4 - Check and interpret. The result is a prediction from the model, not a guarantee for an individual trainee.

Why this matters. Training coordinators use reasonable within-range predictions to plan support.

Common trap. Reporting a bare slope or residual without units, reversing actual minus predicted, or extrapolating beyond the model's evidence.

Worked example 4

Real-life problem. A model predicts 48 packages per hour, but a worker actually processes 53. Find and interpret the residual.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Use residual = actual - predicted and interpret its sign.

Step 3 - Solve step by step.

1. Actual output is 53 packages per hour.
2. Predicted output is 48 packages per hour.
3. Residual = 53 - 48 = +5 packages per hour.
4. The worker processed 5 more packages per hour than the model predicted.

Step 4 - Check and interpret. A positive residual means the point lies above the model line.

Why this matters. Residuals help supervisors see where a general model underpredicts or overpredicts performance.

Common trap. Reporting a bare slope or residual without units, reversing actual minus predicted, or extrapolating beyond the model's evidence.

Worked example 5

Real-life problem. A home-price model built for houses from 900 to 2,400 square feet is used to price a 6,000-square-foot mansion. Evaluate the prediction.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Compare the requested input with the model's observed input range.

Step 3 - Solve step by step.

1. The model's data cover 900 through 2,400 square feet.
2. The mansion's 6,000 square feet lies far outside that range.
3. Using the line there is distant extrapolation.
4. The prediction may be unreliable because luxury features and pricing patterns can differ.

Step 4 - Check and interpret. A calculation can be algebraically correct while its modeling assumption is weak.

Why this matters. Real-estate valuations should use comparable properties rather than extending a local model far beyond its evidence.

Common trap. Reporting a bare slope or residual without units, reversing actual minus predicted, or extrapolating beyond the model's evidence.

Worked example 6

Real-life problem. A scatterplot's residuals are positive at small x, negative in the middle, and positive at large x. Is a straight line suitable?

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Look for random scatter versus a systematic residual pattern.

Step 3 - Solve step by step.

1. The residual signs change in an organized positive-negative-positive pattern.
2. That pattern shows the line misses the data in a curved way.
3. Residuals are not randomly scattered around zero.
4. A nonlinear model may fit better than a straight line.

Step 4 - Check and interpret. A good linear model should not leave a clear shape in the residual plot.

Why this matters. Residual patterns prevent analysts from trusting a simple model that systematically misses real behavior.

Common trap. Reporting a bare slope or residual without units, reversing actual minus predicted, or extrapolating beyond the model's evidence.

Worked example 7

Real-life problem. A line has slope -1.8 when x is outdoor temperature in degrees and y is daily heating cost in dollars. Interpret it.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Use the negative sign and attach the correct input and output units.

Step 3 - Solve step by step.

1. The slope is -1.8 dollars per degree.
2. A one-degree increase in outdoor temperature corresponds to a predicted change of -$1.80 in cost.
3. Thus, predicted daily heating cost decreases by about $1.80 per additional degree.
4. The statement describes the model's association over its relevant range.

Step 4 - Check and interpret. The negative wording agrees with the downward direction of the modeled relationship.

Why this matters. Utilities use temperature-cost models to forecast short-term heating demand.

Common trap. Reporting a bare slope or residual without units, reversing actual minus predicted, or extrapolating beyond the model's evidence.

Worked example 8

Real-life problem. Model A has residuals mostly between -2 and 2; Model B has residuals mostly between -9 and 9 for the same outcome. Which fits more closely?

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Compare typical vertical prediction errors on the same scale.

Step 3 - Solve step by step.

1. Residual magnitude measures distance between actual and predicted values.
2. Model A's errors are generally no more than about 2 units.
3. Model B's errors reach about 9 units.
4. Model A fits the observed data more closely, assuming both are evaluated on the same cases and units.

Step 4 - Check and interpret. Small residuals indicate close predictions but do not automatically prove the model will generalize to new data.

Why this matters. Choosing the better-fitting model can improve forecasts while still requiring validation.

Common trap. Reporting a bare slope or residual without units, reversing actual minus predicted, or extrapolating beyond the model's evidence.

Worked example 9

Real-life problem. A line of best fit predicts 71 at x = 4 and 86 at x = 9. Find the slope between these model points.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Use change in predicted y divided by change in x.

Step 3 - Solve step by step.

1. Change in predicted output: 86 - 71 = 15.
2. Change in input: 9 - 4 = 5.
3. Slope = 15 / 5 = 3.
4. The model output rises by 3 units for each one-unit increase in x.

Step 4 - Check and interpret. Starting at 71 and increasing 3 units for 5 input steps reaches 86.

Why this matters. Analysts can recover a model's rate of change from any two points on its line.

Common trap. Reporting a bare slope or residual without units, reversing actual minus predicted, or extrapolating beyond the model's evidence.

Worked example 10

Real-life problem. Ice-cream sales and sunburn cases have a strong positive association. Explain why the scatterplot does not prove ice cream causes sunburn.

Step 1 - Understand. Identify what is being asked, the relevant group or event, the supplied values, and the units. Do not calculate until these match the wording of the problem.

Step 2 - Plan. Identify a plausible lurking variable and distinguish association from causation.

Step 3 - Solve step by step.

1. The plot shows both variables tend to increase together.
2. Hot, sunny weather can increase both ice-cream purchases and sun exposure.
3. Weather is a plausible confounding or lurking variable.
4. Therefore, correlation alone does not establish that ice cream causes sunburn.

Step 4 - Check and interpret. A causal claim needs a design and evidence that address alternative explanations.

Why this matters. Public conclusions should not turn a predictive pattern into an unsupported cause-and-effect story.

Common trap. Reporting a bare slope or residual without units, reversing actual minus predicted, or extrapolating beyond the model's evidence.

Session synthesis

The central idea of this session is: A linear model is useful when its slope, intercept, predictions, and residuals have sensible contextual meaning within the range supported by the data.

Before moving on, explain one example aloud without looking at its solution. Name the target, justify the method, reproduce the calculation, check the result, and finish with a sentence in context. If any of those five parts is missing, revisit the matching step rather than memorizing only the final number.

Final accuracy checklist

• Did I answer the exact question that was asked?
• Did I use the group, total, interval, or event that belongs in the denominator?
• Did I preserve units and label percentages, percentage points, or rates correctly?
• Is the answer reasonable for the scale and context?
• Does my conclusion match the strength of the data and study design?