Hypothesis not rejected, what does that mean for expected value?

Question

So I have a linear regression analysis where the confidence interval includes 0, therefore the null hypothesis that the intercept is 0 cannot be rejected. Does this mean that the intercept value should not be included in the y = b + mx + error equation when predicting? The x is statistically significant so the slope is not 0. I just don't know what the intercept not being statistically significant means.

Answer 1

I just don't know what the intercept not being statistically significant means

"Statistically significant" is a term relating to the strength of evidence against some particular hypothesis (which, in your case, is the hypothesis that $b = 0$.)

"Not statistically significant" is simply the bare-bones negation of "statistically significant". I.e., it means that the evidence was not "strong enough"... which, frankly, doesn't say much. Notably, the negation of "statistically significant" (i.e. "not statistically significant") is not the same thing as the negation of the hypothesis being "statistically significant". In other words, you do not have evidence of the negation of your hypothesis, you simply don't have evidence for the hypothesis.

However, when building a model, it is typically better to have a simple model rather than a complex one, because model parameters that aren't helpful are harmful. Thus, in order to include parameters in the model, one wants evidence that they are helpful. This suggests a general rule-of-thumb that if the hypothesis that a parameter is non-zero lacks evidence (in the form of a low p-value for the appropriate statistical test), then it may be best to simply leave it out of the model.

So, the way that I would interpret this thing is that "We failed to find evidence that including an intercept term in the model sufficiently improved model performance. Therefore, we opted to use a simpler model that did not include an intercept term"

Note that there are other metrics by which one can measure "evidence that including [some parameter] in the model sufficiently improved model performance", so the above interpretation is only a rule-of-thumb, and should be considered in the real-world context of your problem.