# Estimating intercept from interval for slope and prediction interval

Suppose we have collected some data and formed a simple linear regression model for $$Y$$ versus $$X$$ . Suppose further that $$[a,b]$$ is an $$\alpha$$ % confidence interval for the true slope $$\beta_{1}$$, and $$[c,d]$$ is a $$\beta$$ % prediction interval for $$Y(x_{0})$$ say. Is it possible to get a decent estimate for the true intercept $$\beta_{0}$$ of the model from this information alone? Intuitively, it seems like $$\frac{c+d}{2} - x_{0}\frac{a+b}{2}$$ should be a decent estimate for the intercept but I'm not quite sure how to make this rigorous. I'd appreciate if someone could give some indication if this is true and why it should be true.

• I wonder what "decent" is intended to mean. It strikes me that an estimate could be both accurate and precise, or (alternatively) wildly imprecise, depending on the covariance of the estimated slope and intercept (which are implicitly involved in computing the prediction interval). – whuber Mar 22 at 17:32