I would argue that both sentences are correct and have slightly different shades of meaning.
The teacher has authority to change the data.
"We need to change the data but do not have any authority. At least the teacher has some authority and might be able to change the data."
The teacher has the authority to change the data.
"The teacher has enough authority for the purpose of changing the data."
Please note that the previous sentence is technically another example or "authority to" used without an article.
P.S. There are different approaches to grammar: prescriptive and descriptive. Prescriptivists believe in rules-fist approach and sometimes go a bit too far with it; the other answer is a great example of how it looks. Descriptive approach is about looking what people do and building a theory about how language actually works based on the observations.
P.P.S. We have tests on articles here.