Overview

• This activity will focus on the interpretation of parameters in the context of the problems and on the reasonableness (or unreasonableness) associated with instances of extrapolation.
• Students will use data analysis tools to generate equations for lines of best fit and identify the estimated slope and y-intercept.
• This activity connects back to the correlation coefficient and the method of Least Squares Regression and prepares students for using the least squares regression line to make predictions, residual analysis, and multiple regression.
• [a list of tags like S2, O1, B1, V3] ← Link to EBTP descriptions 

Prerequisite assumptions

Students should be able to do each of the following after completing the What to Know assignment.

• Identify the estimated y-intercept and estimated slope from the equation of the line.
• Interpret the estimated slope in context.

Intended goals for this activity

After completing this activity, students should understand the definition of extrapolation and that the equation of the line of best fit is based on sample data and will change with new datasets.

They should be able to identify the estimated slope and estimated y-intercept given the equation of the line of best fit and interpret the estimated y-intercept in the specific context of a problem.

Synchronous Delivery and Activity Flow

The sample activity delivery below assumes a face-to-face class meeting but can be adapted to a fully online or hybrid delivery by using break-out rooms for pairs and small groups.

Frame the activity (3 minutes)

• Question 1 — Think-Pair-Share S2, C4, V1, V4, O3
  • Use Part A to start a conversation about the importance of looking for trends in quantitative data. If students do not see a connection between the cricket example and the importance of identifying trends, share other examples of situations in which we can identify trends that help us make decisions. For example, a plot of carbon emissions and changes in temperature show a positive trend.
  • Students should identify the explanatory and response variables in Part B on their own. It is important that the students defend their answers. One interpretation is that the researcher gets to choose. Another answer is that it seems unnatural to have the chirps determine the temperature. This could be a good time to plant a seed for “association does not mean causation.”
  • Transition to the in-class activity by briefly discussing the Objectives for the activity.

Activity Flow (20 minutes)

• Question 2 — Working in Groups
  • In Parts A through C, guide students to support their claims using outputs from the Linear Regression tool.
  • Emphasize that every new dataset will generate a different linear equation and that each equation offers different estimates of the parameters, or true population values, of the slope and y-intercept.
  • In Part D, change in estimated temperature for every increase in one chirp per second is estimated “on average,” given all values of chirps per second. This phrase is purposely included. Our least squares regression line estimates the average temperature values for each value of chirps per second. Therefore, the slope of the line tells us how the average temperature changes for a change in the number of chirps.
  • Generally speaking, the least squares regression line connects the points that represent the mean value of temperature for each value of chirps per second, assuming the trend is, in fact, linear. Use Part E to emphasize that each time we repeat the experiment, the observed values of temperature and chirps per second will naturally vary.
  • Part F requires students to extrapolate. Ask them to locate the y-intercept on the graph. Students will notice that this value is not anywhere close to the value on the graph and is consequently outside the range of the [latex]x[/latex] values that were used to create the linear
    model.
• Debriefing Questions 2 and 3 — Whole Class Discussion S4, C3, V1, O1, B2, B4
  • Ensure students understand the dangers involved in extrapolation. Possible talking points:
    •  Extrapolation is the prediction of a response value using an explanatory variable value that is outside the range of the original data. During the discussion, display this definition
      or write it on the board.
    • We should always be cautious regarding the validity of this interpretation—if the range of the explanatory variable, [latex]x[/latex], does not include or come close to 0, then we may get a completely unreasonable interpretation of the y-intercept.
• Question 4 — Students continue in small groups to practice writing interpretations and answering question using statistical terminology. It could be completed for homework if time runs short.

Wrap-up/transition (2 minutes)

• If you did not complete Question 4, the discussion after Questions 2 and 3 will serve as the wrap-up. Otherwise, students could share answers to Question 4.
• Have students refer back to the Objectives and check the ones they recognize.
• Assign the homework or Practice and any What to Know pages for the Forming Connections activities you plan to complete in the next class meeting. C2