# Simple question about linear regression

I am dealing with the following exercise:

Exercise. In a linear regression model of the form $$y = w_1x + w_0$$, if we increase the value of $$x$$ by one unit, what can we expect from the value of of $$y$$?

$$(a)$$ An increase of value equal to $$1$$;

$$(b)$$ An increase of value equal to $$w_0$$;

$$(c)$$ An increase of value equal to $$w_1;$$

$$(d)$$ It is impossible to tell anything about the increase on $$y$$.

My attempt. As an aspirant to be a mathematician, this question leads to one simple line of thinking: if $$x$$ is increased by one unit, one might think of this as a simple variable change of the form $$x \to x+1.$$ If this is case, we obviously expect $$y$$ to suffer an increase of value equal to the parameter $$w_1,$$ hence I would select option $$(c).$$

My concerns. After some further thought, what exactly is "one unit" of the variable $$x$$? As far as I am concerned, this might not be a concept as objeticve as I thought. This makes me think that probably option $$(d)$$ makes some sense aswell.

All the information I have about the question is posted in the Exercise. section. Thanks for any help in advance.

• Thank you @Dave . As I tried to explain in the My attempt. section, if we change the $x$-value to $x+1$, our $y$ becomes $$y = w_1(x+1) + w_0 = w_1x + w_0 + w_1.$$ In others words, this means that there is an increase of $y$ in the value equal to $w_1.$ I was really just wondering, increasing one unit in $x$ means simply having $x+1$ instead of $x?$
– xyz
Mar 15 at 0:06
• What else would it mean to increase a quantity by one unit?
– Dave
Mar 15 at 0:09
• @Dave That's exactly my question (sorry if it's dumb)! Is it possible that there's another meaning than simply increasing $+1?$
– xyz
Mar 15 at 0:12

The answer to the question is certainly $$(c)$$. With a one unit increase in $$x$$, you simply multiply this value by the slope value, and this will yield an increase in $$y$$. So if the equation is:

$$y = .54x + 45$$

This would simplify to an increase of value equal to the slope (the $$w_1$$ in your question):

$$y = .54 + 45$$

Resulting in this $$y$$, a simple $$.54$$ addition to the intercept:

$$y = 45.54$$

To your question about what a one-unit increase in $$x$$ actually means, I suppose that depends on the measurement of the $$x$$ (centimeters, pounds, etc.). However, that doesn't seem to matter here. The units counted are not the same as the type of unit involved. Therefore the answer would still be $$(c)$$.