So I was reading a lot of online articles using the chi-square test to determine the fairness of a die.
Correct me if I am wrong, but isn't the drawbacks of using a goodness of fit test is that you cannot conclude anything about individual sides of die, only if the die was fair or not?
Couldn't we solve this problem using binomial probability similar to a determining fairness of a coin but instead where we use a one-versus all approach? I know what I am writing below is kind of silly and rather pointless; I'm just trying to see if I truly understand the problem.
So instead of one Chi-Square test, we can do 6 tests and running a hypothesis test for each binomial distribution. Let's roll 100 times for each side (total of 600 times):
For side = 1:
$P(1) \approx .1666$
$P(2,3,4,5,6) \approx .8333$
$\mu = np = 100*.1666$
$\sigma^2 = npq = 100*.1666*.8333$
$...$
For side = 6:
$P(6) = .166$
$P(1,2,3,4,5) = .833$
$\mu = np = 100*.1666$
$\sigma^2 = npq = 100*.1666*.8333$
If any of the sides fall outside of the critical region, we reject the null hypothesis, and we can determine which side(s) is most likely biased.
Couple of questions:
- Is this method even valid?
- Would this result give us a similar result to a chi-square test?
- Do we have to do any corrections due to the multiple comparisons problem?