I am working on a research project this summer and I'm a bit confused about type 2 errors. The actual scenario I am testing is nonsense to most people so I'll rephrase the problem in a way that makes more sense.

There is a bag containing many green balls and 1 red ball. A boy comes to play a game where he picks out ballswith replacementand if he picks the red ball he loses. I was not watching the bag before he started playing and he may have put some extra green balls into the bag before beginning to lower his chances of losing.

Even though he might have cheated I let him play anyway until he picks a red ball. After picking the red ball the game is over so no more trials are conducted.

The notation I'm using is this

$\displaystyle \mu\text{ the true chance of picking the red ball before the boy had a chance to interfere (expected to be less than 1/5000)}$

$\displaystyle \bar{x}\text{ the chance of picking a red ball based on trials done by other children who didn't cheat (sample size over 100000)} $

$\displaystyle \theta\text{ the true chance of the boy picking a red ball given that he may have added green balls}$

I mention approximate numbers in case they enable some assumptions to be made.

So far the boy has picked out n balls and has gotten no red ball.

The null hypothesis is that he has not cheated so $\displaystyle \mu=\theta$, the alternative hypothesis is that he has cheated so $\displaystyle \theta < \mu$ I would like to know the probability of type 1 and type 2 errors if I was to either accept or reject the null hypothesis.

My idea on how to do this is the following:

Type 2 error, I decide he has played too long without getting a red ball and has cheated. The probability of him getting this lucky streak is $\displaystyle (1-\mu)^n$, I approximate this with $\displaystyle (1-\bar{x})^n$. When $\displaystyle (1-\bar{x})^n$<0.05 his sample is suspicious and I consider excluding the boy's sample from the study.

Type 1 error, given that I'm considering excluding the boy's sample there is a chance I will make a type 1 error. My idea was to use a t test for this, I would be comparing $\displaystyle \bar{x}$ to the boy's sample which has mean 0 and sample size n. If the probability that $\displaystyle \mu=\theta$ is below 0.05 then I can safely assume he is cheating and reject the null hypothesis

Could I get some feedback on these tests? I'm not sure if they are the right way to go about it.