Can a linear regression be significant if the data is not linear? I performed a linear regression which came out with a significant result however when I checked the scatter-plot for linearity I was not confident that the data was linear.
Are there any other ways to test for linearity without inspecting the scatterplot?
Could the linear regression be significant if it wasn't linear?
[Edited to include scatterplots]




 A: Yes, Aksakal is right and a linear regression can be significant if the true relationship is non-linear. A linear regression finds a line of best fit through your data and simply tests, whether the slope is significantly different from 0. 
Before trying to find a statistical test for non-linearity, I would suggest  reflecting on what you want to model first. Are you expecting a linear (non-linear) relationship between your two variables? What exactly are you trying to uncover? If it makes sense to assume that there is a non-linear relationship as for example between car speed and braking distance, then you can add squared terms (or other transformations) of your independent variable.
Also, a visual inspection of your data (scatterplot) is a very powerful method and an essential first step in your analysis.  
A: Monotonic nonlinear relationships will almost always show up significant when modeling as linear models. If the relationship is nonlinear and not monotonic then it depends on the sample.
Examples of monotonic relationships is logarithm $y=\ln x$ and odd powers such as $y=x^3$. Example of non monotonic relationships are even powers $y=x^2$ and trigonomtric functions such as $y=\sin x$.
For instance, if your sample is for $x\in[-1,1]$, then $y=\sin x$ modeled as $y\sim x$ will likely be significant, see the plot:

However, if your sample is in $x\in[0,\pi]$, then linear modeling will not work at all:

A: I agree with everything Aksakal says.  But as to the first question I think the answer is correlation.  Correlation measures the extent to which there is a linear relationship between the data sets x and y.  
