Basically you want to avoid directly confronting them by implying they are dumb or wrong. Instead, appeal to authority, but in a soft way, and downplay the confidence of the new result.
I had many years of being undiplomatic until I saw a video of Kevin mitnick. His normal nature was quite undiplomatic and fairly autistic, but when he did social engineering, he did an amazing job of telling people what they wanted to hear. Now, when I think something, instead of stating it in a clear and undiplomatic way, I rewrite it in a way that my mental model suggests increases the probability the listeners will change their minds and agree with me. This took a great deal of practice to be able to do in real time.
Point being: Offer your idea during a friendly time where it will not cause the other person to be defensive and then you have to wait. You might find that the next time you bring it up the person is more receptive.