Question

a statistics student wants to determine if there is a relationship between a students number of absences, x, and their grade point average (gpa), y. the given data lists the number of absences and gpas for 15 randomly selected students. using technology, what is the equation for the least - squares regression line? \\( \hat { y } = 3.79 - 0.10 x \\) \\( \hat { y } = 0.10 + 3.79 x \\) \\( \hat { y } = 16.15 - 3.28 x \\) \\( \hat { y } = - 3.28 + 16.15 x \\)

Explanation:

Step1: Recall the form of the least - squares regression line

The equation of the least - squares regression line is \(\hat{y}=b_0 + b_1x\), where \(b_0\) is the y - intercept and \(b_1\) is the slope.

Step2: Analyze the relationship between number of absences (\(x\)) and GPA (\(y\))

As the number of absences (\(x\)) increases, we expect the GPA (\(y\)) to decrease. So the slope \(b_1\) should be negative.

Step3: Check the options

• For the option \(\hat{y}=0.10 + 3.79x\), the slope \(b_1 = 3.79>0\). This implies that as \(x\) (number of absences) increases, \(y\) (GPA) increases, which is not logical.
• For the option \(\hat{y}=-3.28 + 16.15x\), the slope \(b_1=16.15>0\). This also implies a positive relationship between \(x\) and \(y\), which is incorrect.
• For the option \(\hat{y}=16.15-3.28x\), when \(x = 0\) (no absences), \(\hat{y}=16.15\). A GPA of \(16.15\) is not realistic (since GPA is usually on a 4 - point or 5 - point scale).
• For the option \(\hat{y}=3.79-0.10x\), the slope \(b_1=- 0.10<0\) (negative relationship as expected) and the y - intercept \(b_0 = 3.79\) is a reasonable value for a GPA when \(x = 0\) (no absences)

Answer:

\(\hat{y}=3.79 - 0.10x\)