• After introducing the Key Question (“How should colleges evaluate test scores?”), the lesson shifts to a smaller and more accessible data set involving attendance and test scores. Students use this context to learn about linear regression models, predictions, residuals, and residual plots before returning to the Key Question during the discussion. Providing this framing early can help students see how the concepts connect to the larger question and maintain engagement throughout the lesson.
• This lesson builds directly on the previous lesson’s introduction to scatterplots, correlation, and association. The focus now shifts from describing relationships to using and evaluating linear models. Students learn how to interpret a model, make predictions, and assess whether a linear relationship provides a reasonable representation of the data. The mechanics of calculating the least-squares regression line are intentionally deferred to a later lesson so that the emphasis remains on interpretation and model evaluation.

First, download this lesson's Handout Key and read through its Discussion Question section. Then, check out our model discussion norms and the additional background notes below.

• In 2020, the University of California released a report examining the role of standardized tests in admissions. Since then, the university system has adopted a test-blind admissions policy. More recently, faculty have begun reexamining the role of standardized testing, citing concerns about college-level mathematics readiness. In June, the UC Academic Senate officially created a faculty work group to evaluate the return of standardized testing. This evolving discussion provides an authentic context for exploring how statistical models can be used to evaluate the college admissions process.
• The discussion question creates an opportunity to consider how residuals can be used to evaluate student performance relative to expectations rather than relying solely on raw scores. Because residuals measure the difference between an observed value and the value predicted by a model, they provide a way to compare outcomes after accounting for factors included in the model. In this context, residuals represent how much a student overperformed or underperformed relative to the score predicted by their family income. Encourage students to consider both the advantages and limitations of this approach.
• The lesson intentionally avoids presenting a single “correct” answer to the fairness question. Instead, the goal is to evaluate the strengths and weaknesses of a proposed model and consider how different assumptions can influence conclusions.

• The attendance and test score data set provides a simple context for introducing linear regression concepts. Because the relationship is strong and approximately linear, students can focus on interpreting models, making predictions, and understanding residuals without becoming distracted by more complex patterns in the data.
• The University of California admissions context demonstrates how statistical models are used to make decisions in real-world settings. The lesson provides an opportunity to discuss how models can be informative and useful while still having limitations that should be considered carefully.

• Students may wonder why statistics typically writes linear regression models in the form ŷ = a + bx rather than y = mx + b. This notation helps maintain consistency with more complex models, which begin with an intercept term and then add explanatory components.
• It is critical to distinguish between y and ŷ and to use the notation accurately. For a given x-value, y represents the observed value while ŷ represents the value predicted by the model. This distinction becomes especially important when interpreting residuals.
• When working with statistical models, it is important to recognize that models are not perfect representations of reality. Instead, they provide useful approximations based on available information. The aphorism “All models are wrong, but some are useful” provides a helpful summary of this idea.
• Interpolation and extrapolation are not equally reliable. Predictions made within the range of observed x-values rely on relationships supported by the data, while extrapolated predictions assume that the observed trend continues beyond the available data. The obesity example in the lesson provides a useful illustration of why such assumptions can be problematic.
• Residual plots are often more informative than the original scatterplot when evaluating whether a linear model is appropriate. Systematic patterns in the residual plot indicate that important structure remains unexplained by the model, while residuals that appear randomly scattered around zero provide evidence that a linear model is reasonable.