Given a fitted regression model, how to calculate the p-value of a new observation? (apologies in advance if I abuse any terminology)
I have a series of XY data points that I fit with a simple regression and I would like to be able to use this regression to obtain a p-value for the new data point. I have found a couple of sources describing how to calculate a Prediction Interval for Responses to a Particular x which seems to almost answer this question. In fact, I think I can reformulate my question as how do I find the alpha of the smallest prediction interval containing my new data point.
 A: I'm adding this as an answer as the comments are getting long, so that someone else can hopefully provide a better answer.
Basically, it would be trivial to estimate the minimum critical t-value etc off the data you have if you could base the bulk of the calculation on the information from your original regression. For example, E would be estimated as the difference between the predicted value and the actual value of your new data point, then it is just a matter of solving for the critical t-value in the "margin of error" equation you have in your link, then looking at the alpha level for that t-value.
The issue is that the standard error, etc, used in that equation would not take account of your new data points, because it would only be based on the original regression data. You could redo the regression for every new set of data points you receive, which would update the SS etc, then work backwards using the updated information for the equations. This won't be too much of an issue if the new data comes in as a set.
