Confidence intervals for factors+continuous variables

[mtal]

I have

$$y_{ij} = \mu_{i} + \alpha x_{ij} + e_{ij}$$

where $$i = 1, 2,3$$ and $$j = 1, \ldots , r$$.
$$\mu_{i}$$ represents the mean of the data set plus factor levels i , $$\alpha x_{ij}$$ is a continuous variable.

So, the problem is the following:

Construct confidence intervals for $$\mu_1$$ , $$\mu_2 - \mu_1$$ , and $$(\mu_3 - \mu_2) - (\mu_2 - \mu_1)$$.

The original problem consisted of finding conf.intervals for the same things except the continuous variable wasn't in the model.
I have looked around a lot, but I can't find any instructions on confidence intervals when both factors and continuous variables are included.

Help appreciated!

 

[Enuma_Elish]

You can think of it as follows:

Without the x, y = m + e implies m = mean(y).

When you have an x and the equation becomes y = m + ax + e, m = mean(y - ax), where a is the estimated slope coefficient.

EnumaElish