> Is there any data out there to support this claim?

What are you imagining?

The people reporting falsified shipping times on HN are the data you're asking for. I can personally confirm - Amazon "guarantees" two-day shipping for Prime orders, but it will not actually provide two-day shipping, and if you complain their only response is "Well, it got there eventually. What's wrong?"

After enough time where they systematically refused to honor their own purported benefits, I canceled my Prime membership.