• 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, and residuals, so that they can use these concepts to reconsider the Key Question later in the lesson. Providing this framing early – that students will return to the Key Question later in the lesson, after learning some useful tools during the earlier parts of the lesson – can help maintain student engagement throughout the lesson.
• This lesson focuses on using and evaluating linear models. 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. If students ask about the mathematical mechanics of how linear regression models are fit to data, you can reassure them that the topic will be covered in the next lesson.
• In AP Statistics, if students are asked to analyze a residual plot, the plot is almost always provided for them. Therefore, students can focus on understanding how to interpret these graphs rather than learning to construct them.
• When interpreting regression models, encourage students to include context throughout their explanations. Predictions and residuals should all be interpreted using the variables being studied, rather than described only in mathematical terms.

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 academic readiness. In 2026, 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 predictions, 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 based on their family income.
• 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 data displayed at the very beginning of the lesson slide deck are originally from Chetty et al (2025) – a landmark study that provides a uniquely detailed window into the relationships between standardized test scores and student backgrounds.
• The attendance and test score data set provides a simpler context for introducing linear regression concepts. Because the relationship is strong and has relatively few data values, students can focus on interpreting models, making predictions, and understanding residuals. Then, students can apply this understanding to the more complex data set about income and test scores in the Discussion Question.
• Real student-level data test score data is privacy-protected. So, in the Discussion Question, simulated data is provided that closely matches the key summary statistics from the real UC Academic Council report. For the years the report analyzed the SAT, the maximum score on the test was 2400. So, the simulated data uses that scale.

• Students may wonder why statisticians typically write linear regression models in the form ŷ = a + bx rather than y = mx + b. This notation helps maintain consistency with more complex multiple regression models (beyond the scope of this course), which take the form of: ŷ = a + \( b_{1}x_{1} \) + \( b_{2}x_{2} \) + \( b_{3}x_{3} \) + …
• 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 helps reinforce the concept 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.