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I have a simple question about regression.

The situation:
I have three variables $X_1,X_2,X_3$ and a variable I try to estimate $Y$. (In practice I have a list of participants for which the values of all of these variables are known)

Using regression I obtain $$Y= a_0 + a_1X_1 +a_2 X_2 +a_3 X_3 +a_{1,2}X_1\cdot X_2 + a_{1,3} X_1\cdot X_3 + a_{2,3} X_2\cdot X_3 + a_{1,2,3}X_1\cdot X_2\cdot X_3$$

and I get $p$-values for each of these coefficients.

I made these changes:
I decreased all the values of $Y$ by $10$ (because I wanted the values to start at zero). Then I replaced $X_1$ with $X_1-1$ ($X_1$ was the gender, and instead of $1,2$ I wanted it to be $0,1$). Finally I divided $X_2,X_3$ by $40$ to get values between $0$ to $1$.

All these changes also changed the coefficieints $a_0,a_1,...$ (I understand why this is). Positive coefficients stay positive and negative stay negative. But the $p$-value changed and I do not understand why.

Does the $p$-value suppose to change by making these changes? What does it mean?

Basically I lack the understanding myself and I need your help to decide whether I can use the new data (for which the $p$ value is better) to conclude what I want.

Thank you :)

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    $\begingroup$ Ask yourself what the p-values are testing. For instance, in the original model the p-value for $a_1$ was testing $H_0: a_1=0.$ In the second model what is the null hypothesis in terms of the new coefficient? What is it in terms of the original coefficient? $\endgroup$
    – whuber
    Nov 7, 2020 at 19:35

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