a The slope tells us how the dependent variable changes when you move some amount of steps in the positive horizontal direction.
B
b We have been given the r-value.
C
c The residual with the greatest magnitude is the residual that is farthest away from the x-axis.
D
d Substitute x=10 into the model.
E
e Are the residuals evenly scattered around the x-axis?
A
a See solution.
B
b See solution.
C
cGreatest residual: - 6.2
Point: (3,5.96)
D
d 8.34 pounds.
E
e Yes, it is. See solution.
Practice makes perfect
a Let's examine the least squares regression line that Delenn found. We can see that the slope is - 0.55.
y=13.84 - 0.55x
The scatterplot shows the weight of backpacks given the number of quarters of school completed. Therefore, the slope tells us that with each quarter of high school completed, the backpack weight decreases on average by 0.55 pounds.
b The R-squared value tells us how much of the variability in the dependent variable is explained by the independent variable. We know that r= - 0.66, so to obtain R-squared we have to square this value.
R^2= ( - 0.66)^2 ⇔ R^2 ≈ 0.44
An R-squared of 0.44 means that 44 % of the variability in the backpack weight is explained by the number of quarters that has been completed. This means the linear relationship is not that strong.
c Examining the residual plot, we notice that the largest residual is - 6.2 when the number of quarters completed is 3.
There is only one observation that this could be, which is (3,5.96).
d To estimate the weight of a backpack for a student who has completed 10 quarters, we substitute x=10 into the line of best fit and evaluate.
The predicted weight of the backpack is 8.34 pounds.
e If a linear model is the best choice, there should be a random scatter of the residuals about the x-axis. Examining the residual plot, we see that this is the case. Therefore, we can say that a linear model is the best choice. With that said, the model isn't particularly accurate, as the residuals are quite large.
