I'm struggling with the correct way to do hypotheses testing of categorical dummies in regression analysis. Suppose that we are interested in modeling a person's income, and suppose that we have a categorical variable $X= \{\mbox{no college; college; graduate studies}\}$. We may also want to control for, say, age (continuous). What I would like to test here is whether people who completed college have a significantly higher income than those that did not go to college.
We start with a model of the type: $$ \mbox{Income} = \alpha +\beta\cdot \mbox{College} + \gamma\cdot \mbox{Graduate studies} + \lambda \cdot \mbox{Age} + \varepsilon $$
My question pertains to the correct way to test the significance of $\beta$ (or $\gamma$). Continuous variables such as Age are straightforward to test (eg $H_0: \lambda=0$). However, I'm confused about $\beta$ (and $\gamma$) for the following reason. The expected value of income for the `no college' category is $\hat{\alpha}$; $\beta$ and $\gamma$ are the marginal effect of some college and graduate studies, respectively. So when we do hypothesis testing here, we are not comparing $\beta$ and $\gamma$ to a constant (such as zero), but to another point estimate that has a distribution. Therefore, shouldn't we take the distribution of $\alpha$ into account? What is the correct way to test whether category 2 is significantly different from category 1?
Perhaps to make my point clearer, suppose we get: $$\hat{\beta}=10 \mbox{ with a CI: } (5;15)$$ Most people would then claim that $\beta$ is significantly different from zero. But what if we have: $$\hat{\alpha}=30 \mbox{ with a CI: } (10;50)$$ The two distributions would then be overlapping. What I'm struggling to understand is why we do not take into account the distribution of $\hat{\alpha}$ when we test for the effect of a categorical variable. This point becomes evident when we fit a model without intercept. And yet, when I read the literature in my field, I find that people examine the effect of categorical dummies without taking this into account. Is that an error?
Edit
I wasn't clear enough. My question was not about the interpretation of the estimates. The expected effect for the omitted category will be $\hat{\alpha}$ and $\hat{\beta}$ will give us the marginal effect of college compared to the `no college' observations - that is absolutely fine. What I am wondering is about hypothesis testing only. Let's start with the following hypothetical estimate:
$\hat{\beta} = 5$, with a CI of (3;7).
The typical interpretation is that college significantly increases income over 'no studies' (i.e. the constant). That's the classical interpretation I see in the literature (in my field at least).
But then imagine the following estimates for the constant (no studies):
$\hat{\alpha} = 10$, with a CI of (1;19)$
In the latter care, the expected income for somebody with 'no studies' is (1;19) and E[Income] for somebody with college is 15 with a CI of (13;17) (which I obtain by taking the constant + the CI for college). How is it then that the effect of college is considered significant but at same time the distribution of the parameter is entirely contained within the 95% CI of the category to which it is compared?
