I sincerely doubt there's a single person alive that truly believes in the existence of leprechauns, therefore, this will be an excellent example of trying to disprove whether something exists or not.

Hypothesis: I theorize that Leprechauns do not exist

Wait a minute, this hypothesis is not acceptable. Why? Because the hypothesis implies that Leprechauns already exist. Unfortunately, we haven't yet tested the existence of leprechauns using the Scientific Method, so, Leprechaun's cannot yet be considered part of reality, and so they will be impossible to test the non-existence of. Remember, the Scientific Method's golden rule is that nothing exists unless the Scientific Method can support it. However, we can be really clever here and 'invert' the hypothesis into an acceptable form, such as:

New hypothesis: I theorize that Leprechauns might exist

Fantastic, now we have an acceptable hypothesis that doesn't make any assumptions, and also 'indirectly' answers the first hypothesis, because the answers for each hypothesis should be the exact opposite to each other.

You're probably going crazy trying to figure out why the first hypothesis was unacceptable and the second one is acceptable. Well, it all comes down to what the goal of the hypothesis is. The first hypothesis wanted to 'disprove' leprechauns, which implicitly forces us to consider the existence of leprechauns, so, in essence, the first hypothesis actually 'contains' the second hypothesis, which attempts to prove that there 'are' leprechauns. We can't have the first hypothesis just assuming the second hypothesis is already true without it being tested, so our initial step must be to deal with the underlying second hypothesis. The second hypothesis doesn't contain any hidden hypothesis' so it is acceptable as it is.

It's very important to understand that the Scientific Method assumes nothing exists unless it passes the Scientific Method, therefore everything is already considered 'non-existent' until proven it exists. The Scientific Method will only allow our 'knowledge base' to extend from what is already supported by the Scientific Method. By these rules, the first hypothesis is already true by default, unless the results for the second hypothesis can change that.

Color Table:

Strong counter-evidence	Weak counter-evidence	Not useful	Weak evidence	Good evidence	Strong evidence	Not applicable

First-hand Evidence
Type of Evidence	Our instincts consider this to be..	The Scientific Method considers this to be..
Tangibility There are no known leprechauns in captivity. They've never been captured on film, trapped, chronicled, measured, heard, or otherwise, not even a tiny carcass.	..quite obvious this is a fairy tale.	..unfortunate. Would be strong support if we had some physical evidence of this supposedly physical creature.
Influence Rainbows have been witnessed globally, yet paradoxically, nobody has ever been able to encroach upon the rainbow's endpoint successfully in order to locate the purported pot of gold, and hence verify the existence of the leprechaun.	..just a fairy tale for kids. Look at them run for the end of the rainbow, chasing gold.. dumb kids..	..difficult to comprehend how a pot of gold can dictate the endpoints of a rainbow, nor how that in any way contributes to the theory that leprechauns exist. If someone could actually find a pot of gold, there might be more substance to our hypothesis.
Interpretation Leprechauns hide their gold and the only way to find them is to search for the gold at the endpoint of a rainbow.	..a lark. Those kids are still running after it. You can make kids do anything if you fill their head with promises of treasure.	..too generalized. There needs to be much more explanation to the mechanism that drives rainbows to point out the location of a leprechaun's pot of gold, not to mention how that supports the existence of the leprechaun.
Intuition It must be a leprechaun driving those rainbows, so therefore they exist. We're just not fast enough to get to the base of a rainbow to see them with our own eyes before they're gone.	..the 'fun' explanation. The real explanation is the sun shining through the rain.	..an extremely poor explanation when contrasted against the Scientific Method endorsed explanation; that being rain diffracting sunlight into its diverse spectrum of colors, which also makes it impossible to reach the base of since the location of a rainbow is dependent upon the observer and changes as the observer moves. There certainly isn't any mention of a pot of gold in physics.
Desire A big pot of gold...hmmm	..a lot of fun when I was a kid. I chased rainbows for miles looking for gold. Good clean fun, but it's just a story.	..a huge motivator for the kids. They'll believe anything with a prize attached to it. However, no matter how much they want it to be true, this has no influence on our assessment.

Second-Hand Information Credibility
Type of Information	Our instincts consider this to be..	The Scientific Method considers this to be..
Distinguished Source None. All literary references are marked as fiction.	..obvious that its just a story.	..compelling counter-evidence. The sheer litany of resources that have labeled the story of leprechauns as a child's story makes it quite certain to be a fable.
Agenda Source Haven't been able to find a source that seriously purports leprechauns as existing.	..not surprising. That source would have to be crazy to try to pass leprechaun's off as fact	..not applicable.
Mass Support Seems to be a very well-spread story, that most people are familiar with. Still, nobody considers it to be anything more than a story.	..strong counter-evidence. Nobody else believes in leprechauns.	..not relevant either way. Whether everyone believes in leprechauns or not, this type of evidence is meaningless.
Hearsay, Rumor, or Opinion My parents used to tell me about the leprechaun's pot of gold at the end of the rainbow.	..reliable at the time, but now that I'm an adult, I'm obviously quite grateful that someone let me in on the big secret. Imagine chasing imaginary pots of gold at my age, ha!	..inconsequential.

The instincts refuse to give even one iota of credibility to the notions of leprechauns. However, notice that despite how 'obvious' it is that leprechauns are just a fairy tale, the Scientific Method still rigorously and 'fairly' assessed all the evidence before making a determination. The Scientific Method is an impartial machine that feels no shame at looking purely objectively at anything.

Instinctual Conclusion

Leprechauns are just a story.

Scientific Method Conclusion

Extremely improbable that leprechauns exist.

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