Binary variable in regression model (interview question) I got this question, it may be easy but for some reason for me is not:
you are given a binary variable $b\in [0,1]$, that has no predictive power on $y$, but has some on $X$. How would you use it to improve the performance of a regression model $y=\beta X+\epsilon$?"
Any thoughts ? Thank you.
 A: Frame challenge: this is a question to test your ability to deal with a question from a client that uses poor phrasing.
Probably most people in technical consulting roles find themselves dealing with this. My approach has been to keep asking follow-up questions, either to reveal misunderstanding from that client that lead to me getting an opportunity to teach, or getting the client to rephrase the question in a way that makes sense. I'll give an example of how I could see such a conversation going.

Client: "We have this other variable, $b$, that has no ability to predict $y$ but does have some ability to predict $X$. How can we use b to make a better prediction of y than we get from $y = X\beta+\epsilon?$


Consultant: What do you mean that $b$ has no ability to predict $y$ but does have ability to predict $X$?


Client: When we run the regression on $b$, so $y = \beta b + \epsilon$, we get an insignificant $F$-test with $p = 0.06$, but we get $p=0.04$ when we run $X = \beta b +\epsilon$.

At that point, I might see this as an ANCOVA-style problem, where the effect of $X$ on $y$ might wash out the effect of $b$ on $y$ (which is exactly why ANCOVA makes use of the covariate), even if $b$ has an effect on $y$.
