Math 141: Readings for December 5, 2025

Reading

Before coming to lecture today, please read the following material:

• no new reading today.

Homework Assignment

After you’ve completed the reading, but before coming to class, please complete the following homework assignment. Bring your responses to this assignment (handwritten or typed) to class on a piece of paper:

Consider a multiple linear regression model for a sample of 200 trees on Reed’s campus:

\[ y = \beta_o + \beta_1 x_1 + \beta_2 x_2 + \epsilon \]

Here, \(y\) represents the height of a tree, \(x_1\) represents the diameter of the tree, and \(x_2\) takes the value 1 if the tree is evergreen and 0 otherwise. Suppose we fit the model and get the following:

\[ \hat{y} = 10 + 40x_1 - 3x_2. \]

1. Precisely interpret each of the coefficients in the context of the data (for this question, don’t worry about units).

2. Suppose we would like to test if \(\beta_2 = 0\).

   • Does the magnitude of its coefficient help us determine whether or not it is significant? Why or why not?

   • Suppose, after checking and satisfying the necessary assumptions, we find that the observed test statistic, \(t^*\), is -4. Without performing any calculations, is this significant at the \(\alpha = 0.05\) level? Why or why not? Feel free to draw a picture.

   • Using R, calculate the p-value for the test.