I want to make a quantitative statement like "There is a 90% chance that this $X$-$Y$-data follows a linear model (with some noise added on top)". I can't find this kind of statement discussed in standard statistics texbooks, such as James et al.'s "An Introduction to Statistical Learning" (asking as a physicist with rudimentary statistics knowledge).
To be more precise: I'm assuming that some data is generated from $Y = f(X) + \epsilon$, where $f(X)$ is some exact relationship, e.g. the linear model $f(X) = \beta_0 + \beta_1 X$, and $\epsilon$ is noise drawn from a normal distribution with some unknown standard deviation $\sigma$. I want to calculate the probability that some proposed $\hat f(X)$ matches the actual $f(X)$.
I can do a least-squares fit to determine the estimate $\hat f(X)$. Now, if the model is correct ($\hat f(X) = f(X)$), then the residuals of the fit should exactly correspond to $\epsilon$. At the very least, if the data fits the model, there should be no correlation between the residuals and $X$. To be more quantitative, though, I would want to check that the residuals are in fact from a normal distribution with unkown $\sigma$ (although the residual standard error, RSE, will be an estimate for $\sigma$, so I could also assume that $\sigma$ is actually known). Isn't there some way to calculate a p-value for whether some given values (the residuals) are from a given distribution (normal distribution with RSE as the standard deviation)?
I'm not looking for the $R^2$ statistic, which will tell me how linear the data is, but also take into account the noise (larger $\sigma$ will lower the $R^2$ value). In my case, I don't care how noisy the data is, as long as it's normally distributed around the fit $\hat f(X)$.