# Hypothesis Testing with Categorical Variables

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):