check the assumption of normality I have a data and they are definitely not normally distributed. What I want is to test whether the predictor variable is significant or not. So I was wondering is it necessary to check the assumption of normality? Or I should transform the outcome variable and fit the model then check the normality? If the frequently used transformation couldn't meet the  assumption of normality. What should I do?
If I could ignore the assumption,may I still use the coefficient in the output to interpret the model?
one of the qqplot

 A: 
I have data and they are definitely not normally distributed. 

Neither the response nor the predictors in regression need to be normal. The normality of the conditional response (equivalently, the error term) is an assumption for particular forms of inference, but it's not an especially crucial assumption for most of them, most of the time (the impact of violating it may be small). Another alternative is to use different inferential procedures.
Can you say a little more about this distribution? Which variables are you saying are definitely not normal and how do you know?  

may I still use the coefficient in the output to interpret the model?

Certainly you can interpret the coefficient even if you don't have conditional normality. If the relationship is linear and the various sources of bias in the estimate are not present, the estimate of the coefficient should still be consistent, even in the presence of heteroskedasticity. Mere non-normality may be almost a non-issue. (If you care about efficiency it may matter more.)

What I want is to test whether the predictor variable is significant or not. 

The usual test does assume normality of the conditional distribution of the response as mentioned. However, in sufficiently large* samples you can use an asymptotic z-test.
* there's no way to say how big that might be since it depends on a number of things you haven't mentioned.
You might also do something else - there are alternative parametric procedures appropriate for other distributional models as well as several kinds of robust and nonparametric procedures (including permutation tests and bootstrapping) which don't rely on the particular distributional form (or in the robust case, only relatively weakly).

So I was wondering is it necessary to check the assumption of normality? 

Well, it's the conditional distribution that matters (when it matters). You can assess it from residuals (but often it's easy to tell it won't be satisfied without anything more than knowledge of the variables).

Or I should transform the outcome variable and fit the model then check the normality? 

That will change the linearity that you were prepared to assume was there. If that was a good assumption the transformation may be counterproductive, since you'll possibly improve something that you can get around, at the expense of a much more critical assumption.

If the frequently used transformation couldn't meet the assumption of normality. 

I don't know what you mean by "the frequently used transformation"; are you referring to a particular transformation that's frequently used or are you saying that the strategy of transforming the response is frequently used?
