...as in when making an extraordinary claim.

 

Are there historical scientific examples?

relativity is probably the closest example of that.

 

it completely replaced newtonian mechanics(a very well established and accepted theory.) it required a LOT of evidence to be accepted.

 

its not so much you need a bit of shock and awe evidence more than you just need more than would usually be required.

 

say to prove a run of the mill hypothesis that adds to existing theory you need a sample size of 15000(just a random number here, don't put too much significance on it).

 

now, if you came up with a hypothesis that would overturn existing theory completely(and still expalin why the old one worked as it did) then you may need a sample size of 15million or so to convince people.

What springs to my mind is Dirac's prediction of antimatter in a paper in 1928. He suggested that all the particles known at the time have a "mirror" partner. At the time there was no experimental evidence for this, all came from his now famous equation.

 

In 1932 Carl Anderson discovered the antielectron (aka the positron)via studies of cosmic rays.

 

Dirac was awarded a Nobel prize in 1933 for his equation. Anderson was given a Nobel prize in 1936 for his proof of the existence of Dirac's antimatter.

Maybe Young's doublesplit type experiement could be one. The one that shows electrons behave as particles and waves...that helped moved physics into the realms of quantum mechanics (another big jump)

The end of the geocentric model is a good example. Biological evolution might be, too. It's interesting, though, that extraordinary claims that actually turned out to be true are so hard to find. The great majority of progress is incremental, not revolutionary.