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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.

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  • $\begingroup$ 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 $x?$ $\endgroup$
    – xyz
    Commented Mar 15, 2023 at 0:06
  • $\begingroup$ What else would it mean to increase a quantity by one unit? $\endgroup$
    – Dave
    Commented Mar 15, 2023 at 0:09
  • $\begingroup$ @Dave That's exactly my question (sorry if it's dumb)! Is it possible that there's another meaning than simply increasing $+1?$ $\endgroup$
    – xyz
    Commented Mar 15, 2023 at 0:12

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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 $(c)$.

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