Question

Let’s consider a study that followed a randomly selected group of 100 State U students during a two-year period at the school. The study found that a linear relationship exists between the number of hours students spend engaging in social media each week and their cumulative gpa during the two-year period. The model for this relationship can be given by the equation

g?pa = −0.032 × (hours) + 2.944 (a) Interpret the slope of the line in the context of the data.

(b) The residual gpa for a particular student who spent 20 hours per week using social media was found to be 0.476. What was this student’s cumulative gpa during the two-year period?

(c) Would the correlation coefficient for the linear relationship be positive or negative? Explain.

(d) If another study found that the linear correlation coefficient between a student’s gpa and the number of hours spent at the library was r = 0.46, could you conclude that this relationship is stronger than the one between gpa and hours spent on social media? Explain.

Answer 1

Given, Cumulative GPA, = -0.032*(hours per week spent on social media, x) + 2.944

a) Slope of the line = coefficient of x = -0.032

We can observe that the slope is negative

==> As the hours spent on social media increases the predicted cumulative GPA decreases

b) Residual = Observed value - Predicted value

when x = 20 is = -0.032*(20) + 2.944 = 2.304

Given residual = 0.476

==> Observed value = 0.476 + 2.304 = 2.78

Therefore the student’s cumulative GPA during the two-year period is 2.78

c) The correlation coefficient, where b is the slope of the regression curve

We can see that r will have same sign as b

==> The correlation coefficient for the linear relation would be negative

d) Given that the linear correlation coefficient between a student’s GPA and the number of hours spent at the library as r = 0.46

But in order to compare it with the correlation coefficient for linear relation between student’s GPA and the number of hours spent on social media, we need to calculate the latter value

But there is no enough data to calculate the latter correlation coefficient except the slope of the regression curve

Therefore we cannot conclude that the relationship between GPA and hours spent at the library is stronger than the one between GPA and hours spent on social media