I have a data set with a continuous response variable and two categorical covariates. Let's imagine that I worked at an e-commerce company and was trying to regress the revenue we get from each user as a function of customer type ('A', 'B', 'C'), whether they were exposed to a specific treatment ('Exposed', 'Control'), and the interaction between those two variables.
I would fit the following model, in which a type A customer in the Control group would be used as the reference:
model <- lm(revenue ~ 1 + customer_type * treatment, data = users)
In math: $$ \operatorname{revenue}_i=\beta_0+\beta_1\operatorname{TypeB}_i+\beta_2\operatorname{TypeC}_i+\beta_3\operatorname{Exposed}_i+\beta_4(\operatorname{TypeB}\times\operatorname{Exposed})_i+\beta_5(\operatorname{TypeC}\times\operatorname{Exposed})_i $$
The question I'm trying to answer is: is the revenue of type C customers who are exposed to treatment different from the revenue of type C customers who are not exposed to treatment?
The way I tried to figure this out is by deriving what the models would be for the two types of users. That is, for a type C customer who is exposed to treatment, the model becomes
$$ \operatorname{revenue}=\beta_0+\beta_2+\beta_3+\beta_5 $$
and for a type C customer who is not exposed, the model is
$$ \operatorname{revenue}=\beta_0+\beta_2 $$
Thus, the difference between the two is not a single parameter but two ($\beta_3+\beta_5$). This means that there isn't a single t-statistic and associated p-value in the regression output that tests the question I have.
Intuitively, I think this must be possible and I probably have the answer right in front of me. What am I missing? Is the solution to use bespoke contrasts (if so, how would I do that in R?).