Question

In the least-squares line
ŷ = 5 − 8x,
what is the value of the slope?


When x changes by 1 unit, by how much does ŷ change?
a.)When x decreases by 1 unit, ŷ decreases by 8 units.
b.)When x increases by 1 unit, ŷ decreases by −8 units.
c.)When x increases by 1 unit, ŷ decreases by 8 units.
d.)When x increases by 1 unit, ŷ increases by 8 units.

In the least-squares line ŷ = 5 + 6x, what is the marginal change in ŷ for each unit change in x?

Answer 1

Given $ŷ = 5 − 8x,$ the slope is given by $\frac{dŷ}{dx}=\frac{d(5-8x)}{dx}=\frac{d(5)}{dx}−8⋅\frac{dx}{dx}=0−8=−8$. When x increases by 1 unit, ŷ decreases by 8 units. Equivalently, when x decreases by 1 unit, ŷ decreases by 8 units.

Given $ŷ = 5 + 6x$, marginal change in ŷ for each unit change in x is given by $\frac{dŷ}{dx}=\frac{d(6x+6)}{dx}=6\frac{dx}{dx}+\frac{d(5)}{dx}=6+0=6$. Thus, a unit change in x leads to 6 units change in ŷ. That is a unit increase in x leads to 6 units increase in ŷ.