Interpreting the sign of returned coefficients in linear model? I'm having a question regarding, how should the sign of returned coefficients (e.g. by R's lm()) in linear model be interpreted.
Particularly I was doing a model on some kids' test scores and there's a categorical variable motiv $\in \{0,1\}$ where $1$ signifies whether the kid was motivated to study and $0$ signifies whether the kid was not motivated.
I'm using the following model to lm():
fit2 <- lm(dta$X.U.FEFF..mpist. ~ dta$matem + dta$aidink + dta$motiv)

where X.U.FEFF..mpist. is the test score, matem is the kid's mathematics score, aidink is the kid's score in native language and then there's the motiv variable.
lm() returns the following model:
$$mpist =  266.296  + 23.226 matem + 13.725 aidink + (-5.416) motiv$$
one notices that the coefficient for motiv is negative. But this is illogical considering that $motiv == 0$ corresponds to no motivation and $motiv == 1$ corresponds to motivation. So by this, if the kid has been motivated, this would result in a slightly lower test score, but if the kid has not been motivated, this would result in a slightly higher test score.
So how should one interpret a situation like this? Is the sign of the coefficient generated for motiv reasonable?
 A: Do not forget that the model shows the effect of each variable given the others in the model. It is perfectly possible that motivation does affect test score but you cannot see that because you have included two other variables, maths score and language score, which may be affected by motivation so the effect of motivation on test after allowing for maths score and language score may be small. To diagnose further why not see what the mean test score is for each level of motivation?
A: The estimated coefficient is reasonable in sense that it is a value optimal from the model's perspective given your data. You mentioned in comment that it is not statistically significant, what means that it has large errors. The value of this parameter does not seem to have much practical significance since it decreases the initial score of 266.296 (intercept) by just five points -- I guess that this is a negligible effect on the final score?
As about the negative sign and the fact that it is not significant, my first guess would be that simply your motiv variable is a poor one. Motivation is not something of a binary nature, it's simply not that you are motivated, or you aren't, there is a huge gray area in between. How was it measured? Was it a questionary item that kids were asked? Maybe it was a question to their teachers, or parents? If so, maybe it is just teachers who can't judge if kids are motivated, or not...
