Unit 4: Exploring Two-Variable Quantitative Data

Statistical tools allow us to represent and describe patterns in data and to classify departures from patterns. Simulation and probabilistic reasoning allow us to anticipate patterns in data and to determine the likelihood of errors in inference.

Data-based regression models describe relationships between variables and are a tool for making predictions for values of a response variable. Collecting data using random sampling or randomized experimental design means that findings may be generalized to the part of the population from which the selection was made. Statistical inference allows us to make data-based decisions.

Apply concepts and skills involving scatterplots and correlation.

Compute and interpret Least Squares Regression equations that model bivariate data.

Conduct quadratic regression analysis

Students will demonstrate their mastery by performing a complete analysis of a bivariate dataset to determine if a change in one variable can help predict a change in another. They will show they can identify the explanatory and response variables correctly in a given context and justify their choice based on the goals of the study.

A primary way students show understanding is through the construction and interpretation of a Least-Squares Regression Line. They will provide written interpretations of the slope and y-intercept in the specific context of the problem, explaining that for every one-unit increase in the explanatory variable, the response variable is predicted to change by a specific amount. They will also calculate the residual for a specific data point and explain whether the regression model overpredicted or underpredicted the actual value.

Students will further demonstrate their skills by evaluating the "fit" of their linear model. They will create and analyze a residual plot, showing they understand that a random scatter of points in this plot confirms a linear model is appropriate, whereas a distinct curved pattern suggests the relationship is not linear. They will also interpret the correlation coefficient (r) to describe the strength and direction of the relationship and the coefficient of determination ($r^2$) to explain what percentage of the variation in the response variable is accounted for by the linear model.

Finally, students will prove they can identify and describe the impact of outliers and influential points on the regression line. They will explain why a point far from the others might drastically change the slope or the correlation. In their final analysis, they will demonstrate a critical understanding of the "Correlation does not imply Causation" rule, explaining that even a perfect linear relationship does not prove that one variable causes the other due to the potential influence of lurking variables.