# Linear regression - confidence interval for expected difference in $Y$ with respect to unknown values of $X$

Suppose I am given all of the necessary parameters about some linear model, but not the data itself. Namely, I am given $$\hat{\beta}_1,\hat{\beta_0},\bar X, S_e, r^2$$, etc. Also, I know that $$X_1,\dots,X_n$$ are all within the range of $$[40,70]$$. I'm being asked to construct a confidence interval for the expected difference in $$Y$$ over two units of $$X$$. What I'm not sure about is the parameter to be estimated.

I am guessing that a C.I. for $$\mathbb{E}[Y|X=2]$$ is not a good idea, because 2 is not within the range and the intercept will twist the results. I thought about estimating C.I. for $$\mathbb{E}[Y|X=40+2]$$ or $$\mathbb{E}[Y|X=\bar{X}+2]$$, the last one seems more reasonable but I can't think of any justification for it, let alone know whether this is the right approach at all.

Would appreciate any help.

I'd say you want to estimate E[f(x) - f(x+2)] ? Since it's a linear model you only need the slope to know how much will y change for 2 units of x. For the confidence interval, I am not sure, but there's where the SE may come into play

• That turns out to be $\mathbb{E}[2\cdot \hat{\beta}_1]$, so the appropriate C.I. and SE would be for the slope, not for some predicted value, right? the SE, I believe, would be just the same as for estimating $\hat{\beta}_1$, it's just the center of the interval that has doubled. Am I correct? – gbi1977 Feb 17 '19 at 12:31
• From the ci on the slope you can then deduce the ci on y, right? – gsanroma Feb 17 '19 at 12:44
• I'm not sure, you mean consturcting the C.I. for the slope in the standard way, and then double the two ends of the interval? its a different result from the one I mentioned in the previous comment, but here the estimator in the center of the C.I remains unbiased, the manipulation is performed only once the interval is constructed - so it makes more sense, but I want to be sure. – gbi1977 Feb 17 '19 at 13:17